Holography on the Quantum Disk

Motivated by recent study of DSSYK and the non-commutative nature of its bulk dual, we review and analyze an example of a non-commutative spacetime known as the quantum disk proposed by L. Vaksman. The quantum disk is defined as the space whose isometries are generated by the quantum algebra $U_q(\mathfrak{su}_{1,1})$. We review how this algebra is defined and its associated group $SU_q(1,1)$ that it generates, highlighting its non-trivial coproduct that sources bulk non-commutativity. We analyze the structure of holography on the quantum disk and study the imprint of non-commutativity on the putative boundary dual.


Introduction
Everything in quantum gravity should be quantum, including spacetime.A notion of a quantum spacetime is described by a non-commutative geometry i.e. one whose coordinates do not commute with one another.Examples of such spacetimes have appeared in several areas in theoretical physics including string theory [1], matrix models [2] and quantum field theories on non-commutative spaces [3][4][5][6].
Recently, non-commutative geometry has showed up in the bulk picture of double scaled SYK model [7][8][9][10].The double scaling limit is the limit where the number of fermions N and the order of the fermion interaction in the SYK Hamiltonian p are taken to infinity while keeping λ ≡ 2p 2 /N fixed [11].The bulk picture that emerges is described in terms of chords anchored to the boundary [7,8,12].These chords provide an unusual, primitive version of spacetime where distances in the bulk are measured in units of λ, the effective Planck length in the bulk (or ratio between the Planck and AdS scales).Smooth spacetime emerges in the small λ limit in which the number of chords proliferate and smooth spacetime emerges.
At finite temperature of DSSYK, symmetry algebra of the spacetime is a subalgebra of U q (su 1,1 ) [10].The "quantum" aspect of the algebra is the presence of a non-trivial coproduct.
The action of an infinitesimal generator on elements of tensor product of U q (su 1,1 ) modules does not satisfy the Leibniz rule, but instead has a "tailed" action given by where E, K ∈ U q (su 1,1 ).The discreteness of the chord spacetime is intimately related to this coproduct structure.
Surprisingly, this non-trivial coproduct has a remnant in the λ → 0 limit at finite temperature in the regime of large p limit of SYK.As shown in [10], the two point function in this regime is not conformal, yet it is related to a conformal correlator through a non-symmetric coordinate transformation on the coordinates of the two points: where φ ± = ± π 2 + v θ ± ∓ π 2 .The coordinates φ ± are the coordinates on the "fake disk" whose boundary length is β/v.The fake disk coordinates makes the SU (1, 1) symmetry man-ifest.For instance, the correlators invariant under the action of E[O] ≡ cos φ∂ φ O−∆ sin φO.
If we express this symmetry in terms of physical disk coordinates θ ± , the generator E will have to act differently on the O(θ + ) and O(θ − ) due of the asymmetry in the transformation between φ ± and θ ± .However, this asymmetry can be compensated by assigning a non-trivial coproduct to the action of E by introducing another operator P whose actions in terms of physical coordinates is where ) These observations motivate the study of standard examples involving a non-trivial coproduct, hence this paper on the quantum disk and groups.This paper will analyze the structure of holography on a non-commutative version of the hyperbolic disk, also known as the quantum disk that was studied extensively by L. Vaksman [13][14][15][16][17][18].This is a noncommutative spacetime whose symmetry group is SU q (1, 1).
This topic is sufficiently foreign to our community that we felt that we should dedicate a large fraction of this paper to review L. Vaksman's seminal works on the quantum disk.
Hence, sections 2 is a pedagogical review of [13][14][15][16][17][18] describing how to define the quantum disk and its symmetry quantum group.In section 3, we describe the mechanics of holography, extracting boundary correlation functions and induced quantum symmetry transformations of boundary primary operators.In section 4, we study symmetry aspects of the putative boundary dual of the quantum disk.We find the three point function using symmetry and match it to a bulk Witten diagram.

A review of the quantum disk
In this section we describe how to define the quantum disk [14][15][16][17][18]25], a q-deformed version of the hyperbolic disk, as the space whose symmetry group is the quantum group SU q (1, 1) [?, [19][20][21].The coordinates on the quantum disk will be noncommutative, a property that descends from the noncommutativity of the matrix elements of group elements of SU q (1, 1).We will not describe the intrinsic geometric structure of the quantum disk, which is obscured by its noncommutative nature, but rather focus on the properties of functions of its coordinates.
2.1 SU q (1, 1): the symmetry group of the quantum disk To understand the symmetry group of the quantum disk SU q (1, 1), we start with the quantum deformation of the SL(2, C), called SL q (2).Elements of SL q (2) are all 2 × 2 matrices, whose entries satisfy the following relations The combination det q ≡ ad − qbc is called the quantum determinant and belongs to the center of the a, b, c, d algebra.Without the condition det q = 1, (2.1) defines GL q (2).The above relations generate an ideal and are determined by a solution to an R-matrix equation (B.2).We describe this in details in appendix B.
The equation (2.1) is the defining representaion of SL q (2) [23].It is sometimes called its "coordinate representation" since the matrix elements define coordinates parameterizing group elements on the group manifold.Just like normal groups, this group includes the identity element e where a = d = 1, b = c = 0, which satisfies the relations (2.1).Thus, one can define a map ϵ that sets any group element g ∈ SL q (2) to the identity.It is known as the co-unit ϵ(g) = e.
We can define the inverse of the above elements as Formally, we have g The map from a group element to its inverse is known as the antipode S : g → g −1 .Formally, on the entries of matrix defining the group element we have Group multiplication can also be defined for SL q (2).It is a map from the matrix elements of a pair of group elements g 1 and g 2 to their product g 1 g 2 in a way that preserves the ideal (2.1).To define this, first we should think of the matrix elements a, b, c, d as functions on the group manifold that assign a value (abstractly) to each group element, i.e. the matrix elements of a group element g are a(g), b(g), c(g), d(g).Then, the multiplication can be defined as a map × that takes matrix elements of two group elements to matrix elements of a product of these elements Thus we have mapped the matrix element functions a, b, c, d to the functions acting on two copies of the group manifold.This operation of mapping functions on one copy of a space to the functions on two copies is called the coproduct, and is denoted by ∆(•).For SL q (2) we have The coproduct is an algebra homomorphism.
One is often interested in defining functions on the group manifold.This space of functions is a vector space spanned by monomials a i b j c k d l with integer powers and is denoted as O(SL q (2)).The notions of co-unit, antipode and coproduct extend to this algebra in the obvious way.The co-unit implements ϵ : O(SL q (2)) → C, while the coproduct implements ∆ : O(SL q (2)) → O(SL q (2)) ⊗ O(SL q (2)).We can introduce "reversed" maps that are compatible with these maps.Thus, we define as the unit that maps a constant to the identity element of the algebra η : C → O(SL q (2)) (it also maps a constant to the identity of the group SL q (2)) and a multiplication called the product that maps a tensor product to a single copy m : O(SL q (2)) ⊗ O(SL q (2)) → O(SL q (2)) acting as as m(a ⊗ b) = ab.These structures together make O(SL q (2)) a Hopf algebra.
To get SU q (1, 1) we need to impose reality conditions on SL q (2).For q = 1, both are Mobius transformations on the complex plane, but SU (1, 1) preserves the unit disk.
2.2 U q (su 1,1 ): generators of SU q (1, 1) One can define a set of generators on the group manifold of SU q (1, 1) that formally implements an infinitesimal translation.These generators are usually labelled K, E, F and they satisfy an algebra known as U q (su 1,1 ) (more precisely, this is the vector space spanned by polynomials of K, E, F ). Their action on the elements of SU q (1, 1) is defined through the pairing Hence, on a group element we have ⟨K, g⟩ = q 0 0 q −1 , ⟨E, g⟩ = 0 0 The pairing extends to relate all elements of U q (su 1,1 ) to O(SU q (1, 1)), and establishes a duality between the two algebras.To do so, it must satisfy a set of constraints (see appendix (A)).
One constraint is it must respect the algebra relations (or ideal) in (2.1) once the pairing is extended to all elements of O(SU q (1, 1)).For instance, we must have ⟨E, ac⟩ = q⟨E, ca⟩.
There are multiple ways of extending the pairing, one of which is to impose that ⟨E, AB⟩ = ⟨E, A⟩⟨1, B⟩ + ⟨K, A⟩⟨E, B⟩, (2.13) for any A, B ∈ O(SL q (2)).The pairing with the identity implements the co-unit, ⟨1, B⟩ = ϵ(B).These relations define a coproduct for U q (su 1,1 ) given by ∆ provided that the pairing satisfies where X, Y ∈ U q (su 1,1 ).As a result, the pairing can be shown to satisfy The generators K, E, F must satisfy a set of reality conditions compatible with the reality condition of O(SU q (1, 1)).This can be defined using that the pairing satisfies Using the coproduct from K, E, F , the antipode (2.3) and the relations (2.9), implies the following reality conditions Further conditions on the pairing can be used to define a unit, co-unit, antipode, a product for the algebra of K, E, F that makes it a Hopf * -algebra.We review these additional structures in appendix A.2.So far, we have established a duality between two Hopf * -algebras O(SU q (1, 1)) and freely generated algebra of K, E, F .
Finally, the pairing (2.10) is not faithful and has a kernel generated by the relations, or ideal, between K, E, F ,

22)
See appendix A.2.2 for a derivation of this kernel.Imposing these on the algebra of K, E, F defines the algebra U q (su 1,1 ).From this one can deduce a Casimir that commutes with the generators (2.23) The action of the Casimir on tensor product representations descends from the coproduct of the generators, and is given by (2.24)

Coordinates on the quantum disk and their algebra
Having defined the symmetry group of the quantum disk and its associated generators, we turn to defining the quantum disk itself.Recall that the standard hyperbolic disk can be obtained from a quotient of the SL 2 (R) group manifold, see [24].Similarly, as discussed in [16], the quantum disk can be defined as the quotient of SU q (1, 1) by the right action of The action (or coaction) of SU q (1, 1) on the disk correspond to left matrix multiplication t ′ = g × t that maps to the tensor product as (2.29) The elements t ′ ij have the same algebra as t ij .It will be useful later to note that det q (t ′ ) = det q (g ⊗ t) = det q g ⊗ det q t where det q t = t 11 t 22 − qt 12 t 21 .
It is useful to analyze the action of generators on t ij .The action of the generators on t ij is obtained using (2.11) implying the following transformation rules (2.30) To obtain the action of U q (sl 2 ) on arbitrary products and powers of t ij , we follow the same rules as those defined in the pairing in the previous section.The coproduct then implements the action on products of t ij .
FKP: We can also introduce the right action of SU q (1, 1) on the coordinates t ij .Thus, K acts in the following manner (2.33)

Disk coordinates
Now we can implement the quotient to define the disk coordinates z = q −1 t −1 21 t 11 , z * = t 22 t −1 12 , which are indeed invariant under the rescaling of t ij by the action of K FKP:AboveAhmed: We need to define right action of K to this!!The z's should not be invariant under the left action of K. Using the SU q (1, 1) reality conditions t * 11 = t 22 , t * 12 = qt 21 we see that z, z * are conjugate to each other.This also follows from the conjugation relations (2.9) and by conjugating either side of the transformations t ij → t ′ ij .Under the above transformation rules, we have which is a usual Mobius transformation.The tensor product symbol is dropped from the last equation for brevity.As a check, one can show that this mapping preserves the boundary; suppose that w, w * are constructed from the matrix t ′ just as z, z * are constructed from t.
Boundary points in z correspond to zz * = 1 which satisfy det q t = 0 thus implying that det q t ′ = 0 and ww * = 12 .
The action of the generators on z, z * can be deduced from their action on t ij and using the coproduct, which gives (2.36) Again, using the coproduct we can extend this action to powers of z, z * .For "(anti)holomorphic" functions we have (2.37) ) respectively, and then use the remaining relations in (2.25) to commute things around until we have an expression involving only z, z * .The resulting relation is (2.39) This relation can also be obtained using R-matrix methods.As discussed in appendix B, the universal R-matrix for U q (su 1,1 ) is given by which in particular gives where the m : multiplies the elements of the tensor product together 3 .
The algebra of z, z * is denoted as C q [z, z * ].An important element of this algebra is an element y = 1 − zz * that satisfy the following commutation relations zy = q −2 yz, z * y = q 2 yz * . (2.42) It is useful to show connection of C q [z, z * ] with the q-deformed harmonic oscillator.Thus, we notice that â = z thus any element of f ∈ C q [z, z * ] could be thought as an operator acting in the Hilbert space of q-deformed harmonic oscillator.That Hilbert space is spanned by vectors |n⟩ , n ∈ N ≥0 and the action is given by one can show that this mapping is actually an isomorphism and any function ψ(y) of y could be given just by assigning the values of ψ(q 2n ).In some loose sense, the quantum disk is a set of concentric circles at discrete radii of constant y that accumulate up to the unit circle.

Differential q-calculus
Various notions of differential calculus can be extended to the non-commutative setting.
One can define a notion of a differential operator d that satisfies the usual Leibnez rule dz 2 = zdz + dz z.The differential form dz satisfies d(dz) = 0, which follows from d 2 = 0.
The action on z * follows similarly.
The action of the generators of U q (su 1,1 ) on the differentials follows from the Leibnez rule along the commutations above to give The algebra between dz, dz * , z, z * follows from the R-matrix in (2.40).We have The differentials allow us to define differential operators acting on any function h(z, z * ) on the quantum disk through This defines the derivative operators ∂ ∂z , ∂ ∂z * , and holomorphic and anti-holomorphic differentials ∂h, ∂h.This means we can write d = ∂ + ∂.It's not hard to check that these derivatives satisfy the following algebra The Casimir (2.23) can be expressed in terms of these derivatives We check in the appendix (C) that the two sides of this equation have the same action on any monomial z n z * m .
Next we define how to perform integrals over the entire quantum disk.Such an integral can be thought of as a map of functions on the quantum disk to the complex numbers.
Assuming we have a measure dν that invariant under the action of the generators E, F, K, then for any function we must impose for any m, n, l > 0. Now suppose we want to compute the integral of an arbitrary function h(z, z * ).Any such function can be expanded as Note that the terms with extra factors of z, z * , i.e. the first and third terms in the expansion are not invariant under the action of the K and can be expressed as E a f (y) and F b f (y) for some functions f (y), f (y), and hence their integral must vanish. 4What remains is i.e. the integral of a purely radial function.There are two ways of making this formal expression more explicit.The first is to note that the spectrum of y is given by the discrete values q 2n for all n ≥ 0, and hence this integral can be expressed as a sum over h 0 evaluated on these points, namely for some coefficients a n which must be determined.This can be done by considering the (vanishing) integral of F acting on zh 0 (y).Since (2.57) The integral of the first and second terms on the right hand side must vanish, and so we end up with the recursion relation, The value of a 0 is undetermined.However, if we set it to a 0 = π(1 − q 2 ) then we reproduce the standard integral on the classical hyperbolic disk in the limit q → 1, which we will show below.
There's a more intuitive but less precise way of arriving at this result.First we need find a measure of this integral that's invariant under the action of K, E, F .One such measure is the following two-form This is a two form on the disk expressed in terms of the cotangent space element dz∧dz * .Note that (2.48) imply we have dz ∧ dz * = −q −2 dz * ∧ dz. 5 This form motivates the replacement where α is some constant and the measure d q 2 y stands for Jackson integral, namely that we are integrating along y but in discrete steps given by the difference of y at two consecutive points, i.e. d q −2 y = (q 2n − q 2(n+1) ).Then we have This agrees with the (2.56) up to an overall constant.
An additional interesting representation of this integral uses the representation (2.44) is

.62)
This will give an efficient way of computing the integrals on quantum disk.
As a check, we can compare this to the expression the classical hyperbolic disk by taking q → 1.Then we have r 2 = |z| 2 = 1 − y = 1 − q 2n , and hence q 2n − q 2(n+1) ≈ 2rdr and we get which is indeed the radial integral on the hyperbolic disk.
Finally, we discuss two useful relations that could be used in the computation of the integrals.The first is integration by parts that could be expressed as This uses the action of E on the product of two functions via coproduct and the invariance of the integral with respect to the action E. The second is the notion of a delta-function that can be defined as the element (2.65)

Holography
This section will focus on holographic aspects of the quantum disk, focusing on deriving boundary anchored propagators.We will restrict our analysis to scalar fields.These fields will be arbitrary functions of z, z * , and hence are elements of C q [z, z * ].The formulas (2.35, 2.36, 2.37, 2.38) will give us the action of the generators E, F, K on the fields, and the Casimir (2.23, 2.52) will provide the wave equation.

Asymptotic analysis
Suppose we have a field ϕ(z, z * ) on the quantum disk.This field is an element Without loss of generality, we can use the commutation relations of z, z * to express it as where y = 1 − zz * .We demand that the dynamics respects the SU q (1, 1) symmetry of the quantum disk, which implies that its action and equations of motion must be invariant under infinitesimal transformations generated by K, E, F .The simplest U q (sl 2 ) invariant wave equation linear in the field is the sum of the Casimir and mass term where we used the expression of the Casimir in (2.52) in the z, z * coordinates system.This wave equation follows from the action It is instructive to see how to integrate this by parts to express the kinetic term as a square.
With the appropriate measure and definition of □, we write this integral as where in the first step we commute the derivatives and pick up a factor of q −2 .In the second step, we move the measure through to the middle at no expense.In the third, we commute the two factors of the measure and integrate by part using ∂ (f Studying the asymptotic solutions reveals some interesting aspects of fields on the quantum disk.Without any sources on the boundary, we should consider solutions that fall off near the boundary.Without loss of generality, we can consider where we have used that ϕ * = ϕ.Note that on the boundary we can make the coefficient function to be only a function of z since the boundary satisfies zz * = 1.Since y = 0 at the boundary, we can use equation (2.24) to find the value of ∆ as a function of the mass m of the field There are several interesting aspects of this expression.First, the left term is the eigenvalue of the Laplacian (since the solution is an eigenvector asymptotically) and was also obtained in [16].As discussed in [25], to obtain a solution with the boundary condition (3.5) that is real and non-singular everywhere in the bulk, we must pick the principal series representation which sets ∆ = 1 2 + iρ.The eigenvalue becomes and hence the Laplacian is bounded [16, 25] (3.8) The boundedness of the Laplacian implies the theory is finite, at least classically, and in particular has no UV divergences at coincident points.In the q → 1 limit we restore the Breitenlohner-Freedman bound.
The second interesting fact about (3.2) is that the bound on eigenvalues places a bound on masses of (classical) fields on this space!For q ≈ 1, we see that the mass is upper bounded by ln q, the unit of discreteness of the radial direction, which happens to be the Planck scale from the perspective of DSSYK.

Propagators
We will think of the propagator as a function on a tensor product of two copies of the quantum disk with coordinates z, z * and w, w * .Hence, the propagator will be an element of the form The task is to find solutions that satisfy Similar to the classical AdS case, the bulk-to-bulk propagator on the quantum disk has to be a function of a U q (su 1,1 ) invariant distance.This is a function defined on the tensor product of two quantum disks that is annihilated by the actions of E n , F m , (K k − 1) for any n, m, k acting on both coordinates.The distance is given by (the derivation could be found in (A.4)) where the × op is the 'opposite multiplication' symbol indicating that multiplication on the first factor (the quantum disk parameterized by z, z * ) includes a swap: where the expression on the right side of the equal sign involves standard multiplication.
This means the expanded form of the distance is This opposite multiplication is important because it takes U q (su 1,1 ) invariant functions and returning another invariant function, which follows from the following theorem [25].
This theorem is proven in the appendix (A.3).This theorem is important for us since we want to take powers of the invariant distance.The invariant distance raised to an arbitrary positive power (in the opposite multiplication sense) is given by where we used the q-Pochhammer symbol defined as

.15)
See appendix A.4 for a derivation.It will be convenient to write the invariant distance by implementing of the opposite multiplcations as follows where the opposite multiplication takes place only within the square brackets.Inverse powers of the distance will be defined as Note that this expression disagrees with one derived in [16,25]; we found their expression to not be invariant while ours is.Thus, the general propagator is of the form for some coefficients g k .Only negative powers are included since the fields are required to vanish near the boundary (in the absence of sources).
Hence, the boundary limit of the bulk correlation function is of the form Thus we can read off the boundary correlation function using (3.17).Note that the bulk coordinates are noncommutative while their boundary limits are standard commutative variables.Furthermore, the commutativity of the boundary coordinates makes the × op redundant.Hence, the boundary two point function is ⟨O(φ)O(θ)⟩ = (q −2∆ e i(φ−θ) ; q 2 ) −1 ∆ (q −2(∆−1) e −i(φ−θ) ; q 2 ) −1 ∆ , = (e i(φ−θ) ; q 2 ) −∆ (q 2 e −i(φ−θ) ; q 2 ) −∆ . (3.20) As a check, the q → 1 limit indeed limits to the 1-dimensional conformal two point function, As a final comment we can write down the bulk to boundary propagator.By invariance and the boundary condition ϕ ∼ O(z, z * )(1 − zz * ) ∆ , the bulk to boundary propagator must be where w = e iθ .Amusingly, both the bulk to boundary and boundary to boundary propagators were studied by Vaksman [16,25] as structures with interesting transformation properties and not motivated by the context at hand.

q-Conformal Quantum Mechanics
In this section we turn away from the bulk and analyze the properties of the putative boundary quantum mechanical system with SU q (1, 1) symmetry.Note that while the action of the SU q (1, 1) on e iθ at the boundary is the same as that of SU (1, 1) (up to constant factors) they acting differently on powers e iθ n where SU q (1, 1) acts with a nontrivial coproduct.
Boundary primary operators are defined through the extrapolate dictionary from the expansion of bulk operators near the boundary, ϕ ∼ O ∆ (e iθ )y ∆ .This expansion determines the transformation of boundary operator under SU q (1, 1) in the following way.The procedure is to act on the bulk field with a symmetry generator, e.g.E(ϕ) and then perform the same expansion and hence where z → e iθ .The transformations under U q (su 1,1 ) are These will provide Ward identities from which we can extract the correlation functions.Note that the action of the generators on the boundary satisfies the same coproduct as that in the bulk.
The analysis of the correlation functions simplifies considerably when working in the Fourier basis where we expand local operators as The transformation rules (4.3), (4.4),(4.5)imply the Fourier modes transform as

Correlation functions
Summetry under SU q (1, 1) implies that correlation functions are invariant under the action of any of its generators, namely that The strategy we follow in computing the position space correlation functions is by first expanding in the Fourier basis as and then determine the coefficients ⟨O n 1 ...O nm ⟩ by symmetry before performing the sum.
Such an approach was considered before by LeClair and Bernard [26] for SL q (2, R) case.
We'll consider a few cases below for SU q (1, 1) group, the group of isometries of quantum disk.

Two point function
Our task is to determine the Fourier mode overlap ⟨O m O n ⟩.The simplest constraint comes the action of , where we used the coproduct action of , we obtain the recursion relation where This can be solved using the boundary condition ⟨O 0 O 0 ⟩ = 1, and we find the Fourier coefficients We notice that The sum can be recasted in the following compact way And using Ramanujan 1 ψ 1 identity [27] we get Or more compactly Up to the overall constant factor, this is identical to the result obtained from the quantum disk in (3.20) after setting x 1 = e iφ , x 2 = e iθ .
Going back to the bulk picture, one could wonder if the three point function can be obtained from a bulk computation (it must by symmetry, but good to check explicitly).For that we just need to compute the following integral6 and the integral itself are U q (su 1,1 ) invariant.To see that more explicitly, we notice that the integrand is an element of C q [z, z * ] ⊗ C[x 1 , x 2 , x 3 ] and since it is U q (su 1,1 ) invariant we have Checking that the Witten diagram works for general dimension is complicated, but we managed to check it numerically for negative dimensions, ∆ 1,2,3 = −1, −2.

Discussion
We conclude with some outlook, remarks, and some open problems for future work.

q-Conformal quantum mechanics
It would be interesting to revisit the problem of conformal quantum mechanics but where the symmetry group is SL q (2) instead of SL(2, R).In fact, it is simple to generalize the conformal quantum mechanical system [31] to a q-conformal system by replacing the time derivatives with a finite difference derivative, where The operator Q has dimension −1/2, independent of q.It is unclear if this is a feature or a bug, but the theory is non-local in time and perhaps makes more sense in Euclidean time.
Another unclear aspect is whether the time integral should be a regular continuous one or a "Jackson integral" evaluated a discrete time steps of at powers of q 2 .Both choices are consistent with SL q (2) symmetry.

q-Schwarzian action
One could wonder if the quantum disk is a solution to a q-deformed analogue of JT gravity that localizes on q-AdS 2 but with a boundary reparameterization mode.The Lagrangian of this mode would be what one would define as a q-deformed Schwarzian, i.e. a differential (finite difference) operator that's invariant under infinitesimal SL 2 transformation but with a non-trivial coproduct.See [32,33] for proposals on the q-deformed Schwarzian.
The task is to find a (discrete) differential operators that's invariant under the action of U q (su 1,1 ), assuming we define their action on t(u) to satisfy A related problem is to understand how to perform an arbitrary reparameterization of noncommutative coordinates.These transformations would be generated by a q-deformed Virasoro algebra.Studying the central extension of this algebra would be one way of understanding the q-Schwarzian.We leave this for future work.

UV lattice and Renormalization, UV/IR mixing, q affects both
The interplay of between the discreteness of space and the renormalization group is an old and well studied problem, see for instance [34].The upshot is that renormalization group doesn't permit Lorentz invariance to emerge in the IR if it is broken in the UV due to the existence of relevant operators that can be generated under the RG flow.A natural question is whether Lorentz invariance is preserved in the IR if the UV is invariant under a q−deformed version.The analogue for the quantum disk is SU q (1, 1) in the UV and SU (1, 1) in the IR.
One potential hurdle is the problem of UV/IR mixing associated to non-commutative spaces, discussed for instance here [1].In fact, the quantum disk has a hint of this in the spectrum of the Laplacian, where both the IR and UV bounds are controlled by the same paramter q.One would need to compute loop diagrams to truly settle this.

q-Conformal Blocks
Using the ideas similar to the usual conformal field theory we can use the operator product expansion and assosiativity to constrain the possible SU q (1, 1) invariant field theories.For that, we must to study the four-point functions As in the case of the usual CFTs we can not constrain the possible form of this expression, but we can expand it in terms of conformal blocks that satisfy the following equation It would be interesting to study the possible solutions of this equation.

DSSYK at q < 1 and large β
One of the motivations of this work was to study a toy model that resembles the putative bulk dual of DSSYK since it too has a noncommutative nature.A natural question is whether the quantum disk emerges from some regime of DSSYK.
There's some evidence that this might be the case.It was noted in [10] that invariance under U q (su 1,1 ) emerges in the low temperature limit of DSSYK at q < 1.This symmetry fixes the form of the two point function.The exact expression of the two point function was found in [7], so it is simply a matter of taking its low temperature limit.We leave this for future work.

A Introduction to Quantum Groups A.1 General Philosophy
This appendix contains what hopefully will be an accessible and pedagogical review of quantum groups.For that, we will start with the general philosophy or idea that lead to the creation of quantum groups.Usually when we study groups or other classical manifolds we have an idea that it is some sort of geometric object that could be thought as some surface in higher dimensional flat space.Instead, it is more convenient here to use the language of algebraic geometry which analyzes functions on the manifold whose structures are reflected in the algebra of functions.Thus, if we have a topological space M we will denote O(M) as a set of all functions f ∈ O(M), f : M → C on the given topological manifold.This set automatically has the structure of commutative algebra.We have the following operations and special element 1 it is easy to check that 1 is indeed a unit in the algebra O(M).By studying the algebraic properties of this commutative algebra we can understand the geometric properties of the topological manifold M. For instance, the ideals of O(M) will correspond to submanifolds and so on.

A.2 Hopf algebra
Now assume that M is not only just a topological manifold but also a group.Then the then one can check that if we introduce new objects y k = g i k x i they also satisfy the same relation.
On general grounds, the solution of the YB equation is quite complicated, but Drinfeld managed to find a universal solution when R acts on a tensor product of the representations of quantum group [35].For the case at hand where we have V 1 ⊗ V 2 with some action of U q (su 1,1 ) then the R matrix is given by R = exp q 2 (q −1 − q)E ⊗ F q − H⊗H 2 (B.5)

C Casimir and derivatives
Here we show by explicit computation that Consider the action of Casimir operator on the following monomial C q (z n z * m ) = q 3 [n] q 2 [m] q 2 z n+1 z * m+1 + q[n] q 2 [m] q 2 z n−1 z * m−1 − q(1 + q 2 )[n] q 2 [m] q 2 z n z * m = q[n] q 2 [m] q 2 z n−1 1 − (1 + q 2 )zz * + q 2 z 2 z * 2 z * m−1 (C.2) This equation is sometimes written as g i k g j l Rkl mn = Rji rs g r m g s n where R = R • τ and τ is a swap.

3 )
algebra of functions O(M) gets additional structures that encode the group structure of this topological group.Thus, we have now a special element e ∈ M, the identity, and the two additional operations, inverse and product.Evaluating a function on the manifold on the element e gives a map from O(M) to complex functions.Such operations we will denote as the co-unitη : O(M) → C, η(f ) = f (e).(A.2) matrix elements.8For the case at hand we haveThe definition of commutation relations (B.1) allows the definition of the space where the elements of the quantum groups act.Thus, we introduce the coordinate functions x i that satisfy the relation R kl mn x k x l = x m x n , (B.4)