Zeta Function Expression of Spin Partition Functions on Thermal AdS3

We find a Selberg zeta function expression of certain one-loop spin partition functions on three-dimensional thermal anti-de Sitter space. Of particular interest is the partition function of higher spin fermionic particles. We also set up, in the presence of spin, a Patterson-type formula involving the logarithmic derivative of zeta.


Introduction
A symposium in honor of A. Selberg, which commemorated his 70th birthday, was held in Oslo, Norway, during July of 1987.From that event, the volume [1], consisting of 30 invited lectures evolved, included a lecture by S. Patterson in which the construction of Selberg's classic zeta function [2] was extended to cover certain discrete groups (Kleinian groups) Γ acting on real hyperbolic space H n whose fundamental domains, however, were of infinite hyperbolic volume.We denote this Patterson-Selberg (P-S) zeta function for a Kleinian group by Z Γ .
For the special case with n = 3 and with Γ generated by a loxodromic element, it was noticed first by the author (in the 2003 paper [3]) that Z Γ had some relevant connections to the Euclidean BTZ black hole [4].For example, we were able to express BTZ effective action directly in terms of log Z Γ .Moreover, the zeros of Z Γ were shown to correspond to BTZ scattering resonances, in a follow up joint work with P. Perry [5], and black hole entropy connections were described in terms of a "deformation" of Z Γ [6][7][8][9][10].Application of the BTZ scattering in [5] appears in the work of R. Aros and D. Diaz [11,12], for example, where a holographic formula is derived: a formula that relates the determinant of a scattering operator on a space asymptotically anti-de Sitter (AdS) to a "relative" determinant of a Laplacian in the bulk, a result that one can regard as an entry in the AdS/CFTdictionary.
In some later work, A. Bytsenko and M. Guimarães [13,14] (and the author [15], independently) discovered a formula that expresses the one-loop AdS gravity partition function in terms of a quotient of products involving Z Γ ; see Formula (27) of Section 3. It has also been shown in [16] that modular invariant partition functions for an extreme conformal field theory with central charge divisible by eight have an expression in terms of Z Γ .These preliminary remarks (where, again, we have focused only on hyperbolic three-space and the case of a single loxodromic generator of Γ) illustrate the natural appearances of the P-S zeta function Z Γ in the physics literature and its growing applications, of which further details are presented, for example, in the review article [10].
By way of private communication with R. Gopakumar, the author received (among other notes) three pages of "Notes on the heat kernel for AdS 3 ", by J. David, M. Gaberdiel and R. Gopakumar, in which these authors find a product formula for the one-loop partition function for a massive spin field on thermal AdS 3 .They assert that their product "probably has an interpretation as a Selberg zeta function" [17].We show in this paper (in Section 3, Formula (36)) that this indeed is the case.We use, in fact, the central result of these authors in the extended work [18] (Equation (6.9) there) to express, more generally, various one-loop spin partition functions on thermal AdS 3 in terms of the P-S zeta function Z Γ .This includes, in particular, the partition function for the gravitino (the super partner of the graviton) in N = 1 supergravity and the more general case of higher spin fermionic (and bosonic) particles.
We also establish a Gangolli-Patterson-type formula (Formula (49) in Section 4) that relates a certain transform of the heat kernel trace to the logarithmic derivative of Z Γ .The formula offers an interesting contrast to the result known in the spin zero case.
The author extends special thanks to Miss Zijing Zhang for her valued assistance in the preparation of this paper.
2. The Action of SL(2, C) on H 3 , Thermal AdS 3 , and the Patterson-Selberg Zeta Function R, C will denote the field of real, complex numbers, respectively.
denotes real hyperbolic three-space, and where the bar " − " denotes complex conjugation.There are natural embeddings C → H, H 3 → H given by: respectively, for a ∈ C, w ∈ H 3 , i 2 = −1.Using these embeddings, one can define a standard linear fractional action of the complex special linear group G = SL(2, C) on H 3 as follows.For g = a b c d ∈ G, w = (x, y, z) ∈ H 3 , g • w ∈ H 3 is defined by: One computes that g • w is given explicitly by: for: Elements of R 3 , H 3 can also be written as column vectors.In particular, for: the equations in (4) reduce to: We have interest in the case a > 0. For Z = the ring of integers, let: denote the subgroup of G generated by γ (a,b) .If τ = τ 1 + iτ 2 is in the upper 1/2-plane π + , for example (i.e., τ 1 , τ 2 ∈ R, Imτ = τ 2 > 0), then e −πiτ = e a+ib for a = πτ 2 > 0, b = −πτ 1 , and we can form the . Here, the action of Γ τ on H 3 is given by Equation (6).Equivalently, this action is given by Equation (4): γ (πτ 2 ,−πτ 1 ) • w = (u, v, ω) for: which is of the form given in [19][20][21], for example, with Z identified with Γ τ by way of the group isomorphism n → γ n (πτ 2 ,−πτ 1 ) = e −πniτ 0 0 e πniτ (by ( 5)) for n ∈ Z.The Patterson-Selberg (P-S) zeta function corresponding to Γ (a,b) in Equation ( 7) is defined as the Euler product: which is an entire function of z ∈ C. Again, a more general definition is given in [1], for a discrete group Γ acting on H n .There is a function z) .This function, which we denote by log Z Γ (a,b) , is given by: for Rez > 0. Note that e −2an(z±iπ/2a−1) = e −2an(z−1) e ∓inπ = (−1) n e −2an(z−1) , which by Equation (10) gives: We shall also need the logarithmic derivative: for Rez > 0. Similar to Equation ( 11), we have: for Rez > 0.

The Main Result and Some Examples
The central result of [18] is a formula for the heat kernel trace K (s) (τ, τ ; t)(t > 0) for the spin Laplacian ∆ (s) , with spin s, on thermal AdS 3 .This is Formula (6.9) there.A slightly modified version of it is the following, which takes into account the anti-periodicity of half-integral spin particles (like fermions, with s = 1 2 or 3 2 , for example): δ s,0 is zero for s = 0 and one for s = 0.
is the modulus of the boundary torus T 2 of AdS 3 .The notation q = e 2πiτ in Equation (8) of Section 2, which is that of [19,21], differs from the notation q = e iτ in [18]: the τ in [18] is 2πτ in [19,21].Thus, we take here X Γ = Γ \ H 3 as thermal AdS 3 , where Γ ⊂ SL(2, C) is the subgroup generated by the element: In Formula ( 14), the infrared divergent term due to the infinite hyperbolic volume of X Γ is, by convention, neglected.2πτ 1 , 2πτ 2 correspond respectively to the angular potential θ and inverse temperature β.The heat kernel on X Γ is obtained, of course, by averaging over Γ the heat kernel on H 3 (the method of images), and one obtains the trace K (s) (τ, τ ; t) by integration (over X Γ ) along the diagonal.In the spin zero case (s = 0), a proof of Formula ( 14) is given in [23], for example; also, see [24][25][26][27], for example.
We apply Formula (14) in this section to set up a general formula (once and for all) that expresses the effective energy − log (−∆ (s) + M ) in terms of the P-S zeta function.See Theorem 1, which we use to show that various zeta function expressions of one-loop partition functions can be derived, uniformly, by varying the spin and mass parameters s, M .
Start with the standard formula: to compute: Plug in Formula ( 14) and commute the integration with the summation, as one would usually do.In the present case (for the choices A = n 2 τ 2 2 , B = s + 1 + M , with n ≥ 1), Formula (16) gives: for s + 1 + M ≥ 0 (since τ 2 > 0 by definition of τ ).Thus, the main point is to express the sum in Equation ( 18) in terms of Z Γ , for a suitable choice of Γ.
We separate out the integral and half-integral spin cases.If s = m ∈ Z, then for n ≥ 1, (−1) n2s = 1 in Equation ( 20), and if s = (2m+1)/2, then (−1) n2s = (−1) n .In the latter case, the sum in Equation ( 20) is the sum: by Equation (11).In the former case s = m, we have by Equation ( 10) a similar sum of logs, but without the term −iπ/τ 2 .Going back to the definition of z and definition Equation ( 17), we have established: The following formula holds: for half-integral spin s = 1 2 , 3 2 , 5 2 , 7 2 , • • • , where the Patterson-Selberg zeta function Z Γτ is given by definition Equation ( 9) for the choice , C) generated by the element: as in Equations ( 5) and (7).Consequently, we also have (since Note that the Γ τ here differs from the Γ τ in Section 2, which is not an issue.We have focused on the case s = 0 for the applications we have in mind. Since various one-loop partition functions are given in terms of det (−∆ (s) + M ), for suitable s, M , we can use the preceding formula to express these functions in terms of Z Γτ .In what follows are some examples (that draw freely from the results and notation in the literature) where, for convenience, we shall write Γ for Γ τ = Γ (τ 2 /2,τ 1 /2) .From (7.7) of [18], for example, Thus, by Equation ( 25): where, again, we use that Z Γ (z) = Z Γ (z).As indicated in the Introduction, Formula (27) was found by A. Bytsenko and M. Guimarães [13], as well as (independently) by the author; see Theorem 3.26 on page 337 of [15].Equation ( 27) corrects the slight error in the version given on page 337 in [13].
For the Majorana gravitino of N = 1 supergravity, one has by (7.27) of [18], for example, which, by Equation ( 25), can be expressed as: Furthermore, compare Formula (3.11) of [14].Z gravitino 1−loop is also computed "from scratch" in [28].Regarding some remarks in the Introduction, we show, affirmatively, in the next example that the one-loop partition function Z 1−loop on AdS 3 computed in [17] for a spin s field of mass m 0 indeed has an interpretation in terms of a suitable Selberg-type zeta function, as these authors suspected.Their Formula ( 14) is: where k, l ∈ Z, q = e iτ , h s def = (z +s)/2, hs def = (z −s)/2 for z def = 1+ (s − 1) 2 + m 2 o .We show therefore that the product in Equation ( 30) is expressible in terms of the P-S zeta function Z Γ in Equation ( 9) for Γ = Γ (a,b) , where in fact, we take a Then: which simplifies to: Similarly: By definition Equation ( 9) (for any z ∈ C): which, by Equations ( 32) and ( 33), gives for Γ = Γ (a,b) : Therefore, by definition of z, a, b, we can express Formula (30), for Γ = Γ (τ 2 /2,τ 1 /2) , as: (36) Formula ( 29), which involves fermions of spin 1 2 , 2 3 , can be completely generalized, which we now show.In their study of M. Vasiliev's higher spin supergravity on AdS 3 , T. Creutzig, Y. Hikida and P. Rφnne [29] obtain in their Equation (3.7) one of their main results: the computation of the one-loop determinant Z (s) F (in their notation) for higher spin fermionic particles: for q = e 2πiτ .In this notation, which we shall translate to our earlier notation, it is emphasized that the Laplacian ∆ on AdS 3 is restricted to act on transverse, traceless components of spin s + 1 2 , s − 1 2 gauge fields.

A Gangolli-Patterson-Type Formula in the Presence of Spin
In this section, we present a new formula (in Theorem 2) that directly relates a particular integral transform of the heat kernel trace K (s) (τ, τ ; t) in Equation ( 14) to the logarithmic derivative of Z Γτ .In the spin zero case, the formula reduces to Equations ( 50) and (51) below, which is that of [9,23], for example.Furthermore, the formula is, in fact, an exact analogue of R. Gangolli's Formula (2.39) in [22] for the case of the generalized Selberg zeta function attached to a space form X Γ 0 = Γ 0 \ G 0 /K 0 of a rank one symmetric space G 0 /K 0 , where K 0 is a maximal compact subgroup of a non-compact semisimple Lie group G 0 and where Γ 0 ⊂ G 0 is a discrete subgroup, such that Γ 0 /G 0 is compact.Here, we note that for G 0 = SL(2, C), K 0 = SU (2), H 3 (i) = G 0 /K 0 is indeed a rank one symmetric space, but for Γ 0 = Γ τ , Γ 0 /G 0 , however, is not compact.A proof of the quotient structure (i) relies on the action of SL(2, C) on H 3 discussed in Section 1 and can be found in Appendix A2 of [9], for example.
In Theorem 2, we are mainly interested in the case s = 0 : s > 0. In particular, the hypothesis Re(z − 1) 2 + s > 0 holds if Re(z − 1) 2 ≥ 0. For z = x + iy, the latter condition is (x − 1) 2 ≥ y, the graph of this inequality being the set of points (x, y) ∈ R 2 in the two shaded areas below (including the boundaries y = ±(x − 1)):

B
(B for bosons) for quadratic fluctuations of fields about a thermal AdS 3 background: