Overview of High Energy String Scattering Amplitudes and Symmetries of String Theory

In this paper, we studied symmetries of string scattering amplitudes in the high energy limits of both the fixed angle or Gross regime (GR) and the fixed momentum transfer or Regge regime (RR). We calculated high energy string scattering amplitudes (SSA) at arbitrary mass levels for both regimes. We discovered the infinite linear relations among fixed angle string amplitudes and the ifinite recurrence relations among Regge string amplitudes. The linear relations we obtained in the GR corrected the saddle point calculations by Gross, Gross and Mende. In addition, for the high energy closed string scatterings, our results differ from theirs by an oscillating prefactor which was crucial to recover the KLT relation valid for all energies. We showed that all the high energy string amplitudes can be solved using the linear or recurrence relations, so that all the string amplitudes can be expressed in terms of a single string amplitude. We further found that, at each mass level, the ratios among the fixed angle amplitudes can be extracted from the Regge string scattering amplitudes. Finally, we reviewed the recent developments on the discovery of infinite number of recurrence relations valid for all energies among Lauricella SSA. The symmetries or relations among SSA at various limits obtained previously can be exactly reproduced. It leads us to argue that the known SL(K + 3, C) dynamical symmetry of the Lauricella function may be crucial to probe spacetime symmetry of string theory.


. INTRODUCTION AND OVERVIEW
One of the fundamental issues of string theory is its spacetime symmetry structure. It has long been believed that string theory consists of huge hidden symmetries. This is strongly suggested by the UV finiteness of quantum string theory, which contains no free parameter and an infinite number of states. On the other hand, the high energy, fixed angle behavior of string scattering amplitudes was known to be very soft exponential fall-off, while that of a local quantum field theory was power law. Presumably, it is these huge hidden symmetries which soften the UV structure of quantum string theory. In a local quantum field theory, a symmetry principle was postulated, which can be used to determine the interaction of the theory. In string theory, on the contrary, it is the interaction, prescribed by the very tight quantum consistency conditions due to the extendedness of string, which determines the form of the symmetry.
Historically, the first key progress to understand symmetry of string theory was to study, instead of low energy field theory limit, the high energy, fixed angle behavior of hard string scattering (HSS) amplitudes [1][2][3][4][5]. This was motivated by the spontaneously broken symmetries in gauge field theories which were hidden at low energy, but became evident in the high energy behavior of the theory. There were two main conjectures of Gross's [3,4] pioneer work in 1988 on this subject. The first one was the existence of an infinite number of linear relations among the scattering amplitudes of different string states that were valid order by order in string perturbation theory at high energies, fixed angle regime or Gross regime (GR). The second was that this symmetry was so powerful as to determine the scattering amplitudes of all the infinite number of string states in terms of a single dilaton (tachyon for the case of open string) scattering amplitudes. However, the symmetry charges of his proposed stringy symmetries were not understood and the proportionality constants or ratios among scattering amplitudes of different string states were not calculated.
The second key to uncover the fundamental symmetry of string theory was the realization of the the importance of zero norm states (ZNS) in the old covariant first quantized (OCFQ) string spectrum. It was proposed that [6][7][8] spacetime symmetry charges of string theory originate from an infinite number of ZNS with arbitrary high spin in the spectrum. In the late eighties the decoupling of ZNS was also used in the literature to the measure in the study of multi-string vertices by seveal authors and was closely related to the so-called group theoretic approach of stringy scattering amplitudes. This subject will not be covered in this review. For more details see the review in [9]. In the context of σ-model approach of string theory, one turns on background fields on the worldsheet energy momentum tensor T .
Conformal invariance of the worldsheet then requires, in addition to D = 26, cancellation of various q-number anomalies and results to equations of motion of the background fields [10]. It was then shown that [6] for each spacetime ZNS, one can systematically construct a worldsheet (1, 1) primary field δT Φ such that is satisfied to some order of weak field approximation in the σ-model background fields β function calculation. In the above equation, T Φ is the worldsheet energy momentum tensor with background fields Φ and T Φ+δΦ is the new energy momentum tensor with new background fields Φ + δΦ. Thus for each ZNS one can construct a spacetime symmetry transformation for string background fields.
In the even higher mass levels, M 2 = 6 for example, a new phenomenon begins to show up. There are ambiguities in defining positive-norm spin-two and scalar states due to the existence of ZNS in the same Young representations [8]. As a result, the degenerate spin two and scalar positive-norm states can be gauged to the higher rank fields, the symmetric spin four D µναβ and mixed-symmetric spin three D µνα in the first order weak field approximation.
In fact, for instance, it can be shown [11] that the scattering amplitude involving the positivenorm spin-two state can be expressed in terms of those of spin-four and mixed-symmetric spin-three states due to the existence of a degenerate type I and a type II spin-two ZNS.
This stringy phenomenon seems to persist to higher mass levels.
This calculation is consistent with the result in the HSS limit. In fact, it can be shown that in the HSS limit all the scattering amplitudes of leading order in energy at each fixed mass level can be expressed in terms of that of the leading trajectory string state with transverse polarizations on the scattering plane. See Eq.(0.9), Eq.(0.21) and Eq.(0.29) below. One can also justify this decoupling by WSFT to be discussed in section I.D. Finally one expects this decoupling to persist even if one includes the higher order corrections in weak field approximation, as there will be even stronger relations between background fields order by order through iteration.
The calculation of Eq.(0.4) was done in the first order weak field approximation but valid to all energies or all orders in α ′ . A second order weak field calculation implies an even more interesting spontaneously broken inter-mass level symmetry in string theory [12,13]. Some implication of the corresponding stringy Ward identity on the scattering amplitudes were discussed in [12,14] and will be presented in Eq.(0.12). It was then realized that [15,16] the symmetry in Eq.(0.4) can be reproduced from gauge transformation of Witten string field theory (WSFT) [17] after imposing the no ghost conditions. It is important to note that this stringy symmetry exists only for D = 26 thanks to type II ZNS in the OCFQ string spectrum , which is zero norm only when D = 26.
Incidentally, it was well known in 2D string theory that the operator products of the discrete positive norm states ψ + J,M form a w ∞ algebra [18][19][20] The calculation above can be generalized to 2D superstring theory [24].
One can also use ZNS to calculate spacetime symmetries of string on compact backgrounds. The existence of soliton ZNS at some moduli points was shown to be responsible for the enhanced Kac-Moody symmetry of closed string theory. As a simple example, for the case of 26D bosonic closed string compactified on a 2-dimensional torus T 2 ≡ R 2 2πΛ 2 , it was found that massless ZNS (including soliton ZNS) form a representation of enhanced where Λ 2 is a 2-dimensional lattice with a basis R 1 → e 1 √ 2 , R 2 → e 2 √ 2 , and B is the antisymmetric tensor B ij = Bǫ ij . In this calculation one has four moduli parameters R 1 , R 2 , B and → e 1 · → e 2 with → e i 2 = 2. Moreover, an infinite number of massive soliton ZNS at any higher massive level of the spectrum were constructed in [25]. Presumably, these massive soliton ZNS are responsible for enhanced stringy symmetries of the theory.
For the case of open string compactification, unlike the closed string case discussed above, it was found that [26] the soliton ZNS exist only at massive levels. These Chan-Paton soliton ZNS correspond to the existence of enhanced massive stringy symmetries with transformation parameters containing both Einstein and Yang-Mills indices in the case of Heterotic string [12]. In the T-dual picture, these symmetries exist only at some discrete values of compactified radii when N D-branes are coincident [26].
All the above results which are valid to all energies will constitute the part I of this review paper. On the other hand, in part II of this review, we will show that the high energy limit of the discrete ZNS G + J,M in 2D string theory constructed in Eq.(0.7) in part I approaches ψ + J,M in Eq.(0.6) and thus form a high energy w ∞ symmetry of 2D string. This result strongly suggests that the linear relations obtained from decoupling of ZNS in 26D string theory are indeed related to the hidden symmetry also for the 26D string theory.
In part II of this paper, we will review high energy, fixed angle calculations of HSS amplitudes. The high energy, fixed angle Ward identities derived from the decoupling of ZNS in the HSS limit, which combines the previous two key ideas of probing stringy symmetry, were used to explicitly prove Gross's two conjectures [27][28][29][30][31][32]. An infinite number of linear relations among high energy scattering amplitudes of different string states were derived.
Remarkably, these linear relations were just good enough to fix the proportionality constants or ratios among high energy scattering amplitudes of different string states algebraically at each fixed mass level. The first example calculated was the ratios among HSS amplitudes at mass level M 2 = 4 [27,29] (see the definition of polarizations e T and e L after Eq.(0. 16) below) T to Eq.(0.5). Eq.(0.9) is presumably valid order by order in string perturbation theory as we expect the decoupling of ZNS is valid even for string loop amplitudes [33].
Note that Eq.(0.12) is the inter-particle Ward identity corresponding to D 2 vector ZNS in Eq.(0.5) obtained by antisymmetrizing those terms which contain α µ −1 α ν −2 in the original type I and type II vector ZNS [6]. We will use 1 and 2 for the incoming particles and 3 and 4 for the scattered particles. In the Ward identities, 1, 3 and 4 can be any string states and we have omitted their tensor indices for the cases of excited string states.
In the HSS limit, one enjoys many simplifications in the calculation. First, all polarizations of the amplitudes orthogonal to the scattering plane are of subleading order in energy, and one needs only consider polarizations on the scattering plane. Second, to the leading order in energy, e P ≃ e L in the HSS calculation. In the end of the calculation, one ends up with the simple linear equations for leading order amplitudes [27,29] T 5→3 LLT + T 3 (LT ) = 0, (0.14) 10T 5→3 LLT + T 3 T T T + 18T 3 (LT ) = 0, (0.15) 16) where e P = 1 M 2 (E 2 , k 2 , 0) = k 2 M 2 the momentum polarization, e L = 1 M 2 (k 2 , E 2 , 0) the longitudinal polarization and e T = (0, 0, 1) the transverse polarization are the three polarizations on the scattering plane. In Eq.(0.14) to Eq.(0.16), we have assigned a relative energy power for each amplitude. For each longitudinal L component, the order is E 2 and for each transverse T component, the order is E. This is due to the definitions of e L and e T above, where e L got one energy power more than e T . By Eq.(0.15), the naive leading order E 5 term of the energy expansion for T LLT is forced to be zero. As a result, the real leading order term is E 3 . Similar rule applies to T LLT in Eq.(0.14) and Eq.(0. 16). The solution of these three linear relations gives Eq.(0.9). Eq.(0.9) gives the first evidence of Gross conjecture [3,4] on HSS amplitudes.
A sample calculation of scattering amplitudes for mass level M 2 = 4 [29] justified the ratios calculated in Eq.(0.9). Since the proportionality constants in Eq.(0.9) are independent of particles chosen for vertex v 1,3,4 , for simplicity, we will choose them to be tachyons. For the string-tree level χ = 1, with one tensor v 2 and three tachyons v 1,3,4 , all scattering amplitudes of mass level M 2 2 = 4 were calculated to be (s − t channel) with s + t + u = 2N − 8. We thus have justified Eq.(0.9) with T 3 T T T = −8E 9 T (3) sin 3 φ CM . The calculations based on ZNS thus relate [16] gauge transformation of WSFT to high energy string symmetries of Gross. However, in the sample calculation of [5], two of the four high energy amplitudes in Eq.(0.9) were missing, and thus the decoupling of ZNS or unitarity was violated. This is of course due to the unawareness of the importance of ZNS in the saddle-point calculation of [1][2][3][4][5].
In order to apply the saddle-point method, we rewrite the amplitude above into the following form [31,32] T (N,2m,q) (K) = The saddle-point for the integration of moduli, x = x 0 , is defined by f ′ (x 0 ) = 0, (0. 36) and we have It is easy to see that u(x 0 ) = u ′ (x 0 ) = .... = u (2m−1) (x 0 ) = 0, (0. 38) and With these inputs, one can easily evaluate the Gaussian integral associated with the four-point amplitudes We conclude that there is only one independent component of high energy scattering amplitude at each fixed mass level. Based on this independent component of high energy scattering amplitude, one can then derive the general formula of high energy scattering amplitude for four arbitrary string states, and express them in terms of that of tachyons.
All the above calculations can be extended to the case of hard superstring scattering amplitudes which will be discussed in chapter XIII of this review. However, it was found that [34] there were new HSS amplitudes for the superstring case. The existence of these new high energy scattering amplitudes of string states with polarizations orthogonal to the scattering plane is due to the worldsheet fermion exchange in the correlation functions. These worldsheet fermion exchanges do not exist in the bosonic string correlation functions and is, presumably, related to the high energy massive spacetime fermionic scattering amplitudes in the R-sector of the theory.
Obviously, these new high energy amplitudes create complications for a full understanding of stringy symmetry. Nevertheless, the claim that there is only one independent high energy scattering amplitude at each fixed mass level of the string spectrum persists in the case of superstring theory, at least, for the NS sector of the theory [34].
Indeed, it was found [30] that saddle point calculation in [1][2][3][4][5] is only valid for the tachyon amplitude. In general, the results calculated in [1][2][3][4][5] gives the right energy exponent in the scattering amplitudes, but not the energy power factors in front of the exponential for the cases of the excited string states. These energy power factors are subleading terms ignored in [1][2][3][4][5] but they are crucial if one wants to get the linear relations among high energy scattering amplitudes conjectured by Gross.
Interestingly, the inconsistency of the saddle point calculation discussed above for the excited string states was also pointed out by the authors of [35]. The source of disagreement in their so-called group theoretic approach of stringy symmetries stems from the proper choice of local coordinates for the worldsheet saddle points to describe the behavior of the excited string states at high energy limit. It seems that both the ZNS calculation and the calculation based on group theoretic approach agree with tachyon amplitudes obtained in [1][2][3][4][5] (ignore the possible phase factors in the amplitudes to be discussed in the next few paragraphs), but disagree with amplitudes for other excited string states.
The next interesting issues were the calculation of closed string scattering amplitudes and their symmetries in the HSS limit [36]. Historically, the open string four tachyon amplitude in the HSS limit was first calculated in the original paper of Veneziano in 1968. On the other hand, the N -loop closed HSS amplitudes were calculated by the saddle-point method in [1,2]  which is valid for all kinematic regimes and for all string states. This is due to the phase factor sin (πk 2 · k 3 ) in the above equation which was missing in the closed string saddlepoint calculation in [1,2]. One clue to see the origin of this inconsistency is to note that the saddle-point x 0 = 1 1−τ identified for the open string calculation in Eq.(0.37) is in the regime [1, ∞). So only saddle point calculation forĀ (4) open (t, u) is reliable, but not that of A (4) open (s, t) and neither that of closed string amplitude A (4) closed (s, t, u) [36] by the KLT relation. Instead of using saddle-point calculation for the closed HSS amplitudes, the above considerations led the authors of [36] to study the relationship between A (4) open (s, t) andĀ (4) open (t, u) for arbitrary string states in the HSS limit. With the help of the infinite linear relations in Eq.(0. 29), one needs only calculate relationship between s − t and t − u channel HSS amplitudes for the leading trajectory string states. They ended up with the following result in the HSS limit (2006) [36] A (4) open (s, t) = sin (πk 2 .k 4 ) sin (πk 1 k 2 )Ā (4) open (t, u) , (0. 43) which is valid for four arbitrary string states. It is now clear that due to the phase factor in the above equation, the saddle-point calculation of A (4) open (s, t) is not reliable, neither for the closed one A open (s, t) in the HSS limit. The consistent closed string fourtachyon HSS amplitudes can then be calculated by using the KLT relation in Eq.(0.42) to be [36] A (4−tachyon) closed (s, t, u) ≃ sin (πt/2) sin (πu/2) sin (πs/2) (stu) −3 exp − s ln s + t ln t + u ln u 4 (0. 45) The exponential factor in Eq.(0.44) was first discussed by Veneziano [37]. The result for the high energy closed string four-tachyon amplitude in Eq.(0.45) differs from the one calculated in the literature [1,2] by an oscillating factor sin(πt/2) sin(πu/2) sin(πs /2) . One notes here that the results of Eqs.(0.45), (0.44) and Eq.(0.43) are consistent with the KLT formula, while the previous calculation in [1,2] is NOT.
Indeed, one might try to use the saddle-point method to calculate the high energy closed string scattering amplitude. The closed string four-tachyon scattering amplitude is A (4−tachyon) closed (s, t, u) = dxdy exp k 1 · k 2 2 ln |z| + k 2 · k 3 2 ln |1 − z| which is consistent with the previous one calculated in the literature [1,2], but is different from the result in Eq.(0.45). However, one notes that 49) which means that (x 0 , y 0 ) is NOT the local minimum of f (x, y), and one should not trust this saddle-point calculation. There was other evidence pointed out by authors of [36] to support this conclusion. Finally, the ratios of closed HSS amplitudes turned out to be the tensor products of two open string ratios The relationship between s − t and t − u channels HSS amplitudes in Eq.(0.43) was later argued to be valid for all kinematic regime based on monodromy of integration in string amplitude calculation in 2009 [38]. An explicit proof of Eq.(0.43) for arbitrary four string states and all kinematic regimes was given very recently in [39,40].
The motivation for the author in [38] to calculate Eq.(0.43) was different from the discussion above which was related to the calculation of hard closed string scattering amplitudes.
The motivation in [38] was based on the field theory BCJ relation [41] for Yang Recently the mass level dependent of Eq.(0.43) was calculated to be [39,40] A (p,r,q) st ≃ sin π (k 2 · k 4 ) sin π (k 1 · k 2 ) (0.52) by taking the nonrelativistic limit | k 2 | << M S of Eq.(0.43). In Eq.(0.52), B was the beta function, and k 1 , k 3 and k 4 were taken to be tachyons, and k 2 was the following tensor string state V 2 = (i∂X T ) p (i∂X L ) r (i∂X P ) q e ik 2 X (0. 53) where N = p + r + q, M 2 2 = 2(N − 1), N ≥ 2. (0.54) The generalization of the four point function relation in Eq.(0.43) to higher point string amplitudes can be found in [38]. It is interesting to see that historically the four point (high energy) string BCJ relations Eq.(0.43) [36] were discovered even earlier than the field theory BCJ relations Eq.(0.51)! [41].
The ratios calculated in Eq.(0.50) persist for the case of closed string D-particle scatterings in the HSS limit. For the simple case of m = 0 = m ′ , the ratios were first calculated to be − 1 2M q+q ′ [42]. The complete ratios were then calculated through a correspondence between HSS ratios and RSS ratios to be discussed in Eq.(0.76) below, and were found to be factorized [43] (see section XIV.C) It is well known that the closed string-string scattering amplitudes can be factorized into two open string-string scattering amplitudes due to the existence of the KLT formula [44]. On the contrary, there is no physical picture for open string D-particle tree scattering amplitudes and thus no factorization for closed string D-particle scatterings into two channels of open string D-particle scatterings, and hence no KLT-like formula there.
Thus the factorized ratios in HSS regime calculated above came as a surprise. However, these ratios are consistent with the decoupling of high energy ZNS calculated previously in [27-32, 34, 36, 45]. It will be interesting if one can calculate the complete HSS amplitudes directly and see how the non-factorized amplitudes can give the result of factorized ratios.
On the other hand, in contrast to the closed string D-particle scatterings in the HSS limit discussed above, it was shown that, instead of the exponential fall-off behavior of the form factors with Regge-pole structure, the HSS amplitudes of closed string scattered from D24brane, or D-domain-wall, behave as power-law with Regge-pole structure [46]. See Eq. (9.72) and Eq. (9.73) in section IX.A. 4. This is to be compared with the well-known power law form factors without Regge-pole structure of the D-instanton scatterings.
This discovery makes D-domain-wall scatterings an unique example of a hybrid of string and field theory scatterings. Moreover, it was discovered that [46] the usual linear relations of HSS amplitudes at each fixed mass level, Eq.(0.55), breaks down for the D-domain-wall scatterings. This result gives a strong evidence that the existence of the infinite linear relations, or stringy symmetries, of HSS amplitudes is responsible for the softer, exponential fall-off HSS scatterings than the power-law field theory scatterings.
Being a consistent theory of quantum gravity, string theory is remarkable for its soft ultraviolet structure. Presumably, this is mainly due to three closely related fundamental characteristics of HSS amplitudes. The first is the softer exponential fall-off behavior of the form factors in the HSS in contrast to the power-law field theory scatterings. The second is the existence of infinite Regge poles in the form factor of string scattering amplitudes. The existence of infinite linear relations discussed in part II of the review constitutes the third fundamental characteristics of HSS amplitudes.
It will be important to study more string scatterings, which exhibit the above three unusual behaviors in the HSS limit. In section IX.B, we will consider closed string scattered from O-planes. In particular we first calculate massive closed string states at arbitrary mass levels scattered from Orientifold planes in the HSS limit [47]. The scatterings of massless states from Orientifold planes were calculated in the literature by using the boundary states formalism [48][49][50][51], and on the worldsheet of real projected plane RP 2 [52]. Many speculations were made about the scatterings of massive string states, in particular, for the case of Odomain-wall scatterings. It is one of the purposes of section IV.B to clarify these speculations and to discuss their relations with the three fundamental characteristics of HSS scatterings stated above.
For the generic Op-planes with p ≥ 0, one expects to get the infinite linear relations except O-domain-wall HSS. For simplicity, we consider only the case of O-particle HSS [47]. For the case of O-particle scatterings, we obtain infinite linear relations among HSS amplitudes of different string states. We also confirm that there exist only t-channel closed string Regge poles in the form factor of the O-particle scatterings amplitudes as expected.
For the case of O-domain-wall scatterings, we find that, like the well-known D-instanton scatterings, the amplitudes behave like field theory scatterings, namely UV power-law with-out Regge pole. In addition, we find that there exist only finite number of t-channel closed string poles in the form factor of O-domain-wall scatterings, and the masses of the poles are bounded by the masses of the external legs [47]. We thus confirm that all massive closed string states do couple to the O-domain-wall as was conjectured previously [52,53]. This is also consistent with the boundary state descriptions of O-planes. In chapter X, following an old suggestion of Mende [54], we calculate high energy massive scattering amplitudes of bosonic string with some coordinates compactified on the torus [55,56]. We obtain infinite linear relations among high energy scattering amplitudes of different string states in the Hard scattering limit. In addition, we analyze all possible power-law and soft exponential fall-off regimes of high energy compactified bosonic string scatterings by comparing the scatterings with their 26D noncompactified counterparts.
Interestingly, we discover in section X.A the existence of a power-law regime at fixed angle and an exponential fall-off regime at small angle for high energy compactified open string scatterings [56]. These new phenomena never happen in the 26D string scatterings.
The linear relations break down as expected in all power-law regimes. The analysis can be extended to the high energy scatterings of the compactified closed string in section X.B, which corrects and extends the results in [55].
At this point, one may ask an important question for the results of Eqs.(0.9), (0.21), (A. 15) and (0.29) above , namely, is there any group theoretical structure of the ratios of these scattering amplitudes? Let's consider a simple analogy from particle physics. The ratios of the nucleon-nucleon scattering processes (a) p + p → d + π + , (c) n + n → d + π − (0. 56) can be calculated to be (ignore the tiny mass difference between proton and neutron) T a : T b : T c = 1 : from SU(2) isospin symmetry. Is there any symmetry structure which can be used to calculate ratios in Eqs.(0.9), (0.21), (A. 15) and (0.29)? It turned out that part of the answer can be addressed by studying another high energy regime of string scattering amplitudes, namely, the fixed momentum transfer or Regge regime (RR) [57][58][59][60][61][62][63][64][65][66].
In part III of this paper, we will discuss RSS amplitudes and their relations to the fixed angle HSS amplitudes. We will find that the number of RSS amplitudes is much more numerous than that of HSS amplitudes. For example, there are only 4 HSS amplitudes while there are 22 RSS amplitudes at mass level M 2 = 4 [64]. This is one of the reason why decoupling of ZNS in the RR, in contrast to the GR, is not good enough to solve RSS amplitudes in terms of one single amplitude at each mass level.
For illustration and to identify the ratios in Eqs.(0.9) from RSS amplitudes, we will first calculate amplitudes at mass level M 2 = 4 in the RR s → ∞, √ −t = fixed (but √ −t = ∞).
(0. 58) The relevant kinematics are e T · k 1 = 0 (0.59c) and e P · k 3 = 1 Note that in contrast to the identification e P ≃ e L in the HSS limit, e P does not approach to e L in the RSS limit.
We will list the relevant RSS amplitudes at mass level M 2 = 4 which contain polarizations (e T , e L ) only. It turned out that there are eight high energy amplitudes in the RR Among them only four of the above amplitudes are relevant here and can be calculated to be [64] A T T T = t − 9 2 s 3 + 1 4 t 2 + 7 2 t s 2 + (t + 6) 2 2 s , (0.63) and where the kinematic variables (s, t) were used instead of (E, θ) used in the GR.  and similarly for the amplitudes A (T L) and A [T L] in the RR. It is interesting to see that in the GR is of subleading order in energy, while A LT in the RR is of leading order in energy. However, the contribution of the amplitude A LT to A (T L) and A [T L] in the RR will not affect the ratios calculated above.
From the calculation above, it was thus believed that there existed intimate link between high energy string scattering amplitudes in the HSS regime and those in the RSS regime. To study this link and to reproduce the ratios in Eq.(0.29) in particular, one was led to calculate RSS amplitudes for arbitrary mass levels. To simplify the calculation, we use the simple kinematics e T · k 1 = 0 in Eq.(0.29) and the energy power counting of the string amplitudes, and end up with the following rules α T −n : 1 term (contraction of ik 3 · X with ε T · ∂ n X), (0.71) α L −n :      n > 1, 1 term n = 1 2 terms (contraction of ik 1 · X and ik 3 · X with ε L · ∂ n X). The s − t channel scattering amplitudes of this state with three other tachyonic states can be calculated to be [64] A (pn,qm) = − i M 2 (0. 74) In the above, U(a, c, x) is the Kummer function of the second kind. It is crucial to note that c = t 2 + 2 − q 1 , and is not a constant as in the usual case, so U in the above amplitude is not a solution of the Kummer equation. On the contrary, since a = −q 1 an integer, the Kummer function in Eq. (11.25) terminated to be a finite sum.
It can be seen from Eq.(0.74) that the RSS amplitudes with spin polarizations corresponding to Eq.(0.22) at each fixed mass level are no longer proportional to each other. The ratios are t dependent functions and can be calculated to be [64] A (N,2m,q) (s, t) A (N,0,0) (s, t) , (0. 75) where (x) j = x(x + 1)(x + 2) · · · (x + j − 1) is the Pochhammer symbol.
To deduce the link and ensure the following identificationfor the general mass levels suggested by the explicit calculation for the mass level M 2 2 = 4 [64], one needs the following identity where L = 1 − N and is an integer. The identity was proved to be valid for any non-negative integer m and any real number L by using technique of combinatorial number theory [67].
It was remarkable to first predict [64] the mathematical identity above provided by string theory, and then a rigorous mathematical proof followed [67]. It was also interesting to see that the validity of the above identity includes non-integer values of L which were later shown to be realized by Regge string scatterings in compact space [68]. We thus have shown that the ratios among HSS amplitudes calculated in Eqs.(0.9) and (0.29) can be deduced and extracted from Kummer functions [64,69,70] T (N,2m,q) (0.78) All the above calculations so far can be generalized to four classes of superstring Regge scattering amplitudes [65]. See the discussion in chapter XII.
The next interesting issue is to study relations among RSS amplitudes of different string states. To achieve this, one considers the more general RSS amplitudes corresponding to three tachyons and one leading order high energy open string states in the RR at each fixed mass level N = n,m,l>0 np n + mq m + lr l The s − t channel scattering amplitudes of this state with three other tachyonic states can be calculated to be Finally, the amplitudes can be written as two equivalent expressions [71] A (pn;qm; It is easy to see that, for q 1 = 0 or r 1 = 0, the RSS amplitudes can be expressed in terms of In general the RSS amplitudes can be expressed in terms of a finite sum of Kummer functions.
One can then solve these Kummer functions at each mass level and express them in terms of RSS amplitudes. Recurrence relations of Kummer functions can then be used to derive recurrence relations among RSS amplitudes [71]. As an example at mass level M 2 = 4, the recurrence relation leads to the following recurrence relation among Regge string scattering amplitudes In addition, the addition theorem of Kummer function [72] U(a, c, which terminates to a finite sum for a non-positive integer a can be used to derive inter-mass level recurrence relation of RSS amplitudes. By taking, for example, a = −1, c = t 2 + 1, x = t 2 − 1 and y = 1, the theorem gives Note that the last arguments of Kummer functions in the above equation can be different.
can be explicitly demonstrated by using the following recurrence relations of Kummer func-  [62] can also be constructed in this way [73].
Since in general each RSS amplitude was expressed in terms of more than one Kummer function, it was awkward to derive the complete recurrence relations at arbitrary higher mass levels. More recently [74], it was shown that each 26D open bosonic RSS amplitude can be expressed in terms of one single Appell function F 1 . In fact, the s − t channel RSS amplitudes with string state in Eq.(0.79) and three tachyons can be calculated as [74] A (pn;qm; where the Appell function F 1 is one of the four extensions of the hypergeometric function 2 F 1 to two variables and is defined to be where (a) n = a · (a + 1) · · · (a + n − 1) is the rising Pochhammer symbol. Note that when which leads to a recurrence relation for RSS amplitudes at arbitrary mass levels [74] t ′2 A (N ;q 1 ,r 1 ) More higher recurrence relations which contain general number of l ≥ 3 Appell functions can be found in [76].
More importantly, one can show [71,74] that these recurrence relations in the Regge limit can be systematically solved so that all RSS amplitudes can be expressed in terms of one amplitude. All these results seem to dual to high energy symmetries of fixed angle string scattering amplitudes discussed in part II [27-29, 31, 32, 34, 45].
We now proceed to show that the recurrence relations of the Appell function F 1 in the Regge limit can be systematically solved so that all RSS amplitudes can be expressed in terms of one amplitude. As the first step, we note that in [71] the RSS amplitudes was expressed in terms of finite sum of Kummer functions. There are two equivalent expressions [71] as was previously shown in Eq.(0.82). It is easy to see that, for q 1 = 0 or r 1 = 0, the RSS amplitudes can be expressed in terms of only one single Kummer function On the other hand, it was shown in [71] that the Kummer functions ratio is determined and f (α, γ, z) can be calculated by using recurrence relations of U(α, γ, z).
Note in addition that U(0, z, z) = 1 by explicit calculation. We thus conclude that in the Regge limit the Appell functions F 1 (a; 0, b 2 ; c; x, y) and F 1 (a; b 1 , 0; c; x, y) are determined up to an overall factor by recurrence relations. The next step is to derive the recurrence relation which can be obtained from two of the four Appell recurrence relations among contiguous functions.
We can now show that in the Regge limit all RSS amplitudes can be expressed in terms of one single amplitude. We will use the short notation F 1 (a; b 1 , b 2 ; c; x, y) = Eq.(0.101) and the known F 1 (b 1 , 0) and F 1 (0, b 2 ). This process can be continued and one ends up with the result that F 1 (b 1 , b 2 ) are determined for all b 1 , b 2 = −1, −2, −3.... This completes the proof that the recurrence relations of the Appell function F 1 in the Regge limit in Eq.(0.92) can be systematically solved so that all RSS amplitudes can be expressed in terms of one amplitude.
In a very recent paper [40], it was discovered that the 26D open bosonic string scattering amplitudes (SSA) of three tachyons and one arbitrary string state can be expressed in terms of the D-type Lauricella functions with associated SL(K + 3; C) symmetry. As a result, SSA and symmetries or relations among SSA of different string states at various limits calculated previously can be rederived. These include the linear relations conjectured by Gross [3,4]. and proved in [27][28][29][30][31][32]  In addition to the high energy string scatterings discussed in this review, there were other related approaches in the literature discussing higher spin dynamics of string theory. String theory includes infinitely many higher spin massive fields with consistent mutual interactions, and can provide useful hints on the dynamics of higher spin field theory. On the other hand, a better understanding of higher spin dynamics could also help our comprehension of string theory. It is widely believed that the tensionless limit of string [77][78][79][80][81][82][83] is a theory of higher spin gauge fields. In flat spacetime a non-trivial field theory dynamics of the tensionless limit of string theory seems to be ruled out by the theorem of Coleman and Mandula. However, the assumptions of this theorem are violated by the presence of a non-trivial cosmological constant, and one may expect a consistent interacting field theory of higher spins on curved space time. One of the most important explicit and nontrivial construction of interacting higher spin gauge theory is Vasiliev' system in AdS space-time.
In [84], the spectrum of Kaluza-Klein descendants of fundamental string excitations on AdS 5 × S 5 was derived and organized at the higher spin long multiplets of the AdS supergroup SU (2, 2|4) with a rich pattern of shortenings at the higher spin enhancement point.
Furthermore, in the tensionless limit, the field equations from BRST quantization of string theory provide a direct route toward local field equations for higher-spin gauge fields [85].
Recently, in [86], one parameter families of parity violating Vasiliev theory were formu-lated that preserve N = 6 SUSY in AdS 4 . The theory was suggested to be dual to the vector model limit of the N = 6 U(N) k × U(M) −k ABJ theory in the limit of large N and k but finite M. Since the ABJ theory is also dual to type IIA string theory in AdS 4 × CP 3 with flat B-field, it was speculated that the Vasiliev theory must therefore be a limit of this string theory. Roughly speaking, the fundamental string of string theory is simply the flux tube string of the non-Abelian bulk Vasiliev theory. The relations between ABJ vector model,  [87][88][89], vertex operator algebra for compactified spacetime or on a lattice [90][91][92], group theoretical approach of string [35,93].
Another motivation of studying high energy string scattering is to investigate the gravitational effect, such as black hole formation due to high energy string collision, and to understand the nonlocal behavior of string theory. Nevertheless, in [94], it was shown that there is no evidence that the extendedness of strings produces any long-distance nonlocal effects in high energy scattering, and no grounds have been found for string effects interfering with formation of a black hole either.

Stringy symmetries at all energies
In the first part of this review, we discuss stringy symmetries which were calculated to be valid for all energies. These include stringy symmetries calculated by (1) σ-model approach of string theory in the weak field approximation, (2) decoupling of ZNS and stringy Ward identities, (3) Witten's string field theory, (4) Discrete ZNS and w ∞ symmetry of 2D string and (5) Soliton ZNS and enhanced stringy gauge symmetries. We will concentrate on the idea of ZNS and its applications to various calculations of stringy symmetries.
In chapter I we apply ZNS to Sigma model calculation of stringy symmetries [6][7][8]. We calculate generalized stringy symmetries of massive background fields [6,7]. We discover the existence of inter-particle, inter-spin symmetry [6] for higher spin string background fields.
In addition, we demonstrate the decoupling of degenerate positive-norm states by using two approaches, the σ-model calculation [8] and Witten's string field theory [15]. All these results are consistent with calculations of high energy string scattering amplitudes which will be discussed in details in part II and part III. In chapter II, we give a prescription to simplify the calculation of ZNS for higher mass levels [95]. In chapter III, we calculate [23,24] a set of 2D string ZNS with discrete Polyakov momenta and show that its operator algebra forms the w ∞ symmetry algebra of 2D string theory. Incidentally, In chapter V of part II, the corresponding high energy ZNS will be shown to form a high energy w ∞ symmetry [32].
These results strongly suggest that ZNS are symmetry charges of 26D string theory. In chapter IV we calculate soliton ZNS in compact spaces for both closed [25] and open string [26] theories and study their relations to enhanced stringy gauge symmetries.

STRINGY SYMMETRIES
In the first chapter, we review the calculations of string symmetries from ZNS without taking the high energy limit. In the OCFQ spectrum of 26D open bosonic string theory, the solutions of physical state conditions include positive-norm propagating states and two types of ZNS. The latter are [10] Type I :  [6,95] and will be discussed in chapter II.
In the σ-model approach of string theory, one turns on background fields on the worldsheet energy momentum tensor T . Conformal invariance of the worldsheet then requires, in addition to D = 26, cancellation of various q-number anomalies and results to equations of motion of the background fields. A spacetime effective action can then be constructed and used to reproduce string scattering amplitudes. This was a powerful method to study dynamics of the string modes [10]. On the other hand, it was suggested that a spacetime symmetry transformation δΦ for a background field Φ can be generated by a worldsheet where T Φ is the worldsheet energy momentum tensor with background fields Φ and T Φ+δΦ is the new energy momentum tensor with new background fields Φ + δΦ. However, there was no systematic prescription to calculate the worldsheet generator h.
It was then shown that [6] for each spacetime ZNS, one can systematically construct a δT Φ such that In the above equations, A, B, C are positive-norm background fields, θs represent zeronorm background fields, and ∂ 2 ≡ ∂ µ ∂ µ . There are on-mass-shell, gauge and traceless conditions on the transformation parameters θs, which will correspond to BRST ghost fields in a one-to-one manner in WSFT [15]. This will be discussed in section I.D.  (32)) Yang-Mills indices can be constructed in the 10D Heterotic string theories [97].

B. Inter-particle stringy symmetries
It is interesting to see that an inter-particle symmetry transformation for two high spin states at mass level M 2 = 4 can be generated [6] where ∂ µ θ 2 µ = 0, (∂ 2 − 4)θ 2 µ = 0 which are the on-shell conditions of the mixed type I and type II D 2 vector ZNS (1.12) and C (µνλ) and C [µν] are the background fields of the symmetric spin-three and antisymmetric spin-two states respectively at the mass level M 2 = 4. It is important to note that the decoupling of the D 2 vector ZNS, or unitarity of the theory, implies simultaneous change of both C (µνλ) and C [µν] , thus they form a gauge multiplet. This is a generic feature for background fields of higher massive levels in the σ-model calculation of string theory. One might want to generalize the calculation to the second order weak background fields to see the inter-mass level symmetry. This however suffers from the so-called non-perturbative non-renormalizability of 2d σ-model and one is forced to introduce infinite number of counterterms to preserve the worldsheet conformal invariance [98,99].
Note that θ 2 µ in Eqs. (1.11), (1.12) are some linear combination of the original type I and type II vector ZNS calculated by Eqs.(1.1), (1.2). This inter-particle stringy symmetry is consistent with the linear relations among high energy, fixed angle scattering amplitudes of C (µνλ) and C [µν] , which will be discussed in details in part II of the review.

C. Decoupling of degenerate positive-norm states
In the even higher mass levels, M 2 = 6 for example, a new phenomenon begins to show up. Indeed, there are ambiguities in defining positive-norm spin-two and scalar states due to the existence of ZNS in the same Young representations [8]. As a result, the degenerate spin two and scalar positive-norm states can be gauged to the higher rank fields D µναβ and mixed-symmetric D µνα in the first order weak field approximation. Instead of calculating the stringy gauge symmetry at level M 2 = 6, we will only concentrate on the equation of motions. Take the energy-momentum tensor on the worldsheet boundary in the first order weak field approximation to be of the following form.
where τ is the worldsheet time, X ≡ X(τ ). This is the most general worldsheet coupling in the generalized σ-model approach consistent with vertex operator consideration [100,101].
The conditions to cancel all q-number worldsheet conformal anomalous terms correspond to cancelling all kinds of loop divergences [102] up to the four loop order in the 2d conformal field theory. It is easier to use T · T operator-product calculation and the conditions read [8] 2∂ µ D µναβ − D (ναβ) = 0, (1.14a) Here, φ represents all background fields introduced in Eqs. (1.13). It is now clear through  (1.14g) are the gauge conditions for D µναβ and mixed-symmetric D µνα after substituting D 0 µν , D 1 µν and D µ in terms of D µναβ and mixed symmetric D µνα . The remaining scalar particle has automatically been gauged to higher rank fields since Eq.(1.13) is already the most general form of background-field coupling.
This means that the degenerate spin two and scalar positive-norm states can be gauged to the higher rank fields D µναβ and mixed-symmetric D µνα in the first order weak field approximation.
In fact, for instance, it can be explicitly shown [11] that the scattering amplitude involving the positive-norm spin-two state can be expressed in terms of those of spin-four and mixedsymmetric spin-three states due to the existence of a degenerate type I and a type II spintwo ZNS. Although all the four-point amplitudes considered in Ref. [11] contain three tachyons, the argument can be easily generalized to more general amplitudes. This is very different from the analysis of lower massive levels where all positive-norm states seem to have independent scattering amplitudes.
Presumably, this decoupling phenomenon comes from the ambiguity in defining positivenorm states due to the existence of ZNS in the same Young representations. We will justify this decoupling by WSFT in the next section. Finally one expects this decoupling to persist even if one includes the higher order corrections in weak field approximation, as there will be even stronger relations between background fields order by order through iteration.

D. Witten's string field theory (WSFT) calculations
It would be much more convincing if one can rederive the stringy phenomena discussed in the previous sections from WSFT. Not only can one compare the first quantized string with the second quantized string, but also the old covariant quantized string with the BRST quantized string. Although the calculation is lengthy, the result, as we shall see, are still controllable by utilizing the results from first quantized approach in previous sections.
There exist important consistency checks of first quantized string results from WSFT in the literature, e.g. the rederivation of Veneziano and Kubo-Nielson amplitudes from WSFT [103]. In some stringy cases, calculations can only be done in string field theory approach. For example, the pp-wave string amplitudes can only be calculated in the light-cone string field theory [104]. Therefore, a consistent check by both first and second quantized approaches of any reliable string results would be of great importance.
The infinitesimal gauge transformation of WSFT is To compare with our first quantized results in previous sections, we only need to calculate the first term on the right hand side of Eq.(1.15). Up to the second massive level, Φ and Λ can be expressed as where Φ and Λ are restricted to ghost number 1 and 0 respectively, and the BRST charge is The transformation one gets for each mass level are These important observations simplify the demonstration of decoupling of degenerate positive-norm states at higher mass levels, M 2 = 6 and M 2 = 8 more specifically in WSFT.
We will present the calculation for level M 2 = 6. The calculation for M 2 = 8 was discussed in [15]. For M 2 = 4, it can be checked that only C µνλ and C [µν] are dynamically independent and they form a gauge multiplet, which is consistent with result of first quantized calculation presented in the previous sections .
We now show the decoupling phenomenon for the third massive level M 2 = 6, in which Φ and Λ can be expanded as The transformations for the matter part are It can be checked from the above equations that only D µναβ and mixed-symmetric D µνα cannot be gauged away, which is consistent with the result of the first quantized approach in the previous sections. That is , the spin-two and scalar positive-norm physical propagating modes can be gauged to D µναβ and mixed symmetric D µνα . In fact, D 0 µν and D µ can be gauged away by ǫ 0 µνα , ǫ 1 [µν] , ǫ 1 (µν) , ǫ 2 µν and one of the vector parameters, say ǫ 3 µ . The rest, ǫ 4 µ , ǫ 5 µ and ǫ 4 are gauge artifacts of D µναβ and mixed-symmetric D µνα . The transformation for the ghost part are There are nine on-mass-shell conditions, which contains a symmetric spin three, an anti- at least the sum of those at (n − 2)-th and (n − 1)-th massive levels. So positive-norm scalar modes at n-th level, if they exist, will be decoupled according to our decoupling conjecture.
The decoupling of these scalars has important implication on Sen's conjectures on the decay of open string tachyon [106]. Since all scalars on D-brane including tachyon get non-zero vev. in the false vacuum, they will decay together with tachyon and disappear eventually to the true closed string vacuum. As the scalar states together with higher tensor states form a large gauge multiplet at each mass level, and its scattering amplitudes are fixed by the tensor fields, these tensor fields of open string (D25-brane) will accompany the decay process. This means that the whole D-brane could disappear to the true closed string vacuum.

II. CALCULATION OF HIGHER MASSIVE ZNS
Since ZNS are the most important key to generate stringy symmetries, in this chapter we give a simplified method [95] In the OCFQ spectrum, physical states in Eq.(2.1) are subject to the following Virasoro conditions and α 0 ≡ k. Let's first assume we are given positive-norm state solutions of some mass level n. The number of positive-norm degree of freedom at mass level n ( M 2 = 2(n − 1)) is given by Eq.(1.1) and Eq.(1.2) for |x and | x . The only difference is the "mass shift" of L 0 equations.
As is well-known, the L 1 and L 2 equations give the transverse and traceless conditions on the spin polarization. It turns out that, in many cases, the L 1 and L 2 equations will not refer to the L 0 equation or on-mass-shell condition. In these cases, a positive-norm state solution for |Ψ at mass level n will give a ZNS solution L −1 |x at mass level n + 1 simply by taking |x = |Ψ and shifting k 2 by one unit. Similarly, one can easily get a type II ZNS (L −2 + 3 2 L 2 −1 ) | x at mass level n + 2 simply by taking | x = |Ψ and shifting k 2 by two units. For those cases where L 1 and L 2 equations do refer to L 0 equation, our prescription needs to be modified. We will give some examples to illustrate this method. Note that once we generate a ZNS, it soon becomes a candidate of physical state |Ψ to generate two new ZNS at even higher levels.
1. The first ZNS begin at k 2 = 0. This state is suggested from the positive-norm tachyon state |0, k with k 2 = 2. Taking |x = |0, k and shifting k 2 by one unit to k 2 = 0, we get a type I ZNS.
2. At the first massive level k 2 = −2, tachyon suggests a type II ZNS Positive-norm massless vector state suggests a type I ZNS However, massless singlet ZNS Eq.(2.6) does not give a type I ZNS at the first massive level k 2 = −2 since L 1 equation on state Eq.(2.6) refers to L 0 equation, k 2 = 0. This means that L 1 will not annihilate state Eq.(2.6) if one shifts the mass to k 2 = −2.
3. At the second massive level k 2 = −4, positive-norm massless vector state suggests a type II ZNS However, massless singlet ZNS Eq.(2.6) does not give a type I ZNS at mass level k 2 = −4 for the same reason stated after Eq.(2.8). Positive-norm spin-two state at k 2 = −2 suggests a type I ZNS Similar notations will be used in the rest of this paper. Vector ZNS with k 2 = −2 in Eq.(2.8) does not give a type I ZNS for the same reason stated after Eq. (2.8). In this case, however, one can modify |x to be where a, b are undetermined constants. L 0 equation is then trivially satisfied and L 1 , L 2 equations give a : b = 2 : 1. This gives a type I ZNS Similarly, we modify the singlet ZNS with k 2 = −2 in Eq.(9) to be where a, b and c are undetermined constants. L 1 and L 2 equations give 5a + b + 3k 2 c = 0, 5k 2 a + 13b + 3 2 k 2 c = 0. (2.14) For k 2 = −4, we have a : b : c = 5 : 9 : 17 6 . This gives a type I ZNS This completes the four ZNS at the second massive level. It is interesting to note that 4. Similar method can be used to calculate ZNS at level M 2 = 6. We will just list some examples here. They are (from now on, unless otherwise stated, each spin polarization is assumed to be transverse, traceless and is symmetric with respect to each group of indices as in Ref [107]) (2.20) The L 1 and L 2 equations can be easily used to determine a : b : c : d : f = 37 : 72 : 261 : 216 : 450. This gives the type I singlet ZNS

22)
and Note that the modified method was used in Eq.(2.23) and Eq.(2.26).
6. Finally, we calculate general formulas of some zero-norm tensor states at arbitrary mass levels by making use of general formulas of some positive-norm states listed in Ref [107].
a. b.
Note that the Young tabulations of ZNS at level n are subset of the sum of all physical states at levels n − 1 and n − 2.

III. DISCRETE ZNS AND w ∞ SYMMETRY OF 2D STRING THEORY
For the 26D (10D) string theory, it is difficult to do calculations for higher mass string states and extract their symmetry structures which are valid for all energies. This is of course due to the high dimensionality of spacetime. One way to overcome this difficulty has been to probe high energy regimes of the theory and simplify the calculations. This will be done in part II and part III of this review. Another strategy was to study the toy string model, namely, 2D string theory or c = 1 2D quantum gravity. The 2D string theory has been an important laboratory to study non-perturbative information of string theory.
In this chapter, we will derive the w ∞ symmetry structure from the ZNS point of view in the old covariant quantization scheme [23,24]. This is in parallel with the works of [18] and [21,22] where the ground ring structure of ghost number zero operators were identified in the BRST quantization. Moreover, the results we obtained will justify the idea of ZNS used in the 26D (or 10D) theories as discussed previously.
Unlike the discrete Polyakov states, we will find that there are still an infinite number of continuum momentum ZNS in the massive levels of the 2D spectrum and it is very difficult to give a general formula for them just as in the case of 26D theory [12][13][14]. However, as far as the dynamics of the theory is concerned, only those ZNS with Polyakov discrete momenta are relevant. This is because all other ZNS are trivially decoupled from the correlation functions due to kinematic reason. Hence, we will only identify all discrete ZNS or discrete gauge states (DGS) in the spectrum. The higher the momentum is, the more numerous the DGS are found. In particular, we will give an explicit formula for one such set of DGS in terms of Schur polynomials. Finally, we can show that these DGS carry the w ∞ charges and serve as the symmetry parameters of the theory.
A. 2D string theory

ZNS in 2D string theory
We consider the two dimensional critical string action [109] S = 1 8π with φ being the Liouville field. For c = 1 theory Q, which represents the background charge of the Liouville field, is set to be 2 √ 2 so that the total anomalies cancels that from ghost contribution.
For simplicity here we consider only one of the chiral sectors, while the other sector (denoted byz) is the same. The stress energy tensor is If we define the mode expansion of X µ = (φ, X) by , 0) and the zero mode α µ 0 = f µ = (ǫ, p), we find the Virasoro generators The vacuum |0 is annihilated by all α µ n with n > 0. In the old covariant quantization, physical states |ψ are those satisfy the condition L n |ψ = 0 f or n > 0, . We thus no longer restrict ourselves in the material gauge, and the Liouville field φ will play an important role in the following discussions.
In general, there are two types of ZNS in 2D string theory, Type I: Type II: while the type II ZNS is The DGS corresponding to ψ − (type I) The DGS corresponding to discrete momenta of ψ + 3 2 ,± 1 2 are degenerate, i.e., the type I and type II DGS are linearly dependent: There is no pure φ DGS here. In general, the ψ + sector has fewer DGS than the ψ − sector at the same discrete momenta, as a result, the pure φ DGS only arise at the minus sector. This fact is related to the degeneracy of the DGS in the plus sector. Historically the ψ + sector discrete states arise when one considers the singular gauge transformation constructed from the difference of the two plus gauge states [110][111][112][113]115].

Generating the Discrete ZNS
In this section, we will give a general formula for the DGS. In general, there are many DGS for each discrete momentum. The higher the momentum is, the more numerous the DGS are found. We first express the discrete states in Eq.(3.7) in terms of Schur polynomials, which are defined as follows: where S k is the Schur polynomial, a function of {a k } = {a i : i ∈ Z k }. Performing the operator products in Eq.(3.7) , the discrete states ψ ± J,M can be written as

(3.17)
We can write and Taylor expand X(z i ) around z i = 0 In Eq.  We now begin to study the DGS. One first notes that the DGS in Eq.(3.14) can be generated by dz 2πi e − √ 2φ(z) ψ − 1 2 ,± 1 2 (0). In general it is also possible to write down explicitly one of the many ZNS for each discrete momentum in the ψ − sector as follows This is also suggested by the following result [18][19][20] where the r.h.s. is meant to be a DGS. We thus have explicitly obtained a DGS for each ψ − discrete momentum. We stress that there are still other DGS in this sector, for example, the states can be shown to be of dimension 1. Since they are pure φ states, they are also DGS. This expression reminds us of Eq.(3.7). However, there is no SU(2) structure in the φ direction, and the usual techniques of c = 1 2d conformal field theory cannot be applied. The pure φ DGS are only found in the minus sector.
For the plus sector, the operator products of the discrete states defined in Eq.(3.7) form a w ∞ algebra [18][19][20] dz which is the same as ∆(J, M, −i √ 2X(z)) except that the j th row is replaced by

w ∞ Charges and conclusion
It was shown [18][19][20] that the operators products of the states ψ + J,M defined in Eq.(3.7) satisfy the w ∞ algebra in Eq. (3.25). By construction Eq.(3.26) one can easily see that the plus sector DGS G + J,M carry the w ∞ charges and can be considered as the symmetry parameters of the theory. In fact, the operator products of the DGS G + J,M of Eq.(3.26) form the same w ∞ algebra where the r.h.s. is defined up to another DGS. The high energy limit of Eq.(3.26) will be discussed in section V.E of part II of this review.
In summary, we have shown that the spacetime w ∞ symmetry parameters of 2D string theory come from solution of equations Eq.(3.8) and Eq.(3.9). This argument is valid also in the case of 26D (or 10D) string theory although it would be very difficult to exhaust all the solutions of the ZNS [12][13][14]. This difficulty is, of course, related to the high dimensionality of spacetime. The DGS we introduced in the old covariant quantization in this chapter seem to be related to the ghost sectors and the ground ring structure [21,22] in the BRST quantization of the theory.

B. 2D superstring theory
In this section, we will generalize our results in the previous section to N = 1 super-Liouville theory in the worldsheet supersymmetric way [24]. We will work out the DGS of the Neveu-Schwarz sector in the zero ghost picture. We first discuss the N = 1 super-Liouville theory and set up the notations. We then calculate the general formula for discrete positive-norm states in a worldsheet superfield form. Finally a general formula of discrete ZNS or DGS will be presented and w ∞ charges will then be calculated.

2D super-Liouville theory
The N = 1 two dimensional supersymmetric Liouville action is given by [116] S = 1 8π where Φ is the super-Liouville field,Ŷ the superfield curvature, dz = dzdθ and with Bold faced variables denote superfields hereafter. Forĉ = 1 = 2 3 c theory Q, which represents the background charge of the super-Liouville field, is set to be 2 so that the total conformal anomaly cancels that from conformal and superconformal ghost contribution.
The equations of motion show that the left and right-moving components of X µ decouple, and the auxiliary fields F µ vanish. As a result, we need to consider only one of the chiral sectors, while the other (anti-holomorphic) sector has a similar formula. The stress energy tensor is where D = ∂ θ + θ∂ z , and now X µ = X µ (z) + θψ µ (z).
For the Neveu-Schwarz sector, if we define the mode expansion by : α µ,−k α µ k : + 1 2 The vacuum |0 is annihilated by all α µ n and b µ r with n > 0 and r > 0. In the old covariant quantization of the theory, physical states |ψ are those satisfying the conditions   where are zero modes of the level 2 SU(2) κ=2 Kac-Moody algebra inĉ = 1 2d superconformal field theory. Here we note that the NS sector corresponds to states with J ∈ Z while the Ramond sector corresponds to those with J ∈ Z + 1 2 .
To find the explicit expressions for the discrete states, we first define the super-Schur polynomials, where k 2 denotes the integral part of k 2 , as the N = 1 generalization to the Schur polynomial S k , which is defined as (3.42) Note that S k ({i∂ m X/m!}) = S 2k (iX). Direct integration shows that For example, by we have and Performing the operator products in Eq.(3.39) , the discrete states Ψ ± J,M are The vertex operators correspond to the upper components of Eq.(3.49), i.e., . Using Eq.(3.45) and Eq. (3.49) it is found that It can be checked that it satisfies the physical state conditions Eq.(3.37) For M = J − 3, a straighforward calculation gives It is now easy to write down an expression for general M , with S k = S k (−iX(z 0 )) and S k = 0 if k < 0. We will denote the rank (J − M) "primed"- where the RHS is defined up to a DGS.
In general, there are two types of ZNS in the old covariant quantization of the theory, Type I: Type II: while the type II state is As in the bosonic Liouville theory [23], the ZNS corresponding to the discrete momenta of Ψ + 2,±1 are degenerate, i.e., the type I and type II gauge states are linearly dependent: For the minus sector, type I DGS is and type II DGS is Note that 3G −,I 2,±1 − G −,II 2,±1 is a "pure Φ" DGS, similar to the DGS in the bosonic Liouville theory.
We now apply the scheme used in [23] to derive a general formula for the DGS. From Eq.(3.55), the DGS in the minus sector can be written down explicitly as follows (3.63) We thus have explicitly obtained a DGS for each Ψ − discrete momentum. However, there are still other DGS in this sector, for example, the states can be shown to satisfy the physical state conditions. Since they are "pure Φ" states, they are also DGS. For the plus sector, we can subtract two (distinct) positive norm discrete states at the same momentum to obtain a pure DGS As an example, with Eq.(3.65) one finds which is exactly the state we found in Eq.(3.60). We thus have explicitly obtained a DGS for each Ψ + momentum.
where the RHS is defined up to another DGS.
We have demonstrated that the spacetime w ∞ symmetry parameters in the 2D superstring theory come from solution of equations Eq.(3.56) and Eq. (3.57). This phenomenon should survive in the more realistic high dimensional string theory [12,13,15] , although it would be difficult to find the general solution of ZNS (due to the high dimensionality of spacetime). The DGS in the old covariant quantization of the theory is related to the ground ring structure in the BRST approach.

TRY
In this chapter we calculate ZNS in the spectrum of compactified closed and open string theories [25,26]. For simplicity we will only do the calculations on torus compactifications.
The programs can certainly be generalized to more complicated background geometries. We will see that there exist soliton ZNS which generate various enhanced stringy symmetries of the theories.

A. Compactified closed string
In this section, we study soliton gauge states in the spectrum of bosonic string compactified on torus. The enhanced Kac-Moody gauge symmetry, and thus T-duality, is shown to be related to the existence of these soliton ZNS in some moduli points.

Soliton ZNS on
In the simplest torus compactification, one coordinate of the string was compactified on a circle of radius R X 25 (σ + 2π, π) = X 25 (σ, π) + 2πR n (4.1) The single valued condition of the wave function then restricts the allowed momenta to be p 25 = m/R with m, n ∈ Z. The mode expansion of the compactified coordinate for right (left) mover is We have normalized the string tension to be 1 4πT = 1 or α ′ = 2 .The Virasoro operators can be written as and where and by interchanging all left and right mover operators, one gets I.b and II.b states. Type II states are ZNS only at critical space-time dimension. We will only calculate type a states.
Similar results can be easily obtained for type b states. For type I.a state, the m = 0 constraint of Eq.(4.10) gives where N ≡ The basis of Λ D ⋆ is 1 The mode expansion of the compactified coordinates is where O(D, D, Z) is the discrete T-duality group and dim µ = D 2 . To complete the parametrization of the moduli space, one needs to introduce an antisymmetric tensor field B ij in the bosonic string action. This will modify the right (left) momenta to be (4.29) We are now ready to discuss the gauge state. As a first step, we restrict ourselves to moduli space with B ij = 0 . For the type I.a state, the m = 0 constraint of Eq.(4.10) for massless states gives

Massive soliton ZNS
In this section we derive the massive soliton ZNS at the first massive level M 2 = 2. We will find that soliton ZNS exists at infinite number of moduli points. One can also show that they exist at an infinite number of massive level. The existence of these massive soliton ZNS implies that there is an infinite enhanced gauge symmetry structure of compactified string theory. For type I.a state, the m = 0 constraint of Eq.(4.10) gives where we have included a T-duality transformation R → 2 R for some moduli points. Note that Eq.(4.40) tells us that massive soliton ZNS exists at an infinite number of moduli point.   The vertex operators of all soliton ZNS can be easily calculated and written down. Similar results can be obtained for type b ZNS. One can also calculate propagating soliton states by using the same technique. We summarize the moduli points which exist soliton state and soliton ZNS as following: a. Soliton ZNS : b. Soliton states : with mn = 0. For say R = √ 2, one gets M 2 = m 2 2 (n = 0). This means that we have an infinite number of massive soliton ZNS at any higher massive level of the spectrum. One can even explicitly write down the vertex operators of these soliton ZNS. We conjecture that the w ∞ symmetry of 2D string theory [23,109] can be realized in these soliton ZNS. Other moduli points also consist of higher massive soliton ZNS in the spectrum.
Many known spacetime symmetries of string theory can be shown to be related to the existence of ZNS in the spectrum. The Heterotic ZNS for the 10D Heterotic string [12] and the discrete ZNS [23,109] for the toy 2D string are such examples. We have introduced soliton ZNS for compactified closed string in this chapter, and have related them to the enhanced Kaluza-Klein Kac-Moody symmetries in the theory. In many cases, especially for the massive states, it is easier to study stringy symmetries in the ZNS sector than in the propagating spectrum directly.
Since the discrete T-duality symmetry group for bosonic closed string is the Weyl subgroup of the enhanced gauge group, it can also be considered as due to the existence of soliton ZNS. It is not clear whether other discrete duality symmetry group can be understood in this way. Finally, it would be interesting to consider more complicated compactification, e.g. orbifold and Calabi-Yau compactifications and study the relation between soliton ZNS and duality symmetries.

B. Compactified open string
In this section, we study the mechanism of enhanced gauge symmetry of bosonic open string compactified on torus by analyzing the ZNS (nonzero winding of Wilson line) in the spectrum [26]. Unlike the closed string case discussed in the previous section, we will find that the soliton ZNS exist only at massive levels.
These soliton ZNS correspond to the existence of enhanced massive stringy symmetries with transformation parameters containing both Einstein and Yang-Mills indices in the case of Heterotic string [12]. In the T-dual picture, these symmetries exist only at some discrete values of compactified radii when N D-branes are coincident.

Chan-Paton ZNS
We first discuss ZNS of uncompactified open string with Chan-Paton factor and its implication on on-shell symmetry and Ward identity. For simplicity, we consider the oriented U (N) case. The vertex operators of massless gauge state is where λ ∈ U (N) , i ∈ N, j ∈ N and a ∈ adjoint representation of U (N). The on-shell conformal deformation and U (N) gauge symmetry to lowest order in the weak background and δA a µ = ∂ µ θ a (4.52) with T the energy momentum tensor and A a µ the massless gauge field. One can verify the corresponding Ward identity by calculating e.g., 1-vector and 3tachyons four point correlators. The amplitude is calculated to be We now discuss the massive ZNS. The vertex operator of type I massive vector ZNS is We note that the ZNS polarization contains both Einstein and Yang-Mills indices. This is very similar to the 10d closed Heterotic string case [12,13]. The only difference is that in the Heterotic string, one could have more than one Yang-Mills index. The on-shell conformal deformation and the mixed Einstein-Yang-Mills-type symmetry to lowest order weak field approximation are ( − 2) θ a µ = ∂ · θ a = 0 and δM a µν = ∂ µ θ a ν + ∂ ν θ a µ . (4.57) One can also derive the corresponding massive Ward identity by calculating the decay rate of one massive state to three tachyons. The most general amplitude is calculated to be and In Eq.(4.58) ε a etc. are polarizations corresponding to tachyons and ε b µν , ε b µ is polarization of the massive state. The above amplitude satisfies the following ward identity Similar consideration can be applied to the following type II massive scalar gauge state which corresponds to a massive U (N) symmetry.

Chan-Paton soliton ZNS on R 25 ⊗ T 1
We now discuss soliton ZNS on torus compactification of bosonic open string. As is well known, the massless U (N) gauge symmetry will be broken in general after compactification unless N D-branes, in the T-dual picture, are coincident. We will see that when D-branes are coincident, one has enhancement of (unwinding) ZNS and the massless U (N) symmetry will be recovered. These ZNS can be considered as charges or symmetry parameters of U (N) group.
In the discussion of open string compactification, one needs to turn on the Wilson line or nonzero background gauge field in the compact direction. This will effect the momentum in the compact direction, and the Virasoro operators become Note that in Eq.(4.64), α 25 0 ≡ p 25 which also appears in the first term in Eq.(4.63). k is the 25d momentum. θ i , R are the gauge and space-time moduli respectively and l is the winding number in the compact direction. The spectrums of type I and type II ZNS become and For the massless case I = l = 0, one gets N 2 massless solution from equation Eq.(4.65) if all θ i are equal, or in the T-dual picture when N D-branes are coincident. These N 2 massless ZNS correspond to the charges of massless U (N) gauge symmetry. There is no type II massless solution in Eq.(4.66).
We are now ready to discuss the interesting massive case. For M 2 = 2 and general moduli (R, θ i ), 1. I = 1, l = 0, one gets two ZNS solutions from Eq.(4.65): and Now since |θ i − θ j | < 2π, for any given R, there is at most one solution of (|l| , |θ i − θ j |).
One is tempted to consider the case That means in the moduli R = √ 2n, θ i with n ∈ Z + , one has soliton ZNS which imply We would like to point out that similar Einstein-Yang-Mills-type symmetry was discovered before in the closed Heterotic string theory. There, however, one could have more than one Yang-Mills indices on the transformation parameters. For the type II states with Moreover, these linear relations can be used to fix the proportionality constants or ratios among high energy, fixed angle scattering amplitudes of different string states algebraically at each fixed mass level.
The part II of this review is organized as following. Chapter V is one of the main part of this review. We will use three different methods to explicitly prove Gross conjectures [28][29][30][31][32]. These are the decoupling of high energy ZNS, the high energy Virasoro constraints and a saddle-point calculation. In addition, we show that the high energy limit of the discrete ZNS in 2D string theory constructed in part I form a high energy w ∞ symmetry. This result strongly suggests that the linear relations obtained from decoupling of ZNS in 2D string theory are indeed related to the hidden symmetry also for the 26D string theory.
In chapter VI, in addition to analyze ZNS in the helicity basis in the OCFQ string spectrum, we will work out ZNS in the light-cone DDF construction of string spectrum and ZNS in the BRST WSFT [16]. In chapter VII, we discuss hard closed string scatterings [36]. The KLT relation [44] will be extensively used. We also discuss string BCJ relation [38,41,[124][125][126], which is of much interest in the recent development of calculation of field theory scattering amplitudes. In chapter VIII, we calculate four classes of hard superstring scattering amplitudes and derive the ratios among them [34]. In chapter IX, we discuss hard string scattering from D-branes/ O-planes, and closed string decays to open string [42,47,127]. Finally in chapter X, we discuss both hard open and closed string scatterings in the compact spaces [55,56].

V. INFINITE LINEAR RELATIONS AMONG HIGH ENERGY, FIXED ANGLE STRING SCATTERING AMPLITUDES
In this chapter, we will use three different methods to explicitly prove Gross conjectures for 26D bosonic open string theory. We will also show that the high energy limit of discrete ZNS of 2D string constructed in part I form a high energy w ∞ symmetry. We begin with an example [27][28][29] to do the calculation.

A. The first example
For our purpose here, there are four ZNS at mass level M 2 = 4. The complete list of ZNS were calculated in chapter II, and the corresponding Ward identities were calculated to be [14] k µ θ νλ T (µνλ) where θ µν is transverse and traceless, and θ ′ λ and θ λ are transverse vectors. They are polarizations of ZNS. In each equation, we have chosen, say, v 2 (k 2 ) to be the vertex operators constructed from ZNS and k µ ≡ k 2µ . Note that Eq.(5.3) is the inter-particle Ward identity corresponding to D 2 vector ZNS in Eq.(1.12) obtained by antisymmetrizing those terms which contain α µ −1 α ν −2 in the original type I and type II vector ZNS [6]. We will use 1 and 2 for the incoming particles and 3 and 4 for the scattered particles. In Eq.(5.1) to Eq.(5.4),

1,3 and 4 can be any string states (including ZNS) and we have omitted their tensor indices
for the cases of excited string states. For example, one can choose v 1 (k 1 ) to be the vertex operator constructed from another ZNS which generates an inter-particle Ward identity of the third massive level. The resulting Ward-identity of Eq. , T µ χ } is identified to be the amplitude triplet [6] of the spinthree state. T in the CM frame contained in the plane of scattering. They satisfy the completeness relation where µ, ν = 0, 1, 2 and α, β = P, L, T. Diag η µν = (−1, 1, 1). One can now transform all µ, ν coordinates in Eq.(5.1) to Eq.(5.4) to coordinates α, β. For Eq.(5.1), we have θ µν = e µ L e ν L − e µ T e ν T or θ µν = e µ L e ν T + e µ T e ν L . In the high energy E → ∞, fixed angle φ CM limit, one identifies e P = e L and Eq.(5.1) gives ( we drop loop order χ here to simplify the notation) In Eq.(5.9) and Eq.(5.10), we have assigned a relative energy power for each amplitude.
For each longitudinal L component, the order is E 2 and for each transverse T component, the order is E. This is due to the definitions of e L and e T in Eq.(5.6) and Eq.(5.7), where e L got one energy power more than e T . By Eq.(5.9), the E 6 term of the energy expansion for T LLL is forced to be zero. As a result, the possible leading order term is E 4 . Similar rule applies to T LLT in Eq.(5.10). For Eq.(5.2), we have θ ′µ = e µ L or θ ′µ = e µ T and one gets, in the high energy limit, For the D 2 Ward identity, Eq.(5.3), we have θ µ = e µ L or θ µ = e µ T and one gets, in the high energy limit, It is important to note that T [LL] in Eq.(5.13) originate from the high energy limit of , and the antisymmetric property of the tensor forces the leading E 4 term to be zero.
Finally the singlet zero norm state Ward identity, Eq.(5.4), imply, in the high energy limit, One notes that all components of high energy amplitudes of symmetric spin three and antisymmetric spin two states appear at least once in Eq.(5.9) to Eq.(5.15). It is now easy to see that the naive leading order amplitudes corresponding to E 4 appear in Eq.(5.9), (5.11), Eq.(5.13) and Eq.(5.15). However, a simple calculation shows that T 4 LLL = T 4 LT T = T 4 (LL) = 0. So the real leading order amplitudes correspond to E 3 , which appear in Eq.(5.10), Eq.(5.12) and Eq.(5.14). A simple calculation shows that [27,29] Note that these proportionality constants are, as conjectured by Gross [3,4], independent of the scattering angle φ CM and the loop order χ of string perturbation theory. They are also independent of particles chosen for vertex v 1,3,4 . The ratios in Eq.(5.16) should be measurable if the energy scale of string theory is not Planckian. Most importantly, we now understand that the ratios originate from ZNSs in the OCFQ spectrum of the string!
The subleading order amplitudes corresponding to E 2 appear in Eq.(5.9), (5.11), Eq.(5.13) and Eq.(5.15). One has 6 unknown amplitudes and 4 equations. Presumably, they are not proportional to each other or the proportional coefficients do depend on the scattering angle φ CM . We will justify this point later in our sample calculation. Our calculation here is purely algebraic without any integration and is independent of saddle point calculation in [1][2][3][4][5].
It is important to note that our result in Eq.(5.16) is gauge invariant as it should be since we derive it from Ward identities Eq.(5.1) to Eq.(5.4). On the other hand, the result obtained in [5] with T 3
We give one example here [27][28][29] to illustrate the meaning of the massive gauge invariant amplitude. To be more specific, we will use two different gauge choices to calculate the high energy scattering amplitude of symmetric spin three state. The first gauge choice is In the high energy limit, using the helicity decomposition and writing ǫ µνλ = Σ µ,ν,λ e α µ e β ν e δ λ u αβδ ; α, β, δ = P, L, T, we get The second gauge choice is In the high energy limit, similar calculation gives It is now easy to see that the first and second terms of Eq. LLT , was missing in the calculation of Ref [5]. The issue was discussed in details in [30]. To further justify our result, we give a sample calculation in the next section.

A sample calculation of mass level M 2 = 4
In this section, we give a detailed calculation of a set of sample scattering amplitudes to explicitly justify our results presented in the last section. Since the proportionality constants in Eq.(5.16) are independent of particles chosen for vertex v 1,3,4 , for simplicity, we will choose them to be tachyons. For the string-tree level χ = 1, with one tensor v 2 and three tachyons v 1,3,4 , all scattering amplitudes of mass level M 2 2 = 4 were calculated in [14]. They are ( s − t channel only) In deriving Eq.(5.21) to Eq.(5.24), we have made the SL(2, R) gauge fixing by choosing To calculate the high energy expansions (s, t → ∞, s t = fixed ) of these scattering amplitudes, one needs the following energy expansion formulas [29] where ξ = sin φ CM 2 and η = cos φ CM 2 . The high energy expansions of Mandelstam variables are given by We can now explicitly calculate all amplitudes in Eq.(5.16). After some algebra, we get is the high energy limit of with s + t + u = 2N − 8, and we have calculated it up to the next leading order in E. We thus have justified Eq.(5.16) with LT T = T 4 (LL) = 0 as claimed in the previous section. Note that, unlike the leading E 9 order, the angular dependences of E 7 order are different for each amplitudes. The subleading order amplitudes corresponding to T 2 (E 8 order) appear in Eq.(5.9), (5.11), Eq.(5.13) and Eq.(5.15). One has 6 unknown amplitudes. An explicit sample calculation gives which show that their angular dependences are indeed different or the proportional coefficients do depend on the scattering angle φ CM .

Results of mass level M 2 = 6
The calculations for M 2 = 4 in the previous section can be generalized to M 2 = 6 [29].
The calculation was however much more tedious, and to the leading order in energy one ended up with 8 equations and 9 amplitudes. A calculation showed that [29]  It was remarkable to see that the linear relations obtained by high energy limit of stringy Ward identities or decoupling of ZNS were just good enough to solve all the high energy amplitudes in terms of one amplitude! It was even more remarkable to see that the ratios obtained by solving these linear relations matched exactly with the sample calculations for the high energy string amplitudes. However, the calculation soon becomes too complicated to manage when one goes to even higher mass levels. In the next section, we will adopt another strategy to generalize the calculations to arbitrary mass levels.

B. Decoupling of high energy ZNS at arbitrary mass levels
In the following three sections, we will use three methods to generalize our calculations in the previous section to arbitrary mass levels. In this section we will first use method of decoupling of high energy ZNS. We will focus on 4-point functions in this section, although our discussion can be generalized to higher point correlation functions. Due to Poincare symmetry, a 4-point function is a function of merely two parameters. Viewing a 4-point function as the scattering amplitude of a two-body scattering process, one can choose the two parameters to be E (one half of the center of mass energy for the incoming particles i.e., particles 1 and 2 in Fig. 1, and φ (the scattering angle between particles 1 and 3). For convenience we will take the center of mass frame and put the momenta of particles 1 and 2 along the X 1 -direction, with the momenta of particles 3 and 4 on the X 1 − X 2 plane. The momenta of the particles are They satisfy k 2 i = −m 2 i . In the high energy limit, the Mandelstam variables are where E is related to p and q as The polarization bases for the 4 particles are The high energy limit under consideration is Based on the saddle-point approximation of Gross and Mende [1,2], Gross and Manes [5] computed the high energy limit of 4-point functions in the bosonic open string theory. To explain their result, let us first define our notations and conventions. For a particle of momentum k, we define an orthonormal basis of polarizations {e P , e L , e T i }. The momentum polarization e P is proportional to k, the longitudinal polarization e L is the space-like unit vector whose spatial component is proportional to that of k, and e T i are the space-like unit-vectors transverse to the spatial momentum. As an example, for k pointing along the the basis of polarization is where M is the mass of the particle. In general, e T i (for i = 3, · · · , 25) is just the unit vector in the X i -direction, and the definitions of e P , e L and e T 2 will depend on the motion of the particle. For e T 2 , which is parallel to the scattering plane, we denote it by e T (see Fig. 1).
The orientations of e T 2 for each particle are fixed by the right-hand rule, k ×e T 2 = e T 3 , where k is the spatial momentum of 4-vector k. We will use the notation ∂ n X A ≡ e A · ∂ n X for Each vertex is a polynomial of {∂ n X A } times the exponential factor exp(ik · X). Among all possible choices of polarizations for the 4 vertices in a 4-point function, we now argue that only the polarizations L and T need to be considered. The polarization P can be gauged Kinematic variables in the center of mass frame away using ZNS [16]. To see why we can ignore all T i 's except T , we note that a prefactor ∂ n X A can be contracted with the exponent ik · X of another vertex. The contribution of this contraction to a scattering amplitude is proportional to k A ∼ E. If k A = 0 (i.e., if A = L or T ), this is much more important in the high energy limit than a contraction with another prefactor ∂ m X B , which gives η AB ∼ E 0 . Therefore, if all polarizations in all vertices are chosen to be either L or T , the resulting 4-point function will dominate over other choices of polarizations.
The central idea behind the algebraic approach used in [27][28][29] and [30] was the decoupling of ZNS (i.e., the requirement of stringy gauge invariance). A crucial step in the derivation is to replace the polarization e P by e L in the ZNS. It is assumed that, while ZNS decouple at all energies, the replacement leads to states that are decoupled at high energies.

Main results
For brevity, we will refer to all 4-point functions different from each other by a single vertex at the same mass level as a "family". When we compare members of a family, we only need to specify the vertex which is changed.
A 4-point function will be said to be at the leading order if it is not subleading to any of its siblings. We will ignore those that are not at the leading order. Our aim is to find the numerical ratios of all 4-point functions in the same family at the leading order. Apparently, there are more 4-point functions at the leading order at higher mass levels. Our goal may seem insurmountable at first sight.
Saving the derivation for later, we give our main results here. A 4-point function is at the leading order if and only if the vertex V under comparison is a linear combination of vertices of the form where The corresponding states are of the form The mass squared is 2(N − 1). All other states involving α T −2 , α A −3 , · · · are subleading. Using the notation 2 where M = 2(N − 1). This formula tells us how to trade ∂X L and ∂ 2 X L for ∂X T , so that all 4-point functions can be related to the one involving only ∂X T in V 2 . The formula above applies equally well to all vertices.
Since we know the value of a representative 4-point function [27,29,30] 2 More rigorously, V 2 needs to be a physical state in order for the correlation function to be well-defined.
We should keep in mind that our results should be applied to suitable linear combinations of Eq.(5.58), possibly together with subleading states, to satisfy Virasoro constraints.
where T (N) is the high-energy limit of

Decoupling of high energy ZNS
Before we go on, we recall some terminology used in the old covariant quantization. A state |ψ in the Hilbert space is physical if it satisfies the Virasoro constraints To get a rough idea about how each vertex operator scales with E in the high energy limit, we associate a naive dimension to each prefactor ∂ m X A according to the following The reason is the following. Each factor of ∂ m X µ has the possibility of contracting with the exponent ik i · X of another vertex operator so that it scales like E in the high energy limit.
Furthermore, components of the polarization vectors e T and e L scale with E like E 0 and E 1 , respectively.
When we compare vertex operators at the same mass level, the sum of all the integers m in ∂ m X A is fixed. Roughly speaking, it is advantageous to have many ∂X A than having fewer number of ∂ m X A with m > 1. For example, at the first massive level, the vertex operator ∂X T ∂X T e ik·X has a larger naive dimension than ∂ 2 X T e ik·X .
The counting of the naive dimension does not take into consideration the possibility that the coefficient of the leading order term happens to vanish by cancellation. The true dimension of a vertex operator can be lower than its naive dimension, although the reverse never happens.
Through experiences accumulated from explicit computations [27][28][29][30], we find that the highest spin vertex is always at the leading order in its family. Since the naive dimension of this state equals its true dimension, any state with a lower naive dimension than this vertex operator can be ignored. This implies that we can immediately throw away a lot of vertex operators at each mass level, but there are still many left. The problem is that, although there are ∂X T m , it may be possible that having extra factors of ∂X L , which has a higher naive dimension than ∂X T , can compensate the disadvantage of these factors. However, explicit computations at the first few massive levels showed that this never happens.
We will now argue why this is generically true, and show in this section that the only states that will survive the high energy limit at level N are of the form Our argument is essentially based on the decoupling of ZNS in the high energy limit.
Thanks to the Virasoro algebra, we only need two Virasoro operators to generate all high energy ZNS or HZNS. Here M is the mass operator, i.e., M 2 = −k 2 when acting on the state |0, k .
a. Irrelevance of other states To prove that only states of the form Eq.(5.67) are at the leading order, we shall prove that (i) any state which has an odd number of α L −1 is irrelevant (i.e., subleading in the high energy limit), and (ii) any state involving a creation operator whose naive dimension is less than its mode index n, i.e., states belonging to is also irrelevant. We proceed by mathematical induction.
First we prove that any state which has a single factor of α L −1 is irrelevant, and that any state with two α L −1 's is irrelevant if it contains an operator of naive dimension less than its index.
Consider the HZNS L −1 χ where χ is any state without any α L −1 , and it is at level (N − 1). Note that, except α L −1 , the naive dimension of an operator is always less than or equal to its index (we exclude α P −1 as mentioned above). This means that the naive dimension of χ is less than or equal to (N − 1). Since we know that at level N, the state Eq.(5.67) has true dimension N, when computing L −1 χ in the high energy limit, we can ignore everything with naive dimension less than N. This means that we need L −1 to increase the naive dimension of χ by no less than 1. In the high energy limit of L −1 only the first term will increase the naive dimension of χ by 1. All the rest do not change the naive dimension. This means that, to the leading order, This is a state with a single factor of α L −1 and it is a HZNS, so it should be decoupled in the high energy limit. Now consider an arbitrary state χ at level (N − 1) which has a single factor of α L −1 . If χ involves any operator whose naive dimension is less than its index, the naive dimension of χ is at most (N − 1). In the high energy limit except the first two terms, all other terms are irrelevant because they contain a single factor of α L −1 . As the second term has a naive dimension (n − 1) and can be ignored, we conclude that α L −1 χ is irrelevant. Suppose χ is an arbitrary state at level (N − 1) which has 2m factors of α L −1 's. The high energy limit of L −1 χ is given by Eq.(5.73). The second term has (2m − 1) factors of α L −1 and is irrelevant. The rest of the terms, except the first, are irrelevant because they contains at least one operator from the set Eq.(5.70). Hence the first term is a HZNS and is irrelevant.
We have proved our first claim for (m + 1), i.e., a state with (2m + 1) factors of α L −1 decouple at high energies.
Similarly, consider the case when χ is at level (N − 1) and has (2m − 1) factors of α L −1 . Furthermore we assume that it involves operators from the set Eq.
Note that Eq.(5.83) is the high energy limit of the second term of type II ZNS. It is easy to see that the decoupling of (5.81) implies the decoupling of (5.83). So one can neglect the effect of (5.83) even though it is of leading order in energy. It turns out that this phenomena persists to any higher mass level as well. By solving Eqs.
It is easy to see that the decoupling of Eq.
where many terms are omitted because they are not of the form (5.67). This implies that Using this relation repeatedly, we get where the double factorial is defined by (2m − 1)!! = (2m)! 2 m m! . Next, consider another class of HZNS calculated from type II ZNS in Eq.(1.2) Again, irrelevant terms are omitted here. From this we deduce that which leads to Our main result Eq.(5.60) is an immediate result of combining Eq.(5.91) and Eq.(5.94).

C. High energy Virasoro constraints
In this section we will establish a "dual description" of our approach explained above.
The notion dual to the decoupling of high energy ZNS is Virasoro constraints.
Let us briefly explain how to proceed. First write down a state at a given mass level as linear combination of states of the form Eq.(5.67) with undetermined coefficients, which are interpreted as the Fourier components of spacetime fields. Requiring that the Virasoro generators L 1 and L 2 annihilate the state implies several linear relations on the coefficients.
The linear relations can then be solved to obtain ratios among all fields.
To compare the results of the two dual descriptions, we note that the correlation functions can be interpreted as source terms for the particle corresponding to a chosen vertex. Thus the ratios among sources should be the same as the ratios among the fields, since all fields of the same mass have the same propagator. However, some care is needed for the normalization of the field variables. One should use BPZ conjugates to determine the norm of a state and normalize the fields accordingly.

Examples
To illustrate how Virasoro constraints can be used to derive linear relations among scattering amplitudes at high energies, we give some explicit examples in this section. We will calculate the proportionality constants among high energy scattering amplitudes of different string states up to mass levels M 2 = 8. The results are of course consistent with those of previous work [27][28][29] using high energy ZNS.
At the mass level M 2 = 4, the most general form of physical states at mass level M 2 = 4 are given by The Virasoro constraints are and similar amplitudes hereafter can be found in [27][28][29] and the result obtained is consistent with the previous ZNS calculation in [27,28] or Eq.(5.60).
The ratios for M 2 = 6 and M 2 = 8 can be obtained similarly in Appendix A. At M 2 = 6, : − √ 6 9 : which is consistent with the previous ZNS calculation in [29] or Eq.(5.40). It also agree with 3 The normalization factors are determined by the inner product of a state with its BPZ conjugate.
the results of Eq.(5.60) after Young tableaux decomposition. At M 2 = 8, : which can be checked to be remarkably consistent with the results of Eq.(5.60) after Young tableaux decomposition.

General mass levels
In this section we calculate the ratios of string scattering amplitudes in the high energy limit for general mass levels by imposing Virasoro constraints. The final result will, of course, be exactly the same as what we obtained by requiring the decoupling of high energy ZNS.
In the presentation here we use the notation of Young's tableaux.
We consider the general mass level M 2 = 2 (N − 1). The most general state can be written as where 1/ (j m j m j !) are the normalization factors and we defined the abbreviation so that the total mass is N. Since the upper indices µ j −j are symmetric, we used the Young tableaux notation to denote the coefficients in Eq.(5.103). The direct product ⊗ acts on the Young tableaux in the standard way, for example To be clear, for example n = 4, the state can be written as Next, we will apply the Virasoro constraints to the state Eq.(5.103 ). The only Virasoro constraints which need to be considered are with L m the standard Virasoro operator After taking care the symmetries of the Young tableaux, the Virasoro constraints become and A hat on an index means that the index is skipped there (and it should appear somewhere else). In the above derivation we have used the identity for the Young tableaux where σ (i···j) are permutation operators.
a. High energy limit of Virasoro constraints States which satisfy the Virasoro constraints are physical states. What we are going to show in the following is that, in the high energy limit, the Virasoro constraints turn out to be strong enough to give the linear relationship among the physical states. To take the high energy limit for the Virasoro constraints, we replace the indices (µ i , ν i ) by L or T with where M is the mass operator.
To get the ratio for the specific physical states, we make the Young tableaux decomposi- where C l q = q! l!(q−l)! and we have (N − 2q − 2m) T 's and (2m + q − l) L's in the first column, (l) L's in the second column in the second line of the above equation. Therefore, we obtain which is consistent with the ratios Eqs. In previous sections, we have identified the leading high energy amplitudes and derived the ratios among high energy amplitudes for members of a family at given mass levels, based on decoupling principle. While deductive arguments help to clarify the underlying assumptions and solidify the validity of decoupling principle, it is instructive to compare it with a different approach, such as the saddle-point approximation [30]. Therefore, we shall perform direct calculations to check the results obtained above and make comparisons between these two approaches.
In this section, we give a direct verification of the ratios among leading high energy amplitudes based on the saddle-point method. The four-point amplitudes to be calculated consist of one massive tensor and three tachyons. Since we have shown that in the high energy limit the only relevant states are those corresponding to we only need to calculate the following four-point amplitude Notice that here for leading high energy amplitudes we replace the polarization L by P .
Using either path-integral or operator formalism, after SL(2, R) gauge fixing, we obtain the s − t channel contribution to the stringy amplitude at tree level In order to apply the saddle-point method, we need to rewrite the amplitude above into the "canonical form". That is, The saddle-point for the integration of moduli, x = x 0 , is defined by and we have From the definition of u(x), it is easy to see that and With these inputs, one can easily evaluate the Gaussian integral associated with the four-point amplitudes, Eq.(5.122), This result shows explicitly that with one tensor and three tachyons, the energy and angle dependence for the high energy four-point amplitudes only depend on the level N, and we can solve for the ratios among high energy amplitudes within the same family, which is consistent with Eq.(5.60).
We conclude this section with three remarks. Firstly, from the saddle-point approach, it is easy to see why the product of α P −1 oscillators induce energy suppression. Their contribution to the stringy amplitude is proportional to powers of f ′ (x 0 ), which is zero in the leading order calculation. Secondly, one can also understand why only even numbers of α P −1 oscillators will survive for high energy amplitudes based on the structure of Gaussian integral in Eq.(5.122). While for a vertex operator containing (2m ... = u (2m) (x 0 ) = 0, and the leading contribution comes from , this gives zero since the odd-power moments of Gaussian integral vanish. Finally, for the alert readers, since we only discuss the s − t channel contribution to the scattering amplitudes, the integration range for the x variable seems to devoid of a direct application of saddle-point method. We will discuss this issue in chapter VII.

E. 2D string at high energies
Although we have shown that there exist infinitely many linear relations among 4-point functions which uniquely fix their ratios in the high-energy limit, it is not totally clear that there is a hidden symmetry responsible for it. However, we would like to claim that these linear relations are indeed the manifestation of the long-sought hidden symmetry of string theory, and that we are on the right track of understanding the symmetry. To persuade the readers, we test our claim on a toy model of string theory -the 2D string theory.
While the hidden symmetry of the 26D bosonic string theory is still at large, the w ∞ symmetry of the 2D string theory is much better understood. It is known to be associated with the discrete Polyakov states discussed in chapter III of part I of this review. Let us now check whether the w ∞ symmetry is generated by the high energy limit of ZNS. In [23] , explicit expression for a class of discrete ZNS with Polyakov momenta was given in Eq.(3.26).
For illustration, let's repeat it here is defined by a similar expression as Eq.(5.134), but with the j-th row It was shown [23] that ZNS in Eq.(5.133) generate a w ∞ algebra.
In the high energy limit, the factors ∂ k X A are generically proportional to a linear combination of the momenta of other vertices, so it scales with energy E.
. Ignoring the second term in Eq.(5.133) for this reason, we see that these ZNS indeed approach to the discrete states ψ + JM in Eq.(3.20)! Thus, the w ∞ algebra generated by Eq.(5.133) is identified to w ∞ symmetry in Eq. (3.25). This result strongly suggests that the linear relations among correlation functions obtained from ZNS are indeed related to the hidden symmetry also for the 26D strings.
In chapter XV of part III of this review, we will address a similar issue in the RR of 26D string theory, where high energy spacetime symmetry is shown to be related to SL(5, C) of the Appell function F 1 . Although we still do not know what is the exact symmetry group of 26D strings, or how it acts on states, these works shed new light on the road to finding the answers.

VI. ZNS IN DDF CONSTRUCTION AND WSFT
In this chapter, in addition to the OCFQ scheme, we will identify and calculate [16] the counterparts of ZNS in two other quantization schemes of 26D open bosonic string theory, namely, the light-cone DDF [128][129][130] ZNS and the off-shell BRST ZNS (with ghost) in WSFT. In particular, special attentions are paid to the inter-particle ZNS in all quantization schemes. For the case of off-shell BRST ZNS, we impose the no ghost conditions and exactly recover two types of on-shell ZNS in the OCFQ string spectrum for the first few low-lying mass levels. We then show that off-shell gauge transformations of WSFT are identical to the on-shell stringy gauge symmetries generated by two types of ZNS in the OCFQ string theory.
Our calculations in this chapter serve as the first step to study stringy symmetries in lightcone DDF and BRST string theories, and to bridge the links between different quantization schemes for both on-shell and off-shell string theories. In section A, we first review the calculations of ZNS in OCFQ spectrum. The most general spectrum analysis in the helicity basis, including ZNS, is then given to discuss the inter-particle D 2 ZNS [6,8,12,13] at mass level M 2 = 4. We will see that one can use polarization of either one of the two positive-norm states to represent the polarization of the inter-particle ZNS.
In section B, we calculate both type I and type II ZNS in the light-cone DDF string up to mass level M 2 = 4. In section C, we first calculate off-shell ZNS with ghosts from linearized gauge transformation of WSFT. After imposing the no ghost conditions on these ZNS, we can exactly reproduce two types of ZNS in OCFQ spectrum for the first few low-lying mass levels. We then show that off-shell gauge transformations of WSFT are identical to the on-shell stringy gauge symmetries generated by two types of ZNS in the generalized massive σ-model approach [6,8] of string theory.
A. ZNS in the OCFQ spectrum

ZNS with constraints
In the OCFQ spectrum of open bosonic string theory, the solutions of physical states conditions include positive-norm propagating states and two types of ZNSs. The solutions of ZNS up to the mass level M 2 = 4 were calculated in chapter II. We re-list them in the following : Note that there are two degenerate vector ZNSs, Eq.(6.3a) for type II and Eq.(6.3c) for type I, at mass level M 2 = 4. We define D 2 vector ZNS by antisymmetrizing those terms which contain α µ −1 α ν −2 in Eq.(6.3a) and Eq.(6.3c) as following Similarly D 1 vector ZNS is defined by symmetrizing those terms which contain α µ −1 α ν −2 in Eq.(6.3a) and Eq.(6.3c) In general, an inter-particle ZNS can be defined to be D 2 + αD 1 , where α is an arbitrary constant.

ZNS in the helicity basis
In this section, we are going to do the most general spectrum analysis which naturally includes ZNS. We will then solve the Virasoro constraints in the helicity basis and recover the ZNS listed above. In particular, this analysis will make it clear how D 2 ZNS in Eq. (6.4) can induce the inter-particular symmetry transformation for two propagating states at the mass level M 2 = 4.
We begin our discussion for the mass level M 2 = 2. At this mass level, the general expression for the physical states can be written as In the OCFQ of string theory, physical states satisfy the mass shell condition and the Virasoro constraints L 1 |phys = L 2 |phys = 0 which give In order to solve for the constraints Eq.(6.8) and Eq.(6.9) in a covariant way, it is convenient to make the following change of basis, The 2nd rank tensor ǫ µν can be written in the helicity basis Eq.(6.10a) to Eq.(6.10c) as In this new representation, the second Virasoro constraint Eq.(6.9) reduces to a simple algebraic relation, and one can solve it In order to perform an irreducible decomposition of the spin-two state into the trace and traceless parts, we define the following variables We can then write down the complete decompositions of the spin-two polarization tensor as The first Virasoro constraint Eq.(6.8) implies that ǫ µ vector is not an independent variable, and is related to the spin-two polarization tensor ǫ µν as follows Finally, combining the results of Eq.(6.14),Eq.(6.15) and Eq.(6.16), we get the complete solution for physical states at mass level where the oscillator creation operators In comparison with the standard expressions for ZNS in section A, we find that Eq. Eq.(6.17) = 2x We now turn to the analysis of M 2 = 4 spectrum. Due to the complexity of our calculations, we shall present the calculations in three steps. We shall first write down all of physical states (including both positive-norm and ZNS) in the simplest gauge choices in the helicity basis. We then calculate the spin-3 state decomposition in the most general gauge choice. Finally, the complete analysis will be given to see how D 2 ZNS in Eq.(6.4) can induce the inter-particle symmetry transformation for two propagating states at the mass a. Physical states in the simplest gauge choices To begin with, let us first analyses the positive-norm states. There are two particles at the mass level M 2 = 4, a totally symmetric spin-three particle and an antisymmetric spin-two particle. The canonical representation of the spin-three state is usually chosen as where the totally symmetric polarization tensor ǫ µνλ can be expanded in the helicity basis as The Virasoro conditions on the polarization tensor can be solved as follows If we choose to keep the minimal number of L components in the expansion coefficients u ABC for the spin-three particle, we get the following canonical decomposition It is easy to check that the 2900 independent degrees of freedom of the spin-three particle decompose into 24 + 552 + 2024 + 24 + 276 in the above representation.
Similarly, for the antisymmetric spin-two particle, we have the following canonical repre- Rewriting the polarization tensor ǫ [µ,ν] in the helicity basis and solving the Virasoro constraints we obtain the following decomposition for the spin-two state Finally, one can check that the 300 independent degrees of freedom of the spin-two particle decompose into 24 + 276 in the above expression.
For the ZNS at M 2 = 4, we have the following decompositions 1. Spin-two tensor where we have solved the Virasoro constraints on the polarization tensor θ µν The 324 degrees of freedom of on-shell θ µν decompose into 24 + 276 + 24 in Eq.(6.34).

Spin-one vector (with polarization vector
b. Spin-three state in the most general gauge choice In this section, we study the most general gauge choice associated with the totally symmetric spin-three state where Virasoro constraints imply Eq.(6.41a) and Eq.(6.41b) imply that both ε (µν) and ε µ are not independent variables, and Eq.(6.41c) stands for the constraint on the polarization ε µνλ . In the helicity basis, we define Eq.(6.41c) then gives Eliminating u LLP , u LLL and u LLT i from above equations, we have the solution for ε µνλ , ε (µν) and ε µ ε µνλ = u P P P [e P µ e P ν e P λ + 3(e L µ e L ν e P λ + per.)] + u P P L [(e P µ e P ν e L λ + per. Putting all these polarizations back to the general form of physical states Eq.(6.40), we get For the first two terms on the right hand side of Eq.(6.48), we need to make the following replacements. For the positive-norm state |A(ũ) in Eq.(6.29) For the spin-two ZNS |C(θ) in Eq.(6.34), the replacement is given by It is important to note that for the spin-three gauge multiplet, only spin-two, singlet and D 1 vector ZNS appear in the decomposition Eq.(6.48). In the next section, we will see how one can include the missing D 2 ZNS in the analysis.
c. Complete spectrum analysis and the D 2 ZNS After all these preparations, we are ready for a complete analysis of the most general decomposition of physical states at M 2 = 4.
The most general form of physical states at this mass level are given by The Virasoro constraints are The solutions to Eq.(6.52b) and Eq.(6.52c) are given by In contrast to the previous discussion Eq.(6.41a) and Eq.(6.41b) where both ǫ (µν) and ǫ µ are completely fixed by the leading spin-three polarization tensor ǫ µνλ , we now have a new contribution from k ν ǫ [µν] . It will become clear that this extra term includes the inter-particle ZNS D 2 , Eq. The two independent polarization tensors of the most general representation for physical states Eq.(6.51) are given in the helicity basis by A, B, C = P, L, T i ; (6.55) The Virasoro constraint Eq.(6.53) demands that In contrast to Eq.(6.44), the solution to the above equations become It is clear from the expressions above that only U P P L and U (2) P P T i give new contributions to our previous analysis in the last section, so we can simply write down all these new terms Finally, the complete decomposition of physical states Eq.(6.51) in the helicity basis becomes is the inter-particle ZNS introduced in the end of section A with α = −2. Note that the value of α is a choice of convention fixed by the parametrization of the polarizations. It can always be adjusted to be zero. In view of Eq.(6.58) and Eq.(6.59), we see that one can use either V [P A] or U (2) P P A ( A = L, T i ) to represent the polarization of the |D ′ 2 (e A ) inter-particle ZNS. We conclude that once we turn on the antisymmetric spin-two positive-norm state in the general representation of physical states Eq.(6.51), it is naturally accompanied by the D ′ 2 inter-particle ZNS. The polarization of the D ′ 2 inter-particle ZNS can be represented by either V [P A] or U (2) P P A ( A = L, T i ) in Eq.(6.55) and Eq.(6.56). Thus this inter-particle ZNS will generate an inter-particle symmetry transformation in the σ-model calculation considered in chapter I. Note that, in contrast to the high-energy symmetry of Gross, this symmetry is valid to all orders in α ′ .

B. Light-cone ZNS in DDF construction
In the usual light-cone quantization of bosonic string theory, one solves the Virasoro constraints to get rid of two string coordinates X ± . Only 24 string coordinates α i n , i = 1, ..., , 24, remain, and there are no ZNS in the spectrum. However, there existed another similar quantization scheme, the DDF quantization, which did include the ZNS in the spectrum. In the light-cone DDF quantization of open bosonic string [128][129][130], one constructs transverse physical states with discrete momenta , respectively. All other states can be reached by Lorentz transformations. The DDF operators are given by [128][129][130] where the massless vertex operator V i (nk 0 , τ ) =Ẋ i (τ )e inX + (τ ) is a primary field with conformal dimension one, and is periodic in the worldsheet time τ if one chooses k µ = nk µ 0 with n ∈ Z. It is then easy to show that In addition to sharing the same algebra, Eq.(6.65), with string coordinates α i n , the DDF operators A i n possess a nicer property Eq.(6.64), which enables us to easily write down a general formula for the positive-norm physical states as following where | 0, p 0 > is the tachyon ground state and N = m r=1 ri r is the level of the state. Historically, DDF operators were used to prove no-ghost (negative-norm states) theorem for D = 26 string theory. Here we are going to use them to analyses ZNS. It turns out that ZNS can be generated byÃ where A − n is given by It can be shown thatÃ − n commute with L m and satisfy the following algebra We are now ready to construct ZNS in the DDF formalism.
2. M 2 = 2 : One has a light-cone vector A i −1Ã − −1 |0 , which has zero-norm for any D, and two scalars, whose norms are calculated to be For b = 0, one has a "pure type I" ZNS,Ã − −1Ã − −1 |0 , which has zero-norm for any D. By combining with the light-cone vector A i −1Ã − −1 |0 , one obtains a vector ZNS with 25 degrees of freedom, which corresponds to Eq.(6.2b) in the OCFQ approach. For b = 0, one obtains a type II scalar ZNS for D = 26, which corresponds to Eq.(6.2a) in the OCFQ approach.
3. M 2 = 4 : , which has zero-norm for any D. II. Three light-cone vectors, whose norms are calculated to be III. Three scalars, whose norms are calculated to be For c = 0 in Eq.(6.74), one has two "pure type I" light-cone vector ZNS. For e + f = 0 in Eq.6.75), one has two "pure type I" scalar ZNS. One of the two type I light-cone vectors, when combining with the spin-two state in I, gives the type I spin-two tensor which corresponds to Eq.(6.3b) in the OCFQ approach. The other type I light-cone vector, when combining with one of the two type I scalar, gives the type I vector ZNS which corresponds to Eq.(6.3c) in the OCFQ approach. The other type I scalar corresponds to Eq.(6.3d). Finally, for c = 0 and e + f = 0, one obtains the type II vector ZNS for D = 26, which corresponds to Eq.(6.3a) in the OCFQ approach. It is easy to see that a special linear combination of b and e will give the inter-particle vector ZNS which corresponds to the inter-particle D 2 ZNS in Eq. (6.4). This completes the analysis of ZNS for M 2 = 4.
Note that the exact mapping of ZNS in the light-cone DDF formalism and the OCFQ approach depends on the exact relation between operators (Ã − n , A i n , L n ) and α µ n , which has not been worked out in the literature.

C. BRST ZNS in WSFT
In this section, we calculate BRST ZNS in the formulation of WSFT. In addition, we apply the results to demonstrate that off-shell gauge transformations of WSFT are indeed identical to the on-shell stringy gauge symmetries generated by two types of ZNS in the generalized massive σ-model approach [6,8] of string theory. In section I.D [15], the background ghost transformations in the gauge transformations of WSFT [17] were shown to correspond, in a one-to-one manner, to the lifting of on-shell conditions of ZNS in the OCFQ approach. Here we go one step further to demonstrate the correspondence of stringy symmetries induced by ZNS in OCFQ and BRST approaches.
Cubic string field theory is defined on a disk with the action and Φ is the string field with ghost number 1 and b, c are conformal ghosts. Since the ghost number of vacuum on a disk is −3, the total ghost number of this action is 0 as expected.
The string field can be expanded as where the string ground states |Ω are The gauge transformation for string field can be written as where Λ is the a string field with ghost number 0.
For the purpose of discussion in this chapter, we are going to consider the linearized gauge transformation δΦ = Q B Λ, (6.81) where Q B Λ is just the off-shell ZNS. In the following, we will explicitly show that the solutions of Eq.(6.81) are in one-to-one correspondence to the ZNS obtained in OCFQ approach in section VI.A level by level for the first several mass levels.
There is no ZNS in the lowest string mass level with M 2 = −2, so our analysis will start with the mass level of M 2 = 0.
The string field can be expanded as The gauge transformation is then The nilpotency of BRST charge Q B gives which give the on-shell condition k 2 = 0 and the following ZNS This is the same as the scalar ZNS obtained in OCFQ approach.
The string fields expansion are The off-shell ZNS are calculated to be Nilpotency condition requires There are two solutions of Eq.(6.91), which correspond to the type I and type II ZNS, respectively.
1. Type I: in this case D is not restricted to the critical string dimension in Eq.(6.91), i.e. D = 26. Thus The no-ghost conditions of Eq.(6.90) lead to the on-shell constraints The off-shell ZNS in Eq.(6.90) then reduces to an on-shell vector ZNS 2. Type II: in this case D is restricted to the critical string dimension, i.e. D = 26.
Then ǫ 1 can be arbitrary constant. The no-ghost conditions then lead to the on-shell constraints α 2 0 + 2 = 0, (6.96) The second condition can be solved by a special solution which leads to an on-shell scalar ZNS Again, up to a constant factor, the ZNS Eq.(6.95) and Eq.(6.99) are the same as Eq.(6.2b) and Eq.(6.2a) calculated in the OCFQ approach.
The string fields are expanded as The off-shell ZNS are Nilpotency condition requires Similarly, we classify the solutions of Eq.(6.103) by type I and type II in the following: 1. Type I: D = 26. This leads to The no-ghost conditions lead to the on-shell constraints α 2 0 + 4 = 0, (6.105a) There are three independent solutions to the above equations, which correspond to the three type I on-shell ZNS: • Tensor ZNS If we set D = 26, then where θ is an arbitrary constant.
2. Type II: D = 26 in Eq.(6.103), and ǫ 2 , ǫ 3 and ǫ 2 µ are arbitrary constants. The no-ghost conditions lead to the on-shell constraints α 2 0 + 4 = 0, (6.113a) A special solution of above equations is which gives an on-shell vector ZNS For a special value of C = −3i/4, Eq.(6.115) becomes gives Similarly, one can apply the same procedure to type II ZNS in Eq.(6.99), and derive the following type II gauge transformation For the gauge transformation induced by D 2 ZNS in Eq.(6.116), for example, one can use Eq.(6.113a) to Eq.(6.114c) with C = −i/16 to eliminate Eq.(6.120e) to Eq.(6.120k). One can then use the fact that background fields C (µν) and C µ are gauge artifacts of C µνλ in the σmodel calculation, and deduce from Eq.(6.120a) to Eq.(6.120d) the inter-particle symmetry where ∂ λ ǫ The other three gauge transformations corresponding to three other ZNS, the spin-two, D 1 , and scalar can be similarly constructed from Eq.(6.120a) to Eq.(6.120k). One gets Eq.(6.121) to Eq.(6.124) are exactly the same as those calculated by the generalized massive σ-model approach of string theory [6,8].
We thus have shown in this section that off-shell gauge transformations of WSFT are identical to the on-shell stringy gauge symmetries generated by two types of ZNS in the OCFQ string theory. The high energy limit of these stringy gauge symmetries generated by ZNS was recently used to fix the proportionality constants among high energy scattering amplitudes of different string states conjectured by Gross [3,4]. Based on the ZNS calculations in [27][28][29][30] and the calculations in this section, we thus have related gauge symmetry of WSFT [17] to the high energy stringy symmetry conjectured by Gross [1][2][3][4][5].
In conclusion of this chapter, we have calculated ZNS in the OCFQ string, the light-cone DDF string and the off-shell BRST string theories. In the OCFQ string, we have solved the Virasoro constraints for all physical states ( including ZNS) in the helicity basis. Much attention was paid to discuss the inter-particle ZNS at the mass level M 2 = 4. We found that one can use polarization of either one of the two positive-norm states to represent the polarization of the inter-particle ZNS. This justified why one can derive the inter-particle symmetry transformation for the two massive modes in the weak field massive σ-model calculation [6,8].
In the light-cone DDF string, one can easily write down the general formula for all ZNS in the spectrum. We have identified type I and Type II ZNS up to the mass level M 2 = 4.
An analysis for the general mass levels should be easy to generalize.
Finally, we have calculated off-shell ZNS in the WSFT. After imposing the no ghost conditions, we can recover two types of on-shell ZNS in the OCFQ string. We then show that off-shell gauge transformations of WSFT are identical to the on-shell stringy gauge symmetries generated by two types of ZNS in the generalized massive σ-model approach of string theory. Based on these ZNS calculations, we thus have related gauge symmetry of WSFT [17] to the high energy stringy symmetry of Gross [3,4].
VII. HARD CLOSED STRING SCATTERINGS, KLT AND STRING BCJ RELA-

TIONS
In this chapter, we generalize the calculations in chapter V to high energy closed string scattering amplitudes [36]. We will find that the methods of decoupling of high energy ZNS and the high energy Virasoro constraints, which were adopted in chapter V to calculate the  [132,133]. One interesting application of this result is the string BCJ relations [38,41,[124][125][126] which will be discussed in section D. k µ θ ν T µν + θ µ T µ = 0, (7.1a)

2a)
MT 4→2 LL + T 2 L = 0, (7.2b) Note that since T 1 T P is of subleading order in energy, in general T 1 T P = T 1 T L . A simple calculation of Eq.(7.2a) to Eq.(7.2c) shows that [33]  We are now back to the closed string calculation. The OCFQ closed string spectrum at this mass level are ( In addition to the spin-four positive-norm state ⊗ ′ , one has 8 ZNS, each of which gives a Ward identity. In the high energy limit, we have θ µν = e µ L e ν L − e µ T e ν T or θ µν = e µ L e ν T + e µ T e ν L , θ µ = e µ L or e µ T and one replace η µν by e µ T e ν T . In the following, we list only high energy Ward identities which relate amplitudes with even-energy power in the high energy expansion : 1.

B. Virasoro constraints
We consider the mass level M 2 = 8. The most general state is The Virasoro constraints are Taking the high energy limit in the above equations by letting (µ i , ν i ) → (L, T ), and we obtain It is now easy to calculate the general high energy scattering amplitude at theM 2 = 2(N − 1) level where T N (s, t) is the high energy limit of with s + t + u = 2N − 8, and was previously [27,28,30] miscalculated to bẽ One can now generalize this result to multi-tensors. The (s, t) channel of open string high energy scattering amplitude at mass level (N 1 , N 2 , N 3 , N 4 ) was calculated to be [27,28,30] The corresponding (t, u) channel scattering amplitudes of Eqs.(7.31) and (7.33) can be obtained by replacing (s, t) in Eq.(7.32) by (t, u) We now claim that only (t, u) channel of the amplitude, Eq.
in Eq.(7.30) by expanding the Γ function with the Stirling formula However, the above expansion is not suitable for negative real x as there are poles for Γ (x) at x = −N, negative integers. Unfortunately, our high energy limit which we know how to calculate the high energy limit. Note that for the four-tachyon case,Ā open (t, u) = A  One might try to use the saddle-point method to calculate the high energy closed string scattering amplitude. The closed string four-tachyon scattering amplitude is where K = s 8 and f (x, y) = ln(x 2 + y 2 ) − τ ln[(1 − x) 2 + y 2 ] with τ = − t s . One can then calculate the "saddle-point" of f (x, y) to be The high energy limit of the closed string four-tachyon scattering amplitude is then calculated to be which is consistent with the previous one calculated in the literature [1,2], but is different from our result in Eq.(7.45). However, one notes that Finally we calculate the high energy closed string scattering amplitudes for arbitrary mass levels. The (t, u) channel open string scattering amplitude with V 2 = α µ 1 −1 α µ 2 −1 ..α µn −1 | 0, k >, the highest spin state at mass levelM 2 = 2(N −1), and three tachyons V 1,3,4 can be calculated to be In calculating Eq.(7.50), we have used the Mobius transformation y = x−1 x to change the integration region from (1, ∞) to (0, 1). One notes that Eq.(7.50) is NOT the same as Eq.(7.30) with (s, t) replaced by (t, u), as one would have expected from the four-tachyon case discussed in the paragraph after Eq.(7.39) In the high energy limit, one easily sees that Finally the total high energy open string scattering amplitude is the sum of (s, t), (t, u) and (u, s) channel amplitudes, and can be calculated to be open ≃ (−) ΣN i sin (πs/2) + sin (πt/2) + sin (πu/2) sin (πs/2) (stu) exp − s ln s + t ln t + u ln u 2 .
(7.53) By using Eqs.(7.39) and (7.51), the high energy closed string scattering amplitude at mass level (N 1 , N 2 , N 3 , N 4 ) is calculated to be, apart from an overall constant, where T ΣN i (t, u) is given by Eq. which relates field theory scattering amplitudes in the s, t and u channels. In the following, we will discuss the relation for s and u channel amplitudes only. Other relations can be similarly discussed. It was based on the field theory BCJ relation [41]. An explicit proof of Eq.(7.58) for arbitrary four string states and all kinematic regimes was given very recently in [39,40].
Note that for the supersymmetric case, there is no tachyon and the low energy massless by taking the nonrelativistic limit | k 2 | << M S of Eq.(7.58). In Eq.(0.52), B was the beta function, and k 1 , k 3 and k 4 were taken to be tachyons, and k 2 was the following tensor string 7.59) where N = p + r + q, M 2 2 = 2(N − 1).  [34].
In addition, we discover new leading order high energy scattering amplitudes, which are still proportional to the previous ones, with polarizations orthogonal to the scattering plane [34]. These scattering amplitudes are of subleading order in energy for the case of 26D open bosonic string theory. The existence of these new high energy scattering amplitudes is due to the worldsheet fermion exchange in the correlation functions and is, presumably, related to the high energy massive fermionic scattering amplitudes in the R-sector of the theory. We thus conjecture that the validity of Gross's two conjectures on high energy stringy symmetry persists for superstring theory.
A. Decoupling of high energy ZNS We will first consider high energy scattering amplitudes of string states with polarizations on the scattering plane. Those with polarizations orthogonal to the scattering plane will be discussed in section VIII.D. It can be argued that there are four types of high energy scattering amplitudes for states in the NS sector with even GSO parity [34] |n, 2m, Note that the number of α L −1 operator in Eq.(8.2) is odd. In the OCFQ spectrum of open superstring, the solutions of physical states conditions include positive-norm propagating states and two types of ZNS. In the NS sector, the latter are [10] Type I : Eq.(8.11) gives the ratio for states at mass level M 2 = 4 We have used an abbreviated notation for the scattering amplitudes on the l.h.s. of Eq.(8.12).
The HZNS in Eq.(8.11) is the high energy limit of the vector ZNS at mass level M 2 = 2 In fact, in the high energy limit, θ = e L , so |x → (α L −1 ) |0, k and Eq.(8.13) reduces to Eq.(8.11). To calculate the ratio among the high energy scattering amplitudes corresponding to states in Eqs.(8.7) and (8.9), we use the decoupling of the Type II HZNS at mass level Eq. (8.15) gives the ratio Eq. (8.17) gives the ratio On the other hand, Eq.(5.94) gives We conclude that In this section, we will use the method of Virasoro constrains to derive the ratios between the physical states in the NS sector. In the superstring theory, the physical state |φ in the NS sector should satisfy the following conditions: L m |φ = 0, m = 1, 2, 3, · · · , (8.22) In the following, we will use the reduced Virasoro conditions (8.27) and (8.28) to determine the ratios between the physical states in the NS sector in the high energy limit.
To warm up, let us consider the mass level at M 2 = 2 first. The most general state in the NS sector at this mass level can be written as where we use the Young tableaux to represent the coefficients of different tensors. The properties of symmetry and anti-symmetry can be easily and clearly described in this representation.
We then apply the reduced Virasoro conditions (8.27) and (8.28) to the state (8.29). It is easy to obtain which leads to the following equations, To solve the above equation, we first take the high energy limit by letting µ → (L, T ) and The above equations reduce to At this mass level, the terms with odd number of T 's will be sub-leading in the high energy limit and be ignored, the resulting equations contain only terms will even number of T ' as following, The ratio of the coefficients then can be obtained as Now we will consider the general mass level at M 2 = 2(N − 1). At this mass level, the most general state can be written as and we have defined the abbreviation with m j (m r ) is the number of the operator α µ −j ψ ν −r for j ∈ Z and r ∈ Z + 1/2. The summation runs over all possible m j (m r ) with the constrain so that the total mass square is 2 (N − 1).
Solving the constraints (8.27) and (8.28) in the high energy limit, the ratios between the physical states in the NS sector are obtained as (see Appendix B for detail) (8.46) which are exactly consistent with the results obtained by using the decoupling of high energy ZNS in the previous section and the saddle-point calculation which will be discussed in the following section. We shall vary the second vertex at the same level and compare the scattering amplitudes to obtain the proportional constants.

M 2 = 2
The second vertex operators at mass level M 2 = 2, are given by (in the −1 ghost picture), Here we have used the polarization basis to specify the particle spins, e.g.,ψ T ≡ e T µ · ψ µ . To illustrate the procedure, we take the first state, Eq.(8.49), as an example to calculate the scattering amplitude among one massive tensor (M 2 = 2) with one photon (V 1 ) and two tachyons (V 3 , V 4 ). As in the case of open bosonic string theory, we list the contributions of s − t channel only. The 4-point function is given by where we have suppressed the SL(2, R) gauge-fixed world-sheet coordinates x 1 = 0, x 3 = 1, x 4 = ∞. Notice that in both the first and second vertices, it is possible to allow fermion operators ψ µ to have polarization in transverse direction T i out of the scattering plane.
As we shall see in next section that this leads to a new feature of supersymmetric stringy amplitudes in the high energy limit. At this moment, we only choose the polarization vector to be in the P, L, T directions for a comparison with results obtained by the previous two methods.
A direct application of Wick contraction among fermions ψ, ghosts φ, and bosons X leads to the following result where we have used the short-hand notation, (3, 4) ≡ k 3 · k 4 . Based on the kinematic variables and the master formula for saddle-point approximation, where u 0 , f 0 , u ′ 0 , f ′′ 0 , etc, stand for the values of functions and their derivatives evaluated at the saddle point f ′ (x 0 ) = 0. In order to apply this master formula to calculate stringy amplitudes, we need the following substitutions (α ′ = 1/2) where θ is the scattering angle in center of momentum frame and the saddle point for the integration of moduli is x 0 = 1 1−τ . In the first scattering amplitude corresponding to Eq.(8.49), we can identify the u(x) function as Equipped with this, we obtain the high energy limit of the first amplitude, Here we can identify the u(x) function for saddle-point master formula, Eq.(8.54) One can check that u II (x 0 ) = u ′ II (x 0 ) = 0, and Thus, the amplitude associated with the massive state, Eq.(8.50), is given by In the third case, after replacing the second vertex operator in Eq.(8.52) by Eq.(8.51), we get the Wick contraction The high energy limit of this amplitude, after applying the master formula of saddle-point approximation, is  Since the applications of saddle-point approximation is essentially identical to previous cases, we simply list the results of our calculations Combining these results, we conclude that the ratio between the M 2 = 4 vertices is given by which agrees with Eq.(8.20).

GSO odd vertices at M 2 = 5
In addition to the stringy amplitudes associated with GSO even vertices we have calculated in the previous sections, we can also apply the same method to those associated with the GSO odd vertices. While it is a common practice to project out the GSO odd states in order to maintain spacetime supersymmetry, it turns out that we do find linear relation among these amplitudes. This seems to suggest a hidden structure of superstring theory in the high energy limit.
To see this, we examine the vertices of odd GSO parity, at the mass level M 2 = 5. Based on the power-counting rule as in the bosonic string case, we can identify the relevant vertices and the associated vertex operators as follows To calculate 4-point functions, we can fix the first vertex (V 1 ) to be a tachyon state in the −1 ghost picture, 73) and the third and the fourth vertices to be tachyon state in the 0 ghost picture, as Eq.(8.48).
Since the applications of saddle-point approximation is essentially identical to previous cases, we simply list the results of our calculations It is worth noting that in the second calculations, we need to include both u ′′ (x 0 ) and u (3) (x 0 ) terms of the first order corrections in saddle-point approximation, Eq.(8.54), to get the correct answer.
Combining these results, we conclude that the ratio between the M 2 = 3 vertices is given by ) |0, k = 2M 2 : 1 = 10 : 1.  Remarkably, the final answer is polarizations e T i are the same as that of polarization e T up to a sign. Let's consider the third example to justify this point. It is straightforward to show the following in the open string theory [134,135].
It would be of crucial importance to calculate high energy massive fermion scattering amplitudes in the R-sector to complete the proof of Gross's two conjectures on high energy symmetry of superstring theory. The construction of general massive spacetime fermion vertex, involving picture changing, will be the first step toward understanding of the high energy behavior of superstring theory.

IX. HARD STRING SCATTERINGS FROM D-BRANES/O-PLANES
In this chapter, we study scatterings of bosonic closed strings from D-branes [42] in section A, and O-planes [47] in section B. In particular, we will discuss hard strings scattered from D-particle [42] and D-domain-wall [46]. We will also study hard strings scattered from Oparticle [47] and O-domain-wall [47]. In addition, in section C, we calculate the absorption amplitudes [127] of a closed string state at arbitrary mass level leading to two open string states on the D-brane at high energies.

A. Scatterings from D-branes
In this section we study the general structure of an arbitrary incoming closed string state scatters from D-brane and ends up with an arbitrary spin outgoing closed string states at arbitrary mass levels [42]. The scattering of massless string states from D-brane has been well studied in the literature and can be found in [53,[136][137][138][139][140] Here we extend the calculations of massless closed string states to massive closed string states at arbitrary mass levels. Since the mass of D-brane scales as the inverse of the string coupling constant 1/g, we will assume that it is infinitely heavy to leading order in g and does not recoil. We will first show that, for the (0 → 1) and (1 → ∞) channels, all the scattering amplitudes can be expressed in terms of the beta functions, thanks to the momentum conservation on the D-brane.
Alternatively, the Kummer relation of the hypergeometric function 2 F 1 can be used to reduce the scattering amplitudes to the usual beta function [42]. After summing up the (0 → 1) and (1 → ∞) channels, we discover that all the scattering amplitudes can be expressed in terms of the generalized hypergeometric function 3 F 2 with special arguments, which terminates to a finite sum and, as a result, the whole scattering amplitudes consistently reduce to the usual beta function.
For the simple case of D-particle, we explicitly calculate [42] high energy limit of a set of scattering amplitudes for arbitrary mass levels, and derive infinite linear relations among them for each fixed mass level. Since the calculation of decoupling of high energy ZNS remains the same as the case of scatterings without D-brane, the ratios of these high energy scattering amplitudes are found to be consistent with the decoupling of high energy ZNS in Chapter V. The cases of RR strings scattered from D-particle will be discussed in chapter XIV where the complete ratios among GR scattering amplitudes will be calculated.
We will first begin with the simple case of tachyon to tachyon scattering and then generalize to scatterings of states at arbitrary mass levels. The standard propagators of the left and right moving fields are In addition, there are also nontrivial correlator between the right and left moving fields as well X µ (z)X ν (w) = −D µν log (z −w) as a result of the boundary condition at the real axis. Propagator Eq.(9.3) has the standard form Eq.(9.1) for the fields satisfying Neumann boundary condition, while matrix D reverses the sign for the fields satisfying Dirichlet boundary condition. We will follow the standard notation and make the following replacement which allows us to use the standard correlators Eq.(9.1) throughout our calculations. As we will see, the existence of the Propagator Eq.(9.3) has far-reaching effect on the string scatterings from D-brane.

Tachyon to tachyon
In this section, we consider the tachyon to tachyon scattering amplitude To fix the SL (2, R) invariance, we set z 1 = iy and z 2 = i. Introducing the SL (2, R) Jacobian we have, for the (0 → 1) channel, In the above calculations, we have defined a 0 = k 1 · D · k 1 = k 2 · D · k 2 , (9.9) b 0 = 2k 1 · k 2 + 1, (9.10) c 0 = 2k 1 · D · k 2 + 1, (9.11) so that and −k 2 We have also used the integral representation of the hypergeometric function 13) and the following identity which we discuss in the section IX A 5. In addition, the momentum conservation on the D-brane D · k 1 + k 1 + D · k 2 + k 2 = 0 (9.15) is crucial to get the final result Eq.(9.8). Finally, by using change of variablet = t 2 , Eq.(9.8) can be further reduced to the beta function where we have omitted an irrelevant factor.
For the (1 → ∞) channel, we use the change of variable y = 1+t 1−t and end up with the same result since N = 0 for the case of tachyon.
Alternatively, one can use the Kummer formula of hypergeometric function to reduce Eq.(9.7) to the final result Eq.(9.16). In this calculation, we have used the Kummer condition γ = 1 + α − β, (9.20) which is equivalent to the momentum conservation on the D-brane Eq.(9.15).

Tensor to tensor
In this section, we generalize the previous calculation to general tensor to tensor scatterings. In this case, we define where n a , n b and n c are integer and where k 2 1 = 2(N 1 − 1) and N 1 is now the mass level of k 1 . After a similar calculation as the previous section, it is easy to see that a typical term in the expression of the general tensor to tensor scattering amplitudes can be reduced to the following integral Similarly, for the (1 → ∞) channel, one gets The sum of the two channels gives where the generalized hypergeometric function 3 F 2 is defined to be Note that the energy dependence of the prefactor 4 (2i) k 1 ·D·k 1 +k 2 ·D·k 2 in the scattering ampli-

Hard strings scattered from D-particle
In this section, we will calculate the high energy limit of string scattered from D-brane [42]. In particular, we will calculate the ratios among scattering amplitudes of different string states at high energies. For our purpose here, for simplicity, we will only consider the string states with the form of Eq.(5.67) with m = 0 (the ratios for the m = 0 case will be discussed in chapter XIV). The reason is as following. It was shown that [27][28][29][30] the leading order amplitudes containing this component will drop from energy order E 4m to E 2m , and one needs to calculate the complicated naive subleading contraction terms between ∂X and ∂X for the multi -tensor scattering in order to get the real leading order scattering amplitudes. For our closed string scattering calculation here, even for the case of one tachyon and one tensor scattering, one encounters the similar complicated nonzero contraction terms in Eq.(9.3) due to the D-brane. So we will omit high energy scattering amplitudes of string states containing this (α L −1 ) 2m component. On the other hand, we will also need the result that the high energy closed string ratios are the tensor product of two pieces of open string ratios [36].
To simplify the kinematics, we consider the case of D0 brane or D-particle scatterings [42]. The momentum of the incident particle k 2 is along the −X direction and particle k 1 is scattered at an angle φ. We will consider the general case of an incoming tensor state and an outgoing tachyon state. Our result can be easily generalized to the more general two tensor cases. The kinematic setup is For the scattering of D-particle D ij = −δ ij , and it is easy to calculate e T · k 2 = e L · k 2 = 0, (9.34) and The high energy scattering amplitude is then calculated to be Set z 1 = iy and z 2 = i to fix the SL(2, R) invariance, we have .
(9.45) Now in the high energy limit, the master formula Eq.(9.27) reduces to where n a = 2n − 2 (q + q ′ ) , (9.47) The total high energy scattering amplitude can then be calculated to be (9.53) The high energy limit of the beta function is Finally we get the high energy scattering amplitudes at mass level M 2 = 2(n − 1) where the high energy limit of B a 0 + 1, b 0 +1 2 is independent of q + q ′ . We thus have explicitly shown that there is only one independent high energy scattering amplitude at each fixed mass level. It is a remarkable result that the ratios − 1 2M q+q ′ for different high energy scattering amplitudes at each fixed mass level is consistent with Eq.(5.60) for the scattering without D-brane as expected. In general, for an incoming tensor state and an outgoing tensor state scatterings, the ratios are .
Finally, one notes that the exponential fall-off behavior in energy E is hidden in the high energy beta function. Since the arguments of Γ(a 0 + 1) and Γ(a 0 + b 0 2 + 3 2 ) in Eq.(9.55) are negative in the high-energy limit, one needs to use the well known formula to calculate the large negative x expansion of these Γ functions, and obtain the Regge-pole structure [36] of the amplitude. This is to be compared with the power-law behavior with Regge-pole structure for the D-domain-wall scattering to be discussed in the next section.

Hard strings scattered from D-domain-wall
We have shown, in the last section, that the linear relations for string/string scatterings persist for the string/Dp-brane scatterings with p 0. In particular, the linear relations for the D-particle scatterings [42] were explicitly demonstrated. All the high energy string/Dpbrane scattering amplitudes with p 0 behave as exponential fall-off as was claimed in [53,[136][137][138][139][140] In this section, in contrast to the common wisdom, we show that [46], instead of the exponential fall-off behavior of the form factors with Regge-pole structure, the high energy scattering amplitudes of string scattered from D24-brane, or Domain-wall, behave as power-law with Regge-pole structure. This is to be compared with the well-known powerlaw form factors without Regge-pole structure of the D-instanton scatterings to be discussed in Eq.(9.74) below.
This discovery makes Domain-wall scatterings an unique example of a hybrid of string and field theory scatterings. Our calculation will be done for bosonic string scatterings of arbitrary massive string states from D24-brane. Moreover, we discover that the usual linear relations [42] of high energy string scattering amplitudes at each fixed mass level, Eq.(9.55), breaks down for the Domain-wall scatterings [46]. This result gives a strong evidence that the existence of the infinite linear relations, or stringy symmetries, of high energy string scattering amplitudes is responsible for the softer, exponential fall-off high energy string scatterings than the power-law field theory scatterings.
We consider an incoming tachyon closed string state with momentum k 1 and an angle of incidence φ and an outgoing massive closed string state α T −1 In the high energy limit, the angle of incidence φ is identified to the angle of reflection θ, and e P approaches e L , k 1 , k 2 ≃ E. For the case of Domain-wall scattering Diag D µν = (−1, 1, −1), and we have The scattering amplitude can be calculated to be (9.64) Set z 1 = iy and z 2 = i to fix the SL(2, R) gauge, and include the Jacobian d 2 z 1 d 2 z 2 → 4 (1 − y 2 ) dy, we have, for the (0 → 1) channel, F i in Eq.(9.67) is the high energy limit of the generalized hypergeometric function At this stage, it is crucial to note that in the high energy limit for the Domain-wall scatterings. As a result, F i reduces to the form of Eq.(9.68), and depends on the energy E. Thus in contrast to the generic Dp-brane scatterings with p ≥ 0, which contain two independent kinematic variables, there is only one kinematic variable for the special case of Domain-wall scatterings. It thus becomes meaningless to study high energy, fixed angle scattering process for the Domain-wall scatterings. As we will see in the following calculation, this peculiar property will reduce the high energy beta function in Eq.(9.65) from exponential to power-law behavior and, simultaneously, breaks down the linear relations as we had in Eq.(9.55) for the D-particle scatterings.
Finally, the scattering amplitude for the (0 → 1) channel can be calculated to be (similar result can be obtained for the (1 → ∞) channel) where (α) n ≡ Γ(α+n) Γ(α) for integer n. On the other hand, since the argument of Γ(a 0 + 1) in Eq.(9.72) is negative in the high energy limit, we have, by using Eq.(9.13) and Eq.(9.70), Note that the sin (πa 0 ) factor in the denominator of Eq.(9.73) gives the Regge-pole structure, and the energy dependence E −2(n−1) gives the power-law behavior in the high energy limit.
As a result, the scattering amplitude for the Domain-wall in Eq.(9.72) behaves like power-law with the Regge-pole structure.
The crucial differences between the Domain-wall scatterings in Eq.(9.72) and the Dparticle scatterings (or any other Dp-brane scatterings except Domain-wall and D-instanton scatterings) in Eq.(9.55) is the kinematic relation Eq.(9.70). For the case of D-particle scatterings [42], the corresponding factors for both F i in Eq.(9.67) and the fraction in Eq. (9.71) are independent of energy in the high energy limit, and, as a result, the amplitudes contain no q + q ′ dependent energy power factor. So one gets the high energy linear relations for the D-particle scattering amplitudes. On the contrary, for the case of Domain-wall scatterings, both F i in Eq.(9.68) and the fraction in Eq.(9.71) depend on energy due to the condition Eq.(9.70). The summation in Eq.(9.71) is then dominated by the term i = q + q ′ , and the whole scattering amplitude Eq.(9.72) contains a q + q ′ dependent energy power factor. As a result, the usual linear relations for the high energy scattering amplitudes break down for the Domain-wall scatterings.
It is crucial to note that the mechanism, Eq.(9.70), to drive the exponential fall-off form factor of the D-particle scatterings to the power-law one of the Domain-wall scatterings is exactly the same as the mechanism to break down the expected linear relations for the domain-wall scatterings in the high energy limit. In conclusion, this result gives a strong evidence that the existence of the infinite linear relations, or stringy symmetries, of high energy string scattering amplitudes is responsible for the softer, exponential fall-off high energy string scatterings than the power-law field theory scatterings.
Another interesting case of D-brane scatterings is the massless form factor of scatterings On the other hand, since t is large negative in the high energy limit [36], the application of Eq.(9.13) to Eq.(9.74) produces no sin (πa 0 ) factor in contrast to the Domain-wall scatterings. So there is no Regge-pole structure for the D-instanton scatterings. We conclude that the very condition of Eq.(9.70) makes Domain-wall scatterings an unique example of a hybrid of string and field theory scatterings.

A brief review of 2 F 1 and 3 F 2
In this section, we review the definitions and some formulas of hypergeometric function 2 F 1 and generalized hypergeometric function 3 F 2 which we used in the text. hypergeometric functions form an important class of special functions. Many elementary special functions are special cases of 2 F 1 . The hypergeometric function 2 F 1 is defined to be (α, β, γ constant) The hypergeometric function 2 F 1 is a solution, at the singular point x = 0 with indicial root r = 0, of the Gauss's hypergeometric differential equation with indicial root r = 1 − γ can be expressed in terms of 2 F 1 as following (γ = integer) Other solutions of Eq. (9.77), which corresponds to singularities x = 1, ∞, can also be expressed in terms of the hypergeometric function 2 F 1 . The following identity which we used in the text can then be derived.

F 1 has an integral representation
which can be used to do analytic continuation. Eq.(9.80) with x = −1 was repeatedly used in the text in our calculations of string scattering amplitudes with D-brane.
There exists interesting relations among hypergeometric function 2 F 1 with different arguments where x = ϕ(t) is an algebraic function with degree up to six. As an example, the quadratic transformation formula can be used to derive the Kummer's relation , (9.83) which is crucial to reduce the scattering amplitudes of string from D-brane to the usual beta function.
In summing up the (0 → 1) and (1 → ∞) channel scattering amplitudes, we have used the master formula In Eq.(9.84), B is the beta function and 3 F 2 is the generalized hypergeometric function, which is defined to be For those arguments of 3 F 2 in Eq. (9.84), the series of the generalized hypergeometric function 3 F 2 terminates to a finite sum. For example, (9.86)

B. Scatterings from O-planes
Being a consistent theory of quantum gravity, string theory is remarkable for its soft ultraviolet structure. This is mainly due to two closely related fundamental characteristics of high-energy string scattering amplitudes. The first is the softer exponential fall-off behavior of the form factors of high-energy string scatterings in contrast to the power-law field theory scatterings. The second is the existence of infinite Regge poles in the form factor of string scattering amplitudes. The existence of infinite linear relations discussed in part II of the review constitutes the third fundamental characteristics of high energy string scatterings.
In the previous section, we showed that these linear relations persist [42] for string scattered from generic Dp-brane [136] except D-instanton and D-domain-wall. For the scattering of D-instanton, the form factor exhibits the well-known power-law behavior without Regge pole structure, and thus resembles a field theory amplitude. For the special case of Ddomain-wall scattering [53], it was discovered [46] that its form factor behaves as power-law with infinite open Regge pole structure at high energies. This discovery makes D-domainwall scatterings an unique example of a hybrid of string and field theory scatterings.
Moreover, it was shown [46] that the linear relations break down for the D-domain-wall scattering due to this unusual power-law behavior. This result seems to imply the coexistence of linear relations and soft UV structure of string scatterings. In order to further uncover the mysterious relations among these three fundamental characteristics of string scatterings, namely, the soft UV structure, the existence of infinite Regge poles and the newly discovered linear relations stated above, it will be important to study more string scatterings, which exhibit the unusual behaviors in the high energy limit.
In this section, we calculate massive closed string states at arbitrary mass levels scattered from Orientifold planes in the high energy, fixed angle limit [47]. The scatterings of massless states from Orientifold planes were calculated previously by using the boundary states formalism [48][49][50][51], and on the worldsheet of real projected plane RP 2 [52]. Many speculations were made about the scatterings of massive string states, in particular, for the case of O-domain-wall scatterings. It is one of the purposes of this section to clarify these speculations and to discuss their relations with the three fundamental characteristics of high energy string scatterings.
For the generic Op-planes with p ≥ 0, one expects to get the infinite linear relations except O-domain-wall scatterings. For simplicity, we consider only the case of O-particle scatterings [47]. For the case of O-particle scatterings, we will obtain infinite linear relations among high energy scattering amplitudes of different string states. We also confirm that there exist only t-channel closed string Regge poles in the form factor of the O-particle scatterings amplitudes as expected.
For the case of O-domain-wall scatterings, we find that, like the well-known D-instanton scatterings, the amplitudes behave like field theory scatterings, namely UV power-law without Regge pole. In addition, we will show that there exist only finite number of t-channel closed string poles in the form factor of O-domain-wall scatterings, and the masses of the poles are bounded by the masses of the external legs [47]. We thus confirm that all massive closed string states do couple to the O-domain-wall as was conjectured previously [52,53].
This is also consistent with the boundary state descriptions of O-planes.
For both cases of O-particle and O-domain-wall scatterings, we confirm that there exist no s-channel open string Regge poles in the form factor of the amplitudes as O-planes were known to be not dynamical. However, the usual claim that there is a thickness of order √ α ′ for the O-domain-wall is misleading as the UV behavior of its scatterings is power-law instead of exponential fall-off.

(9.92)
For simplicity, we are going to calculate one tachyon and one massive closed string state scattered from the O-particle in the high energy limit. One expects to get similar results for the generic Op-plane scatterings with p ≥ 0 except O-domain-wall scatterings, which will be discussed in the next section. For this case D µν = −δ µν , and the kinematic setup are where e P , e L and e T are polarization vectors of the tensor state k 2 on the high energy scattering plane. One can easily calculate the following kinematic relations in the high energy limit e T · k 2 = e L · k 2 = 0, (9.98) We define (9.108) and the Mandelstam variables can be calculated to be In the high energy limit, we will consider an incoming tachyon state k 1 and an outgoing tensor state k 2 of the following form For simplicity, we have omitted above a possible high energy vertex (α L −1 ) r ⊗ (α L −1 ) r ′ [42,55]. For this case, with momentum conservation on the O-planes, we have The high energy scattering amplitude can then be written as One can easily see that We will choose to calculate A 1 and A 2 . For the case of A 1 , we have To fix the modulus group on RP 2 , choosing z 1 = r and z 2 = 0 and we have Similarly, for the case of A 2 , we have The scattering amplitude on RP 2 can therefore be calculated to be The integral in Eq.(9.122) can be calculated as following where we have used the following identities of the hypergeometric function F (α, β, γ, x) To further reduce the scattering amplitude into beta function, we use the momentum conservation in Eq.(9.113) and the identity (1 + α) F (−α, 1, 2 + β, −1) + (1 + β) F (−β, 1, 2 + α, −1) We finally end up with string Regge poles exist in the form factor. We will see that the fundamental characteristics of O-domain-wall scatterings are very different from those of O-particle scatterings as we will now discuss in the next section.

Hard strings scattered from O-domain-wall
For this case the kinematic setup is e T = (0, sin θ, cos θ) , (9.131) In the high energy limit, the angle of incidence φ is identical to the angle of reflection θ and Diag D µν = (−1, 1, −1). The following kinematic relations can be easily calculated e T · k 2 = e L · k 2 = 0, (9.134) e T · k 1 = −2k 1 sin φ cos φ ∼ −E sin 2φ, (9.135) e T · D · k 1 = 0, (9.136) e T · D · k 2 = 2k 2 sin φ cos φ ∼ E sin 2φ, (9.137) We define 144) and the Mandelstam variables can be calculated to be The first term of high energy scatterings from O-domain-wall is The second term can be similarly calculated to be The scattering amplitudes of O-domain-wall on RP 2 can therefore be calculated to be By using the similar technique for the case of O-particle scatterings, the integral above can be calculated to be One thus ends up with where we have used c 0 ≡ 1 2 (M 2 1 + M 2 ) in the high energy limit. It is easy to see that the larger the mass M of the external leg is, the more numerous the closed string poles are. We thus confirm that all massive string states do couple to the O-domain-wall as was conjectured previously [52,53]. This is also consistent with the boundary state descriptions

C. Hard closed strings decay to open strings
In this section, we calculate the absorption amplitudes [127] of a closed string state at arbitrary mass level leading to two open string states on the D-brane at high energies.
The corresponding simple case of absorption amplitude for massless closed string state was calculated in [141] (The discussion on massless string states scattered from D-brane can be found in [53,[136][137][138][139][140]). The inverse of this process can be used to describe Hawking radiation in the D-brane picture.
As in the case of Domain-wall scattering discussed above, this process contains one kinematic variable (energy E) and thus occupies an intermediate position between the conventional three-point and four-point amplitudes. However, in contrast to the power-law behavior of high energy Domain-wall scattering which contains only one kinematic variable (energy E), its form factor behaves as exponential fall-off at high energies.
It is thus of interest to investigate whether the usual linear relations of high energy amplitudes persist for this case or not. As will be shown in this section, after identifying the geometric parameter of the kinematic, one can derive the linear relations (of the kinematic variable) and ratios among the high energy amplitudes corresponding to absorption of different closed string states for each fixed mass level by D-brane. This result is consistent with Our final results, however, will remain the same for arbitrary two open string excitation at high energies. Conservation of momentum on the D-brane implies where D µν =diag{−1, 1, −1, 1}. It is crucial to note that, in the high energy limit, k c = k op and the scattering angle θ is identical to the incident angle φ. One can calculate which will be useful for later calculations. We define the kinematic invariants s ≡ 4k 1 · k 2 = 2M 2 1 + 2M 2 2 + 2 (k 1 + k 2 ) 2 = −2 (4 + t) , (9.166) and calculate the following identities Note that there is only one kinematic variable as s and t are related in Eq.(9.166) [141]. On the other hand, since the scattering angle θ is fixed by the incident angle φ, φ and θ are not the dynamical variables in the usual sense.
Following Eq.(9.112), we consider an incoming high energy massive closed state to be [42,46] The amplitude of the absorption process can be calculated to be x, i} to fix the SL(2, R) gauge and use Eq.(9.161-9.164), we have By using the binomial expansion, we get Finally, to reduce the integral to the standard beta function, we do the linear fractional transformation x 2 = 1−y y to get In addition to an exponential fall-off factor, the energy E dependence of Eq. . This is because for the absorption process we are considering, there is only one kinematic variable and the usual Ward identity calculations do not apply. To compare Eq.(9.173) with the "ratios" of the Domain-wall scattering [46] T (n,0,q;n,0,q ′ ) T (n,0,0;n,0,0 Note that since the scattering angle θ is fixed by the incident angle φ, φ is not a dynamical variable in the usual sense. Another way to see this is through the relation of s and t in Eq.(9.166). We will call such an angle a geometrical parameter in contrast to the usual dynamical variable. This kind of geometrical parameter shows up in closed string state scattered from generic Dp-brane (except D-instanton and D-particle) [42,46]. This is because one has only two dynamical variables for the scatterings, but needs more than two variables to set up the kinematic due to the relative geometry between the D-brane and the scattering plane at high energies.
We emphasize that our result in Eq.(9.173) is consistent with the coexistence [46] of the linear relations and exponential fall-off behavior of high energy string/D-brane amplitudes.
That is, linear relations of the amplitudes are responsible for the softer, exponential fall-off high energy string/D-brane scatterings than the power-law field theory scatterings.

X. HARD SCATTERINGS IN COMPACT SPACES
In this chapter, following an old suggestion of Mende [54], we calculate high energy massive scattering amplitudes of bosonic string with some coordinates compactified on the torus [55,56]. We obtain infinite linear relations among high energy scattering amplitudes of different string states in the Gross kinematic regime (GR). This result is reminiscent of the existence of an infinite number of massive ZNS in the compactified closed and open string spectrums constructed in chapter IV [25,26].
In addition, we analyze all possible power-law and soft exponential fall-off regimes of high energy compactified bosonic string scatterings by comparing the scatterings with their 26D noncompactified counterparts. In particular, we discover in section X.A the existence of a power-law regime at fixed angle and an exponential fall-off regime at small angle for high energy compactified open string scatterings [56]. These new phenomena never happen in the 26D string scatterings. The linear relations break down as expected in all power-law regimes. The analysis can be extended to the high energy scatterings of the compactified closed string in section X.B, which corrects and extends the results in [55].
A. Open string compactified on torus

High energy Scatterings
We consider [56] hard scatterings of 26D open bosonic string with one coordinate compactified on S 1 with radius R. As we will see later, it is straightforward to generalize our calculation to more compactified coordinates. The mode expansion of the compactified coordinate is where K 25 is the canonical momentum in the X 25 direction Note that l is the quantized momentum and we have included a nontrivial Wilson line with U(n) Chan-Paton factors, i, j = 1, 2...n., which will be important in the later discussion.
The mass spectrum of the theory is where we have defined level mass asM 2 = 2 (N − 1) and N = k =0 α 25 −k α 25 k + α µ −k α µ k , µ = 0, 1, 2...24. We are going to consider 4-point correlation function in this chapter. In the center of momentum frame, the kinematic can be set up to be [55]  Note that The center of mass energy E is defined as (for large p, q) (10.10) We have where s, t and u are the Mandelstam variables with Note that the Mandelstam variables defined above are not the usual 25-dimensional Mandelstam variables in the scattering process since we have included the internal momentum K 25 i in the definition of k i . We are now ready to calculate the high energy scattering amplitudes.
In the high energy limit, we define the polarizations on the scattering plane to be where the fourth component refers to the compactified direction. It is easy to calculate the following relations In this chapter, we will consider the case of a tensor state [55] α T However, for our purpose here and for simplicity, we will not consider the general vertex in this chapter. The s − t channel of the high energy scattering amplitude can be calculated to be (We will ignore the trace factor due to Chan-Paton in the scattering amplitude calculation . This does not affect our final results in this chapter)

(10.25)
After fixing the SL(2, R) gauge and using the kinematic relations derived previously, we have where B(u, v) is the Euler beta function. We can do the high energy approximation of the gamma function Γ (x) then do the summation, and end up with

Classification of Compactified String Scatterings
It is well known that there are two kinematic regimes for the high energy string scatterings in 26D open bosonic string theory. The UV behavior of the finite and fixed angle scatterings in the GR is soft exponential fall-off. Moreover, there exist infinite linear relations among scatterings of different string states in this regime [27-32, 34, 36, 42, 45, 127]. On the other hand, the UV behavior of the small angle scatterings in the Regge regime is hard power-law.
The linear relations break down in the Regge regime. As we will see soon, the UV structure

a. Gross Regime -Linear Relations
In the Gross regime, p 2 ≃ q 2 ≫ K 2 i and p 2 ≃ q 2 ≫ N, Eq.(10.27) reduces to For each fixed mass level N, we have the linear relation for the scattering amplitudes with coefficients consistent with our previous results [27-32, 34, 36, 42, 45, 127]. and define the "26D scattering angle" φ as following   We now consider the second possible regime for the case of φ = finite, namely The last choice for the power-law regime is It is easy to show that this is indeed a power-law regime.
The last kinematic regime for the case of small angle φ ≃ 0 scattering is Compactified 24D (or less) scatterings For this case, we need to introduce another parameter to classify the UV behavior of high energy scatterings, namely the angle δ between K 1 and K 2 , K 1 · K 2 = |K 1 | |K 2 | cos δ. Similar results can be easily derived through the same method used in the compactified 25D scatterings. The classification is independent of the details of the moduli space of the compact spaces. We summarize the results in the following finite Exponential fall-off K 2 i ≃ p 2 ≃ q 2 ≫ N and q K 1 = −p K 3 K 2 i ≫ p 2 ≃ q 2 ≫ N and cos δ = 0

B. Closed string compactified on torus
In this section, we consider hard scatterings of 26D closed bosonic string [55] with one coordinate compactified on S 1 with radius R. As we will see later, it is straightforward to generalize our calculation to more compactified coordinates.

Winding string and kinematic setup
The closed string boundary condition for the compactified coordinate is (we use the notation in [10]) X 25 (σ + 2π, τ ) = X 25 (σ, τ ) + 2πRn, (10.47) where n is the winding number. The momentum in the X 25 direction is then quantized to be K = m R , (10.48) where m is an integer. The mode expansion of the compactified coordinate is The left and right momenta are defined to be 52) and the mass spectrum can be calculated to be (10.53) where N R and N L are the number operators for the right and left movers, which include the counting of the compactified coordinate. We have also introduced the left and the right level masses as Note that for the compactified closed string N R and N L are correlated through the winding modes.
In the center of momentum frame, the kinematic can be set up to be where p ≡ |p| and q ≡ |q| and With this setup, the center of mass energy E is One can easily calculate the following kinematic relations where the left and the right Mandelstam variables are defined to be We are now ready to calculate the string scattering amplitudes. Let's first calculate the where we have used α ′ = 2 for closed string propagators we obtain where we have used M 2 iL,R = −2 for i = 1, 2, 3, 4. In the above calculation, we have used the following well known formula for gamma function . (10.80)

High energy massive scatterings for general N R + N L
We now proceed to calculate the high energy scattering amplitudes for general higher mass levels with fixed N R + N L . With one compactified coordinate, the mass spectrum of the second vertex of the amplitude is We now have more mass parameters to define the "high energy limit". So let's first clear and redefine the concept of "high energy limit" in our following calculations. We are going to use three quantities E 2 , M 2 2 and N R + N L to define different regimes of "high energy limit". See FIG. 4. The high energy regime defined by E 2 ≃ M 2 2 ≫ N R + N L will be called Mende regime (MR). The high energy regime defined by E 2 ≫ M 2 2 , E 2 ≫ N R + N L will be called Gross region (GR). In the high energy limit, the polarizations on the scattering plane for the second vertex operator are defined to be where the fourth component refers to the compactified direction. One can calculate the following formulas in the high energy limit e T · k 1L = e T · k 1R = 0, (10.89) which will be useful in the calculations of high energy string scattering amplitudes.
For the noncompactified open string, it was shown that [31,32,45], at each fixed mass level M 2 op = 2(N − 1), a four-point function is at the leading order in high energy (GR) only for states of the following form where N 2l + 2q, l, q 0.To avoid the complicated subleading order calculation due to the α L −1 operator, we will choose the simple case l = 0. We made a similar choice when dealing with the high energy string/D-brane scatterings [42,46,127]. There is still one complication in the case of compactified string due to the possible choices of α 25 −n and α 25 −m in the vertex operator. However, it can be easily shown that for each fixed mass level with given quantized and winding momenta ( m R , 1 2 nR), and thus fixed N R + N L level, vertex operators containing α 25 −n orα 25 −m are subleading order in energy in the high energy expansion compared to other choices α T −1 (α T −1 ) and α L −2 (α L −2 ) on the noncompact directions. In conclusion, in the calculation of compactified closed string in the GR, we are going to consider tensor state of the form (10.92) at general N R + N L level scattered from three other tachyon states with N R + N L = 0.
Note that, in the GR, one can identify e p with e L as usual [27][28][29]. However, in the MR, one can not identify e p with e L . This can be seen from Eq.(10.85) to Eq.(10.88). In the MR, instead of using the tensor vertex in Eq.(10.92), we will use as the second vertex operator in the calculation of high energy scattering amplitudes. Note also that, in the MR, states in Eq.(10.93) may not be the only states which contribute to the high energy scattering amplitudes as in the GR. However, we will just choose these states to calculate the scattering amplitudes in order to compare with the corresponding high energy scattering amplitudes in the GR.
The high energy scattering amplitudes in the MR can be calculated to be After the standard SL(2, C) gauge fixing, we get By using Eqs.(10.85) to (10.90), the amplitude can be written as where, as in the calculation of section B.2 for the GR, we have used Eq.(10.78) to do the integration. It is easy to do the following approximations for the gamma functions One can now do the double summation and drop out the M iL,R terms to get

The infinite linear relations in the GR
For the special case of GR with E 2 ≫ M 2 2 , one can identify q with p, and the amplitude in Eq.(10.98) further reduces to It is crucial to note that the high energy limit of the beta function with s + t + u = 2n − 8 is [27,28] where we have calculated the approximation up to the next leading order in energy E. Note We see that there is a m R , 1 2 nR dependence in the sin(πs L /2) sin(πt R /2) sin(πu L /2) factor in our final result. This is physically consistent as one expects a m R , 1 2 nR dependent Regge-pole and zero structures in the high energy string scattering amplitudes.
In conclusion, in the GR, for each fixed mass level with given quantized and winding momenta m R , 1 2 nR (thus fixed N L and N R by Eq.(10.53)), we have obtained infinite linear relations among high energy scattering amplitudes of different string states with various (q L , q R ). Note also that this result reproduces the correct ratios − 1 2M 2 q L +q R obtained in the previous works [42,46,127]. However, the mass parameter M 2 here depends on Presumably, the infinite linear relations obtained above can be reproduced by using the method of high energy ZNS, or high energy Ward identities, adopted in the previous chapters [27-32, 34, 36, 42, 45, 127]. The existence of Soliton ZNS at arbitrary mass levels was constructed in chapter IV [6]. A closer look in this direction seems worthwhile. In the chapter, however, we are more interested in understanding the power-law behavior of the high energy string scattering amplitudes and breakdown of the infinite linear relations as we will discuss in the next section.

Power-law and breakdown of the infinite linear relations in the MR
In this section we discuss the power-law behavior of high energy string scattering amplitudes in a compact space. We will see that, in the MR, the infinite linear relations derived in section B.4 break down and, simultaneously, the UV exponential fall-off behavior of high energy string scattering amplitudes enhances to power-law behavior. The power-law behavior of high energy string scatterings in a compact space was first suggested by Mende [54].
Here we give a mathematically more concrete description. It is easy to see that the "power law" condition, i.e. Eq.(3.7) in Mende's paper [54], turns out to be we define the following "super-highly" winding kinematic regime Note that all m i were chosen to vanish in order to satisfy the conservations of compactified momentum and winding number respectively [55].  and λ 2 = − 2p n 2 R the same. This completes the discussion of power-law regime at fixed angle for high energy compactified closed string scatterings. The "super-highly" winding regime derived in this section is to correct the "Mende regime" discussed in [55]. The regime defined in Eq.(10.109) is indeed exponential fall-off behaved rather than power-law claimed in [55].

Stringy symmetries of Regge string scattering amplitudes
In this part of the review, we are going to discuss string scatterings in the high energy, fixed momentum transfer regime or Regge regime (RR) [57][58][59][60][61][62]. See also [142][143][144]. We will see that Regge string scattering amplitudes contain information of the theory in complementary to string scattering amplitudes in the high energy, fixed angle or Gross regime(GR). The UV behavior of high energy string scatterings in the GR (hard string scatterings) is well known to be very soft exponential fall-off, while that of RR is power-law. There are some other fundamental differences and links between the calculations of Regge string scatterings and hard string scatterings, which we list below : A. The number of high energy scattering amplitudes for each fixed mass level in the RR is much more numerous than that of GR calculated previously [64]. F. All the RR amplitudes can be expressed in terms of one single Appell function F 1 [74].
This result enables us to derive infinite number of recurrence relations among RR amplitudes at arbitrary mass levels, which are conjectured to be related to the known SL(5, C) dynamical symmetry of F 1 .
The part III of the paper is organized as following. In chapter XI we calculate Regge string scattering amplitudes in terms of Kummer functions [64]. We then prove a set of Stirling number identities [67] and use them to reproduce ratios among hard string scattering amplitudes in the GR discussed in chapter V. Finally we calculate recurrence relations among Regge string scattering amplitudes, and use them to prove Regge stringy Ward identities or decoupling of ZNS in the RR for the first few mass levels [71]. In Chapter XII, we generalize the calculations to four classes of Regge superstring scattering amplitudes and reproduce the ratios calculated in chapter VIII. In addition, discover new high energy superstring scattering amplitudes with polarizations orthogonal to the scattering plane [65].
In Chapter XIII we generalize the calculation of four tachyon BPST vertex operator [62] to the general high spin cases [73], and derive the recurrence relations among these higher spin BPST vertex operators. In Chapter XIV we discuss higher spin Regge string states scattered from D-particle [43]. We will obtain the complete GR ratios, which include a subset calculated in section IX.A.3, from Regge string states scattered from D-particle.
In addition, we will see that although there is no factorization for closed string D-particle scattering amplitudes into two channels of open string D-particle scattering amplitudes, the complete ratios are factorized which came as a surprise.
In chapter XV we discover that all RR amplitudes can be expressed in terms of one single Appell function F 1 [74]. More general recurrence relations among RR amplitudes will be derived. We will also discuss the SL(5, C) dynamical symmetry of F 1 which is argued to be closely related to high energy spacetime symmetry of 26D bosonic string theory.

XI. KUMMER FUNCTIONS U AND PATTERNS OF REGGE STRING SCAT-TERING AMPLITUDES (RSSA)
In this chapter, we first calculate a subclass of Regge string scattering amplitudes (RSSA) and expressed them in terms of Kummer functions of the second kind [69,70]. We then prove in section B a set of Stirling number identities [67] and use them to reproduce ratios among hard string scattering amplitudes calculated in chapter V. In section C, we show that all RSSA are power law behaved, and the exponents of the power law are universal and are independent of the mass levels [64]. In section D, we calculate the most general RSSA and derive recurrence relations among them [71]. We show that, for the first few mass levels, the decoupling of ZNS in the RR or Regge stringy Ward identities can be derived from these recurrence relations [71]. This shows that, in contrast to the GR considered in chapter V, recurrence relations are more fundamental than Regge stringy Ward identities. Finally we prove that all RSSA can be solved by these recurrence relations and expressed in terms of one single RSSA [71].

A. Kummer functions and RSSA
We now begin to discuss high energy string scatterings in the RR. That is in the kinematic regime s → ∞, As in the case of GR, we only need to consider the polarizations on the scattering plane, which is defined in Appendix C. Appendix C also includes the kinematic set up. Instead of using (E, θ) as the two independent kinematic variables in the GR, we choose to use (s, t) in the RR. One of the reason has been, in the RR, t ∼ Eθ is fixed, and it is more convenient to use (s, t) rather than (E, θ). In the RR, to the lowest order, Eqs.(C.13) to (C.18) reduce to and Note that e P does not approach to e L in the RR. This is very different from the case of GR. In the following discussion, we will calculate the amplitudes for the longitudinal polarization e L . For the e P amplitudes, the results can be trivially modified. We will find that the number of high energy scattering amplitudes for each fixed mass level in the RR is much more numerous than that of GR calculated previously. On the other hand, it seems that the saddle-point method used in the GR is not applicable in the RR. We will first calculate the string scattering amplitudes on the scattering plane e L , e T for the mass level M 2 2 = 4. In the mass level M 2 2 = 4 (M 2 1 = M 2 3 = M 2 4 = −2), it turns out that there are eight high energy amplitudes in the RR The s − t channel of these amplitudes can be calculated to be

9)
10) (11.11) and One important observation for high energy amplitudes in the RR is for those amplitudes with the same structure as those of the GR in Eq.(5.67). For these amplitudes, the relative ratios of the coefficients of the highest power of t in the leading order amplitudes in the RR can be calculated to be

13)
14) which reproduces the ratios in the GR in Eq. (5.16). Note that the symmetrized and antisymmetrized amplitudes are defined as T [T L] = 1 2 T T L − T LT ; (11.17) and similarly for the amplitudes A (T L) and A [T L] in the RR. Note that T LT ∼ (α L −1 )(α T −2 )|0 in the GR is of subleading order in energy, while A LT in the RR is of leading order in energy.
However, the contribution of the amplitude A LT to A (T L) and A [T L] in the RR will not affect the ratios calculated above. As we will see next, this interesting result can be generalized to all mass levels in the string spectrum.
We first calculate high energy string scattering amplitudes in the RR for the arbitrary mass levels. Instead of states in Eq.(5.67) for the GR, one can argue that the most general string states (ignore the e P amplitudes) one needs to consider at each fixed mass level N = n,m nk n + mq m for the RR are These RR amplitudes are good enough to reproduce the GR ratios calculated previously. By the simple kinematics e T ·k 1 = 0, and the energy power counting of the string amplitudes, we end up with the following rules to simplify the calculation for the leading order amplitudes in the RR: n > 1, 1 term n = 1 2 terms (contraction of ik 1 · X and ik 3 · X with ε L · ∂ n X). (11.20) The s − t channel scattering amplitudes of this state with three other tachyonic states can be calculated to be (11.21) The Beta function above can be approximated in the large s, but fixed t limit as follows where (a) j = a(a + 1)(a + 2)...(a + j − 1) (11.23) is the Pochhammer symbol. The leading order amplitude in the RR can then be written as which is UV power-law behaved as expected. The summation in Eq.(11.24) can be represented by the Kummer function of the second kind U as follows, Finally, the amplitudes can be written as (11.26) In the above, U is the Kummer function of the second kind and is defined to be It is crucial to note that c = t 2 + 2 − q 1 , and is not a constant as in the usual case, so U in Eq.(11.26) is not a solution of the Kummer equation. This will make our analysis in the next section more complicated as we will see soon. On the contrary, since a = −q 1 an integer, the Kummer function in Eq.(11.25) terminated to be a finite sum. This will simplify the manipulation of Kummer function used in this chapter.

B. Reproducing ratios among hard scattering amplitudes
It can be seen from Eq.(11.24) that the Regge scattering amplitudes at each fixed mass level are no longer proportional to each other. The ratios are t dependent functions and can be calculated to be [64] , (11.29) where (x) j = x(x+1)(x+2) · · · (x+j −1) is the Pochhammer symbol which can be expressed in terms of the signed Stirling number of the first kind s (n, k) as following To ensure the identification for the general mass levels  (11.30) where L = 1 − N and is an integer. Similar identification can be extended to the case of closed string as well [36]. For all four classes of high energy superstring scattering amplitudes, L is an integer too [65]. A recent work on string D-particle scattering amplitudes [43] also gives an integer value of L. Note that L affects only the sub-leading terms in O Here we give a simple example for m = 3 [65] 6 j=0 Mathematically, Eq.(11.30) was exactly proved [64,65] for L = 0, 1 by a calculation based on a set of signed Stirling number identities developed recently in combinatorial theory in [145].
For general integer L cases, only the identity corresponding to the nontrivial leading term (2m)! m! (−t ′ ) −m was rigorously proved in [65], but not for other "0 identities". A numerical proof of Eq.(11.30) was given in [65] for arbitrary real values L and for non-negative integer m up to m = 10. It was then conjectured that [65] Eq.(11.30) is valid for any real number L and any non-negative integer m. It is important to prove Eq.(11.30) for any non-negative integer m and arbitrary real values L, since these values can be realized in the high energy scattering of compactified string states, as was shown recently in [68]. Real values of L appear in string compactifications due to the dependence on the generalized KK internal

Proof of the new Stirling number identity
We now proceed to prove Eq.(11.30) [67]. We first rewrite the left-hand side of Eq. (11.30) in the following form Thus c (l, j − i) = 0 only if j ≥ i and l ≥ j − i. We can rewrite G (m, i) as where we have defined The recurrence of the signless Stirling number identity c (k + n, k) = (n + k − 1) c (n + k − 1, k) + c (n + k − 1, k − 1) (11.39) leads to the equation with the initial value The first couple of C n (x) can be calculated to be Now by induction, it is easy to show that and f n (1) = (2n − 1)!!. (11.44) In order to prove Eq.(11.35), we note that (−1) N S N (p) is the coefficient of x N +p in the 45) which is obviously zero for N ≥ p + 1. This proves S N (p) = 0 for N ≥ p + 1 and thus Eq. (11.35).
In order to prove the first identity in Eq.(11.34), we first note that the above argument remains true for i = m and 0 ≤ p < i. So Eq.(11.34) corresponds to the case p = i = m. By using Eq.(11.36), we can evaluate (11.46) Eq.(11.46) corresponds to the coefficient of x 2m in the function By Eq. (11.44), this coefficient is This proves Eq.(11.34). We thus have completed the proof of Eq.(11.30) for any non-negative integer m and any real value L.
It was remarkable to first predict [64] the mathematical identities in Eq.(11.30) provided by string theory, and then a rigorous mathematical proof followed [67]. It was interesting to see that the validity of Eq.(11.30) includes non-integer values of L which were later realized by Regge string scatterings in compact space [68].

Subleading orders
In this section, we calculate the next few subleading order amplitudes in the RR for the mass level M 2 2 = 4, 6 [64]. We will see that the ratios in Eq. We will extend the kinematic relations in the RR to the subleading orders. We first express all kinematic variables in terms of s and t, and then expand all relevant quantities in s : (11.53) e T · k 1 = 0. (11.54) A key step is to express the scattering angle θ in terms of s and t. This can be achieved by One can then calculate the following expansions for the mass level M 2 2 = 4 to subleading orders in s in the RR. These are . We conjecture that these ratios persist to all energy orders in the Regge expansion of the amplitudes. This is consistent with the results of GR by taking both s, −t → ∞. For the mass level M 2 2 = 6 [29], the amplitudes can be calculated to be

67)
. (11.70) In the above calculations, as in the case of M 2 2 = 4, we have ignored a common overall factor which will be discussed in the next section. Note that the ratios of the coefficients in the leading order t for the energy orders s 4 , s 3 , s 2 reproduced the GR ratios in Eq.(5.16).
However, the subleading terms for orders s 1 , s 0 contain no GR ratios. Mathematically, this is because the highest power of t in the coefficient functions of s 1 is 4 rather than 5, and those of s 0 is 4 rather than 6. This is because the power of t in the kinematic relation Eq. (11.59) can be as high as one wants if one goes to subleading orders, while that of Eq.(11.58) is not. The sin θ factor in Eq.(11.59) contributes terms of higher order powers of t, while cos θ factor in in Eq.(11.58) does not. This can be seen from the kinematic relation in Eq. (11.56).
In general, one can easily show that the sin θ factor will contribute only for the even mass

C. Universal power law behavior
In the discussion of the last section, we ignored an overall common factor Γ(−1−s/2)Γ(−1−t/2) Γ(u/2+2) of the amplitudes for mass levels M 2 2 = 4, 6. We paid attention only to the ratios among scattering amplitudes of different string states. In this section, we calculate the high energy behavior of string scattering amplitudes for string states at arbitrary mass levels in the RR.
The power law behavior ∼ s α(t) of the four-tachyon amplitude in the RR is well known in the literature. Here we want to generalize this result to string states at arbitrary mass levels.
We can use the saddle point method to calculate the leading term of gamma functions in the RR Thus, the overall s-dependence in the amplitudes is of the form A (kn,qm) ∼ s α(t) (in the RR) (11.72) where α(t) = α(0) + α ′ t, α(0) = 1 and α ′ = 1/2. (11.73) This generalizes the high energy behavior of the four-tachyon amplitude in the RR to string states at arbitrary mass levels. The new result here is that the behavior is universal and is mass level independent. In fact, as a simple application, one can also derive Eq. (11.72) directly from Eq.(11.26) by using (in the RR) (11.74) We conclude that the well known ∼ s α(t) power-law behavior of the four tachyon string scattering amplitude in the RR can be extended to high energy string scattering amplitudes of arbitrary string states.

D. Recurrence relations of RSSA
To discuss relations among RSSA, one need to consider the complete RR string states [71]. The complete leading order high energy open string states in the Regge regime at each fixed mass level N = n,m,l>0 np n + mq m + lr l are The case for q m = 0 has been calculated previously in [64,65] We stress that the inclusion of both α P −m and α L −l operators in Eq.(11.75) will be crucial to study Regge string Ward identities to be discussed in the later part of this chapter. It is also important to discuss the conformal invariant property of high energy string scattering amplitudes [65]. The momenta of the four particles on the scattering plane are , +q cos φ, +q sin φ (11.79) where p ≡ |p|, q ≡ |q| and k 2 i = −M 2 i . The relevant kinematics are (11.81) and e T · k 1 = 0, e T · k 3 ≃ − √ −t (11.82) wheret andt ′ are related to t by finite mass square terms Note that, unlike the case of GR, here e P does not approach to e L in the RR. The Regge string scattering amplitudes can then be explicitly calculated to be (11.84) In the second equality of the above equation, we have dropped the first term in the bracket with power of p n , and the first terms in the brackets with powers of q m and r l for m, l > 1.
These terms lead to subleading order terms in energy in the Regge limit [64,65]. Now the beta function in Eq. (11.84) can be approximated in the RR by [64,65] B − s 2 where (a) j = a(a + 1)(a + 2)...(a + j − 1) is the Pochhammer symbol. Finally we arrive at the amplitude with two equivalent expressions It is interesting to note that the Regge behavior is again universal and is mass level independent as in the case of previous section [64] B −1 − s 2 , −1 − t 2 ∼ s α(t) (in the RR) (11.88) where α(t) = α(0) + α ′ t, α(0) = 1 and α ′ = 1/2. That is, the well known ∼ s α(t) powerlaw behavior of the four tachyon string scattering amplitude in the RR can be extended to arbitrary higher string states. This result will be used to construct an inter-mass level recurrence relation for Regge string scattering amplitudes later in Eq.(13.56).

Recurrence relations and RR stringy Ward identities
In this section, we first discuss Regge stringy Ward identities derived from Regge string ZNS (RZNS) for mass level M 2 = 2 and 4 [71]. We will see that, unlike the case for the Eq.(11.86) and Eq.(11.87) with a = −q 1 (or −r 1 ) a non-positive integer, one can use recurrence relations to solve all U(−q 1 , c, x) functions algebraically and thus determine all Regge string scattering amplitudes at arbitrary mass levels algebraically up to multiplicative factors [71]. We stress that for general values of a, the best one can obtain from recurrence relations is to express any Kummer function in terms of any two of its associated function (see the appendix D).
There are 9 Regge string amplitudes for the mass level M 2 = 2, and (11.101) Therefore Eq.(11.100) is equivalent to Finally one can use Eq.(11.95) andt = t to prove Eq. (11.102). This completes the proof of Regge stringy Ward identities for mass level M 2 = 2 by using recurrence relations of Kummer functions.
We will give a brief description for the case of mass level M 2 = 4. There are 22 Regge string amplitudes for the mass level M 2 = 4, A P P P , A P P L , A P P T , A P LL , A P LT , A P T T , To fix the notation, we adopt the convention of mass ordered in the α α −n operators, for example,  [27][28][29] (Correspondingly the creation operators α P −n and −α L −n are identified, where the sign comes from the difference between the timelike and spacelike directions specified by the metric of the scattering plane η µν = diag (−1, 1, 1) .), and take high energy fixed angle limit to get three Ward identities in leading order energy [27][28][29] T LLT + T (LT ) = 0, (11.111) 10T LLT + T T T T + 18T (LT ) = 0, (11.112) T LLT + T T T T + 9T [LT ] = 0, (11.113) which can be easily solved to get [27][28][29] T We can now turn to prove Regge stringy Ward identity Eq. (11.103). We first note that Eq.(11.104) and Eq.(11.105) implies that Eq.(11.103) is equivalent to 25A P P P − 75A P P + 50A P = 0. (11.127) The three terms in Eq.(11.127) divided by the beta function are Therefore we want to show Using Eq.(11.96), we obtain can be similarly proved.

Solving all RSSA by Kummer recurrence relations
We observe that the recurrence relations of Kummer functions are more powerful than Regge stringy Ward identities in relating Regge string scattering amplitudes. This is indeed the case as we will show [71] now in the following that all Regge string scattering amplitudes can be algebraically solved by using recurrence relations up to multiplicative factors in the first line of Eq.(11.86) (or Eq.(11.87)).
Secondly, we want to show that each Kummer function in the summation of Eq. (11.87) can be expressed in terms of Regge string scattering amplitudes. To show this, we first consider r 1 = 0 amplitudes in a fixed mass level and a fixed q 1 with no summation over Kummer functions. These amplitudes contain only one Kummer function. Then let us take the amplitude with the maximum p 1 . By decreasing p 1 and increasing r 1 by 1, we can create an amplitude with two Kummer functions in the same mass level and the same q 1 . The first one of the two Kummer functions is the one appeared in the previous amplitude with r 1 = 0, so we can write the second Kummer function in terms of the two amplitudes, one with r 1 = 0 and the other with r 1 = 1.
By decreasing p 1 and increasing r 1 by 1 again, we can create an amplitude with three Kummer functions in the same mass level and the same q 1 . The first two of the three Kummer functions is the ones appeared in the previous two amplitudes, so we can write the third Kummer functions in terms of the three amplitudes. We can repeat this process until p 1 = 0. In this way, we can express all the Kummer functions in Eq. (11.87) in terms of the RR amplitudes.
In the following, as an example, let us illustrate the above process for the mass level The recurrence relation leads to the following recurrence relation among Regge string scattering amplitudes [71] M √ −tA P P P − 4A P P T + M √ −tA P P L = 0. For the third example, we construct an inter-mass level recurrence relation for Regge string scattering amplitudes at mass level M 2 = 2, 4. We begin with the addition theorem of Kummer function [72] U(a, c, which terminates to a finite sum for a non-positive integer a. By taking, for example, a = −1, c = t 2 + 1, x = t 2 − 1 and y = 1, the theorem gives in its coefficients and are in general independent of Regge stringy Ward identities. The dynamical origin of these recurrence relations remain to be studied. These recurrence relations among Regge string scattering amplitudes are dual to linear relations or symmetries among high energy fixed angle string scattering amplitudes discovered previously [27-29, 31, 32, 45].
Recently, five-point tachyon amplitude was considered in the context of BCFW application of string theory in [146]. It will be interesting to consider both RR and GR of higher spin five-point scattering amplitudes.
Note that, in order to simplify the notation, we have only shown the second state of the four point functions to represent the scattering amplitudes on both sides of each equation above. This notation will be used throughout the paper whenever is necessary. Eqs.(12.1) to (12.4) are thus the SUSY generalization of Eq.(5.60) for the bosonic string. There are much more high energy fermionic string scattering amplitudes other than states we will consider in this chapter.
We stress that, in addition to high energy scatterings of string states with polarizations orthogonal to the scattering plane considered previously in the GR [34], there are more high energy string scattering amplitudes with more worldsheet fermionic operators b P,T − n 2 in the string vertex.
The first RR scattering amplitude we want to calculate corresponding to state in Eq.
where we have dropped out an overall factor. In Eq.
In Eq.(12.7), e P ·k 1 x 2 is of subleading order in the RR and 1 x is the ghost contribution. The second term of Eq.(12.8) vanishes due to the SL(2, R) gauge fixing x 1 = 0, x 2 = x, x 3 = 1 and x 4 = ∞. The first term of Eq.(12.8) vanishes due to e T 1 · e P 2 = 0. The amplitude then reduces to The Beta function above can be approximated in the large s, but fixed t limit as follows where (a) j = a(a + 1)(a + 2)...(a + j − 1) (12.13) is the Pochhammer symbol. The leading order amplitude in the RR can then be written as which is UV power-law behaved as expected. The summation in Eq. (12.14) can be represented by the Kummer function of the second kind U as follows, Finally, the amplitudes can be written as There are some important observations for the high energy amplitude in Eq. (12.16). First, the amplitude gives the universal power-law behavior for string states at all mass levels A (N,2m,q) 1 ∼ s α(t) (in the RR) (12.17) where α(t) = a 0 + α ′ t, a 0 = 1 2 and α ′ = 1/2. (12.18) This generalizes the high energy behavior of the four massless vector amplitude in the RR to string states at arbitrary mass levels. Second, the amplitude gives the correct intercept a 0 = 1 2 of fermionic string. Finally, the amplitude can be used to reproduce the ratios calculated in the GR as we will see in section E.
Note that this is the only case with odd integer 2m + 1. The RR scattering amplitude corresponding to state in Eq.(8.2) can be written as where we have dropped out an overall factor. The amplitude can be calculated to be We then do an approximation for beta function similar to the calculation for A (N,2m,q) 1 and end up with (12.21) Note that there are two terms in Eq. (12.21), and the first argument of the U function a = −1 − 2m is odd. These differences will make the calculation of the ratios in the next section more complicated. Finally, the amplitude gives the universal power-law behavior for string states at all mass levels with the correct intercept a 0 = 1 2 of fermionic string.
The third RR scattering amplitude corresponding to state in Eq.(8.1) is where we have dropped out an overall factor. The scattering amplitude can be calculated to be We then do an approximation for beta function similar to the calculation for A (N,2m,q) 1 and end up with In this case there are again two terms as in the amplitude A 2 but with an even argument a = −2m. Finally, the amplitude gives the universal power-law behavior for string states at all mass levels with the correct intercept a 0 = 1 2 of fermionic string.
The fourth RR scattering amplitude corresponding to state in Eq.(8.4) is where we have dropped out an overall factor. The scattering amplitude can be calculated to be With a similar approximation for the beta function, we get Again the amplitude gives the universal power-law behavior for string states at all mass levels with the correct intercept a 0 = 1 2 of fermionic string. In the next section we are going to use the four amplitudes calculated in this section to extract ratios calculated in the fixed angle regime.

E. Reproducing ratios among hard SUSY scattering amplitudes
In the bosonic string calculation discussed in chapter XI [64], we learned that the relative coefficients of the highest power t terms in the leading order amplitudes in the RR can be used to reproduce the ratios of the amplitudes in the GR for each fixed mass level. In this section, we are going to generalize the calculation [65] to four classes of fermionic string states for arbitrary mass levels. We begin with the first amplitude of Eq.(12.16).

Ratios for
It is important to note that there are no linear relations among high energy string scattering amplitudes, Eq.(12.16), of different string states for each fixed mass level in the RR.
In other words, the ratios A where we have used Eq.(11.3a) to replace t byt. If the leading order coefficients in Eq. (12.28) extracted from the amplitudes in the RR are to be identified with the ratios calculated in the GR in Eq.(12.1), we need the following identity (12.30) where L = N + 1 and is an integer. This identity was proved in [67].
The ratios A can be calculated to be The bracket in the above equation can be simplified by dropping out the subleading order terms in the calculation, and one obtains where we have dropped out the subleading order terms in the second equality of the calculation. Finally, the ratios can be calculated to be

Ratios for
The ratios A We thus have succeeded in extracting the Ratios of high energy superstring scattering amplitudes in the GR from the high energy superstring scattering amplitudes in the RR. In the next section, we will study the subleading order amplitudes.

Subleading order amplitudes
In this section, we calculate the next few subleading order amplitudes in the RR for the mass levels M 2 2 = 2(N + 1) = 4, 6. The calculation for M 2 2 = 8 can be found in [65]. The relevant kinematic can be found in the Appendix C. We will see that the ratios derived in the previous section persist to subleading order amplitudes in the RR. , and it is easy to check that these are the same as the ratios in the fixed angle limit. Moreover, the ratios persist in the second subleading order terms s 0 t 3 as 1 4 : 1 32 : − 1 16 . The ratios terminate to this order. We can also compare the ratios among different worldsheet fermionic states but with the same mass level M 2 2 = 4. We have the expansions: The ratios of the leading order coefficients are proportional to that of state |2, 0, 0 |b T  They again match with the ratios in the fixed angle limit. One can also find that the second subleading order ratios are the same 1 4 : 1 16 : − 1 8 : 1 32 . Again the ratios terminate to this order.
The ratios of the leading order coefficients are given by , respectively. Again they agree with the ratios in the fixed angle limit. We expect that the ratios persist to all orders in the expansions.

TORS
In this chapter, we study higher spin Regge string scattering amplitudes from BPST vertex operator approach [73]. Note that in the original BPST paper [62], the authors calculated the case of four tachyon closed string and thus Pomeron vertex operators. Here, for simplicity, we will calculate higher spin BPST vertex operators at arbitrary mass levels of open bosonic string. The calculation can be easily generalized to closed string case.
We find that all BPST vertex operators can be expressed in terms of Kummer functions of the second kind. We can then derive infinite number of recurrence relations among BPST vertex operators of different string states. These recurrence relations among BPST vertex operators lead to the recurrence relations among Regge string scattering amplitudes discovered in chapters XI and XV. [71,74].

A. Four tachyon scattering
We will calculate high energy open string scatterings in the Regge Regime Note that the convention for s and t adopted here is different from the original BPST paper in [62].
We first review the calculation of tachyon BPST vertex operator [62]. The s − t channel of open string four tachyon amplitude can be written as Since s → ∞, the integral is dominated around ω = 1. Making the variable transformation ω = 1 − x, the integral is dominated around x = 0, we obtain Alternatively, the integral in A can be expressed as A = dω e ik 1 X(0) e ik 2 X(ω) e ik 3 X(1) e ik 4 X(∞) . (13.5) One can calculate the operator product expansion (OPE) in the Regge limit e ik 2 X(w) e ik 3 X(z) ∼ |w − z| k 2 ·k 3 e i(k 2 +k 3 )X(z)+ik 2 (w−z)∂X(z)+··· .
This means In evaluating Eq.(13.5), one can instead carry out the ω integration first in Eq.(13.6) at the operator level to obtain the BPST vertex operator [62] V BP ST = dωe ik 2 X(ω) e ik 3 X(1) which leads to the same amplitude as in Eq.(13.4) (13.8) B. Higher spin BPST vertex

A spin two state
It was shown [64,65,71] that for the 26D open bosonic string states of leading order in energy in the Regge limit at mass level M 2 2 = 2(N − 1), N = n,m,l>0 np n + mq m + lr l are of the form Eq. (11.75). In this section, we first consider a simple case of a spin two state α P −1 α P −1 |0 corresponding to the vertex ∂X P 2 e ik 2 X (ω). The four-point amplitude of the spin two state with three tachyons can be calculated by using the conventional method .
The momenta of the four particles on the scattering plane are 10) where p ≡ |p|, q ≡ |q| and k 2 i = −M 2 i . The relevant kinematics in the Regge limit are [64,65,71] (13.14) and wheret andt ′ are related to t by finite mass square terms By using Eq.(13.14), one easily see that the three terms in Eq.(13.9) share the same order of energy in the Regge limit. We stress that this key observation on the polarizations for higher spin states was not discussed in [62,147].

(13.18)
One can carry out the ω integration in Eq. (13.18) at the operator level to obtain the BPST vertex operator which leads to the same amplitude . (13.20) Note that the three terms in Eq. (13.19) lead to the three terms respectively in Eq. (13.20) with the same order of energy in the Regge limit.

Higher spin states
We now consider the higher spin state which corresponds to the vertex The four-point amplitude of the above state with three tachyons was calculated to be (from now on we set M 2 = M) [64,65,71] where U is the Kummer function of the second kind. One can calculate the OPE in the Regge limit (1 − ω) k 2 ·k 3 −N +j e ikX(1)−ik 2 (1−ω)∂X(1) (13.27) where N = n,m (np n + mq m ). We can carry out the ω integration in Eq. (13.27) to obtain the BPST vertex operator V (pn;qm) (1) . (13.28) One notes that, in Eq.(13.28), M∂X (1) · e P = k 2 · ∂X(1) and the summation over j can be simplified. The BPST vertex operator can be further reduced to V (pn;qm) pointed out in section XI.C [64] and will be crucial to derive inter-mass level recurrence relations among BPST vertex operators to be discussed later.  From any two of these six relations the remaining four recurrence relations can be deduced.
The confluent hypergeometric function U(a, c, x) with parameters (a±m, c±n) for m, n = 0, 1, 2...are called associated functions. Again it can be shown that there exist relations between any three associated functions, so that any confluent hypergeometric function can be expressed in terms of any two of its associated functions.
For the next example, we construct an inter-mass level recurrence relation for BPST vertex operators at mass level M 2 = 2, 4. We begin with the addition theorem of Kummer function [72] U(a, c, which terminates to a finite sum for a non-positive integer a. By taking, for example, a = −1, c = t 2 + 1, x = t 2 − 1 and y = 1, the theorem gives [71] U −1, Eq.(13.55) leads to an inter-mass level recurrence relation among BPST vertex operators [73] M(2)(t + 6)V  [71]. The corresponding r 1 for each BPST vertex operator are (0, 1, 2, 3). Here we use a new notation for BPST vertex operator, for exam-  (1) e P ·∂X(1) From the above equations, one can easily see that U 0,  (1) . To derive an example of recurrence relation, one notes that Eq.(13.32) gives (13.66) which leads to the recurrence relation among BPST vertex operators Other recurrence relations of Kummer functions can be used to derive more recurrence relations among BPST vertex operators. For example, Eq.(13.32) gives a recurrence relation of U 0, t 2 + 1, t 2 − 1 and its associated functions U 0, t 2 − 1, t 2 − 1 and U 0, which leads to the recurrence relation among BPST vertex operators

E. Arbitrary mass levels
In this section, we solve the Kummer functions in terms of the highest spin string states scattering amplitudes for arbitrary mass levels. The highest spin string states at the mass level M 2 = 2 (N − 1) are defined as where only α −1 operator appears. The highest spin string states BPST vertex operators can be easily obtained from Eq.(13.45) as  Let us consider, for example, the recurrence relation the above recurrence relation becomes Plug the Kummer functions Eq.(13.74) into the above recurrence relation, we obtain the recurrence relation among BPST vertex operators at general mass level N where we have defined As an example, at the mass level M 2 = 4 with q 1 = r 1 = 0, we get

XIV. REGGE STRING SCATTERED FROM D-PARTICLE
In this chapter we study [43] scattering of higher spin closed string states at arbitrary mass levels from D-particle in the RR. The scattering of massless string states from D-brane was well studied in the literature and can be found in [53,[136][137][138][139][140] Since the mass of Dbrane scales as the inverse of the string coupling constant 1/g, it was assumed that it was infinitely heavy to leading order in g and did not recoil.
We will extract the complete infinite ratios in Eq.(5.60) among high energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes.
The complete ratios calculated by this indirect method include a subset of ratios in Eq.(9.55) calculated previously by direct fixed angle calculation [42].
More importantly, we discover that the RR amplitudes calculated in this chapter for closed string D-particle scatterings can NOT be factorized and thus are different from amplitudes for the high-energy closed string-string scattering calculated previously [36,148].
GR Amplitudes for the high-energy closed string-string scattering calculated in chapter VII can be factorized into two open string scattering amplitudes by using a calculation [148] based on the KLT formula [44]. Similarly the RR closed string-string amplitudes [36] can be factorized too. Presumably, this non-factorization is due to the non-existence of a KLTlike formula for the string D-brane scattering amplitudes. There is no physical picture for open string D-particle tree scattering amplitudes and thus no factorization for closed string D-particle scatterings into two channels of open string D-particle scatterings.
However, surprisingly, we will find [43] that in spite of the non-factorizability of the closed string D-particle scattering amplitudes, the complete ratios derived for the fixed angle regime are found to be factorized. These ratios are consistent with the decoupling of high-energy ZNS calculated in Eq.(5.60) of chapter V. [27-32, 34, 45].

A. Kinematics Set-up
In this chapter, we consider an incoming string state with momentum k 2 scattered from an infinitely heavy D-particle and end up with string state with momentum k 1 in the RR.
The high energy scattering plane will be assumed to be the X − Y plane, and the momenta are arranged to be k 1 = (E, k 1 cos φ, −k 1 sin φ) , (14.1) and φ is the scattering angle. For simplicity, we will calculate the disk amplitude in this paper. The relevant propagators for the left-moving string coordinate X µ (z) and the right- where matrix D has the standard form for the fields satisfying Neumann boundary condition, while D reverses the sign for the fields satisfying Dirichlet boundary condition. Instead of the Mandelstam variables used in the string-string scatterings, we define c 0 ≡ 2k 1 · D · k 2 + 1 = 2 E 2 + k 1 k 2 cos φ + 1, (14.10) so that The normalized polarization vectors on the high energy scattering plane of the k 2 string state are defined to be [27][28][29] 14) (14.15) e T = (0, 0, 1). (14.16) One can then easily calculate the following kinematics e T · k 2 = 0, e T · D · e P = e P · D · e T = 0, which will be useful in the amplitude calculation in the next section.

B. Regge String D-particle scatterings
We now begin to calculate the scattering amplitudes. For simplicity, we will take k 1 to be the tachyon and k 2 to be the tensor states. One can easily argue that a class of high energy string states for k 2 in the RR are [64,65] |p n , p ′ n , q m , q ′ m = where M 2 2 = (N − 2).

An example
Before calculating the string D-particle scattering amplitudes for general cases, we take an example and illustrate the method of calculation. We consider the case As we will see in the next section, the string D-particle scattering amplitudes with the general We start with the procedure in [44] to treat the vertex operator corresponding to the state (14.21). V = i 6 ε µ 1 ···µ 6 : ∂X µ 1 ∂X µ 2 ∂ 2 X µ 3 e ik 2 X (z) : :∂X µ 4∂X µ 5∂ 2X µ 6 e ik 2X (z) : = i 6 : ∂X T ∂X P ∂ 2 X P e ik 2 X (z) : :∂X T∂X P∂2X P e ik 2X (z) : P ∂ 2 X P (z) : linear terms (14.26) In the last equation, we have introduced the dummy variables ε (1) associated with the non-vanishing component ε T P P T P P of the polarization tensor and written the operator in the exponential form. "linear terms" indicate that we take the sum of the terms linear in all of ε (1) P . This sum can be rephrased as the coefficient of the product ε P because we set the dummy variables to be 1 at the end of calculation. The string D-particle scattering amplitudes can be calculated to be linear terms (14.27) To fix the SL(2, R) modulus group on the disk, we set z 1 = 0 and z 2 = r, then d 2 z 1 d 2 z 2 = d (r 2 ) . By using Eq. (14.17), the amplitude can then be reduced to Here the Pochhammer symbol is defined by (x) y = Γ(x+y) Γ(x) , which, if y is a positive integer, is reduced to (x) y = x(x + 1)(x + 2) · · · (x + y − 1). From the Regge behavior Eq.(14.29), we see that increasing one power of 1/r in the integrand results in increasing one-half power of a 0 .
Thus we obtain the following rules to determine which terms in the exponent of Eq. (14.28) contribute to the leading behavior of the amplitude: We can now drop the subleading terms in energy to get where [· · · ] ǫ T P T P in the second line and [· · · ] ǫ P P in the third line indicate that we take the P and ε The explicit form of the amplitude for the current example is

General cases
Now we move on to general cases. The vertex operator corresponding to a general massive state with d left-modes and d ′ right-modes is of the following form. p n + q n , d ′ = n>0 p ′ n + q ′ n (n 1 , n 2 , · · · , n d+d ′ ) =    · · · , m, · · · , m pm , · · · , n, · · · , n qn , · · · , m ′ , · · · , m ′ For the calculation of the correlator involving the operator Eq.(14.36), we introduce parameters associated with the polarization tensor and exponentiate the kinematic factors.
where "linear terms" means the terms linear in all of ε (n) T i , and ε ′(m) P j . Below we use symbols like (the meanings of these symbols are not unique.) and do not write the normal ordering symbol : : to avoid messy expressions.
The string D-particle scattering amplitudes of these string states can be calculated to be where only linear terms are taken in the expansion of the exponential (in the sense of Eq.(14.37)). In Eq.(14.40), we have used the simplified notation ε (n) T j ≡ ε T , j = 1, 2, ...p n , n ∈ Z + for the spin polarizations, and similarly for the other polarizations. Note that there will be terms corresponding to quadratic in the spin polarization. After fixing the SL(2, R) modulus group on the disk, we set z 1 = 0 and z 2 = r, then d 2 z 1 d 2 z 2 = d (r 2 ) . By using Eq. (14.17), the amplitude can then be reduced to where only linear terms are taken in the expansion of the exponential. We can see also that most of the terms in the forth and fifth lines of the exponent are discarded as subleading. If we start with the terms consisting of only the factors coming from the first three lines, the other terms are obtained by series of replacements of two factors in them with one factors coming from the forth and fifth lines, and for each of the replacements we can see how it changes the power of energy. We do not need to calculate the infinite number of derivatives. For each differentiation the increase of the power of 1/r is less than or equal to 1, while the powers of 1/r in the first three lines increase with n, n ′ , m or m ′ , which implies that if one term in the forth or fifth line is discarded, the terms with higher n, n ′ , m, m ′ in the same line are also discarded. The sequences of those discarded terms start at (n, n ′ ) = (1, 1), (m, m ′ ) = (1, 2), and (m, m ′ ) = (2, 1). In this way, we can see that only the terms with m = m ′ = 1 in the fifth line contribute to the leading behavior.
Thus we obtain the generalization of Eq.(14.31) (1 − r 2 ) 2 ε P q 1 P q ′ 1 (14.42) where the symbols ε ··· are similar to the ones in Eq. (14.31) and indicate that we take the coefficients of the products of the dummy variables in the exponents. ( ε and has no analog in any of the previous works. It will play a crucial role in the following calculation in this paper.
For further calculation, we first note that exp ε P q 1 Thus the amplitude can be further reduced to identity of the Kummer function 2 2mt−2m U −2m, t 2 + 2 − 2m,t 2 14.48) to get the final form of the amplitude Note that the amplitudes in Eq. (14.49) can NOT be factorized into two open string Dparticle scattering amplitudes as in the case of closed string-string scattering amplitudes [36,148].
An interesting application of Eq.(14.49) is the universal power law behavior of the amplitudes. We first define the Mandelstam variables as s = 2E 2 and t = −(k 1 + k 2 ) 2 . The second argument of the beta function in Eq. (14.49) can be calculated to be where we have used Eq.(14.9) and M 2 2 = (N − 2). The amplitudes thus give the universal power-law behavior for string states at all mass levels where m, m ′ , q and q ′ are non-negative integers. We can take the following values It is now easy to calculate the RR ratios for each fixed mass level A (N ;2m,2m ′ ;q,q ′ ) A (N,0,0,0,0) = (i) which is a b 0 -dependent function.
Before studying the fixed angle ratios for string D-particle scatterings, we first make a pause to review previous results on string-string scatterings.  [27][28][29][30][31][32]45]. It was discovered that there was an interesting link between high energy fixed angle amplitudes T and RR amplitudes A. To the leading order in energy, the ratios among fixed angle amplitudes are φ-independent numbers, whereas the ratios among RR amplitudes are t-dependent functions. However, It was discovered [64] in chapter XI.B that the coefficients of the high energy RR ratios in the leading power of t can be identified with the fixed angle ratios, namely [64] lim To ensure this identification, one needs the following identity [64,65,67,148] 2m (14.60) where L = 1 − N and is an integer. Note that L effects only the subleading terms in scattering amplitudes considered in chapter VIII, both the corresponding RR amplitudes and the complete ratios of the leading (in t) RR amplitudes considered in chapter XII can be calculated [65]. For the fixed angle regime [34], the complete ratios can be calculated by the decoupling of high energy ZNS. It turns out that the identification in Eq.(14.59) continues to work, and L is an integer again for this case [65].

c. Compactified open string
For compactified open string scatterings, both the amplitudes and the complete ratios of leading (in t) RR can be calculated [68]. For the fixed angle regime or GR, the complete ratios can be calculated by the decoupling of high energy ZNS.
The identification in Eq.(14.59) continues to work. However, only a subset of scattering amplitudes corresponding to the case m = 0 was calculated. The difficulties has been as following. First, it seems that the saddle-point method is not applicable here. On the other hand, it was shown that [27][28][29][30] the leading order amplitudes containing (α L −1 ) 2m component will drop from energy order E 4m to E 2m , and one needs to calculate the complicated naive subleading order terms in order to get the real leading order amplitude. One encounters this difficulty even for some cases in the non-compactified string calculation. In these cases, the method of decoupling of high energy ZNS was adopted.
It was important to discover [68] that the identity in Eq. All other high energy string scatterings calculated previously [64,65,148] correspond to integer value of L only.

d.
Closed string For closed string scatterings [148], one can use the KLT formula [44], which expresses the relation between tree amplitudes of closed and two channels of open string (α ′ closed = 4α ′ open = 2), to simplify the calculations. Both ratios of leading (in t) RR amplitudes and GR amplitudes were found to be the tensor product of two ratios in Eq.(14.59), namely [148]  We now begin to discuss the RR closed string, D-particle scatterings considered in this chapter.

Closed string D-particle scatterings
a. m = m ′ = 0 Case In chapter IX [42], the high energy scattering amplitudes and ratios of fixed angle closed string D-particle scatterings were calculated only for the case m = m ′ = 0. For nonzero m or m ′ cases, one encounters similar difficulties stated in the paragraph before Eq.(14.61) to calculate the complete fixed angle amplitudes. A subset of ratios can be extracted from Eq.(9.55) and was found to be [42] T (N,0,0,q,q In view of the non-factorizability of Regge string D-particle scattering amplitudes calculated in Eq. (14.49), one is tempted to conjecture that the complete ratios of fixed angle closed string D-particle scatterings may not be factorized. But on the other hand, the decoupling of high energy ZNS seems to imply the factorizability of the fixed angle ratios.
b. General case We can show explicitly that the leading behaviors of the inner products in Eq.(14.41) involving k 1 , k 2 , e T , e P and D are not affected by the replacement of e P with e L if we take the limit b 0 → ∞ after taking the Regge limit. Therefore we proceed as in the previous works on Regge scattering. The calculation for the complete ratios of leading (in  It is well known that the closed string-string scattering amplitudes can be factorized into two open string-string scattering amplitudes due to the existence of the KLT formula [44]. On the contrary, there is no physical picture for open string D-particle tree scattering amplitudes and thus no factorization for closed string D-particle scatterings into two channels of open string D-particle scatterings, and hence no KLT-like formula there. Here what we really mean is: two string, two D-particle scattering in the limit of infinite D-particle mass. This can also be seen from the nontrivial string D-particle propagator in Eq.(14.6), which vanishes for the case of closed string-string scattering.
Thus the factorized ratios in high energy fixed angle regime calculated in the RR in In addition, we show that these recurrence relations in the Regge limit can be systematically solved so that all RSSA can be expressed in terms of one amplitude. All these results are dual to high energy symmetries of fixed angle string scattering amplitudes discovered previously in chapter V [27-32, 34, 36, 45].

A. Appell functions and RSSA
The leading order high energy open string states in the Regge regime at each fixed mass level N = n,m,l>0 np n + mq m + lr l are [71,74] |p n , q m , r l = where p ≡ |p|, q ≡ |q| and k 2 i = −M 2 i . The relevant kinematics in the Regge regime are e T · k 1 = 0, e T · k 3 ≃ − √ −t (15.8) The s − t channel one higher spin and three tachyons string scattering amplitudes in the Regge limit can be calculated as A (pn;qm;r l ) = 1 0 dx x k 1 ·k 2 (1 − x) k 2 ·k 3 · e P · k 1 x − e P · k 3 1 − x q 1 e L · k 1 x + e L · k 3 1 − x where (a) n = a · (a + 1) · · · (a + n − 1) is the rising Pochhammer symbol. Note that when a or b(b ′ ) is a non-positive integer, the Appell function truncates to a polynomial. This is the case for the Appell function in the RSSA calculated in Eq. (15.14) in the following Alternatively, it is interesting to note that the result calculated in Eq.(15.14) can be directly obtained from an integral representation of F 1 due to Emile Picard (1881) [149] which is consistent with the result calculated in Eq. (15.14). It is important to note that although F 1 in Eq.(15.14) is a polynomial in s, the result in Eq.(15.14) is valid only for the leading order in s in the Regge limit. Note that in contrast to the previous calculation [71] in Eq.(11.86) and Eq.(11.87) where a finite sum of Kummer functions was obtained, here we get only one single Appell function in Eq. (15.14). This simplification will greatly simplify the calculation of recurrence relations among RSSA to be discussed in the next section. Indeed, for those highest spin string states at the mass level M 2 2 = 2 (N − 1) the string amplitudes reduce to which can be used to solve easily the Appell function F 1 in terms of the RSSA A (N ;q 1 ,r 1 ) .
We now proceed to show that the recurrence relations of the Appell function F 1 in the Regge limit in Eq.(15.14) can be systematically solved so that all RSSA can be expressed in the recurrence relations of the Appell function F 1 in the Regge limit in Eq.(15.14) can be systematically solved so that all RSSA can be expressed in terms of one amplitude.

C. Higher recurrence relations
With the result calculated in Eq.(15.14), one can easily derive many recurrence relations among RSSA at arbitrary mass levels. For example, the identity in Eq. (15.29) leads to √ −t A (N ;q 1 ,r 1 ) + A (N ;q 1 −1,r 1 +1) − M 2 A (N ;q 1 −1,r 1 ) = 0, (15.30) which is the generalization of Eq.(11.152) discussed in chapter XI [71] for mass level M 2 2 = 4 to arbitrary mass levels M 2 2 = 2(N − 1). Incidentally, one should keep in mind that the recurrence relations among RSSA are valid only in the Regge limit. We give one example to illustrate the calculation. By using Eq. x (x − y) yF 1 (a; b 1 , b 2 + 3; c; x, y) = 0, (15.34) which leads to a recurrence relation for RSSA at arbitrary mass levels t ′2 A (N ;q 1 ,r 1 ) A (N ;q 1 ,r 1 +3) = 0. (15.35) More higher recurrence relations which contain general number of l ≥ 3 Appell functions can be found in [76].
Since it was shown that [75] the Appell function F 1 are basis vectors for models of irreducible representations of sl(5, C) algebra, it is reasonable to believe that the spacetime symmetry of Regge string theory is closely related to SL(5, C) non-compact group. In particular, the recurrence relations of RSSA studied in this chapter are related to the SL(5, C) group as well. Further investigation remains to be done and more evidences need to be uncovered.
Proof: In the high energy limit, we only need to consider the leading energy terms.
We solve the above equations, the ratios between the physical states in the NS sector in the high energy limit are given as In the RR, however, the Regge stringy Ward identities or decoupling of ZNS in the RR turned out to be not good enough to solve all the Regge scattering amplitudes algebraically.
This is due to the much more numerous Regge string scattering amplitudes than those in the GR at each fixed mass level. In this appendix, we list all ZNS for M 2 = 2 and 4 and calculate their Regge limit which we use in the text to demonstrate the calculation. At the first massive level k 2 = −2, there is a type II ZNS and a type I ZNS [θ · α −2 + (k · α −1 )(θ · α −1 )] |0, k , θ · k = 0. (E.4) In the Regge limit, the polarizations of the 2nd particle with momentum k 2 on the scattering plane used in the text were defined to be e P = 1 M 2 (E 2 , k 2 , 0) = k 2 M 2 as the momentum polarization, e L = 1 M 2 (k 2 , E 2 , 0) the longitudinal polarization and e T = (0, 0, 1) the transverse polarization which lies on the scattering plane. η µν = diag (−1, 1, 1). The three vectors e P , e L and e T satisfy the completeness relation η µν = α,β e α µ e β ν η αβ where µ, ν = 0, 1, 2 and α, β = P, L, T and α T −1 = µ e T µ α µ −1 , α T −1 α L −2 = µ,ν e T µ e L ν α µ −1 α ν −2 etc. In the Regge limit, the type II ZNS in Eq.(E.3) gives the Regge string zero-norm state Note that the norms of Regge "zero-norm" states may not be zero. For instance the norm of Eq.(E.5) is not zero. They are just used to produce Regge stringy Ward identities Eq.(11.91), Eq.(11.89) and Eq. (11.90) in the text.
This completes the four ZNS at the second massive level M 2 = 4.
In the Regge limit, the scalar ZNS in Eq.(E.8) gives the RZNS For the type I spin two ZNS in Eq.(E.9), we define θ µν = α,β e α µ e β ν u αβ , symmetric and transverse conditions on θ µν implies u αβ = u βα ; u P P = u P L = u P T = 0. (E.13) Naively, the traceless condition on θ µν implies u P P − u LL − u T T = 0. (E.14) However, for the reason which will become clear later that one needs to include at least one component u N N perpendicular to the scattering plane and modify Eq.(E.14) to u P P − u LL − u T T − u N N = 0. (E. 15)