Quantum exponentials for the modular double and applications in gravity models

In this note, we propose a decomposition of the quantum matrix group SL$_q^+(2,\mathbb{R})$ as (deformed) exponentiation of the quantum algebra generators of Faddeev's modular double of $\text{U}_q(\mathfrak{sl}(2, \mathbb{R}))$. The formula is checked by relating hyperbolic representation matrices with the Whittaker function. We interpret (or derive) it in terms of Hopf duality, and use it to explicitly construct the regular representation of the modular double, leading to the Casimir and its modular dual as the analogue of the Laplacian on the quantum group manifold. This description is important for both 2d Liouville gravity, and 3d pure gravity, since both are governed by this algebraic structure. This result builds towards a $q$-BF formulation of the amplitudes of both of these gravitational models.

At the level of the action, JT gravity with negative cosmological constant can be reformulated in terms of a BF model based on the sl(2, R) structure [16,17,18,19].One is hence led towards understanding amplitudes in JT gravity using the formalism of 2d BF theory, which is essentially an application of representation theory of the underlying algebraic structure.In particular, for the solution of amplitudes of 2d BF-models for group G, a crucial role is played by the group representation matrices R mn (g) = ⟨R, m| g |R, n⟩.This is a direct consequence of the Peter-Weyl theorem implying the R mn (g) form an orthogonal basis for L 2 (G).These objects can physically be viewed as two-sided quantum wavefunctions m n R that diagonalize the quadratic Casimir, which for the models of interest is equal to the Hamiltonian of the system.This is in parallel to the solution strategy of 2d Yang-Mills models in the older literature [20,21,22].
In some physical situations, the indices m and/or n at the boundaries are fixed and constrained to diagonalize one of the generators X of the algebra g.The result is then a coset.E.g. if the ket is constrained as X|R, n⟩ = 0 for some generator X ∈ g, we obtain the right coset G/U (1) where the 1-parameter subgroup generated by X is modded out by right-multiplication.If both bra and ket are constrained as such, we have the double coset U (1)\G/U (1).
As mentioned, for lower-dimensional gravity with negative cosmological constant, the sl(2, R) algebra is the relevant group-theoretical structure: (1.1) For any representation, the representation matrix R(g), g ∈ SL(2, R), can be written in the Gauss-Euler decomposition: R(g) = e γF e 2ϕH e βE , γ, ϕ, β ∈ R. (1.2) Restricting the range of the coordinates γ and β to R + , R(g) belongs to the positive subsemigroup SL + (2, R).
The asymptotic AdS boundary conditions [23,24] imply that the bra and ket of the relevant representation matrix elements are eigenstates of F † and E respectively (with eigenvalue ν and µ respectively that we will not further specify in this work). 1Hence the left-and rightmost element in the Gauss decomposition (1.2) are diagonalized on a two-boundary state and the important part of the matrix element reduces to only the Cartan element e 2ϕH insertion as R νµ (ϕ) ≡ ⟨ν|e 2ϕH |µ⟩ with principal series representation label j = −1/2 + ik, k ∈ R + .This representation-theoretic object R νµ (ϕ) is called a Whittaker function [26,27,28,29] or mixed parabolic matrix element, since its bra and ket diagonalize different parabolic generators.As the wavefunction on a slice that connects two asymptotic boundaries, it is the centerpiece of gravitational amplitudes in 2d JT gravity since it is used to compute the disk amplitude and boundary correlation functions [30,25], see [31] for the application to N = 1 JT supergravity, and [32] for the N = 2 case.
However, JT gravity is not an isolated datapoint; it is instead connected to other exactly solvable gravitational models.We have seen in [33,34] that 2d Liouville gravity has a structure of the amplitudes that mirrors the JT case, but where the representation theoretic objects are q-deformed and doubled into the (tensor product) quantum algebra U q (sl(2, R))⊗U q(sl(2, R)), where q = e πib 2 and q = e πib −2 . 2 JT gravity then emerges in the classical b → 0 scaling regime. 3 Likewise, in [42], see also [43], we presented arguments that 3d aAdS gravity in its interior is governed by representation theoretic objects consistent with a q-BF model based on the same structure U q (sl(2, R))⊗U q(sl(2, R)), where the parameter b is related to the AdS length 2G N in the semi-classical regime. 4It was shown in the same work that taking b → 0 combined with low temperature compared to the AdS length ℓ, results in two copies of the JT description.
The combination of the quantum algebras with deformation parameter q and q is called the modular double, since the parameter b gets mapped to 1/b, a modular Stransform. 5 The modular double of U q (sl(2, R)) was first introduced by L. Faddeev in 1999 [44] and received considerable attention since then.It was linked to 2d Liouville CFT and its fusion and braiding matrices in the seminal works [45,46], its defining properties were analyzed in more detail in [47,48,49], it was related to Teichmüller theory [50,51], and has been extended by now to general split real quantum groups in a body of work [52,53,54].
The importance of this quantum algebra cannot be overstated in lower-dimensional gravitational physics.The reason is that the quantum algebra that is related6 to the Virasoro algebra, is precisely the modular double U q (sl(2, R))⊗U q(sl(2, R)).E.g. the Virasoro modular S-matrix precisely matches with the Plancherel measure of the principal series representations of the modular double, which form a closed set of representations under tensor product.
And in fact, one defines the modular double as the algebraic object that has precisely only these irreducible representations [48].Since the Virasoro symmetry algebra is the governing principle for both 3d pure gravity, 2d Liouville gravity, and 2d JT gravity (as a limiting form of both of these), all of these gravitational models are governed in one way or another by this same underlying group-theoretic object. 7ur main goal in this note is to understand how the analogue of the Gauss-Euler structural decomposition of (1.2) works for this modular double algebraic structure relevant for 2d Liouville gravity and 3d gravity.Our proposal is the formula: where g b is Faddeev's (non-compact) quantum dilogarithm [58].The remaining notation will be explained in the main text.Finding such a formula is important since it is a step towards the ill-understood q-BF formulation (see e.g.[59,60,61]) of the amplitudes in both 2d Liouville gravity and 3d pure aAdS gravity.This note is organized as follows.In section 2 we motivate and argue for the above formula (1.4) for the Gauss decomposition of the modular double.In section 3 we use this decomposition to explicitly compute the hyperbolic representation matrix element, finding agreement with an earlier determination by I. Ip, and providing evidence for our proposal.Section 4 describes some first applications in terms of the Casimir eigenvalue equation(s), the regular representation, and the interpretation of representation matrices as gravitational wavefunctions.Appendix A provides a detailed technical description how the proposal (1.4) is a manifestation of Hopf duality applied directly to the modular double.A technical comment is made in appendix B.

Towards a Gauss-Euler decomposition of the modular double
If we write a generic 2 × 2 quantum group element g as the quantum group SL q (2) is defined and parametrized by the variables (A, B, C, D) satisfying: or, equivalently, in terms of the non-commutative coordinates (γ, ϕ, β) satisfying For the case of the quantum group SL q (2), the Gauss-Euler decomposition in any representation is known, with a formula analogous as in (1.2):8 R(g) = e γF q −2 e 2ϕH e βE q 2 . (2.5) This was first proven by C. Fronsdal and A. Galindo in [62], and further analyzed and extended in [63,64,65,66].The generators E, F and H are representation matrices satisfying the dual U q (sl(2)) quantum algebra: For the two parabolic generators E and F , the matrix exponential is q-deformed.The q-exponential is defined by its series expansion as: in terms of q-numbers [n] q .The equation (2.5) is important since it relates the quantum algebra (2.6) with the associated dual quantum group SL q (2), in a similar way as ordinary exponentiation relates a classical Lie algebra with a Lie group.So far we have worked with the complex forms of these objects.Real forms of these quantum groups have been classified (see e.g.[67]) and consist of the compact real form SU q (2), the non-compact form SU q (1, 1) where q ∈ R, and the non-compact form SL q (2, R) with |q| = 1.The compact real form SU q (2) is of no interest in gravity.The non-compact form SU q (1, 1) is relevant for double-scaled SYK, but not for 3d gravity and Liouville gravity.We henceforth focus solely on the real form SL q (2, R).The reality condition in this case is enforced by the existence of a *-relation for which Next, we work towards the modular double of SL q (2, R).The quantum algebra of the modular double U q (sl(2, R))⊗U q(sl(2, R)) and its representations are constructed as follows.The irreducible representations of the modular double are continuous, and the generators can be represented as self-adjoint finite-difference operators acting on L 2 (R) as follows [45,46,68]: where T a f (t) = f (t + a) is a shift operator.The representation label is α = Q/2 + is for real s and Q = b + b −1 .The algebra U q (sl(2, R)) is generated by the basis elements F l K m E n , l, n ∈ N, m ∈ Z.One supplements to these three generators, the three dual generators ( K, Ẽ, F ) defined in terms of the same relations (2.8) upon replacing q → q (b → b −1 ).The dual algebra is generated by the dual basis elements F l Km Ẽn , l, n ∈ N, m ∈ Z.The two pairs of generators (K 2 , E, F ) and ( K2 , Ẽ, F ) commute with each other as can be readily checked.At the level of algebras, this means that the total algebra is a tensor product of both separate algebras, hence the notation U q (sl(2, R))⊗U q(sl(2, R)).The resulting quantum algebra has by construction a symmetry b → b −1 , unlike a single copy of U q (sl(2, R)), reflected in the Gauss decomposition of the dual quantum group (2.5).Finally, for the modular double it is natural to rescale the parabolic generators as since these parabolic generators have the crucial transcendental property [48]: (2.10)So far, the discussion was at the quantum algebra level.To pass to the modular double of the quantum group SL q (2, R), we will have to further adjust the q-exponentials in (2.5) to implement the b → b −1 symmetry.In the spirit of Faddeev's original work [44], it is clear that the correct replacement will be to use Faddeev's quantum dilogarithm g b (defined below) in place of the q-exponentials.We hence propose the following Gauss-Euler decomposition: as the bridge between the quantum algebra U q (sl(2, R))⊗U q(sl(2, R)) and the modular double quantum group SL + q (2, R).The triple (γ, ϕ, β) satisfy the same commutator relations (2.4) as before, and where γ and β are naturally restricted to a positive spectrum, since they appear as arguments within g b which incorporates non-polynomial powers of its argument in its definition.This explains the superscript + for the associated quantum group SL + q (2, R).Let us motivate the proposal (2.11) in more detail.The relevant deformed exponential is Faddeev's quantum dilogarithm g b (x), defined as [58,69]: (2.12) It satisfies the properties: ) The first two properties identify g b (x) as a quantum exponential function.The last one implements the self-duality (b → b −1 ) at the level of the quantum group representation matrices in (2.11).The function g b has the following inverse Mellin transform in terms of the double sine function S b (x): ) (2.17) We will evaluate these integrals by contour deforming to the left half-plane where we pick up the residues of all poles of the S b -function.The S b -function has known double sets of poles with residues: where (n, m) ∈ N 2 (including zero), and where in the last line we introduced the "symmetric" q-number as: So the g b -function has the double series expansion: Using the identity: we can rewrite this suggestively as a product of two q-exponentials as: (2.22) Using the rescaled generators (2.9) and the transcendental relations (2.10), we have: containing simultaneously both a generator from the quantum group and its dual, in the same exponentiated form as in (2.5).Plugging this in (2.11), we can expand which contains a product of five exponentials.
Comparing to (2.5), we note that there are factors of i inserted in the deformed exponentials.This is no surprise; we have explained in earlier work [25,31] that the eigenvalues of the E and F generators acquire additional factors of ±i when transferring from the full group SL(2, R) to the positive semigroup SL + (2, R), which is a reflection of this property on the undeformed q → 1 limit.
Notice the Cartan generator H is not doubled in (2.25).This is because, after setting ϕ → bϕ such that e 2ϕbH = e 2ϕb −1 H , it is b → b −1 invariant and hence "serves both quantum groups" in Faddeev's wording [44].A perhaps more natural choice of parametrization of the modular double is to hence rescale ϕ → bϕ, γ, β → γ b , β b , leading to the representation matrix: where one has the tantalizingly simple non-commutativity relations: We develop the interpretation of the formula (2.25) in terms of duality of Hopf algebras in Appendix A, and argue in more detail why a representation of the modular double quantum algebra maps into a representation of the associated dual matrix quantum group.This is what ordinarily constitutes a proof of the above formula (2.25), see [62].However, one can read our arguments there also as an interpretation on what the modular double quantum group is concretely as a Hopf algebra.
In the next section, we apply (2.11) directly to compute some important representation matrix elements, matching them with earlier results and hence providing indirect evidence for the validity of (2.11).

q-Representation matrix elements
As mentioned in the Introduction, representation matrix elements R mn (g) can be written in a bra-ket notation as ⟨m| g |n⟩.Different bases for the bra or ket indices m and n respectively can then be related by inserting complete sets of states as We will use the greek ν and µ indices to represent so-called parabolic indices, diagonalizing the parabolic generators F † and E of (1.1) respectively.The latin s 1,2 indices denote hyperbolic indices diagonalizing the hyperbolic generator K = q H instead.The above change-of-basis relation is well-known for classical Lie groups [70], but here we explore it for the modular double quantum group at hand.

Warm-up: Whittaker function
As warm-up for the generic representation matrix element of the next section, we compute here the Whittaker function for the modular double, where g is in the Cartan subgroup.This result has been known for some time due to work by S. Kharchev, D. Lebedev and M. Semenov-Tian-Shansky [47], but we perform the computation in a slightly different way utilizing (3.1), which will allow us to leverage it towards computing the generic hyperbolic matrix element in the next subsection.In gravity, the Whittaker function describes the two-sided wavefunction as described in the Introduction.
In the current set-up, g = e 2ϕH .The hyperbolic generator K = T ib/2 = q H where H = 1 2πb ∂ t , can be diagonalized as and eigenvalue Hψ(t) = isψ(t), where we have set x = e 2πbt .These modes are orthonormal: The resulting hyperbolic matrix element with only the Cartan generator inserted is given by ⟨s 1 | e 2ϕH |s 2 ⟩ = e 2is 1 ϕ δ(s 1 − s 2 ). (3.5) The Whittaker vector ϕ µ (t) simultaneously diagonalizing E and Ẽ, was determined in [47]. 10Note that this system is consistent, with these precise prefactors, due to the transcendental relation (2.10), implying we are in fact diagonalizating e and ẽ ≡ e 1/b 2 .Using the parametrization (2.8) of the quantum algebra, the Whittaker vector is where α = Q/2 + is and where the integration contour C is along the real axis above the pole at the origin ζ = 0.
From (3.3) and (3.7) we can compute the overlap between these states as It is useful to compare this overlap to its classical limit.To find it, we rescale where the new µ and k are fixed as b → 0. Using S b (bx) x−1/2 Γ(x), we find the classical b → 0 overlap: Likewise, for the conjugate of the Whittaker vector simultaneously diagonalizing F † and F † :12 the expression: leading to the overlap: with analogous classical b → 0 limit (after suitable rescalings again): Inserting the different ingredients (3.5), (3.8) and (3.13), we obtain: indeed matching with the Whittaker function of [47] in the specific case where ϵ = 0.13

Hyperbolic representation matrix element
We start anew with the equality but now insert the full quantum group element g in the form of our proposal (2.11).
On the LHS of (3.16), the ket and bra diagonalize the rescaled self-adjoint parabolic generators e and f with eigenvalue ν and µ respectively.So the LHS can be simplified into where we inserted the Whittaker function (3.15).
To evaluate the RHS of (3.16), we can use the expressions (3.8) and (3.13): ) Inserting these in (3.16), and performing the integral transforms (ranging only over positive values of µ and ν): Hence we obtain the hyperbolic representation matrix element: Evaluating the integrals using the Mellin transform of (2.16), we finally obtain (after dropping the tildes): with the explicit normalization factor 2 (s 2 −s 1 ) . (3.24) We used the following equality: The quantity γe 2ϕ β is sometimes called the hyperbolic element [70], and is the only combination of the coordinates the integral in (3.23) depends on.
One can see here that if we picked the "wrong" deformed exponentials from (2.5) to compute the representation matrix element, we would produce instead q-Gamma functions as a result of the µ-and ν-integrals (see Appendix B).
In gravity, the hyperbolic representation matrix element is relevant for describing fully internal wavefunctions (whose endpoints are not on holographic boundaries).

Comparison to earlier work
The correctness of our result can be appreciated by comparing (3.23) to an earlier determination of this hyperbolic representation matrix element using a different technique to which we turn next.In particular, our result (3.23) is to be compared to the expression (7.35) of I. Ip [53].We first rewrite our expression slightly.Using the substitution iζ = −iτ − α and the identity S b (x) = 1/S b (Q − x), we can rewrite (3.23) as with a new normalization factor Ñ that we will not track explicitly.

.31)
The q-hypergeometric function F b is: (3.32) Plugging these in, the explicit matrix element of [53] becomes: . 14 We set s there → −bs1, α there = −bs2 and l there = −α. 15These arise in the b-binomial theorem.For u, v positive self-adjoint operators satisfying uv = q 2 vu, we have: Finally, using the identifications we see that (3.33) matches with (3.27), up to the prefactor Ñ that does not depend on the coordinates (γ, ϕ, β).

Some applications
In this section, we deduce some properties of the representation matrix elements, that are in part of direct relevance for gravitational calculations.These results are made technically possible by virtue of the explicit formula (2.11).

Casimir difference equation(s)
In the undeformed set-up, the sl(2, R) Casimir operator is diagonalized by the irreducible representation matrix elements.Mixed parabolic representation matrix elements lead to a simplified Casimir equation which is just the Liouville equation. 16The q-analogue of the latter was shown in [47] to be satisfied by the Whittaker function (3.15).As an application of our proposal (2.11) and calculational procedure, we will here derive the Casimir eigenvalue equations that the hyperbolic q-representation matrix elements (3.23) satisfy.
The modular double quantum algebra has two Casimir operators.The Casimir operator corresponding to U q (sl(2, R)) commutes with all generators E, F, H and Ẽ, F , H. It is given by the expression (up to a choice of normalization): There is also a dual Casimir operator coming from U q(sl(2, R)): We now apply the Casimir operator Ĉ within the representation matrix element as On the one hand, the Casimir is proportional to the unit matrix in the principal series representations (since it is irreducible), so we have by explicitly computing Ĉ using (2.8): On the other hand, we can manipulate it into a difference operator that acts on the representation matrix as follows.We use (3.16) once again and compute the LHS with the insertion of Ĉ: ⟨ν| g Ĉ |µ⟩.The desired hyperbolic representation matrix element then follows again by Laplace transforming as in (3.20).We hence first evaluate ⟨ν| g Ĉ |µ⟩.
In inserting the Casimir operator (4.1), the term with the Cartan generator q ±(2H+1) just contributes a linear combination of shift operators on the ϕ-coordinate as: The F E term in (4.1) gives after commuting the F past g and using that bra and ket diagonalize F † and E respectively as in (3.6) and (3.11), an insertion of This term can be rewritten in terms of q-derivatives acting on the group element itself using the identities: which are directly derived using (2.16), (2.17), and represent the fact that g b is a qexponential function. 17Here we have used the textbook q-derivative (for suitable choice of q), defined as The integral transformations (3.20) can then be done immediately.Likewise, we have the dual identities: that can be used to derive the dual Casimir equation.We end up with the Casimir difference equation and its dual: Let us make some comments.
• When using b → b −1 to find the second equation, one has to make sure to transform the coordinates (γ, ϕ, β) in the correct way.Alternatively, one can work with the invariant rescaled coordinates defined around (2.26).
• Since the coordinates (γ, ϕ, β) are non-commutative, care has to be taken in the ordering of the different factors as written.In particular, we need to order the coordinates as (γ, ϕ, β) as done in the decomposition of g (2.11) and explicitly in R s 1 s 2 (g) as e.g.done in (3.22) and (3.23).Our way of writing these equations implies that the γ-derivative acts from the left, the β-derivative acts from the right, and the e −2ϕ factor needs to be applied in the "middle" of g.
• It is instructive to explicitly check that (3.23) satisfies these difference equations (4.12).For the second term of (4.12), one uses the properties: such that the second term causes an effective shift (γe 2ϕ β) iζ/b → (γe 2ϕ β) iζ/b−1 in the integrand of (3.23).Using then a contour shift iζ → iζ +b and the defining shift properties of the double sine function S b (x + b ±1 ) = 2 sin πb ±1 x S b (x), one can explicitly show that (4.12) is satisfied.The dual equation is checked analogously.
• In general, solving difference equations has an enormous ambiguity since the values of the unknown function are only related at discrete points.This is reflected in the presence of arbitrary periodic functions (sometimes called "quasi-constants") in the general solution.However, assuming b 2 is irrational, the pair of difference equations (4.12) associated to the modular double leads to a "dense" covering of the (γ, ϕ, β) coordinate regions by combining back-and-forwards shifts of both b and b −1 .

Regular representation of the modular double quantum group
The Casimir eigenvalue equation of an ordinary Lie group G is the result of decomposing the regular representation into its irreducible components.Here we use (2.11) to directly construct the regular representation of the modular double from first principles, and show that it indeed leads to the pair of Casimir equations (4.12).The left-regular representation of any Lie group G is defined by acting on the set of functions in L 2 (G) as: Infinitesimally, this group action leads to a differential operator Li , defined by the relation Li or: Analogously, one defines the right-regular realization as: leading to Ri g = gX i .
For quantum groups, we can directly work with (4.17) and (4.19) as defining the left-and right-regular realization in terms of difference operators Li and Ri . 19For concreteness, we focus on (4.19), and collect the results on the left-regular realization at the end.
To extend this definition to the modular double of a quantum group, we use the "doubled" group element g parametrized in (2.11).The quantum algebra generators are (E, F, H) and ( Ẽ, F , H) for the two copies.We will prove the following statement: The regular representation of the modular double of the quantum group SL q (2, R), defined through either (4.17) or (4.19), is equal to the modular double of the regular representation.
Let's start with the element E in the right-regular realization (4.19).We want to find the operator RE such that This operator as written acts from the right on any expression, which we depict by the arrow on top.Next let's look at the Cartan element K = q H in the form: so that we read off: where we used and defined the scaling operator R β a f (β) ≡ f (aβ).Finally, the hardest generator is We use the property and obtain 21 The Casimir operator can be evaluated and is of the form 22 20 We refrain from putting an arrow on T ϕ since this does not matter when acting on g. 21As an example of how one works with expression such as these, we work out the first term in detail: 22 Care has to be taken for the swapped ordering in which the operators are applied from the right of the expression.In particular, the q-derivative appears a priori on the left of the second term, but can be pulled through in a second step to match with the written expression.
If one instead defines the operators such that they appear directly ordered in the correct place in the expression, as in the previous subsection, the expression could be written as (4.32) Analogously we have: For the last one, we need now the dual identity: which finally leads to the expression: This results in the dual Casimir on the second line of (4.12).
The Casimir operator in the regular representation of the algebra is interpreted as the analogue of the Laplacian on the quantum group manifold.In the case of the modular double quantum group, there are two Casimir operators that are b → 1/b dual to each other.Representation matrix elements are simultaneous eigenfunctions of both, as we have explicitly checked for the hyperbolic representation matrix element in equation (4.12), but can also be explicitly seen in the simpler case of the Whittaker function R νµ (ϕ) (3.15) by checking explicitly that it satisfies: The three generators (4.21), (4.23) and (4.28) are the same as those found by using the SL q (2) Gauss-Euler decomposition (2.5), as we computed explicitly in [41]. 23aking their generator duals b → 1/b (and being careful about the scaling of the coordinates (γ, ϕ, β) as before), we immediately obtain (4.32), (4.33) and (4.35) without any calculation.We hence conclude that the modular double of the regular representation equals the regular representation of the modular double, as defined and determined in this subsection.
For completeness, we collect the generators of the left-regular realization (4.17): where we used the identities:

.44)
This leads to the same Casimir operator (4.30) and its modular dual.

One-sided wavefunctions and gravitational interpretation
As a final application, we write down an expression for the representation matrix with on the left a hyperbolic eigenstate, and on the right a generalized parabolic eigenstate as follows.The right boundary state is generalized into the one-parameter family of states [47]:

.45)
Setting α 2 = 0 reduces the right boundary state to the one studied before, but we choose to be slightly more general here.Going through an analogous computation as before where we set ϕ µ,α 2 (t) ≡ ⟨t| µ, α 2 ⟩ and use we arrive at the single-sided representation matrix: with the explicit normalization factor The choice α 2 = ± 1 2 is prefered in the context of Liouville gravity, but we leave it arbitrary here.Moreover, for physics applications one might want to set β = 0 and consider the quantum subgroup generated by γ, ϕ only, corresponding to an intermediate case between the Whittaker function only depending on ϕ, and the full irrep matrix element depending on all three coordinates γ, ϕ, β.Concretely, this just means one deletes the last factor of g * b (.) in (4.47) since g b (0) = 1.Such one-sided wavefunctions are of interest when describing the Hilbert space exterior to a black hole as follows.Starting with a two-sided Hilbert space (and wavefunction), one can attempt to split the Hilbert space into a left (L) piece and a right (R) piece.However, in gauge theories and gravity alike, such a splitting cannot be done directly due to the non-local constraints acting on physical states in the Hilbert space [72,73,74,75].Factorization can be achieved by enlarging the Hilbert space and allowing surface charges at the splitting (or entangling) surface.In lower-dimensional gauge theory models, this is done explicitly by factorizing using the defining property of a representation: where the gravitational wavefunction ψ ab (g) ∼ R ab (g).This factorizes a two-sided wavefunction into the product of wavefunctions with the index c living at the splitting surface.
The splitting surface itself is also the entangling surface or a black hole horizon according to an observer whose observations are restricted to a single side.Exploiting the fact that lower-dimensional gravitational models have a gauge theoretic description, we investigated a similar factorization in several gravity models in earlier work [25,42].It turned out that in both these cases, the splitting index c has to be a hyperbolic index s as discussed here.Hence in the case when the underlying group theoretic structure is the modular double U q (sl(2, R))⊗U q(sl(2, R)), the above one-sided wavefunctions (4.47) are precisely these split gravitational wavefunctions, with one asymptotic index and one index on the entangling surface.Moreover, the hyperbolic index s is then labeling edge state degrees of freedom that are inaccessible for an outside fiducial or one-sided observer, an observer whose observations are restricted to information living on just this one side.The set of all possibilities for s describes the different black hole microstates that are consistent with its macroscopic properties (i.e. its total mass), encoded in the Casimir eigenvalue.This is the picture we advocated for in [25,42].We summarize the gravitational interpretation of these different gravitational wavefunctions in figure 1.

Concluding Remarks
We have provided evidence that matrix exponentiation to go from the quantum algebra to the quantum group can be shown to work (in the sense of (1.4)) for the modular double U q (sl(2, R))⊗U q(sl(2, R)).This is particularly useful since this is how concrete calculations of BF amplitudes are done when there are boundary conditions that restrict some of the generators as discussed in the Introduction.Moreover, this calculational technique allows us to relate the two previously known representation matrix elements of SL + q (2, R): Ip's hyperbolic representation matrix element [53], written in our parametrization in (3.23), and the Whittaker function (3.15) [47].We applied our proposal to show that the representation matrix elements are simultaneous solutions to two Casimir eigenvalue equations (4.12), and we constructed a representation matrix element that mixes a hyperbolic and parabolic index (4.47),relevant to describe one-sided wavefunctions in lower-dimensional gravity models.Structurally, we have constructed the regular representation of the modular double, reproducing both Casimir operators, and showed that it is equal to the modular double of the regular representation.More abstractly, in Appendix A we have embedded and interpreted our proposal in terms of Hopf duality between the modular double quantum algebra and the (Hopf) dual matrix quantum group.
Moreover, the presented technique looks amenable to supersymmetrization to find the N = 1 hyperbolic representation matrix element starting with the known Whittaker function recently determined [33].These would be of importance for amplitudes in 2d N = 1 Liouville supergravity and 3d N = 1 supergravity.
We can prove this last formula directly by contour deforming the s-integral to the left half plane, and picking up all poles of the Γ-function with residue Res s=−n Γ(s) = (−) n /n!, so the RHS indeed becomes The q-exponential introduced above has an analogous inverse Mellin transform in terms of a q-Gamma function: which we define by this relation.The q-Gamma function is meromorphic on the complex plane with simple poles at s = −n, n ∈ N, with residues Res s=−n Γq (s) = (−) n /[n] q !. Indeed, contour deforming the RHS to the left half-plane, we similarly get: which is the q-exponential function.This Γq function satisfies its defining relation: This holds for the "compact" q-deformation.We however are interested in the "noncompact" case.Instead of the q-Gamma function, the relevant quantity is then the double sine function S b (x).

Figure 1 :
Figure 1: Different gravitational wavefunctions (blue) and the modular flow (red) relevant for each of them in the gravitational application.Dashed diagonal lines are black hole horizons.Hyperbolic s-labels are attributed to fixed points of the modular flow (black hole horizons), and count black hole microstates.Left: two-sided wavefunction (two holographic boundaries) (3.15).Middle: one-sided wavefunction (one holographic boundary) (4.47).Right: Interior wavefunction (no holographic boundaries) (3.23).