Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed $D=4$ quantum inhomegeneous conformal Hopf algebras $\mathcal{U}_{\theta }(su(2,2)\ltimes T^{4}$) and $\mathcal{U}_{\bar{\theta}}(su(2,2)\ltimes\bar{T}^{4}$), where $T^{4}$ describe complex twistor coordinatesand $\bar{T}^{4}$ the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently we introduce the quantum deformations of $D=4$ Heisenberg-conformal algebra (HCA) $su(2,2)\ltimes H^{4,4}_\hslash$ ($H^{4,4}_\hslash=\bar{T}^4 \ltimes_\hslash T_4$ is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length $\lambda_p$ will be called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We shall describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and by the quantization map in $H_\hslash^{4,4}$. We introduce as well generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra $\mathcal{U}_\theta(su(2,2)\ltimes T^4).$


Towards Quantum Gravity
One can distinguish two basic levels in quantization procedure of physical models describing contemporary fundamental interactions: (i) The first level can be called quantum-mechanical with canonically quantized phase space coordinates and possible presence of classical gravity only as a static background.
On such a level, we find all familiar relativistic quantum field theories, e.g., QED and QCD (fields quantized, space-time geometry flat and Minkowskian).
(ii) The second level also has quantized gravity and noncommutative space-times (all fields, including gravity and space-time geometry are quantized).
Quantum gravity (QG) remains a subject of rather hypothetical models (see, e.g., [1,2,3,4]), however it is mostly agreed that QG effects require at ultra-short distances the replacement of classical Einsteinian space-time by quantum noncommutative space-time geometry (see [5]). The QG-generated noncommutativity corrections appear as proportional to the powers of Planck mass mp or inverse powers of Planck length λp.
where c is the light velocity and G is the gravitational Newton constant. The QG origin of Planck length can be seen from Formula (1), with simultaneous presence of ℏ and G.
In order to study algebraically the QG modifications of the space-time geometry in Special Relativity, one can look at the λ-dependent deformations U λ (P 3,1 ) of the Poincaré algebra P 3,1 = o(3, 1) ⋉ P 3,1 , where P 3,1 denotes the four-momenta sector and λ describes an elementary length parameter which can be fixed λ = λp. Further, we consider the Minkowski space-time coordinates xµ ∈ M 3,1 together with covariantly acting Poincaré symmetry and introduce the semi-direct product algebra where P 3,1 ⋉M 3,1 describes the relativistic phase space P 3,1 = (M 3,1 ; P 3,1 ), which after the first quantization level is endowed with relativistic Heisenberg algebra structure. Such algebra A , also called Heisenberg-Lorentz algebra, can be further deformed into quantum algebra U λ (A ), which, to describe quantum symmetry, should have Hopf algebra or Hopf-algebroid structure. We stress that only U λ (P 3;1 ) can be introduced as Hopf algebra, i.e., Drinfeld quantum symmetry group [6,7]. If = 0. the relativistic quantum phase space P 3;1 implies that U λ (A ) has the algebraic structure of Hopf algebroid [8,9,10,11,12,13]. Such class of deformations of algebra (2) provides so-called DSR (Doubly Special Relativity) algebra describing quantum space-times with covariantly acting quantum symmetry. We use the original name for DSR algebras [14,15,16,17], however some authors use the name DSR for "Deformed Special Relativity', which has a vague informative content.
The name "doubly" is due to the dependence on two parameters: c (light velocity) and λp (Planck length, or mp ∼ (lp) −1 ). The first parameter, c, appears in the physical basis of the relativistic classical algebra A and the second parameter, λ, determines the QG-induced modification of the algebraic structure (2).
It was argued already in the 1930s [18] that QG models should at the basic level depend on three fundamental nonvanishing constants, c, ℏ and G, where G can be replaced by λp or mp (see (1)); if the cosmological constant or de Sitter radius of the Universe is finite, it introduces additional geometric parameter. The variant of DSR algebra with additional de Sitter radius as additional geometric parameter was called Triply Special Relativity (TSR); see [19].
The model of quantum space-time symmetries, which was an inspiration for introducing DSR algebras, is provided by the κ-deformed Poincaré-Hopf algebra [20,21] with semi-direct product structure presented in [22] in so-called bicrossproduct basis.
Our aim is to describe some class of quantum-deformed twistors and provide the counterpart of DSR algebra in the noncommutative framework of quantum twistors. It should be recognized here that there are already several interesting papers dealing with quantum deformations of twistors and their geometries (see, e.g., [23,24,25,26,27,28,29]).
We recall that the points zµ of complex Minkowski space-time are specified in T 4 by two-dimensional planes with twistor coordinates, (πα, ωα), satisfying the Cartan-Penrose incidence relation The quantum-mechanical twistorstA,tA on first basic quantization level are provided by the oscillator-like canonical commutation relations (CCR) [23,30].
One can calltA the twistor coordinates andtA the twistor momenta; they introduce the twistorial analog of the relativistic quantum-mechanical phase-space algebra for conformal-covariant twistorial models. We recall that one can obtain the twistor realization of D = 4 conformal algebra o(4, 2) ≃ su(2, 2) given by the bilinear products of Quantum-Mechanical (QM) twistorstA,tA (see also Section 3.1). The twistors tA ∈ T 4 satisfying the Cartan-Penrose incidence relations (7) provide the geometric alternative for the description by complex Minkowski space-time geometry. The real Minkowski coordinates xµ(zµ = xµ + iyµ) are obtained if the 2 × 2 Hermitian matrix zα β is parameterized as follows In such a case, one gets from (7) that (t, t) = 0, i.e., the real Minkowski coordinates have as twistor counterparts the null twistor planes (so-called α-planes), with vanishing norm (6). For the discussion of QG effects, it is more appropriate and realistic to consider twistors corresponding to curved space-time. The simplest examples of nonflat space-times are the ones with constant curvature R or cosmological constant Λ = ± 1 R 2 , where Λ > 0 for de Sitter and Λ < 0 for anti-de Sitter geometries. In such a case, the standard quantization relations (8) and (9) for twistors should be modified, with respective deformation determined by the Λ-dependent antisymmetric constant second rank twistor IAB, called in twistor theory the infinity twistor (see, e.g., [30]). In the basis (4), it is given by the following formula (we choose further Λ > 0) One can introduce in twistorial phase space (T 4 ,T 4 ) the deformation of Poisson structure described by the following complex-holomorphic (2, 0) symplectic two-form [27] generated by the holomorphic (1, 0) Liouville one-form Ω1 = IABt A dt B , where Ω2 = dΩ1. In dual twistor spaceT 4 , the complex-anti-holomorphic (0, 2) symplectic two-form (we denotetA ≡ t † A = ηABt B ) (13) is complex-conjugated to (12), which leads to the relation It appears that, within the framework of Hopf-algebraic quantum deformations of inhomogeneous conformal algebras su(2, 2), one gets separately the deformations of twistorst A ∈ T 4 andt ∈T 4 , which lead to the quantum-mechanical (first level) quantization of symplectic structures (12) and (13). The deformed twistors obtained by the quantization of symplectic Poisson structures (12)- (14) are called the palatial twistors [27]. After the quantization procedure, one gets the holomorphic and anti-holomorphic noncommutativity relations modifying (9) as follows (further, in many formulae, we put = c = 1).
In the concluding Section 4, we present an outlook, with directions for possible future research.
The coset (27) parameterizes complex 2-planes in T 4 which are determined by non-parallel pairs of intersecting twistors t i A (i = 1, 2; A = 1, . . . 4; (t 1 , t 2 ) = 0) and satisfy the pair (7) of Cartan-Penrose incidence relations. The complex Minkowski coordinates z µ = 1 2 (σ µ )α β zα β are expressed by the pair of intersecting twistor coordinates t i A = (π i α , ωα i ) as follows The primary aim of the Penrose program during the last fifty years was to encode any curved Einsteinian space-time structure in geometric twistorial framework; in particular, it was important to find the vocabulary permitting to translate any general relativity solution in space-time into the twistorial language. This goal was however achieved only partially, with modest hopes that the program of finding the twistor formulation of general relativity theory will be fully successful. However, in last decade, Roger Penrose became inspired by the idea that perhaps it is an easier task to construct the twistorial noncommutative version of quantum gravity. Such a view, conceptually attractive, however still faces the basic question of how the appropriate formulation of quantum gravity in the space-time picture would look. On the twistorial side, some first steps towards the construction of twistorial quantum gravity model were provided by Penrose (see also [29]).

Twist-Deformed Inhomogeneous Conformal Hopf Algebras and Holomorphic/Anti-Holomorphic Quantum Twistors
Our first task is to show how the relations (15) can be obtained in the framework of quantum deformations of inhomogeneous D = 4 conformal algebras, with the respective holomorphic twistor coordinates tA ∈ T 4 or anti-holomorphic twistorial complex momentatA ∈T 4 . For such a purpose, we consider the pair of semi-dual Hopf algebras H0 ≡ U(isu(2, 2)) = U(su(2, 2)⋉T 4 ) andH0 ≡ U(īsu(2, 2)) = U(su(2, 2)⋉T 4 ), with Hermitianconjugated generators in T 4 andT 4 , but with the same twistorial realization of D = 4 conformal subalgebra su(2, 2). Subsequently, one gets the holomorphic and anti-holomorphic palatial twistors if the Hopf algebras H0 andH0 are twisted, respectively by the following pair of twists, where further, in Section 3.1 we justify the same dependence of twists (29) and (30) from the elementary length parameter λ.
In the general case, the antisymmetric numerical tensor ΘAB can be chosen as complex, but, in the case of Palatial twistors, because ΘAB = IAB, they are real. The pair of twists (29) and (30) are dual under the twistorial Born mapt which, as the twistorial counterpart of the Born duality map xµ ↔ pµ [39,40], interchanges the twistorial momentatA and the twistorial coordinateŝ tA.
where A is the H-module algebra. We recall that, in the case of noncommutative θµν-deformed quantum space-times and quantum four-momenta (see (19) and (20)), one gets analogously in coproducts ∆F (Mµν) and ∆F (Mµν ) the additional terms which are linear in θµν and bilinear in four-momenta (for twist (16)) or bilinear in space-time coordinates (for twist (17)); these terms are needed for the twisted quantum Poincaré invariance of the algebraic relations (19) and (20) (see also [33]). For coordinate and momenta twistors, one can consider the quantum covariance under two different inhomogeneous twisted conformal Hopf algebras UF (isu(2, 2)) and UF (īsu(2, 2)), but they can be mapped into each other if we supplement the twistorial Born map (31) with the following exchange relation (ΘAB in the general case are complex, but it should be observed that θµν in both Formulae (19) and (20) is real and not changing under the map (49)).
one usually assumes that g = βl 2 p (where β is a dimensionless constant) and one can consider Snyder model as the first historical example of DSR model. In the Hopf-algebraic framework of quantum groups, the general DSR algebra can be described as quantum algebra U λ (ADSR), where ADSR is given by the Formula (2).
If we wish to introduce the twistorial counterpart of DSR theory, described by the corresponding class of twistorial DSR algebras, one should replace the algebra ADSR (see (2)) by the algebra AT DSR where H 4,4 =T 4 ⋊ ℏ T 4 denotes the quantum twistorial phase space, described by the twistorial oscillators algebra (see (8) and (9)). Subsequently, one can introduce the twistorial DSR (TDSR) algebra as described by the following quantum deformations: In Formula (53), we use the particular case of semi-direct product, called smash product (see, e.g., [46,47]) of the Hopf algebra H and its module algebra If h, h ′ ∈ H and a, b ∈ A, the multiplication rule in H is described by the which uses as input the coalgebraic sector in H. One can propose two ways of constructing TDSR algebra (53), in analogy with the two ways of describing the relativistic quantum NC phase space in Snyder model (see [48,49,50]): (1) by proposing the quantum twistorial map as given by Formulae (35) and (36) (further in this section we link such a map with a cochain twist quantization); (2) by calculating for the quantum Hopf algebras U λ (isu(2, 2)) and U λ (īsu(2, 2)) the Heisenberg double construction, which provides the generalized twistorial quantum phase space spanned by the quantum symmetry generators and the dual conformal quantum matrix group coordinates (for κ-deformed Poincaré-Heisenberg double see [51,11]; for θµν-deformed Poincaré-Heisenberg double, see [52,53]).

Twist Deformation of Twistors by Drinfeld Twist
One can construct the Drinfeld twist (see, e.g., [38,55,54]) by multiplication of two-cocycle twists (29) and (30) in various ways, related by BCH-type formulas. Such twist can be used for twist quantization of the Heisenberg-conformal algebra (52), which becomes a quasi-bialgebroid described by the smash product of the conformal su(2, 2) and canonical twistorial Heisenberg algebra as the su(2, 2) module.
In Formula (64), the exponential factor is dimensionless. Because F does not satisfy the two-cocycle condition, the resulting twisted coproducts are not coassociative and the twist quantization will generate the quasibialgebroid structure. One gets the twist-deformed quantum twistor variablesξR = (tA,tA)(R = 1 · · · 8) as describing the twist quantization of Hopf algebra modulê If in (65), we insert (34) and (64), then calculate the contribution generated by the linear λ-term in f = ln F, we get i.e., the linear term in the λ −1 power expression gives the quantization maps analogous to (35) and (36). Taking only the linear term into consideration, we obtain the following modification of twistorial CCR (see (8) and (9) The relations (66)-(69) without higher order terms in λ −1 can be treated as describing the quantization map for ΘAB-deformed twistorial Heisenberg algebra.

Heisenberg Doubles and Generalized Twistorial Quantum Phase Space
It can be shown that, in the D = 4 space-time framework, both (4 + 4)dimensional quantum phase space as well as the (10+10)-dimensional one which also contains the Lorentz sector can be described as the Heisenberg doubles, providing various generalizations and extensions of Heisenberg algebra.
The Heisenberg double is a special example of the smash product (54), when A is identified with the dual Hopf algebra H ⋆ . In such a case, the nondegenerate bilinear Hopf pairing < ·, · >: H ⊗ A → C between two Hopf algebra H and H ⋆ is used, with the following action H ⊲ H ⋆ h ⊲ a = a (1) < h, a (2) > .
Subsequently, one can write in accordance with the action (48) on the Hopf algebra module. One can derive in H the cross relations between the algebraic sectors of H and which completes the multiplication table in H ⊗ H ⋆ . In applications of the Heisenberg double H ⋉ H ⋆ to physical models, the Hopf-algebra H usually describes the generalized quantum momenta, while the dual Hopf algebra H ⋆ provides the sector of generalized quantum positions. In the four-dimensional space-time approach, one obtains the generalized (10 + 10)-dimensional quantum phase space expressed as the Heisenberg double H (P) (H = U(ĝ) denotes the enveloping Hopf-Lie algebra and H ⋆ = C(G) is the Hopf algebra of functions on the Lie group manifold G): where iô(3, 1) describes Poincaré algebra and IO(3, 1) the dual Poincaré group. In the quantum case, e.g., in applications to QG, both Hopf algebras in (75) can be quantum-deformed in a way preserving the Hopfalgebraic duality property, e.g., by twisting or κ-deformation [11,51,52].
In the twistor approach, one can choose as the generalized twistorial quantum phase space the following Heisenberg double where the Planck length plays the role of dimensionfull deformation parameter. In particular, one can consider in (76) the twist deformations with twists F,F (see (29) and (30)). If we observe that the twistorstA,tA as well as the twists (29) and (30) are related by the Born map (see (31)), we obtain the following table of four Hopf algebras, describing possible twist-deformed inhomogeneous D = 4 quantum conformal symmetries and D = 4 inhomogeneous quantum conformal groups Born duality Born duality UF λ (īsu(2, 2)) Hopf ⇄ duality CF λ (ISU (2, 2)).
If the Hopf algebras U(isu(2, 2)), U(īsu(2, 2)) are twist-deformed, the cotwist-deformed algebras C(ĪSU (2, 2)), C(ISU (2, 2)) provide noncommutative matrix entries of quantum SU (2, 2) group. The noncommutativity of matrix group elements g A B ∈ SU (2, 2) is determined by RTT relations (see, e.g., [26,7]) where the R-matrix is expressed by the following cotwist formula The cotwist F AC BD dual to the twist F is determined by the following evaluation map: The noncommutative multiplication formula of cotwisted SU (2, 2) matrix elements is given by the formula Because the twistor coordinates and momenta T 4 ,T 4 as well as the complex Minkowski space-time coordinates are expressed by the element of SU (2, 2) group (see (25)- (27)), the NC multiplication rule (82) defines the noncommutativity of cotwist-deformed twistor coordinatestA,tA as well as the complex quantum Minkowski coordinatesẑα β . Further, using the cotwist-deformed multiplication rule (82), one can also derive the cotwist deformation of incidence relations (7).

Outlook
The aim of this paper is the presentation of some aspects of the NC framework for quantum-deformed twistors. Our inspiration came from the paper by Penrose [27] who under the name of palatial twistors introduced the "physical" class of ΘAB-deformed dS (de Sitter) twistors, with the parameters θAB determined geometrically by real de Sitter infinity twistor IAB (see (11)). In our scheme, we reduce the multiparameter deformations effectively to the one-Parametric ones by using the geometric degree of freedom which describes the variable Planck length or variable Planck mass. As an example, one can provide the generalized κ-deformations depending on the constant four-vector aµ, generating the following aµ-dependent quantum space-times [60]: where aµ = λa The following are some directions in which one can continue the studies presented in this paper: 1. If we consider the twistor correspondence with complexified spacetimes, one should introduce the pair of dual twistors (tA, wA; tAw A = 0) called ambitwistors, not linked by complex conjugation (Hermitian conjugation in quantized case), which provide the description of complex null geodesies in complexified Minkowski space M C [61,62,63]. In such a case, if w A = (λ α , µα), one can introduce the symplectic 2-form (see (13)) (84) where λ = 1 r 2 appears as the second cosmological constant. In such a case, the curvatures R and r associated with twistors tA and wA can be different; in particular, if R ≫ r ,they may provide the tool to describe two de Sitter geometries characterizing the cosmological macroscopic distances and the ultrashort Planckian ones. It appears that the duality map tA ↔ wA, which implies the interchange relation R ↔ r, can be linked with Born duality relation (see, e.g., [39,65,64,40]). One can speculate that the presence of the pair of dual radii (r,R) in ambitwistor framework can lead to the description of quantum effects simultaneously at ultrashort (radius r) and at macroscoping (radius R) cosmological distances.
2. The D = 4 twistorial construction presented here can be quite easily generalized to D = 3 and D = 6 twistors, described by the D = 3 and D = 6 conformal groups Sp(4; R) ≃ O(3, 2) and Uα(4; H) ≃ O(6, 2). We add that the D = 4 conformal group SU (2, 2) can also be described as the antiunitary one Uα(4; C) [66,67]. In such a way, we deal with the antiunitary family of groups Uα(4; F), where field F = R, C, H. In addition, since the 1970s, supertwistors [68] have been studied, which are a well recognized tool in the studies of superparticles, superstrings, supersymmetric gauge theories and supergravity.
4. Recently, the twistorial field-theoretic approach to formulate gauge theories and gravity in twistor space has been promoted (see, e.g., [76,77]), with the dynamics described by twistorial actions. By using local twistor geometry, one obtains in a natural way conformal gravity [78]; the twistorial model of Einstein gravity with non-zero cosmological constant can also be obtained by embedding into twistorial conformal gravity [76,77]. The formulation of QG in twistorial framework, by analogy with the approach presented in [5], may require as well the noncommutative twistorial quantum geometry.
The paper was supported by Polish National Science Center, project 2017/27/B/ST2/01902.