Dynamical actions and q-representation theory for double-scaled SYK

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SU$_q(1,1)$. We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SU$_q(1,1)$, whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the $q$-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.


Introduction
The SYK model [1,2] has attracted enormous attention in the field ever since the realization [2,3] that it is arguably the simplest quantum mechanical system with a gravitational dual. In particular, the SYK model is a quantum mechanical system of N fermions with a p-local Hamiltonian with (Gaussian) For fixed finite p, the low-energy physics is governed by Schwarzian quantum mechanics [2,3], which is in turn gravitationally dual to JT gravity [6][7][8][9][10] upon introducing near AdS 2 boundary conditions. 2 This duality has taught us a lot about quantum black holes and quantum gravity in AdS 2 . However, it also has its shortcomings, one of the most notable ones being that JT gravity is not UV complete (since it describes only the IR limit of SYK). The holographic bulk dual of the full SYK model is unknown at this time, and remains one of the big unsolved problems in this field. Some proposals have been made, see for instance [12][13][14][15], but no unified and accepted conclusion has been reached.
Fortunately, besides the low energy limit of SYK, there are other interesting limits. In particular one interesting regime called double-scaled SYK [16][17][18] (or DSSYK for short) is obtained by sending p Ñ 8 whilst simultaneously sending N Ñ 8 with q (0 ă q ă 1) defined in (1.1) held fixed. The JT or Schwarzian regime is recovered by afterwards sending q Ñ 1´whilst zooming in on low energies.
This begs the question whether perhaps DSSYK also has an understandable bulk description, some type of minimal UV completion of JT gravity which captures some of the more stringy features of the full SYK model [3]. It has been pointed out in recent work [4] that the answer -at least to some degree -is yes. 3 The quantum amplitudes calculated in [17,18] use so-called "chord diagrams", 4 where the emerging bulk picture that appears is that quantum mechanically one can think of the states of the bulk Hilbert space (on any slice between two asymptotic boundaries) as states with a fixed number of chords n. As such, quantum mechanically one can roughly think of spacetime as being "discretized" at a fundamental level. 1 Our convention for q differs from the one used for instance in [4] by q 2 " q there , we follow the group theory literature [5]. 2 For a recent review on JT gravity, see e.g. [11]. 3 For other interesting recent work involving DSSYK and chords see for instance [26][27][28][29][30][31][32][33][34][35][36][37][38].
Although this chord picture is attractive, it still leaves us slightly unsatisfied. Namely, chords are a highly quantum mechanical picture, and much of our insights and intuition in physics arises from quantizing some classical dynamical system. The goal of our work is to build towards a bulk description of this kind.
Classically JT gravity has a first-order formulation as an SLp2, Rq BF theory, derived in the early literature in [39][40][41][42], and more recently in e.g. [43,44]. At the quantum level, the relation holds up to subtleties related to the precise choice of algebraic structure [21,23,24,45,46]. 5 The SLp2, Rq symmetry allows one to derive WdW wavefunctions and various correlation functions, via representation theory techniques [23]. BF theory is topological -it can be reduced entirely to boundary dynamics. As we quickly review in section 2, for JT gravity this boundary dynamics is a constrained particle on an SLp2, Rq group manifold, where the constraints arise by imposing asymptotic (nearly) AdS 2 boundary conditions [8][9][10]. These effectively reduce the dynamics to Schwarzian quantum mechanics.
In this work, we study some aspects of the representation theory of the quantum group SU q p1, 1q. We reproduce the two-sided wavefunction and correlation functions of DSSYK by implementing the correct boundary conditions on the constructed representation matrices. We clarify the relevant techniques in section 3.
We then derive a path integral description (involving in particular a classical action) for a particle traveling on the SU q p1, 1q (quantum) group manifold, which we supplement with constraints generalizing the asymptotic AdS 2 boundary conditions. The resulting restricted boundary action is dubbed the q-Schwarzian and derived in section 4. By construction, quantization of this system reproduces DSSYK amplitudes (representation theory of SU q p1, 1q).
This description should enable one to reverse engineer a topological bulk theory akin to BF gauge theory, whose boundary dynamics is precisely the q-Schwarzian. Such a model would then be the firstorder formulation of the bulk dual to DSSYK. Building in part on [47,48], we propose in section 5.1 that the bulk model in question is a particular "Poisson sigma model" [49], which can be rewritten as a 2d dilaton gravity [50,51] with potential V pΦq " sinp2 log qΦq log q , (1.2) which results in classical solutions with dS 2 and AdS 2 regions. It would be interesting to proceed along this lines to check the relation between dS 2 quantum gravity and DSSYK [27,34,35,52,53]. 6 We will present more details on this bulk description elsewhere [54].
This web of dualities is summarized below diagrammatically in (2.25), (1.8), (2.22) and (1.7). 5 These subtleties will not be important for our main story as they are mainly there for JT and Liouville gravity, whereas DSSYK seems to avoid them automatically. 6 Or perhaps we should call it dSSYK?

Summary and structure
The remainder of this work is structured as follows.
In section 2 we review and rephrase the structure of amplitudes in JT gravity that follow from the SLp2, Rq BF description, and review how DSSYK amplitudes have a similar structure [17].
In section 3 -the technical heart of this work -we construct the right-and left regular realization of the quantum group SU q p1, 1q and use this to compute observables in the Hamiltonian description of a quantum particle on SU q p1, 1q. We then constrain this system, and show that the amplitudes of the resulting constrained quantum mechanics are equivalent to those of DSSYK.
In section 4 we construct a continuum dynamical system describing a particle on SU q p1, 1q, which gives rise to the aforementioned Hamiltonian system upon using canonical quantization, and impose the constraints (3.34) which lead to a q-Schwarzian phase space path integral (4.21) for the case with one asymptotic boundary, and to a q-Liouville phase space path integral for the case of a Cauchy slice with two asymptotic boundaries: DϕDp ϕ expˆiˆdtˆp ϕ ϕ 1`p 1´e´2 ϕ q e´i log q p ϕ plog qq 2`e i log q p ϕ plog qq 2˙˙. (1.3) In the concluding section 5 we propose a (topological) gravitational bulk dual that is equivalent to these boundary theories. We'll compare our realizations of section 3 with the ones that appeared in [37], and discuss potential generalizations.
We provide more mathematical details on quantum group SU q p1, 1q and its relation to U q psup1, 1qq in Appendix A.
Our main technical results are the following: • The (quantum) group manifold SU q p1, 1q is a non-commutative three-dimensional space with Gauss coordinates pγ, ϕ, βq. The non-commutativity arises from quantization; the fields pγ, ϕ, βq in the path integral are ordinary classical fields. In fact, via a simple coordinate transformation in phase space, one can obtain an equivalent quantum mechanical description with commutative coordinates. Upon constraining the system one finds that γ and β are redundant, hence one can gauge-fix γ " β " 0, resulting in the two-dimensional phase space path integral (1.3).
• The Laplacian on this quantum group manifold can be written down explicitly as the Casimir in the regular representation of the quantum algebra. It is physically interpreted as the Hamiltonian (time-independent) Schrödinger operator whose corresponding classical action describes a particle moving on the quantum group. Importantly, this Hamiltonian is non-Hermitian. Upon implementing the boundary conditions, the Hamiltonian becomes the (non-Hermitian) Wheeler-de Witt Hamiltonian describing evolution of two-boundary (or wormhole) gravitational wavefunctions.
• Our choice of Haar measure on SU q p1, 1q, upon gauge-fixing to β " γ " 0, fixes the inner product on wavefunctions which reproduces the correct DSSYK amplitudes (3.60). Here the left eigenfunctions [56] of the Hamiltonian xθ 1 |ny differ from the right eigenfunctions by en important factor pq 2 ; q 2 q n . This inner product has the important feature that it samples the wavefunction at discretely separated points, which avoids infinite degeneracies in the physical Hilbert space and effectively discretizes the chord number n. This discretization is hence a quantum effect.
Our current understanding of the dual descriptions of DSSYK in this language can be summarized by the following picture for the two-sided model: q-Liouville (4.19) particle on SU q p1, 1q (4.12) and the one-sided model: q-Schwarzian (4.21) particle on SU q p1, 1q (4.12) g " ? holography q´L H L F fixed bdy condition ?
The leftmost part of this diagram (the holographic bulk) and how to transfer in detail to the boundary description (the middle of the diagram) will not be addressed here, and is not fully developed yet. We make some comments in section 5.1, but postpone a deeper study to future work [54].

Representation theory for JT gravity
One can rewrite the Euclidean JT action (we'll set 8πG N " 1) in first-order variables [39][40][41] including the boundary term [21] as with A and χ valued in the slp2, Rq algebra: and with the trace in the 2d representation.
The boundary conditions equate χ and A, and additionally constrain one component of both fields: gravitationally corresponding to fixing the total boundary length ℓ " β{ε in terms of the renormalized length β, and the dilaton asymptotics Φ| BM " 1{2ε. For the second boundary condition (2.4), the precise value we choose for χ F at the boundary is somewhat arbitrary, as long as it is non-zero. This can be appreciated as follows. As in the earlier literature [57], one can perform a gauge transformation in the bulk BF model which has support on the boundary, changing χ Ñ b´1χb and A Ñ b´1Ab with b " e log aH . This causes χ F 's value at the boundary to multiply with a. This is usually done in the aAdS context to rescale the boundary condition to a natural value, e.g. iχ F | BM " 1. Here though, for comparison with DSSYK later on, it is more insightful to not do this and leave it as (2.4).
One proceeds by path integrating out χ in the bulk BF action of (2.1). This localizes the model to flat SLp2, Rq connections A " dgg´1. Plugging this back into (2.1), and using only the first boundary condition in (2.3), the model reduces to the dynamics of a non-relativistic particle moving on the group manifold: Now consider the specific case of a manifold M that is (locally) a strip with two timelike boundaries.
The boundary action contains naively two pieces, on each part of the boundary: These are however not totally independent due to a common zero-mode. The more precise derivation leads to a path integralˆd where one mods out constant common transformations g 1,2 " g 1,2 h, h P G. This represents two twisted versions of I, with an integral over the twist/defect parameter λ, which we will not specify further (for details we refer e.g. to appendix C of [23]). Using path integral manipulations, one can rewrite this as a single particle on a group model as 8 dλˆm od G Dg 1 Dg 2 e´I λ rg 1 s´I λ rg 2 s "ˆDg e´I rgs , g " g 2 g´1 1 .
As a phase space integral, and using the explicit coordinatization (2.10), we can writê This system has a Hamiltonian that is equal to the Laplacian on the group manifold SLp2, Rq, which is mathematically given by the quadratic Casimir L C in the regular representation. The latter is diagonalized by the irrep matrix elements, which hence form the physical wavefunctions of the model.
Let us work this out more explicitly. A SLp2, Rq group element can be parameterized by the Gauss decomposition with coordinates (γ, ϕ, β): g " e γF e 2ϕH e βE , (2.10) In these coordinates, one can write down the left-regular representation of first-order differential oper- 8 To see this [57], one uses the field redefinition pg1, g2q Ñ pg, πgq with g " g2g´1 1 and πg " g1g´1 2 Bug2g´1 1`B ug1g´1 1 , after which the πg path integral decouples and is Gaussian. The remaining action is again the particle on a group action for g. One way to quickly appreciate this procedure is to realize that this is the dimensional reduction of the well-known argument on how to transfer from two chiral WZW models into a single non-chiral WZW model. The dimensional reduction of chiral and non-chiral WZW model both yield the particle on a group system. ators: and analogously the right-regular realization which we will not write down explicitly. The quadratic Casimir in this representation is a quadratic differential operator: Next we implement the asymptotic boundary condition (2.4). Since both g 2 and g 1 are subject to this constraint, one finds that g is subject to two constraints (left transformations on g 1 become right transformations on g): Let us first circle back to the argument that one can rescale the precise value of the eigenvalue. From the Gauss decomposition (2.10), we can derive the mathematical identity: gpγ, ϕ`log a, βq " e γF e 2pϕ`log aqH e βE " b e γF a e 2ϕH e βEa b, (2.14) where b " e log aH as before. Now, changing g Ñ b´1gb´1 induces a gauge transformation on A because A " dgg´1 Ñ b´1Ab. This means that rescaling the eigenvalues of E and F by a is gauge-equivalent to changing the origin of the ϕ-coordinate on the group manifold. This will be explicit in the equations that follow.
We can write down a constrained version of the action (2.9) by plugging the asymptotic constraints (2.13) into (2.9). From a quantum mechanics perspective, these are first-class constraints (their Poisson bracket vanishes) [58]. The path integral description (2.9) then reduces tô where both β and γ have become redundant fields, and can be gauge-fixed to β " γ " 0. This last equation is the Liouville quantum mechanics description of the Schwarzian model (see for instance [59,60]). Liouville QM has the famous KPZ shift property where we can define ϕ r " ϕ`log ε to absorb the 1{ε 2 prefactor from the potential term, confirming the argument above in eq. (2.14). The physical wavefunctions of this constrained quantum mechanical system are thus the Hamiltonian eigenfunctions:´1 The spectral density then follows from the orthogonality relation of the wavefunctions (up to constant These wavefunctions of the "boundary theory" in fact have a direct bulk interpretation as wavefunctions of the gauge-invariant data: the line integral of A along a spatial slice (evaluated on the flat connection A " dgg´1): Both labels of the representation matrix are constrained to i simply because we impose the boundary conditions (2.4) at both boundaries x 1 and x 2 .
Natural operator insertions in this model (2.15) have zero charges (so as not to disrupt the boundary conditions [23]): In the BF language, these correspond to a boundary-anchored Wilson line. Operator matrix elements are readily explicitly computed as [23] . (2.21) The above relation between JT on an interval (times time) and Liouville quantum mechanics (2.15) can be summarized pictorially as Liouville QM (2.15) particle on SLp2, Rq (2.9) One can also write down a single-sided path integral. The boundary condition (2.4) then imposes only L F "´i ε "´ip γ . The phase space path integral (2.9) becomeŝ This is the Schwarzian phase space path integral [9]. Indeed, integrating out first p ϕ , and then p β , one recovers the more familiar Schwarzian path integral O ∆ "˜ε 2 9 β 1 9 β 2 pβ 1´β2 q 2¸∆ , (2.26) which is directly related to the bare (unrenormalized) geodesic length ℓ computed in an AdS 2 geometry between two boundary endpoints on the wiggly boundary curve: ℓ " log pβ 1´β2 q 2 ε 2 9 β 1 9 β 2 . (2.27) Comparing (2.20) with (2.26) and (2.27) directly shows that we identify ϕ " ℓ{2. The quantity 2ϕ r " 2ϕ`2 log ε introduced earlier is then precisely the renormalized geodesic distance, as the subscript r already suggested. Working only with renormalized coordinates and renormalized operator insertions, we would strip off the ε 2∆ prefactor in (2.21), which is the usual choice one makes when discussing the correlation functions in JT gravity [11].

Amplitudes of DSSYK
The amplitudes of the double-scaled regime of the SYK model (DSSYK) were computed in [17,18] using a technique based on so-called chord diagrams; these calculations are nicely summarized and reviewed in several places [4,26] and we will merely state the results.
In DSSYK people have computed correlators of specific operators involving the product of s fermions where s " ? N for N Ñ 8 [4,17,18]. The answer for a single such operator pair is with the explicit expression for the operator "matrix element": For the purposes of this work we want to point out [17] that these amplitudes have the same structure as the JT amplitudes. In particular, upon taking the scaling limit: the spectral density (2.28) of DSSYK reduces to the JT density (2.18). 11 It was moreover shown in [17] in eq. (5.13) that the JT scaling limit, upon including some spurious prefactors, of (2.32) becomes . (2.34) This matches the JT matrix element (2.21) with the identification Notice the appearance of ε 2∆ , illustrating that the JT limit of DSSYK produces the bare (unrenormalized) quantities.
This motivates (as alluded to in section 1) to search for a representation theoretic interpretation of DSSYK amplitudes, and the associated boundary path integral formulations which generalize the Schwarzian (2.23) and Liouville models (2.15).

Representation theory for DSSYK
In this section we demonstrate how the amplitudes of double-scaled SYK derived in [17,18], are composed from the representation theory of the quantum group SU q p1, 1q. Aspects of the associated algebra U q psup1, 1qq were discussed in [4,17,18,61], and recently by [37], where they try to find a realization of the algebra such that it can reproduce the transfer matrix of DSSYK.
We will derive some representation theory of SU q p1, 1q in a systematic manner by first defining and explicitly constructing the right-and left-regular realizations of the quantum group of interest.
Our strategy contains two key points: 1. We parametrize our quantum group element using the (not widely appreciated) Gauss decomposition [62][63][64]. This formula is the bridge between the infinitesimal level (algebra) and the global level (group), which is key in deriving (and solving) the quantum mechanics of a particle on the manifold SU q p1, 1q.
2. From the regular representation, the Casimir eigenvalue problem is solved by the representation matrices. These physically represent the wavefunctions of this quantum mechanics. Given this interpretation, we will have the full machinery of usual group theory at our disposal. In particular, we can find a natural inner product (specified by the Haar measure), and we know that operator insertions in this theory are representation matrices.
The first point is physically crucial. The Gauss decomposition allows one to easily implement Brown-Henneaux [65] asymptotically AdS 2 boundary conditions [19,21,23,[43][44][45][46]57] in the JT model. We will impose similar boundary conditions that will allow us to make contact with the DSSYK model.
The second point enables us to pinpoint the embedding in group theory of the bilocal operators in DSSYK [17], and the inner product allows us to carefully compute correlators, which we match with the known DSSYK answers in section 3.4. On a technical level, one important step in the calculation is to properly deal with the fact that the Casimir (or Hamiltonian) is a non-Hermitian operator, therefore the left-and right eigenstates (wavefunctions) are different [56,66]. 12 We now define the quantum group SU q p1, 1q, and derive the regular realizations of the associated algebra U q psup1, 1qq.

Left-and right-regular realizations
The quantum group SU q p1, 1q is a deformation of the Lie group SUp1, 1q. Similarly, one can also perform a deformation at the level of the (universal enveloping) algebra U q psup1, 1qq. The generators E, F and The Fronsdal-Galindo result for the Gauss-decomposition of SU q p1, 1q [62,63] relates these two concepts together: g " e γF q´2 e 2ϕH e βE q 2 , 0 ă q ă 1 , providing a q-deformation of the usual exponentiation that transfers between Lie algebra and Lie group.
The q-exponential function is defined as We provide more details and background on (3.2) (which is key for what follows) in appendix A.
It is convenient to also introduce the exponentiated operator K " q H . The "quadratic" Casimir of the algebra U q psup1, 1qq is (up to an overall constant) which one checks explicitly by calculating rC, X A s " 0 for X A " E, F or H.
We want to construct the regular realizations of the algebra U q psup1, 1qq that follow from the Gauss decomposition (3.2). The idea is that for any one of the generators X A " E, F, H one can write X A g as a differential operator in the coordinates γ, ϕ, β acting on g. We exemplified this already for SLp2, Rq around (2.11). By construction, these differential operators (which we call L A ) have the same algebra (3.1) as X A . We should think of these L A as naturally acting on functions on the (quantum) group manifold f pgq, where they correspond with an infinitesimal action: 13 In the non-commutative setting, one should think about the functions f pgq as defined by their Taylor coefficients f nmp in an ordered expansion in γ, ϕ and β, eliminating by hand ordering ambiguities 14 Now we obtain explicit expressions for these operators. By definition we search a differential operator Introducing the standard notion of a q-derivative and rescaling operator: such that Using the algebra of coordinates (3.4) and derivatives (3.8), one finds that the rescaling operators R β a 13 This sign is necessary for LA to have the same algebra as XA. 14 We should note that the non-commutativity of the coordinates also implies a non-trivial algebra of the derivatives with respect to the coordinates. Namely, besides the obvious one immediately also derives " Another useful fact is Another similar identity for β also holds. and R γ a commute with everything (except β respectively γ). This makes them convenient operators for what follows. Next, we calculate L H . Using HF " F pH´1q one obtains Hγ n F n "´nγ n F n`γn F n H, such that L H¨e γF q´2 e 2ϕH e βE q 2 "´H e γF q´2 e 2ϕH e βE q 2 " γ d dγ e γF q´2 e 2ϕH e βE q 2´e γF q´2 H e 2ϕH e βE q 2 . (3.14) This results in which happens to be identical to the classical SLp2, Rq case (2.11). In terms of the exponentiated operator K " q H , one finds where we defined the shift operator: The final generator takes some more work. Repeatedly using the commutator of E and F one eventually Furthermore by carefully commuting coordinates and generators passed each other at each step we find Combining this with the second piece of (3.18) one arrives at These expressions indeed satisfy the correct algebra (3.1): The Casimir operator can now be explicitly computed by inserting these expressions in (3.5), leading to: Mathematically, this difference operator is the Laplacian on the (quantum) group manifold SU q p1, 1q. 15 Physically, this Casimir L C is the Hamiltonian H " L C of quantum mechanics on SU q p1, 1q, and the Casimir eigenvalue problem is the (time-independent) Schrödinger equation of this dynamical system, which we will study in more detail in section 4.
One can repeat the logic for the right-regular representation, defined by: Again, by construction, the differential operators R A satisfy the same algebra (3.1) as the generators X A and generate infinitesimal transformations on functions on SU q p1, 1q.
Explicit expressions are derived in a similar manner, e.g.: Because of the non-commutative nature of the coordinates pγ, ϕ, βq, it is natural to define the rightregular realization as differential operators that act from the right as 16 The downside is that one has to go through the mental gymnastics of reading from right to left. One eventually finds: which mirrors the the expressions for L A . We will compare this realization with the (ad hoc) realization used in [37] in section 5.2. For ease of calculations, it is sometimes more convenient to have all operators acting from one side. This leads to the equivalent rewriting of (3.26): One can check explicitly that these generators R A satisfy the algebra (3.1), the Casimir is identical to (3.22) L C " R C " H, and that they commute with the left-regular realization L A : The latter property is an immediate consequence of the definition of these operators since L A¨RB¨f pgq "

Gravitational matrix elements
In this section we compute "gravitationally constrained" representation matrix elements of SU q p1, 1q.
Let us first explain what we mean by "gravitationally constrained" representation matrix elements, and why they are physically relevant.
We are interested in solving a quantum mechanical system in the coordinates system pγ, ϕ, βq with Hamiltonian H " L C (3.22). This system has a left-right SU q p1, 1q symmetry with generators L A , R A and identical Casimirs L C " R C " H. As usually in quantum mechanics, this means the spectrum is organized in (irreducible) representations of SU q p1, 1q. Indeed, one can simultaneously diagonalize H and one generator from each set pL A , R B q, thus we have the set of eigenstates |θ, µ 1 µ 2 y with We can translate this into difference equations on L 2 pGq (as in (3.9)) by defining f θ µ 1 µ 2 pgq " xg|θ µ 1 µ 2 y, leading to the eigenvalue equations The point is that the solutions to these equations are representation matrix elements of SU q p1, 1q: Indeed, using the definitions of the left-and right-regular realizations (3.6) and (3.23), the eigenvalue equations become which we see are indeed solved by the representation matrix elements (3.31): f θ µ 1 µ 2 pgq " R θ, µ 1 µ 2 pgq.
As we quickly reviewed in section 2.1, in the SLp2, Rq BF formulation of JT gravity, special cases of these wavefunctions have a bulk interpretation as the gravitational WdW wavefunction on an interval stretching between two boundaries with asymptotically AdS 2 boundary conditions [23-25, 45, 60]: imposing the gravitational constraints (2.13) leads to the constrained wavefunctions R E, ii pϕq (2.17): Here we seek the generalization of those asymptotically AdS 2 boundary conditions to the DSSYK context. Contrary to the JT case, we do not have a first principles derivation in gravity, since we do not yet know the gravitational dual of DSSYK. The purpose (in part) of this section is to claim the generalization of the group theoretic formulation (2.13) of the gravitational boundary conditions to the DSSYK context and check that with those constraints, quantum mechanics on SU q p1, 1q reproduces (the amplitudes of) DSSYK. Once a first-order bulk theory is identified that reduces to quantum mechanics on SU q p1, 1q, one could take these group theoretic constraints and translate them back to gravitational boundary conditions in the bulk gravitational dual of DSSYK.
We claim that the correct generalization of (2.13) is 17 It may seem strange to have the Cartan elements q L H respectively q´R H appear on the RHS of (3.34), but in fact this is consistent and necessary. Physically, in quantum mechanics one can constrain any combination of operators as long as the constraint commutes with the Hamiltonian H. 18 Mathematically, such types of constraints were shown to be required to define constrained representation matrices for higher rank quantum groups [69]. 19 For rank one (our case), it is still a possibility one can introduce. For instance, in a quantum integrability context, this freedom can result in different quantizations of the same underlying classical system [70].
Notice that in the q Ñ 1´classical limit, the constraints (3.34) reduce to (2.13), with the identifi- Analogous to the discussion around (3.31), one can solve these constraints by proposing that the 17 These constraints were inspired by a similar discussion in [55], where it was observed that the amplitudes of 2d Liouville gravity have an embedding in the representation theory of the so-called modular double Uqpslp2, Rqq b U r q pslp2, Rqq. In that context it would also be interesting to understand how the constraints translate back to gravitational boundary conditions. Since in that case both sides of the duality are better understood it would be a good place to gain momentum for tackling the current DSSYK problem. 18 One checks explicitly using (3.13), (3.15) and (3.27) that the two constraints in (3.34) commute with one another and with the Hamiltonian (3.22), though this is automatic from the definition. 19 This requires in fact a further slight generalization where one uses a 2-parameter generalization as q α 1 L H and q α 2 R H . The resulting irrep matrix elements only depend on the difference α1´α2. We chose simply the most symmetric option.
wavefunctions are particular representation matrix elements where the left-and right eigenstates of the respective constraints should now satisfy To determine the gravitational matrix elements, we then solve the Casimir equation (3.22), which for our purposes can be conveniently rewritten as The resulting representation matrices R θ ii pgq are still a function of all coordinates pγ, ϕ, βq. However, the dependence on both γ and β is relatively simple since it factors out. The "reduced" matrix element which you obtain by setting γ " β " 0 are called Whittaker functions [71][72][73] in the mathematical literature. We claim that those Whittaker functions are actually the physical wavefunctions of our constrained system and study them from hereon.
Physically, the simplest way to prove this step is to use the "constrain first" approach to quantization of constrained quantum mechanical systems with some first class (gauge) constraints [58]. 20 With the purpose of identifying a dynamical action and path integral description, we will discuss the classical phase space, the constraints and the resulting gauge freedom of our system in section 4. The analysis in section 4 confirms that indeed one can use the gauge freedom of the gravitational boundary conditions to fix With this constraint the gravitational matrix elements become xϕ|θy " R θ ii pe 2ϕH q " xθ i| e 2ϕH |θ iy .

41)
20 Representation theory corresponds with a "quantize first" approach. Unfortunately this is often the more complicated trajectory, numerous examples in gauge theory illustrate the efficiency of "constrain first" quantization. The most known example is the solution of Chern-Simons theory [74,75], but also 2d Yang-Mills [76][77][78][79] and 3d gravity [80,81] are noteworthy. The point is that "quantize first" can quickly becomes tedious, one needs a nice inner product and compute the overlaps of all states, identify all null states and mod them out. These final two steps sound much simpler than they are in practice, for instance even in 2d Maxwell this takes some work [82]. In our set-up, we have an inner product (see section 3.3) so in principle one could go through this. However, it is inefficient therefore we will not do so.
such that the Schrödinger equation H¨f pgq " E f pgq becomes q xϕ`log q|θy`pq´1´qe´2 ϕ q xϕ´log q|θy " pq´q´1q 2 Epθq xϕ|θy . (3.42) As a difference equation, this Schrödinger equation only relates the wavefunction at points with discrete separation in ϕ. Therefore there is an (uncountable) infinite degeneracy for each Epθq in the solution space. Indeed, for any solution xϕ|θy of (3.42), the function xϕ|θy f periodic pϕq, with f periodic pϕ`log qq " f periodic pϕq, is also a solution to (3.42). This is a general feature of working with difference equations instead of differential equations, and requires additional input in our model to address.
In our case, the resolution is that xϕ|θy f periodic pϕq and xϕ|θy are actually physically indistinguishable.
Indeed, as we will see in the next section 3.3, the natural inner product on SU q p1, 1q we will use (the Haar measure) essentially samples the wavefunctions at discrete locations only, which in our β " γ " 0 constraint system boils down to the equidistant sampling ϕ "´n log q . To match with the physics of DSSYK, this step is crucial, since the discretized coordinate n plays the role of the integer number of chords [4,37] which represents a discretization of bulk spacetime. In our framework, this discretization arises from quantum mechanics. We will also see this in section 4, where we write classical actions for the continuous fields γ, ϕ and β. 21 With this input, the difference equation (3.42) has a unique solution for each Epθq up to an "initial" condition which we will take to be x0|θy " 1 . Now we can finally solve the Schrödinger equation for the physical data, namely the values of the wavefunctions xn|θy at the discrete sampling points. From (3.42) one finds q xn´1|θy`pq´1´q 2n`1 q xn`1|θy " pq´q´1q 2 Epθq xn|θy , (3.46) with solution for the gravitational matrix element 22 xn|θy " R θ ii pnq " q n pq 2 ; q 2 q n H n pcospθq|q 2 q , Epθq " 1 pq´1{qq 2 2 cospθq . (3.48) In this reduced recursion relation (3.47), the Hamiltonian H reduces to the transfer matrix T of DSSYK, which is discussed very nicely in [4,18]. 23 These wavefunctions xn|θy are known to be the eigenfunctions of T in DSSYK [4,18].
The functions (3.48) are the bulk wavefunctions, obtained in the chords language in [4,18]. What we are shooting for eventually is to identify them as the WdW wavefunctions of a bulk gravitational theory (1.7) with a first-order formulation that reduces to quantum mechanics on SU q p1, 1q:

Inner product
In order to compute observables and construct a Hilbert space, we ought to define an inner product on states xψ 1 |ψ 2 y "ˆpdgq q xψ 1 |gy xg|ψ 2 y . For classical Lie groups the natural inner product to be used is the Haar measure dg, which is uniquely defined (up to a prefactor) by the property that it is left-right invariant, i.e. for any group element ĥ dg f ph´1 gq "ˆdg f pgq "ˆdg f pg hq .
(3.47) 23 There are minor differences between H and T , having to do with the normalization of the states xn| -which in our case is determined by the Haar measure in section 3.3. These differences are physically irrelevant, as we will recover the DSSYK amplitudes (the physical data) in section 3.4. One should include an overall factor in our H if we want to precisely match H eigenvalues with the DSSYK energies, see footnote 10.
For quantum groups we will work with the infinitesimal version of this constraint. In particular, we will fix the Haar measure pdgq q on SU q p1, 1q by imposinĝ pdgq q L E¨f pgq " 0 ,ˆpdgq q L F¨f pgq " 0 , (3.52) with identical constraints for R E and R F . Consistency with the algebra (3.1) then implies furthermorê Our treatment here was inspired by [83]. For SLp2, Rq with Gauss decomposition (2.10), imposing these constraints indeed gives the known Haar measure dg " dγ e 2ϕ dϕ dβ.
We claim that the correct Haar integration on SU q p1, 1q for our model iŝ Here the integrals are so-called Jackson integrals: from which the second relation follows, namely that the Jackson integral is the inverse of the q-derivative pd{dxq a , meaning the Jackson integral of pd{dxq a f pxq vanishes. With this property it is obvious that since L E "´pd{dγq q´2 (3.13). Similarly, reordering the Taylor series of R F¨f pγ, ϕ, βq as in (3.56) cancels the T φ log q prefactor in (3.27), thus one finds that R F acts purely as pd{dβq q 2 on the β dependence in the Jackson integral, so this vanishes. To verify the L E constraint (3.52) one writes (3.20) L E¨f pγ, ϕ, βq " ÿ n,m,p f nmp γ n e´2 ϕ ϕ mˆd dβ˙q 2 β p`γ f pγ, ϕ`log q, βq´f pq 2 γ, ϕ´log q, βq q´q´1 . It is interesting to try to reverse this logic and derive the Haar measure (3.56) from these constraints.
Usually, the Haar measure is determined up to an overall multiplicative constant. However even when taking this into account, the solution is not unique (unlike for classical groups). The integration we have presented in equation (3.56) is the "roughest" sampling which is a solution; all other solutions are more "dense" samplings pdγq q´2 {a pde 2ϕ q q 2{b pdβq q 2{c for integers a, b, c, which even includes the limiting classical measure dγ e 2ϕ dϕ dβ. In our set-up, we choose the "roughest" sampling as the correct inner product, to avoid infinitely degenerate Hilbert spaces as discussed around (3.43). 25 After gauge-fixing β " γ " 0 (3.39) the Haar integral (3.56) implements the inner product as xθ 1 |θ 2 y "ˆ8 0 pde 2ϕ q q 2 xθ 1 |ϕy xϕ|θ 2 y "`8 ÿ n"´8 q´2 n xθ 1 |ny xn|θ 2 y , where precisely the sampling (3.43) rolls out and where xn|θ 2 y is the wavefunction (3.48).

Some amplitudes
In order to compute the overlap xθ 1 |θ 2 y we must still determine the left eigenvector xθ 1 |ny of the Casimir (3.41). Indeed, if we impose that H |θ 2 y " 2 cospθ 2 q |θ 2 y , xθ 1 | H " 2 cospθ 1 q xθ 1 | , then we are guaranteed orthogonality xθ 1 |θ 2 y " δpθ 1´θ2 q . This would be completely standard were it not for the fact that H in (3.41) is a non-Hermitian operator. This is perhaps most evident when we write T ϕ log q as in (3.17), but see also later (4.18) which is manifestly "complex". 26 The result is that the left-eigenvectors are not just (the complex conjugate of) the right-eigenvectors a well-known fact in the condensed matter literature [56] when one tries to diagonalize non-Hermitian Hamiltonians. Since we have an inner product (3.60) we can derive the equation for xn|θ 1 y: Note the (subtle) difference in prefactor with the right wavefunction (3.48). This difference is important because now one indeed finds explicitly orthogonal states (this final sum is computed in [18], and again we dropped overall constants not fixed by the current arguments) xθ 1 |θ 2 y "`8 ÿ n"0 q´2 n xθ 1 |ny xn|θ 2 y "`8 ÿ n"0 1 pq 2 ; q 2 q n H n pcospθ 1 q|q 2 qH n pcospθ 2 q|q 2 q " δpθ 1´θ2 q pe˘2 iθ 1 ; q 2 q 8 , (3.67) which reproduces the known DSSYK spectral density [16,18] ρpθq " pe˘2 iθ ; q 2 q 8 .

69)
26 Again it is interesting to compare to SLqp2, Rq in which case these translation operators are in the imaginary direction and thus Hermitian. The Casimir is Hermitian, and the left-and right wavefunctions are just complex conjugates [55].
again in line with the DSSYK results of [16,18]. The state n " 0 reflects the fact that in some eventual bulk calculation we start our bulk evolution from a point, evaluating the representation matrix element on the identity g " 1, where indeed (3.2) puts n " 0 [23]. In the chord language it means we start from a no-chords state [4] (a different way of saying we contract the bulk Cauchy slice to a point): x0| e´β H |0y " β initial configuration n " 0 or g " 1 final configuration n " 0 or g " 1 time flow (3.70) One of the elements that is still missing is how operators in DSSYK are embedded in representation theory and quantum mechanics on SU q p1, 1q. We recall from ( Any such operator can be decomposed into operators with fixed Casimir eigenvalue. We can find operators with these eigenvalues by considering discrete series representation matrix elements of SU q p1, 1q (for which the Casimir can be found in (see (G.8) in [26]) 27 O ∆ " x∆ 0| g |∆ 0y " R ∆ 00 pgq .
where we parametrized the Casimir eigenvalue as: The matrix elements become manifestly independent of γ and β: R ∆ 00 pgq " R ∆ 00 pϕq. Since we are ultimately only interested in inner products of such operators, and since such inner products (3.60) sample only at ϕ "´n log q we are only interested in the operator at these points R ∆ 00 pnq. It is easy 27 The states |θy correspond with continuous series representations ∆ " 1{2`is with θ "´2 log q s.
These results pave the way for identifying a bulk description of these operators. For instance, in BF theory you obtain this operator by inserting a Wilson line in the path integral [23]: which in turn for SLp2, Rq (see for instance [24]) is the path integral of a particle with m 2 " ∆p∆´1q propagating through the JT gravity bulk.
In summary, we have shown that DSSYK amplitudes follow from quantum mechanics on SU q p1, 1q with constraints (3.34). This embedding allows us to write up Schwarzian-type path integral descriptions in section 4, and helps in identifying a first order (gauge theory-like) bulk gravitational description of DSSYK.

q-Liouville and q-Schwarzian boundary actions for DSSYK
In this section, we will construct classical dynamical (phase space) actions involving ordinary commutative fields, which upon quantization give rise to the non-commutative Hamiltonian quantum systems described in section 3 (which as we showed describe DSSYK). This classical dynamical system is a consistent description of what one would call a "particle on SU q p1, 1q" (with constraints). The resulting actions can then be used in a path integral description to compute the same amplitudes which we computed in section 3 (there from a Hamiltonian point of view).

Non-commutative coordinates from canonical quantization
We will ad hoc construct a classical phase space, which upon quantization produces the Hamiltonian systems of section 3. We consider a classical 6-dimensional phase space with coordinates γ, ϕ, β, p γ , p ϕ , p β equiped with the symplectic two-form These coordinates are not Darboux (canonical) coordinates of the symplectic manifold. However, one can find canonical coordinates by transferring to new coordinates γ, φ, β, p γ , p φ , p β where we only change the ϕ-coordinate as: This symplectic structure implies the usual Poisson brackets tγ, p γ u " 1, tβ, p β u " 1, tϕ, p ϕ u " 1 , supplemented by tγ, ϕu " i log q γ , tp γ , ϕu "´i log q p γ , tβ, ϕu " i log q β , tp β , ϕu "´i log q p β . Now we perform canonical quantization by the usual prescription: The point that we want to make is that ordinary canonical quantization, after the change of coordinates (4.2), gives rise quantum mechanically to non-commutative coordinates γ, ϕ, β with algebra (3.4), and the algebra of derivatives (3.7) and (3.8). In particular canonical quantization (4.5) implies rγ, p γ s " iℏ , rφ, p φ s " iℏ , rβ, p β s " iℏ . When one defines according to (3.4) the new coordinate ϕ " φ`log q pγB γ`β B β q, one finds immediately from (4.6) the coordinate algebra rγ, ϕs "´log q γ , rβ, ϕs "´log q β , rβ, γs " 0 , (4.7) which are precisely the relations (3.4) for the non-commutative coordinates of SU q p1, 1q. The algebra with the derivatives (3.7) and (3.8) follows in the same manner. Here we have rescaled q in the following manner log q quant " ℏ log q class . Such scaling is required to describe the classical phase space underlying the quantum Hamiltonian system of section 3. From here on, we put ℏ " 1 in all equations to streamline notation.
This procedure should come as no surprise to readers familiar with quantum mechanics on noncommutative spaces [84]. The usual approach is to consider the non-commutative algebra of coordinates (and their derivatives) and find the Darboux basis of canonical coordinates. The result is an "ordinary" system with commutative coordinates, and eigenstates of the Hamiltonian can be expressed as wavefunctions of those ordinary commutative coordinates.

Path integral description of a particle on SU q p1, 1q
To construct the classical action for particle on SU q p1, 1q, one first needs to construct the corresponding classical symmetry generators on phase space. Consider the following set of such classical currents 2i log q γ , e L " e´2 ϕ e i log q p´p ϕ`2 γpγ q e 2i log q βp β´1 2i log q β´γ e i log q p ϕ e 2i log q pγpγ´p ϕ q´1 2i log q . th L , e L u " e L , th L , f L u "´f L , te L , f L u " q 2ih L´q´2 ih L 2i log q , (4.9) which is indeed the classical equivalent of the U q psup1, 1qq algebra (3.1). Similarly the currents f R "´e´2 ϕ e 2i log q γpγ´1 2i log q γ´β e 2i log q pp ϕ´β p β q´1 2i log q , e R "´e´i log q p ϕ e 2i log q βp β´1 2i log q β .
(4.10) satisfy (4.9) and are the classical limits of the right regular realization or U q psup1, 1qq (3.27).
To obtain a system which has these currents generating symmetries, we should choose a Hamiltonian which has vanishing Poisson brackets with all of them. There is only one independent possibility: H " ef`ˆs inplog q hq log q˙2 "´1 4 e i log q p ϕ plog qq 2´1 4 e´i log q p ϕ plog qq 2´1 4 e´2 ϕ e´i log q p ϕ 1 βγ 1 plog qq 2 pe 2i log q βp β´1 qpe 2i log q γpγ´1 q , (4.11) which is the classical limit of the SU q p1, 1q Casimir (3.22). As a consistency check, one can take the classical limit q Ñ 1´of all currents as well as the Hamiltonian to recover the particle on SLp2, Rq, discussed in section 2.1. Using the canonical coordinates (4.2), it is elementary to write up the Feynman phase space path integral description for this model: e´2 φ e´i log q pφ 1 βγ 1 plog qq 2 pe´2 i log q βp β´1 qpe´2 i log q γpγ´1 q˙˙, (4.12) which should be considered the generalization of (2.9) to SU q p1, 1q.
This system is periodic in the momentum variable p φ with period 2π log q , consistent with the quantized system effectively being defined on some φ-lattice with spacing log q. 28 Likewise, the system is periodic in p log β " βp β and p log γ " γp γ , consistent with an effective lattice spacing 2 log q for the coordinates log β and log γ in the quantum theory, as indeed suggested by the Haar measure (3.54).
This theory will result in the Hamiltonian system described in section 3 upon quantizing. Indeed, (as we already learned in section 4.1) the coordinates γ, ϕ, β and their derivatives acquire the correct non-commutative structure upon canonical quantization. Furthermore, the currents a L , a R become the quantum generators L A , R A .

Constraints and the q-Liouville path integral
Let us now mimic the analysis for JT gravity on an interval with two boundaries of section 2.1, resulting in a generalization of the Liouville action (2.15). For this we need to impose the gravitational boundary conditions (3.34), which we reproduce here for convenience Here we want to impose these as classical constraints on phase space. The associated classical constraints can be written as ψ L " 1 γ pe´3 i log q γpγ´e´i log q γpγ q e i log q pφ{2 " i , ψ R " 1 β pe i log q βp β´e´i log q βp β q e´i log q pφ{2 "´i .
(4.14) 28 In contrast for Liouville gravity the e i log q pφ factors are replaced by real exponentials resulting in no φ-periodicity.
By construction (as the classical limit of (4.13)), or via explicitly computing Poisson brackets, one sees that these constraints have vanishing Poisson brackets with one another and with the Hamiltonian tψ L , Hu " 0 , tψ R , Hu " 0 , tψ L , ψ R u " 0 , This makes sense, as the constrained Hamiltonian also only depends on p φ and this second coordinate.
However, the remaining two coordinates (in the 4d reduced classical phase space) are gauge modes and non-physical. They should be gauge-fixed, and one simple gauge choice is to fix β " γ " 0 (3.39).
Renaming the physical coordinates as p ϕ and ϕ (for notational comfort) one recovers a constrained simple Hamiltonian: 29´4 plog qq 2 H " p1´e´2 ϕ q e´i log q p ϕ`e i log q p ϕ . This path integral describes the two-sided dynamics from the boundary perspective.
By diagonalizing the Hamiltonian of this dynamical system, we obtain the wavefunctions (3.48) but without the factor q n . This was to be expected, and is similar to the difference between K 2iE 1{2 p2e´ϕq and e´ϕK 2iE 1{2 p2e´ϕq in the JT story, as the Haar measure (3.54) has a factor q´2 n whereas the path integral naturally comes with a flat measure. 29 With the gauge-choice γ " β " 0 this is actually precisely what the physical coordinates become.

The q-Schwarzian path integral
Finally we reproduce the analogue of the Schwarzian phase space path integral (2.23) for which we only impose one constraint ψ L " i. Again, we can gauge-fix to γ " 0 which reduces the Hamiltonian to 30 H "´1 4 e i log q p ϕ plog qq 2´1 4 e´i log q p ϕ plog qq 2`1 2 log q e´2 ϕ e´i log q 3p ϕ {2 e 2i log q βp β´1 2i log q β . (4.20) The generalization of the Schwarzian path integral (2.23) to the DSSYK context is then e´i log q pφ plog qq 2 1 2 log q e´2 ϕ e´i log q 3p ϕ {2 e 2i log q βp β´1 2i log q β˙˙. xϕ, β|θ sy " R θ is pϕ, βq. This picture corresponds to the "angular" slicing discussed for JT gravity and BF gauge theory in [45].
The one-sided wavefunctions (4.22) will have a bulk interpretation as having support on the black hole horizon at the "inner" endpoint [45,85], with a frozen particle on SU q p1, 1q describing edge modes s on the horizon. It would be interesting to make this more explicit.
Finally, we remark that that whereas the two-sided q-Liouville action (4.19) can be derived directly from a chords description [4], the same cannot be said about the one-sided q-Schwarzian description (4.21), for which one really must start from the embedding in the full particle on SU q p1, 1q description (4.12).

Concluding remarks
We conclude this work with several comments.
First we'll propose a candidate gravitational path integral dual to our story (and hence to DSSYK). 30 One technical comment is that this depends slightly on the choice of α1 and α2 discussed in footnote 19. An independent calculation of the wavefunctions (4.22) of this q-Schwarzian theory should show which choice matches with DSSYK. The only effect would be changing the prefactor of i log q p ϕ in the exponential in the final term.
Then we will compare our realizations of section 3 with the ones of [37]. Finally we briefly discuss some potential generalizations.

Towards a gravitational dual of DSSYK
For the bulk gravitational dual we propose the following first-order formulation, most commonly known as a Poisson-sigma (gauge theoretical) model consisting of one-forms A B and scalars χ B . The target space is 3-dimensional (the indices B, C can take three values), and we will label the components as A 0 , A 1 , A H and χ 0 , χ 1 , χ H . The Poisson matrix takes the form P H0 pχq " χ 1 , P H1 pχq " χ 0 , P 01 pχq "´s inp2 log qχ H q 2 log q .
We choose the boundary term as which forces upon us the boundary equation of motion Hpχq , (5.4) which should be considered the non-linear generalization of (2.3).
One can show that this model is topological with only six total degrees of freedom [49] in classical phase space, because one can integrate out the bulk values of A t and mod out by redundancies associated with this constraint. The Hamiltonian of the theory comes only from the boundary term in (5.1). One furthermore finds that half of the classical phase space variables are (suppose the boundary is at x " 0) χ B p0q, with the following Poisson brackets [49] tχ H p0q, χ 0 p0qu " χ 1 p0q , tχ H p0q, χ 1 p0qu " χ 0 p0q , tχ 0 p0q, χ 1 p0qu "´s inp2 log qχ H p0qq 2 log q .
These are precisely the Poisson brackets of our SU q p1, 1q currents e, f and h (4.9) with the identifications and moreover the Hamiltonian due to (5.3) is precisely the classical SU q p1, 1q Hamiltonian (4.11) H " ef´c osp2 log qhq 2 log q .
This guarantees that the quantization of this system matches with our boundary path integral (4.12).
One can rewrite (5.1) as a dilaton gravity model by introducing A B " pe 0 , e 1 , ωq and χ B " pX 0 , X 1 , Φq expˆiˆMˆΦ dω`X a de a´1 2 V pΦq e 0^e1`X 0 ω^e 1`X 1 ω^e 0˙`i S bdy˙, (5.8) which after integrating out the torsion constraints attains the second-order form 31 Remarkably this very model was also suggested as a candidate bulk description in section 4.3 of [47] and in [48] based on completely different arguments using the pG, Σq formulation of the SYK model [16].
We want to make two more comments on this dilaton gravity model. Firstly, we note the structural similarity with our boundary actions (4.19), (4.21). In particular, note the periodicity in Φ of the dilaton potential. This stems from the periodicity in our boundary Hamiltonian, which ended up discretizing spacetime into chords on the quantum mechanical level. See the discussion around (3.43).
Secondly, the classical solution of the sine dilaton gravity model has on-shell Ricci scalar 33 Choosing a radial coordinate in the bulk as r " Φ [86], we see that the radial region r " 0 is essentially AdS 2 with a curvature R "´2. Perhaps more tantalizing is the fact that for r "´π 2 log q one finds a radial region of approximately constant positive curvature R "`2, dS 2 . 34 So this model provides us with an opportunity to prove in more detail the dictionary between DSSYK and dS 2 physics proposed in [27,34,35,52,53]. This is exciting as no (top-down) models of dS quantum gravity are well understood 31 One then also deduces from (5.4) the boundary conditions generalizing the JT boundary conditions ωt`2Φ " 0 relating the length (or curvature) and boundary value of the dilaton. If the boundary value of the dilaton were fixed for instance to Φ " π{2 log q, this conditions says we are at the location of some static observer (pode/antipode) in the R " 2 region where ωt " 0. 32 We are being slightly schematic here, since we did not take into account the boundary term (5.3), nor did we investigate the translation of the constraints (3.34) into gravitational boundary conditions. We leave a more detailed analysis to the future [54]. 33 In DSSYK, one typically considers the coupling constant´1{ log q " N {p 2 of order 1 [16]. Upon rescaling the dilaton by log q one sees that this is indeed also the coupling constant in our gravitational theory (5.10). This suggests the theory should be under semiclassical control when considering´log q ! 1. Indeed [38], the DSSYK amplitudes have non-trivial and interesting semiclassical approximations when taking´log q ! 1 but keeping the energy finite (if the energy is taken to zero simultaneously, one recovers JT gravity). One would hope to reproduce these results from dilaton gravity semiclassics using (5.10). For q or order 1, quantum fluctuations will be large.
to date, therefore this deserves significantly more study.
The corresponding classical solution for the metric is of the form ds 2 "´1 f prq dt 2`f prq dr 2 , f prq " 2plog qq 2 cosp2 log q r h q´cosp2 log q rq , (5.12) which has a black hole horizon at r h and a cosmological horizon at π{ log q´r h .The curvature at both horizons is R "´2 cosp2 log q r h q. Depending on r h (set by the total energy in the spacetime), we have two cases: 1. For r h ą π 4 log q both horizons and the intermediate regions have positive cosmological constant R ą 0. In the regime r h "´π{2 log q the entire space between the horizons has constant curvature R "`2, and one would recover dS 2 JT gravity. In light of [27,34,35,52,53] it would make sense if this would correspond with boundary conditions where β " 0. We leave a further investigation to future work [54].
2. For r h ă π 4 log q the horizons are in a region with R ă 0, however there is still a positively curved region R ą 0 in between the horizons. The AdS 2 JT black hole is recovered by r ! 1{ log q whilst zooming in on the near horizon region r h ! 1{ log q. This corresponds with zooming in on low energies (deep IR in the bulk). Here we indicated the black hole horizon Φ " r h (full lines) and the cosmological horizon (dotted lines).
It is worthwhile to contrast this discussion with the semi-classical geometry distilled in [38] from the single Liouville field g describing the double-scaled regime of the pG, Σq formulation of SYK [16]. That Liouville geometry leads to AdS 2 which gets corrected by loop effects to even more negative values of the Ricci scalar (using the Ricci scalar computed with the quantum expectation value of the metric).
This seems at odds with our current geometrical interpretation. However, this Liouville geometry is not the same as the metric in the dilaton-gravity description. Indeed, this puzzle has been encountered before in the context of Liouville gravity and the minimal string [90,91], where the actual metric g µν and dilaton field Φ are orthogonal linear combinations of the Liouville field g and the matter sector field.

Comparison with recent work on non-commutative AdS 2
Let us compare our right regular representation (3.26) with related recent literature on the quantum group symmetry of DSSYK. In particular, in [37] the authors study non-commutative versions of AdS 2 and how the symmetry quantum group acts on it. They use two languages: that of a discretized lattice, and of non-commutative variables. In terms of the latter, our right regular realization is similar to their eq. (7.2) by identifying their generators A 1 , B 1 , C 1 , D 1 as 35 This structural observation is important to facilitate generalization of their goals to related models, and to provide additional mathematical and physical motivation for their construction.

Factorization across bulk entangling surfaces
A further question one can ask is whether we can factorize the bulk Hilbert space using this (quantum) group theoretic approach [60]. In [45], such an argument was presented for JT gravity, and in [64,85] this was extended to pure AdS 3 gravity. As alluded to at the end of section 4, technically, one needs to understand more general representation matrix elements that represent one-sided wavefunctions which connect the holographic boundary to the black hole horizon. In this work, we have computed the twosided wavefunction R θ ii pgq, whereas the one-sided wavefunction R θ is pgq and the interior wavefunction R θ ss pgq were not studied. Finding expressions for these seems to be within reach by solving the Casimir eigenvalue problem. In particular, understanding the physical meaning of such a factorization map in this non-local context of DSSYK would be interesting.

Comparison with Liouville gravity and beyond
It was argued in [55,90,91] that Liouville gravity (and the minimal string) can be formulated in terms of a dilaton gravity model like (5.10), but with a sinh dilaton potential, whose amplitudes are governed by the quantum group SLq p2, Rq. There is hence a clear parallel to this DSSYK story. However, there are also noteworthy technical differences.
For the quantum group relevant to Liouville gravity the Haar measure used when defining the inner product is continuous (see section 2.3 of [55]), unlike in our case where is is discrete (3.54).
Nevertheless the Casimir equation is still a difference equation, which leads to the question how the "infinite degeneracy problem" discussed around (3.42) is resolved in that case. It turns out that the two-boundary wavefunctions of Liouville gravity are the simultaneous solution of two incommensurate 35 These values of a and r a are also the ones that they argue are relevant to match directly with the DSSYK problem. We also need the coordinates z there " β and H there " e ϕ , and a slight change of convention for the non-commutative coordinate algebra pβ, ϕ, γq, which in their case satisfy the modified relation e ϕ β " q 2 βe ϕ .
Casimir difference equations, (3.22) and its dual related by b Ñ 1{b [64] with q " e πib 2 . 36 These two equations lead to a unique solution and hence no infinite degeneracy (but for a different reason than in the DSSYK case). Also the Casimir operator is Hermitian for Liouville gravity, so left-and right-eigenfunctions are related simply by complex conjugation.
It is relatively straightforward to modify the boundary action analysis of section 4 to the situation of Liouville gravity. Given these three datapoints (JT, DSSYK, Liouville gravity), it is tempting to wonder whether one can generalize our boundary action description further to arbitrary dilaton potentials V pΦq.
Notice that the exponentials in (5.15) for the fermionic generators did not get q-deformed because their series expansion truncates after the linear term in any case. It would be interesting to prove the expression (5.15) directly, and then follow the main line of this work to find the analogous decomposition 36 Note that the analysis of section 4 does not include this (quantum) modular duality at the level of the classical action, which is as expected. This mirrors an old subtlety in 2d Liouville CFT, where the classical action has no obvious b Ñ 1{b symmetry, but it is still present in quantum amplitudes (and exploited in the conformal bootstrap when using Teschner's trick [92]). See for instance [93][94][95]. 37 There does not seem to be notation to distinguish this from the other real forms of the complex group OSpqp1|2q. In terms of the quantum algebra, the real forms were classified [97] and parallel the classification of the real forms of SLqp2q. This real form was denoted there as Uqpospp1|2, Rqq (option (10a) in their Table 1).
of the N " 1 DSSYK amplitudes.
Generalizing to N " 2 requires OSp q p2|2, Rq. It would likewise be interesting to match the qrepresentation theory of this quantum group with N " 2 DSSYK amplitudes.

Acknowledgments
We A Quantum group SU q p1, 1q and its Gauss decomposition In the main text, we studied the representation theory of quantum group SU q p1, 1q. Here we give a bit more background, and comments on its relation with the quantum algebra U q psup1, 1qq.

A.1 Hopf duality and the Gauss decomposition
In this subsection, we provide some background and details on (3.2) which we hope is sufficient to guide the reader through the actual proof [62]. To do this, it is important to clarify the interrelation between the two different Hopf algebras SU q p1, 1q and U q psup1, 1qq.
The quantum group SU q p1, 1q (which in some literature is also called the coordinate Hopf algebra O q pSU p1, 1qq) is a Hopf algebra generated by four generators a, b, c, d, modulo the following relations [5] ab " qba , cd " qdc , ac " qca , bd " qdb , bc " cb , ad´da " pq´q´1qbc , The second line is called the quantum determinant, and one can check that it commutes with all the generators of the algebra. The co-product on this Hopf algebra is defined as and by construction looks like how one would perform the ordinary matrix product of p2ˆ2q-matrices.
For our purpose, it is convenient to use the related coordinates for the same algebra: a " e ϕ , b " e ϕ β , c " γe ϕ , d " e´ϕ`γe ϕ β .
An a priori unrelated Hopf algebra is U q psup1, 1qq [5,17,61], which is the algebra generated by three generators E, F, H, modulo the following relations It can be thought of as a deformation of the universal enveloping algebra of sup1, 1q. The co-product on this Hopf algebra is defined as: with K " q 2H . Now, these two Hopf algebras are actually closely related in a way that mimics the relation between the Lie algebra and Lie group. The relation is through the mathematical construction of Hopf duality.
We first discuss this in a general setting and come back to the concrete case of SU q p1, 1q and U q psup1, 1qq below.
Let us denote a basis for the two Hopf algebras by P α and X α respectively, where α runs through all basis elements. 38 The dual pairing of Hopf algebras is defined [5] by a bilinear mapping x . , . y for which the dual basis elements of both algebras P α and X α are related as xP α , X β y " δ β α . Moreover, it has to satisfy the additional duality properties x∆pP α q, X a b X b y " xP α , X a X b y, xP α P β , X a y " xP a b P b , ∆pX a qy, where one defines the operation x . , . y on the tensor product as: x . b . , . b . y " x . , . y x . , . y. The duality properties (A.8) encode that multiplication and co-multiplication of both Hopf algebras get swapped under duality. Indeed, writing generically for structure coefficients E αβ γ , F γ αβ , H αβ γ and G α βγ , one finds that (A.8) leads to the equalities: matching the co-product of one algebra to the product of the other and vice versa. Now, for matrix quantum groups such as SU q p1, 1q one defines the co-product (A.3) to match with how ordinary matrix multiplication would occur between 2 matrices with entries a, b, c, d and a 1 , b 1 , c 1 , d 1 , which mutually commute. Denoting the basis elements of this coordinate Hopf algebra as P α , the corresponding elements of the product matrix have the form where the structure coefficients E αβ γ (A.9) of the co-product were used. The prime denotes an entry taken from the second matrix in the matrix multiplication. For example for SU q p1, 1q with co-product (A.3), e.g. the basis element a 2 b of the product matrix becomes paa 1`b c 1 q 2 pab 1`b d 1 q.
We now define two objects g 1 , g 2 as: and read this as an expansion in "coordinates" P α , P 1 α and "generators" X α . Their product is of the form: where we used the product of the X α "generator" algebra (A.10). The coordinates of the product matrix g 1 g 2 , ř α,β E αβ γ P α P 1 β , have precisely the form of (A.12) and hence the g i 's form a (co-)representation of the matrix quantum group.
We now go through the above construction with the following logic [62]. Given a representation and a Hopf algebra generated by the X α , we can directly bootstrap the dual Hopf algebra and basis P α (using the duality properties (A.8)). From this we can write down explicitly the g i which hence form a (co-)representation of the dual matrix quantum group.
Let us explicitly apply this strategy to the case at hand of SU q p1, 1q and U q psup1, 1qq. The Hopf algebra U q psup1, 1qq is spanned by the basis elements X α " F n H k E m where α labels the multi-index pn, k, mq. The dual Hopf algebra SU q p1, 1q is spanned by γ a p2ϕq b β c . We need to change basis and find the (a priori unknown) dual basis P α , satisfying xP α , X β y " δ β α . Their precise form can be found by using the precise quantum algebra relations (A.6) and the duality conditions (A.8) leading to [62]: This results in the decomposition (3.2): g " ÿ α P α X α " e γF q´2 e 2ϕH e βE q 2 , 0 ă q ă 1 , (A. 16) where the (q-)exponentials are automatically obtained upon summing over all indices pn, k, mq. The first equality (definition) shows that g is a specific element of both dual Hopf algebras SU q p1, 1q and U q psup1, 1qq (depending on one's perspective on what to call "coordinates" and "generators"), and not directly a generalization of a Lie group.
However, because of the above general property (A.14), g can also be thought of as a (co-)representation of the matrix quantum group SU q p1, 1q, which is made more suggestive by the Gauss-like decomposition in the second equality. The object g (A.16) has many good properties, for example when we take q Ñ 1, it goes back to the undeformed Gauss decomposition. Most importantly for our application, this expression for g allows us to construct representation theory of U q psup1, 1qq on SU q p1, 1q, because as shown in the main text, the generators of U q psup1, 1qq acting on g can be represented by q-calculus [5] on the non-commutative coordinates pγ, ϕ, βq.

A.2 Gauss decomposition for pragmatists
To give people a practical feeling how this Gauss decomposition and Hopf algebra can be viewed as a generalization of Lie groups into "noncommutative" geometries, we talk about the two-dimensional fundamental (co-)representation, where the quantum group SU q p1, 1q is written as two-by-two matrices [5,63] g The non-trivial statement in this definition is that this composes well under standard matrix multiplication: if g 1 and g 2 are defined with non-commutative entries a 1 , a 2 etc. as above and g 1¨g2 "˜a 12 b 12 c 12 d 12¸, (A.18) then a 12 , b 12 , c 12 , d 12 satisfy the same algebra (A.1) as a, b, c and d.
As in the classical case, it is convenient to parameterize the quantum group manifold in different ways. In particular, we choose g "˜e ϕ e ϕ β γe ϕ e´ϕ`γe ϕ β¸( A. 19) where the non-commutative coordinates pγ, ϕ, βq satisfy (3.4), which one can check is equivalent to (A.1) and (A.2). And this is actually the basis transformation we take in (A.4). Now we connect SU q p1, 1q to the q-deformed algebra U q psup1, 1qq. In the defining two-dimensional representation we have In the defining representation, this identity is somewhat trivial since both the quantum algebra and the q-exponentials are the same as in the undeformed q Ñ 1 case.
Checking that the middle entry of (A.21) (the Gauss decomposition) is the correct generalization to other representations takes more work [62,[98][99][100][101], but the idea is simple. For any finite dimension n, one can construct pnˆnq-dimensional matrices E, F and H which form a representation of the quantum algebra (3.1) (see e.g. [5]). Then one checks that the resulting pnˆnq-dimensional matrix g with entries that are polynomials in γ, e ϕ and β still composes properly under matrix multiplication, using the algebra of coordinates (3.4). 39 This fixes the Taylor series in F , H and E of the middle entry in (A.21) and eventually leads to the q-exponentials. 40 The case n " 3 is rather pedagogically detailed in [63].

A.3 Real forms and complex conjugation
Finally a comment about SU q p1, 1q versus SL q p2, Rq. By definition, SU q p1, 1q corresponds to the real form of the aforementioned quantum group with 0 ă q ă 1 real and SL q p2, Rq is the real form of the quantum group with |q| " 1. Classically, there is a distinction between SUp1, 1q and SLp2, Rq by the way their elements behave under complex conjugation. This carries over to the q-deformed setup, and in fact determines the range of q in these definitions.
In particular for SU q p1, 1q one defines "complex conjugation", by introducing a˚operation as: a˚" d , b˚" qc , c˚" q´1b , d˚" a . The non-trivial point is that this definition is consistent with the algebra (A.1) only when q˚" q. The 39 Which means checking it implies the co-product (A.3) with (A.4). 40 In fact, since F and E are lower-resp. upper triangular, the Taylor series of the q-exponentials in the dimension n irrep truncates at the n th term, allowing a clean ab initio determination of the series of the q-exponential order by order as one increases n.
defining 2d representation has the following property in common with the classical SUp1, 1q: which means one should not think of β, e ϕ and γ as simply "real coordinates", unlike for the quantum group SL q p2, Rq. This discussion about "complex conjugation" plays no role in the main text, we present it just to resolve potential confusions the reader might have about the distinction between SU q p1, 1q versus SL q p2, Rq. 41 B Details on N " 1 quantum supergroup OSp q p1|2, Rq The U q pospp1|2, Rqq quantum superalgebra is generated by three elements H|F`, F´with q H F˘" q˘1 2 F˘q H , tF`, F´u " q 2H´q´2H q´q´1 . (B.1) The bosonic generators E˘are defined through tF˘, F˘u "˘1 2 E˘, and are obsolete in the universal enveloping algebra, which is why one usually does not write them explicitly. They are important when parametrizing the (Hopf) dual quantum supergroup, as we show below.
One readily checks that the matrix generators in the fundamental representation satisfy (B.1) H " 41 Representation theory of Uqpslp2, Rqq is actually far more common in gravitational contexts, see for instance [55,64] for relations with the minimal string and the ubiquitous appearance of Uqpslp2, Rqq in the context of 3d gravity [19,21,85,[102][103][104].