Matrix integrals $\&$ finite holography

We explore the conjectured duality between a class of large $N$ matrix integrals, known as multicritical matrix integrals (MMI), and the series $(2m-1,2)$ of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an $S^2$ topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an $S^2$ and a $T^2$ topology using BRST cohomology considerations. Matrix integrals support this finiteness.


Introduction
Being 0 + 0 systems, matrix integrals are of a more finite nature than large N quantum field theories traditionally explored in holography. In this work we explore, discuss and review in detail a particular class of matrix integrals, known as multicritical matrix integrals (MMI) [1,2]. MMI are built out of a single Hermitian N × N matrix organised in an even polynomial of order 2m with (m − 1) free parameters (couplings). Despite being constructed from a single matrix, MMI admit (m − 1) distinct critical exponents in the leading order planar expansion, which are encoded in the non-analytic behaviour of the matrix integral as a function of its couplings. In the large N limit and upon tuning the couplings to a set of special values, MMI are conjectured to be dual to the series (2m − 1, 2) of minimal models [22,23] -which we denote by M 2m−1,2 -on a fluctuating background. This can be described by coupling M 2m−1,2 to two-dimensional quantum gravity, where the theory of quantum gravity at hand upon fixing the Weyl gauge is given by Liouville theory [40]. In part, our work extends to general m the analysis and approach of [2] which dealt specifically with the case m = 3.
For m ≥ 2, M 2m−1,2 is a non-unitary two-dimensional CFT consisting of (m−1) distinct Virasoro primaries, each accompanied with an infinite tower of Virasoro descendants. The conformal dimensions of the Virasoro primaries are increasingly negative, with the highest being the vanishing conformal dimension of the identity operator. While the norm of the Virasoro primaries of M 2m−1,2 is positive, the norm of the Virasoro descendants is negative, leading to the non-unitarity of the models. We note that the non-unitary minimal models M 2m−1,2 are related to integrable lattice models. The Lee-Yang singularity [29] characterising the zeroes of the partition function of the Ising model in an imaginary magnetic field in the thermodynamic limit has been identified with M 5,2 [25]. On the other side M 7,2 has been conjectured [30] to correspond to the tricritical phase (the crossing point of the three lowest energy levels) of a generalisation of the Ising model with three state classical variables, known as the Blume-Capel model [31].
As explored extensively throughout this work, an important piece of evidence in establishing the conjectured duality between MMI and Liouville theory coupled to M 2m−1,2 is the matching of critical exponents between MMI and the continuum theory. We uncover the relation between the continuum theory on an S 2 topology and the explicit form of the MMI through its (finite) coupling dependence in the leading order planar expansion, as well as its perturbative multi-vertex expansion.
A more expansive and in some ways orthogonal direction has been pursued in [41][42][43][44][45]. The authors compare correlation functions of integrated operators (correlation numbers) to analogous quantities in the matrix integral. In our analysis of the partition function of the continuum theory we are turning on a single operator at finite coupling of the minimal model, the calculation of correlation numbers involves turning on multiple operators with an infinitesimal coupling.
MMI have also made an appearance, see e.g. [47,48], in the context of JT gravity [46]. In that context the continuum theory is studied on manifolds with boundary as compared to our analysis on S 2 and more generally on compact Riemann surfaces.
One of the key motivations of our work is the existence of a semiclassical limit exhibited by Liouville theory coupled to M 2m−1,2 . This is the large m limit, and was first observed in [36]. Specifically, upon fixing the area of the physical metric, restricting to an S 2 topology, and turning on only the identity operator of M 2m−1,2 one finds a round two-sphere geometry as the saddle point solution. This is the geometry of Euclidean two-dimensional de Sitter space. Two-dimensional de Sitter space supports finiteness [61][62][63], and its conjectured entropy is finite [64,65].

Outline
In section 3 and 4 we study the diagrammatic expansion of MMI, providing new combinatorial expressions for Feynman diagrams whose vertices emanate an arbitrary even number of edges. As an example there are 243180821057048715513033825248570669471308484796973569520429442294243 32116879409838986729881600000000000000000 diagrams consisting of fifteen distinct vertices emanating an even number between four and thirty-two edges. We provide a concrete framework identifying each of the (m−1) distinct planar critical exponents of the MMI in section 5. Geometrically these critical exponents are living on distinct fine-tuned "hypersurfaces" in coupling space. We match the critical exponents of the MMI to those of the continuum theory of Liouville theory coupled to M 2m−1,2 in section 6. This matching comes with an important subtlety. Whereas for unitary minimal models the identification of critical exponents in the matrix integral with critical exponents of the emergent large N continuum theory uses the KPZ relation [37][38][39], the minimal models at hand are non-unitary and require a generalisation of the KPZ formula [2,7]. In section 7 we consider the operator content of M 2m−1,2 on a fluctuating background. On a fluctuating background the number of operators of the M 2m−1,2 is subject to the Virasoro constraints.
Gauge fixing to the Weyl gauge further introduces the bc-ghost system. Initiated by work of Lian-Zuckerman (LZ) [53] and subsequent work [54][55][56] it was observed that the resulting BRST cohomology admits an infinite number of operators with non-vanishing ghost number and matter and Liouville descendants. The infinite set of LZ operators is still much smaller than the infinite tower of Virasoro descendants arising for each primary operator of M 2m−1,2 on a fixed sphere. As a consequence of the Riemann-Roch theorem we do however infer that LZ operators do not lead to additional critical exponents on an S 2 topology. This may render the non-unitarity of M 2m−1,2 on S 2 less severe. On the other hand the LZ operators contribute to the torus partition function [51,52], which we match to the leading non-planar result of the MMI. We observe that the partition function on S 2 dominates (in absolute value) over the partition function on T 2 only for a sufficiently large cosmological constant Λ, whereas for small Λ > 0, the partition function on T 2 dominates. We do not yet have a clear understanding of this phenomenon but it would be interesting to explore its consequences for the Hartle-Hawking picture [49,50]. Some more open questions we present in section 8.

Multicritical matrix integrals
In [2,7,18] it has been conjectured that a certain class of matrix integrals -known as multicritical matrix integrals [1] -in the large N limit and upon tuning certain couplings are dual to twodimensional quantum gravity coupled to M 2m−1,2 . We explore this conjecture by drawing explicit connections between the (m − 1) primaries of M 2m−1,2 and properties of the multicritical matrix integral.
We consider the matrix integral with α ≡ (α 2 , . . . , α m ) ∈ R m−1 . We will denote the set of numbers α as the couplings of the polynomial (2.3). We highlight that the number of free parameters α is equal to the number of primaries of M 2m−1,2 . Upon diagonalisation of M , we can analyse the planar contribution of (2.1) in the large N limit using a saddle point approximation. This reduces the exponent of (2.1) to where we assumed that the eigenvalues λ ∈ spec(M ) are distributed in the interval [−a, a] and ρ (m) ext (λ, α) is the eigenvalue density obtained as the solution of The prime indicates a derivative with respect to λ. For more details we refer to [13,15,16]. Another important quantity is the resolvent [4] Sending N → ∞ the sum can be replaced by an integral, where each eigenvalue is weighted by its average density For a higher order even polynomial it is convenient to express the resolvent as [8] From the definition of the resolvent (2.7) one obtains its large z scaling R(z) ∼ 1/z reducing (2.8) to the condition 1 (2.11) B(n, 1/2) denotes the beta function. We will call the condition N m (α) = 0 the normalisation condition for our matrix integral. For a particular choice of α we can turn the normalisation condition into (u − 4m) m = 0 , (2.12) in other words u = 4m is an m th order zero. The values of α leading to this behaviour can be easily obtained by recursively solving the discriminant of N m (α) = 0 [1,11]: (2.13) Finally we define the expectation value of the loop operator W [8], which is related to the resolvent (2.7) through a Laplace transform (2.14) We use (2.9) to obtain the large z expansion of the resolvent for the multicritical matrix integrals (2.3): Using (2.14), we relate the large z expansion of the latter to the small expansion of the loop 1 To evaluate (2.10) the following integral identity is useful: operator. For small we find where we defined As a final remark we note that evaluating (2.7) close to the real axis z = x ± i we obtain the important relations Combining res a with the definition of the resolvent (2.7) and the integral identity (2.9) we obtain the extremal eigenvalue density for V m (λ, α) where we used that the hypergeometric 2 F 1 (a, b; c; z) can be written as a polynomial as soon as either a or b become non-positive integers. At the multicritical point α c ≡ (α m,c ) (2.13) and close to the boundary of the eigenvalue interval where u = 4m (2.12) the density scales as ρ (m) ext (z, α c ) ∝ (4m − z 2 ) (2m−1)/2 generalising the well-known exponent 3/2 [4] in the quartic m = 2 model. Using (2.16) we obtain the planar on-shell action (2.4) for arbitrary m: At the critical point we find where H n denotes the n th harmonic number. The subscript c indicates that we zoom into criticality (2.13). At large m (2.21) scales as where γ denotes the Euler-Mascheroni constant and B k the k th Bernoulli number.

Planar diagrams with a single vertex
In this section we discuss the diagrammatic expansion of the matrix integral (2.1). We expand the normalisation condition (2.11) and the planar on-shell action (2.20) for small couplings α.
For the m = 2 model with a single coupling this was first explored by [4]. To account for the two indices of the matrix M one uses the 't Hooft double line notation [3]. We will denote the resulting diagrams as ribbon diagrams. Whereas for the m = 2 model one only encounters ribbon diagrams whose vertices emanate four edges, for the multicritical matrix integrals (2.3) we have to deal with vertices emanating an arbitrary even number of edges [13].

An m = 2 refresher
Before delving into the multi-coupling perturbation theory we quickly review the m = 2 case. For more details we refer to [16]. For m = 2 we have the polynomial (2.3) Normalisation of the eigenvalue density implies the vanishing of N 2 (α 2 ) in (2.11) with solutions u (2) ± given by u Of the two solutions, only u + is well-behaved near α 2 = 0. The other solution u − exhibits a pole at α 2 = 0, and is ordinarily discarded. Nonetheless, it is worth noting that knowledge of the residue of the pole at α 2 is sufficient to reconstruct u − . We also note that (3.2) exhibits a non-analytic behaviour close to α 2 = −1/12 which we recognise as the m = 2 multicritical point (2.13). For α 2 < α (2) 2,c the normalisation condition has no real solution. We now discuss the m = 2 model from a perturbative perspective in small α 2 . For α 2 > α (2) 2,c we obtain the small α 2 expansion of u Inspecting the above expression, one recognises the Catalan numbers Further to this, we see that the critical value α 2,c = −1/12 controls the radius of convergence of the power series. At large k, the summand in (3.3) goes as ∼ k −3/2 (α 2 /α (2) 2,c ) k . This behaviour encodes the fact that there is a square root non-analyticity in the solution u (2) + and as we shall soon see, it is intimately related to the growth of planar diagrams. Defining is the generating function of the connected planar bubble diagrams generated by the matrix integral with a quartic interaction. From a small α 2 expansion of Explicitly the propagator is given by 2 (4α 2 ) counts planar diagrams with four edges emanating from each vertex, with the shift α 2 → 4α 2 accounting for the 1/4 weighting each vertex. We have The summand in (3.5) scales as ∼ k −7/2 (α 2 /α 2,c ) k at large k. This behaviour encodes the growth of discrete Riemann surfaces with a fixed number of vertices k [5].

Binomial matrix integrals
We discuss the polynomials [13,14] V n (M, α n ) ≡ 1 2 For n = 2 the above polynomial is equal to the m = 2 multicritical polynomial (3.1) discussed in the last section. By setting all of the couplings but α n to zero in (2.19) we obtain the normalisation condition For α n =α n,c , whereα (3.10) has a second order zero at u = 4n/(n − 1), whereas for any other non-vanishing value of the couplingα n , (3.10) has n distinct solutions. We further note that For small α n only one of the solutions of (3.10) can be uniformly approximated by a perturbative expansion which is a power series in the coupling α n .
To obtain the leading expression in the perturbative expansion (3.10) we set α n = 0, however this prevents us from obtaining the other n − 1 solutions. Solutions which cannot be obtained in a perturbative expansion when setting the perturbation parameter to zero are discussed within the field of singular perturbation theory.
Singular perturbation theory. To recover the perturbative expansion of the n − 1 solutions of (3.10) singular for α n → 0 we start by rescaling u → α −ν n u, ν ∈ R + . For the case at hand (3.10) we obtain the rescaled equation For small α n and 0 < ν < 1/(n − 1) or ν > 1/(n − 1) (3.13) we only obtain the trivial solution u = 0. We are left with two special points ν ∈ {0, 1/(n − 1)}, where we find a non-trivial solution for u. The perturbative solution for ν = 0 is the regular solution u (n) . 2 A superscript indicates that u solves (3.10). For ν = 1/(n − 1) we obtain solved by the n − 1 roots of unity We then obtain the n distinct perturbative expressions approximating all n solutions of (3.10).
To discuss the perturbative analysis of (3.10) we take the regular solution u (n) . Its small α n expansion reads α ñ α n,c , n ≥ 3 .

Planar diagrams with multiple vertices
In this section we consider the diagrammatic multi-vertex expansion of the multicritical matrix integrals (2.3). We discuss the m = 3 and m = 4 (2.3) cases in some detail since the normalisation condition (2.11) for these matrix integrals is a cubic and a quartic polynomial whose roots admit explicit expressions. For m ≥ 5 the normalisation condition is a quintic or higher polynomial, and in general not solvable by radicals. The general m case can be dealt with from a perturbative perspective by employing the Lagrange inversion theorem [13].
Two-variable perturbation theory. The solution of N 3 (α 2 , α 3 ) = 0 regular near the origin in coupling space is Performing the substitution k = k 1 + 2k 2 , n = k 1 + k 2 we find the single sum expression Along the path γ (3) we recover (4.9), as shown in appendix E. Equation (4.11) provides a perturbative expansion regular for small couplings α 2 and α 3 . Depending on the range of the couplings (4.11) arises from a different solution (4.2) of the normalisation condition N 3 (α 2 , α 3 ) = 0. Switching for simplicity to polar coordinates (α 2 , α 3 ) = (r cos φ, r sin φ), we observe that the function is well behaved and real near the origin.
On-shell action for m = 3. We define Using (2.20) for m = 3 and the regular solution (4.10) we obtain the small α 2 , α 3 expansion It is convenient to perform again the substitution k = k 1 + 2k 2 , n = k 1 + k 2 leading to Evaluating (4.15) at the multicritical point α 3,c = 1/270 (2.13) we recover the value of the on-shell action at criticality (2.21). Note that in the definition of F (0) 3 (α 2 , α 3 ) in (4.13) we subtract the Gaussian term which evaluates to 3/4. In appendix E we show that the summand scales as ∼ k −10/3 , which differs from the ∼ k −7/2 encountered in (3.5). F To see some of these coefficients explicitly we expand the exponential in (2.1) (4.17) leading to the graphical representation of the propagator, the quartic and the sextic vertex In fig. 4 we show the diagrams contributing to O(α 2 α 3 ) in (4.16) . , , ,
Three-variable perturbation theory. Solving (4.18) and expanding the regular solution for α 2 , α 3 and α 4 close to zero we obtain Performing the substitution k = k 1 + 2k 2 + 3k 3 , n = k 1 + k 2 + k 3 , l = k 2 + k 3 According to this conjecture (4.22) reduces to (4.20) along γ (4) . Equation (4.22) provides a perturbative expansion regular for small couplings α. Depending on the range of the couplings it arises from a different solution (B.1) of the normalisation condition N 4 (α) = 0, as we elucidate further in appendix B.
On-shell action for m = 4. We define .

m ≥ 5 analysis
For general m ≥ 5 we cannot solve the normalisation condition (2.11) by radicals. Instead we conjecture generalisations of (4.10) and (4.21) for the normalisation condition and the expressions (4.14) and (4.25) for the on-shell action relying on numerical results.
Single-variable perturbation theory. We start by parametrising a path connecting the origin in coupling space to the multicritical point (4.28) This leads to the regular solution of N m (α) = 0 convergent for |t| ≤ 1. For large k the summand scales as ∼ Multi-variable perturbation theory. For general small couplings α the perturbative expansion of the regular solution reads The above expression accounts for mixing of the couplings. Note that if we set all but one of the couplings (e.g. α ) to zero (4.30) reduces to (3.17). We conjecture that this reduces to (4.29) along the path γ (m) . Similarly to the case m = 3 and m = 4 we believe that also for m ≥ 5 there exists a smooth function which leads to (4.30) when approached from different directions in coupling space. We note that we can also express (4.30) in terms of incomplete exponential Bell polynomials [20]. Using the Lagrange inversion theorem (for an introduction see e.g. [19]) to solve the normalisation condition perturbatively (2.11) we obtain wheref k = 0 for k ≥ m; k ( ) ≡ k(k + 1) · · · (k + − 1) denotes the rising factorial and Y n,k (x 1 , . . . , x n−k+1 ) are the incomplete exponential Bell polynomials, defined recursively through The summation is over all sequences j 1 , j 2 , . . . j n−k+1 of non-negative integers subject to the conditions (4.33) The Bell polynomial encodes information on the partitions of a set. Y n,k (x 1 , . . . , x n−k+1 ) tells us how many partitions with block size between 1 and (n − k + 1) a set with n elements can have when divided into k blocks. As an example reflects that the set y ≡ {y 1 , y 2 , y 3 , y 4 } can be divided into blocks of size 2 in two different ways. We can have 3 mutually, non-overlapping subsets, each consisting of a block of size two. Additionally we have 4 different ways to break y into a size 1 block and a size 3 block.

Non-analytic behaviour of multicritical matrix integrals
In this section, we uncover the non-analytic behaviour of the planar on-shell action (2.20) as a function of its couplings near the multicritical point (2.13).
An m = 2 refresher. For m = 2 the polynomial (2.3) reduces to the quartic polynomial From (2.8) it is straightforward to obtain the resolvent and using res a (2.18) we find the eigenvalue distribution (2.5): Combining the above leads to the planar on-shell action (2.20) with ω (2) n defined in (2.17). Using u where the subscript n.a. indicates the leading non-analyticity. The leading non-analytic behaviour, encoded in the critical exponent 5/2, characterises the particular universality class associated to the m = 2 (5.1) and binomial matrix integrals (3.9) and is intimately related to the exponent 7/2 we observed in (3.5) at large k.

Critical exponents for
The resolvent and the eigenvalue density are given by For the m = 3 multicritical matrix integral, in contradistinction to the m = 2 model (5.1), we obtain two different non-analyticities -one along a fine-tuned path, another one along a generic path in coupling space.
Generic path. Further to this, one can uncover another critical exponent by zooming into criticality while adding a linear deformation to one of the couplings. This leads to the ansatz The solution u is expanded up to order O( ) to avoid the appearance of spurious non-analyticities. Expanding the action around (5.10) we find The critical exponent is given by 7/3, which again differs from the 5/2 critical exponent of the m = 2 model. We believe that there are no other critical exponents near the multicritical point for m = 3 in the leading order planar expansion.

Critical exponents for general m
Since for m ≥ 4 the normalisation condition is a higher order polynomial, we now outline a perturbative approach for the fine-tuned path.
Fine-tuned path. We would like to deform the couplings near the multicritical point α c (2.13) in the following manner α = α c + s , u = 4m + x , (5.12) where x and are small parameters, and s ≡ (s 2 , . . . , s m ) ∈ R m−1 . Expanding the normalisation condition (2.11) we find where H (j) m (hypersurfaces) are defined as We are thus led to the following ansatz 16) giving rise to the normalisation condition We refer to appendix C for a proof of (5.17). Recalling the discussion near (5.9) for m = 3, to ensure that we obtain the correct non-analytic behaviour for F It can be checked that adding subleading corrections to more than one of the α n does not lead to additional non-analyticities.
We now determine the non-analytic behaviour of F  2 ] is anx-independent expression whose explicit form we present in appendix D. Takingx =x * to be a solution of the normalisation condition N Example m = 3. Let us take The parameter x is itself small and fixed in terms of s 2 , s 3 , and through N 3 (α) = 0. For generic values of s 2 and s 3 , expanding the normalisation condition N 3 (α) = 0 for small leads to the following three solutions We thus recover the non-analytic behaviour observed in (5.10).
Adding further subleading terms will not change the leading non-analytic behaviour. Setting s 2 = 1 ands = ±1/5 we recover (5.8). Expanding N 3 (α 2 , α β 3 ) for small , we find (5.19) Evaluating the action in a perturbative expansion along (5.25), we obtain (5.21) 27) recovering (5.9) for s 2 = 1 ands = ±1/5. From the above expression we infer that for vanishings the coefficient multiplying the leading non-analyticity vanishes, since its solution inx =x agrees with the solution of the leading order term in (5.26). The non-analytic behaviour in (5.27) is robust against other deformations of the ansatz (5.25).
Generic path for general m. In addition to the (m − 2) critical exponents in (5.21) the matrix integrals (2.1) exhibit one more multicritical exponent.
To obtain (5.4) for m = 2 we observed the reaction of the action when allowing α 2 to slightly deviate from its critical value α 2,c (2.13). We can generalise this for the polynomials V m (M, α) for m ≥ 3. Here we observe the reaction of the action when allowing one arbitrary coupling to deviate away from the multicritical point. In other words we consider where WLOG α m deviates away from its critical value α Upon choosing oneÃ (n) theC for 2 ≤ ≤ 2m − 1 are fixed uniquely. WLOG we chooseÃ (m) and find The leading non-analyticity is thus given by with β 3 given by Example m = 3. For the multicritical matrix integral with m = 3 we make the ansatz 2,c , α 3 = α 3,c + , u = 12 +Ã 1/3 + . This is a consequence of our particular choice of deformation (5.15). For m prime the full set of critical exponents are elements of Q + /Z. For non-prime m some of the critical exponents are integers. Whether or not we should refer to these integers as critical exponents is a subtle matter. 5 A point of concern for integer critical exponents is that they may be sensitive to analytic redefinitions of the couplings. Summary. In summary, the set of (m − 1) critical exponents for the m th multicritical matrix integral (2.3) are given by m/(m − r ), r = 1, . . . , m − 2 (5.21) and 2 + 1/m (5.32). 5 It can often happen that when a critical exponent is naïvely integer valued there is in fact a logarithmic dependence on the coupling. Logarithmic behaviour is also present for the critical exponent of a two-matrix model [9] whose continuum description has been argued to be the free fermion coupled to two-dimensional gravity. It is relatively straightforward to prove that our integer valued critical exponents are indeed integers exhibiting no logarithmic dependence on the coupling.

Critical exponents in the continuum picture
In section 5 we uncovered a set of non-analyticities arising from the deformation of multicritical matrix integrals (2.3) slightly away from the multicritical point (2.13). We showed that the m th multicritical matrix integral has (m−1) distinct non-analyticities (5.21) and (5.32). In this section we will uncover the same non-analyticities within the continuum picture of M 2m−1,2 coupled to two-dimensional quantum gravity.

A minimal model refresher
Minimal models are two-dimensional CFTs characterised by two coprime integers (p, p ) with p, p ≥ 2 and WLOG we assume p > p . 6 We will denote the (p, p ) minimal model by M p,p . The central charge of M p,p is given by Each M p,p has a finite number of conformal primaries O r,s whose (holomorphic) conformal dimension is given by , r = 1, . . . , p − 1 , s = 1, . . . , p − 1 .  In what follows we will focus on M 2m−1,2 with m ≥ 2. The general expression for their central charge is The number of conformal primaries is n 2m−1,2 = (m − 1), and their holomorphic dimensions are given by , r = 1, . . . , m − 1 . Although c (2m−1,2) grows at large m, it has been argued [21,57] that a better measure of the number of degrees of freedom is captured by We note that c (2m−1,2) eff < 1 goes to one in the large m limit.

Critical exponents
To compute critical exponents associated to a given conformal field theory, we consider the partition function of the theory deformed by a small amount of a particular conformal primary O ∆ . We first discuss critical exponents for CFTs on a fixed background, and then proceed to a fluctuating background.
2d CFT on a fixed flat background. From the perspective of a path-integral, we would like to compute where denotes the size of the flat square on which the CFT resides. The conformal primary O ∆ has dimension (∆,∆) and for simplicity we take ∆ =∆. The dimensionful scales of the problem are the volume of space 2 , the ultraviolet length scale uv , and the coupling λ ∆ whose holomorphic scaling dimension is ∆ λ = ∆ − 1. Following the line of argumentation from the scaling hypothesis Z[λ ∆ , ] = Z[q −∆ λ λ ∆ , q ], q ∈ R + [28] we would like the UV independent part of log Z[λ ∆ , ] to be extensive in the volume. Given that Z N is a normalisation constant independent of λ ∆ . Notice that the critical exponent associated to the identity operator is simply ν 0 = 1. Furthermore, for ∆ > 1, corresponding to an irrelevant O ∆ , the critical exponent would be negative.
2d CFT on a fluctuating background. We now consider a two-dimensional CFT with central charge c m < 1 coupled to two-dimensional gravity. Integrating over all metrics renders the extensivity condition of the scaling hypothesis somewhat subtle. In the Weyl gauge the twodimensional metric is chosen to be g ij = e 2bϕg ij . The problem then maps to studying the matter CFT with central charge c m , trivially coupled to a Liouville CFT with central charge c L = 26−c m , and the bc-ghost system with central charge c g = −26 [40]. The Liouville action is given by [40] S L [ϕ, Λ] = 1 4π whereg ij is taken to be the round metric on S 2 such that R[g ij ] = 2 and Λ ≥ 0 is the cosmological constant. Moreover, The residual gauge invariance in the Weyl gauge enforces that all operators of the combined theory are spinless primaries with conformal dimension ∆ = 1. In the trivial ghost sector this is achieved by dressing the matter primaries of weight ∆ m by a Liouville operator of weight ∆ L = 1 − ∆ m .
Unitary 2d CFT on a fluctuating background. We now specify to a unitary two-dimensional CFT with c m ∈ (0, 1). The simplest critical exponent corresponds to the matter identity whose coupling is Λ. The partition function of interest is 7 We indicate the partition function on a fluctuating background by Z. A simple derivation of the above follows from performing a shift in ϕ [38,39]. Due to the Liouville dressing, the critical exponent of the identity is no longer simply given by ν 0 = 1 (6.10), but rather [37] (6.14) The critical exponent for ν grav informs us how to modify the scaling behaviour of length upon coupling to gravity. On a fixed background the total scaling dimension of a length scale is minus one, whereas now we must take it to be ν grav /2. ν grav is also known as the string scusceptibility. Now, rather than the identity we consider turning on a matter conformal primary O ∆ . The partition function of interest, in the Weyl gauge, becomes where we set Λ = 0 since we are interested in turning on O ∆ alone. We further have 16) which ensures that the matter operator is dressed appropriately. We note that b = σ ∆=0 . Upon shifting ϕ → ϕ − (log λ ∆ )/2σ ∆ , [38,39] and noting that the path-integration measure over ϕ is invariant under such shifts, it is straightforward to deduce (6.17) where N is a λ ∆ independent normalisation. For more details we refer to [16]. In effect, one is replacing σ ∆=0 with σ ∆ min (6.13). Note that Turning on other operators O r,1 while setting Λ min = 0 leads to A fixed "area" perspective. In order to compare to the perturbative discussion of section 4 it proves instructive to consider the gravitational path integrals with a constraint fixing the total area of space to a fixed value υ. This can be achieved by inserting a δ-function inside of the gravitational path-integral (6.9). For two-dimensional conformal field theories with c m < 1, we have where N is independent of υ and Λ. Integrating Z area [υ] against υ, we recover Z[Λ] (6.13). For c m = 0, we note that 1 + Q/b = 7/2, the value observed in (3.5). For c m = c (2m−1,2) in (6.4), we find instead 1 + Q/b = 3/2+m. Let us now consider those non-unitary minimal models whose lowest weight operator O min is different from the identity. We can also consider fixing

Comparison to matrix integrals
At this stage it behooves us to compare our results to those of the multicritical matrix integrals. We take inspiration from 't Hooft's diagrammatic picture [3], whereby the perturbative diagrams of the matrix integrals correspond to discretised Riemann surfaces. Care must be taken in identifying the appropriate quantities between the matrix integrals and the continuum picture.
M 3,2 on a fluctuating background. Let us begin by discussing the simplest case, namely m = 2. In this case the matrix diagrammatics (3.5) indicates that the dependence on the number of vertices k goes as ∼ k −7/2 (α 2 /α (2) 2,c ) −k at large k. One is motivated to identify the number of vertices, k, with the area of the surface in the continuum picture. Both are extensive quantities sensitive to the total number of points on the surface. In doing so, one finds a match between the behaviour of the fixed area partition function (6.23) and the matrix diagrammatics. This suggests that the identification of k in the matrix diagrammatics and υ in the continuum is indeed sensible. 8 Going from the diagrammatics to the critical exponent is simply a matter of integrating (summing) over υ (k), and identifying Λ ∝ (α 2 − α (2) 2,c ).
M 2m−1,2 on a fluctuating background. We would like to compare the asymptotics at large vertex number from the multicritical matrix diagrammatics to the continuum picture. Part of the issue is that there are multiple couplings, and consequently multiple paths in coupling space to reach the multicritical point. Along the path (4.28) which simultaneously tunes several couplings, the growth of vertices goes as ∼ k −(3+1/m) t k (4.37). Recalling (6.25) and noting that for general m, 1 + Q/σ min = (3 + 1/m), we find evidence that such a tuning corresponds to fixing the extensive quantity (6.24), rather than the area (6.23). The remaining task is to identify Λ min , and the additional (m − 2) couplings λ ∆ r,1 from the perspective of the multicritical matrix integral. This is precisely the problem of non-analyticities solved in section 5. The non-analyticities found in the m th multicritical matrix integral correspond to the values Q/σ min (6.20) and σ min /σ ∆ r,1 , r = 1, . . . , m − 2 (6.22) arising from M 2m−1,2 on a fluctuating background. We thus identify Λ min = in (5.32), λ ∆ r,1 = σ min /Q and r = m − r − 1 in (5.21). We observe that σ min /Q is independent of r. Further to this, our hypersurface equation (5.15) provides the detailed relation between the matrix deformation and the corresponding matter primary.

Remarks on a Hilbert space
In this section we remark on the Hilbert space of M 2m−1,2 coupled to two-dimensional gravity, and its manifestation from the matrix integral perspective.

S 2 considerations
On a fixed background, M 2m−1,2 has a finite number of primaries equal to n 2m−1,2 = (m−1), each accompanied by an infinite tower of descendants. On a fluctuating background these operators must satisfy constraints arising from the diffeomorphism invariance. Additionally we need to consider the contribution from the Liouville and bc-ghost sector.
Concretely we must identify the set of BRST invariant operators. This was examined in early work of Lian-Zuckerman (LZ) [53] and subsequent work [54][55][56]. Under the assumption that the Liouville sector can be treated as a linear dilaton theory, it was noted that the BRST cohomology comprises of an infinite collection of operators. In particular LZ operators have a non-trivial ghost number and generally contain matter and Liouville descendants. Though also infinite, this infinity is far smaller than the infinite operator content of the matter theory on a fixed background arising from the Virasoro descendants. The origin of these operators is intimately connected to the presence of null operators in the Liouville sector [34] 9 and the matter sector. In conformal gauge ds 2 = e 2bϕ(z,z) dzdz the LZ operators are given by where t ∈ Z and ± denote the particular LZ operator. The holomorphic conformal dimensions of these operators are given by where Q = b+b −1 and LZ operators are graded by the ghost number. The LZ weights for M 2m−1,2 are given by ∆ LZ r,± (t) = A r,± (t) − ∆ r,1 − 1 , where A r,± (t) are given by [24] A r, The anti-holomorphic conformal dimension has to be equal to the holomorphic conformal dimen-sion. The argument t is related to the ghost number, whereas the subscript ± indicates whether the ghost number is even (+) or odd (−). Example where combining (6.12) with c (3,2) (6.7) we have b = 2/3. Besides the operator (7.6) there exists another operator R LZ 1,+ (−1) with σ LZ = 2/b. To show the BRST invariance of these operators, it is useful to recall the (holomorphic) BRST current [58] J BRST = c T ϕ + 1 2 : c T g : where T ϕ and T g are the Liouville and ghost stress tensor respectively T ϕ = −(∂ϕ) 2 + Q∂ 2 ϕ , T g =: (∂b)c : −2∂ (: bc :) . (7.8) In particular we find [54] δR LZ 1, where L n ,L n are the Virasoro generators, satisfying In other words the BRST variation leads to a null operator.
We note that we consider the stress tensor of the Liouville action arising when assuming it is a linear dilaton theory. This is justified when calculating the critical exponents (6.18) for which we set the cosmological constant to zero. The BRST transformation of a LZ operator can produce null operators in the matter or Liouville sector [54], which must subsequently be set to zero.
One may ask whether the LZ operators contribute additional critical exponents for the theory on S 2 . By the Riemann-Roch theorem, non-vanishing bc-correlation functions on a compact Riemann surface with Euler characteristic χ require [58] n c − n b = 3 2 χ . (7.11) For S 2 we have χ = 2. Given that LZ operators have a non-trivial ghost number, generically different from n c − n b = 3, we expect no new critical exponents from the LZ operators on an S 2 topology.
Assuming that some form of the operator-state correspondence holds for S 2 we are thus led to conclude that the associated Hilbert space is finite-dimensional. This might be related to observations on de Sitter space [61,62].
Contrarily to S 2 the torus T 2 has Euler characteristic χ = 0. As a consequence of the Riemann-Roch theorem (7.11) we thus infer that the LZ operators contribute to the torus partition function.

T 2 considerations
On the cylinder, the Hilbert space H T 2 lives on spatial S 1 constant time slices. States |Ψ ∈ H T 2 in the trivial ghost sector living on these spatial slices are subject to the Virasoro constraints where L tot n andL tot n are the Virasoro generators for the matter and Liouville sector. The first equation in (7.12) is what replaces the Hamiltonian constraint in canonical quantum gravity [59], while the second replaces the spatial diffeomorphism constraint. The above equations are the state analog of the constraint that vertex operators with trivial ghost contribution must have ∆ ≡ ∆ L + ∆ r,1 = 1 and ∆ =∆. Other states in H T 2 , as first pointed out by Lian-Zuckerman [53], may also include non-trivial ghost excitations.
One way to characterise H T 2 is through the torus partition function [51,52]. For fixed modular parameter τ = τ 1 + iτ 2 the states in the BRST cohomology contribute where q = e 2πiτ 2 and we used (7.4). The overall shift encodes the Casimir energy from the ghost, Liouville and matter sector. The τ 1 -independence of Z fixed [τ 2 ] is due to the diffeomorphism constraint (7.12). What remains to be done is integrate over the modular parameter τ 2 and the zero modes of the Liouville sector F is the fundamental domain of the modular group. The power of τ 2 is fixed by modular invariance and stems from the various zero modes in the bc-ghost and Liouville sector. The logarithm in Λ stems from the volume of the Liouville zero mode, and essentially encodes the fact that the Liouville interaction imposes a cutoff in the Liouville field space. Evaluating T [Λ min ] leads to [51] T Comparing to the first non-planar contribution of the matrix integral as presented in appendix A we see that under the identification Λ min = the results agree (A.12). In this way the LZ states appear in the leading non-planar contribution of the m th multicritical matrix integral.

Discussion and open questions
We summarise some open questions and speculative remarks.
Large m and Euclidean dS 2 . In [36] it has been observed that upon coupline M 2m−1,2 to gravity whilst fixing the area υ and turning on only the identity operator of M 2m−1,2 exhibits a saddle point solution in the large m limit. This saddle point solution is the round metric on S 2 , which is Euclidean dS 2 . Motivated by this, this work provides the basis to understand this observation from the matrix integral point of view. Recalling (5.21) for r = m − 2 we recover Zamolodchikov's continuum critical exponent from a matrix integral perspective. In the large m limit and upon tuning the couplings to the multicritical point (2.13) the polynomial V m (λ, α) 3) reduces to 10 [11] lim m→∞ V m (λ, α c ) = 1 2 λ 2 2 F 2 1, 1; Moreover the width of the eigenvalue distribution (2.19) scales with m and so becomes unbounded in the large m limit. We further remark that the most fine-tuned path where we switch on all the hypersurfaces and s ∈ H As postulated in [65], the logarithm of the Euclidean gravitational path-integral on a compact manifold for theories admitting a sphere saddle gives a semiclassical expansion of the entropy of the corresponding de Sitter solution. The presence of a semiclassical limit allows one to interpret the details of its expansion in terms of the classical saddle. The fixed area partition function on a genus h = 0 surface (6.23) for large negative c m can be written as where υ 0 is a reference area. As noted in [16], to leading order the logarithmic term resembles an entanglement entropy for a two-dimensional CFT with central charge c m [66][67][68], with the subleading corrections corresponding to contributions stemming from the CFT being coupled to dynamical gravity. Moreover from the matrix perspective 2ϑ = log N 2 suggesting that the size N of the matrix also has an entropic interpretation.
Non-unitarity & torus Hilbert space. As a consequence of the Riemann-Roch theorem (7.11) the LZ operators contribute on T 2 . In particular this implies that we have to deal with descendants of the Virasoro primaries of M 2m−1,2 . These are negative norm operators. The consequences of the non-unitarity from both the point of view of the MMI and the point of view of the continuum theory remain to be explored. This allowed us to explicitly determine the radius of convergence and for m ≥ 3 using γ (m) (4.28) we observed the critical exponent F (0) m,n.a. (α) γ (m) ∼ 2+1/m (4.37). However introducing the single 10 We would like to acknowledge Jorge Russo for useful discussions.
parameter t ∈ [0, 1] connecting the origin in coupling space to the multicritical point prevents from observing the other (m − 2) critical exponents m/(m − r ), r = 1, . . . , m − 2, from a diagrammatic perspective. It would be interesting to uncover these.
Disk topology. In [32][33][34][35] Liouville theory was studied on the disk topology. Upon taking the semiclassical limit b → 0 the Liouville action admits a saddle point solution for ϕ. From the perspective of the physical metric g ij = e 2bϕg ij this can be interpreted as the hyperbolic metric on the Poincaré disk (the Euclidean AdS 2 black hole) Since coupling M 2m−1,2 to gravity implies b = 2/(2m − 1) (6.12) this semiclassical limit corresponds to a large m limit. It would be interesting to explore relations between this saddle and recent discussions on JT gravity and matrix integrals [47]. For Dirichlet boundary conditions ϕ diverges at the boundary of the disk and the relation to matrix integrals was studied in [10]. It would be interesting to generalise this to the multicritical case. For Neumann boundary conditions one needs to further add the boundary term to the bulk action, where h is the induced metric on S 1 and K is the extrinsic curvature at the boundary. The comparison to the matrix integral uses either the resolvent R(z) or the loop operator W [8]. The boundary cosmological constant Λ B is connected to z. For multicritical matrix integrals R(z) is given by (2.15), W is given by (2.16).
Hartle-Hawking & topology. As a final remark, it is interesting to note that the partition function Z[Λ] on S 2 only dominates (in absolute value) over the partition function T [Λ] on T 2 for sufficiently large Λ, while for small enough Λ > 0, the T 2 partition function dominates. This is also true for higher genus partition functions. It would be interesting to understand if this has any consequences for the Hartle-Hawking picture [49,50]. Further to this, being matrix integrals rather than matrix path integrals there is no a priori indication for the existence of Hilbert space from the matrix integral perspective. It would be interesting to uncover a Lorentzian picture directly from the matrix integral [60].

A Non-planar contribution
To compare to the log-divergence in (7.15) we need to go beyond the planar approximation of the large N limit of F m (α) (4.35). Whereas the planar contribution is obtained from a large N saddle point approximation, to find non-planar contributions one needs to make use of other techniques. We will use the method of orthogonal polynomials [12]. We will only provide minimalistic details, for a more detailed explanation of this method we refer for example to [16]. Two polynomials are said to be orthogonal with respect to a weight function w(x) if they satisfy ortho a : dx w(x) p n (x)p m (x) = h n δ m,n (A.1) In addition to (A.1), orthogonal polynomials satisfy the three-term recurrence relation ortho b : x p n (x) = A n p n (x) + S n p n+1 (x) + R n p n−1 (x) for n > 0 , where A n , S n , and R n are some real constants. Focusing on monic polynomials we obtain [12] 1 where we highlight the coupling dependency of h n (A.1) and R n (A.2) explicitly.
An equation similar to (A.6) can be obtained for m ≥ 3 upon choosing ω(x) = e −N Vm(x,α) , m ≥ 4 (A.1). Our final ingredient will be the Euler-Maclaurin formula In the above, f (x) is a 2p times continuously differentiable function, R N is a remainder term scaling as O(1/N 2p+1 ), and the B 2n denote the Bernoulli numbers. Applying the Euler-Maclaurin formula to and expanding (A.4) in inverse powers of N , we find 1 Note that we encounter an ambiguity in choosing r 0 (x, α) since it is the solution of an m th order polynomial. We pick the solution yielding the on-shell value (2.21) when evaluating the O(N 0 ) integral along γ (m) . Evaluating the second line in the above expression is in general difficult however to get the coefficient of the log-divergence we only care about the first integral. To further simplify our analysis we zoom into the multicritical point α c (2.13). The non-analytic behaviour of (A.11) occurring for α = α c close to the upper boundary, equals the non-analyticity observed upon considering small deformations away from the multicritical point only after evaluating the integral. We obtain for m = 3 and m = 4 As a final remark we note that F (1) m (4α 2 , . . . , 2mα m ) counts leading non-planar diagrams. More explicitly it counts diagrams whose vertices are emanating four or 2m edges and which can fit on a surface of genus one. As an example, a perturbative analysis of (A.11) using r 0 (x, α) and r 2 (x, α) for m = 3 easily reveals In this section, we discuss the solutions of the normalisation condition (4.18) for the m = 4 case: (B.2) In the above, we have defined Much of our interest lies in a solution of N 4 (α) = 0 that is regular in a small neighbourhood ℵ 0 around the origin of coupling space α = 0. To analyse the problem, we can consider approaching α = 0 uniformly in all directions, and exploring the behaviour of the various solutions throughout ℵ 0 . An exhaustive analysis reveals that one must keep track of the various signs of α and the special combination (α 3 + 14α 4 ). The term (α 3 + 14α 4 ) is already revealed in the form of ∆ 0 and can be seen to carry through into the more involved building blocks such as Q and S. For instance, near ℵ 0 we have We find the following combination of solutions to be smooth near ℵ 0 where we have introduced the notation Further properties of N 4 (α) = 0 The solutions (B.1) also reveal additional information. For instance, expanding the discriminant D 4 (B.7) at small α 2 and α 3 , we identify α 4 = −27/8960 as the special valueα 4,c in (3.11). Similarly, expanding the discriminant D 4 at small α 3 and α 4 , we identify α 2 = −1/12 as the special value α 2,c in (2.13). Finally expanding for small α 2 and α 4 reveals α 3 = −2/135 as the special valueα 3,c (3.11). Near (α 2 , α 3 , α 4 ) = (0, 0, −27/8960), (α 2 , α 3 , α 4 ) = (0, −2/135, 0) as well as (α 2 , α 3 , α 4 ) = (−1/12, 0, 0), ∆ 1 remains non-vanishing such that the non-analytic behaviour of the solutions u (4) is that of a square root. Expanding the discriminant D 4 near α 2 = −1/8, reveals α 3 = 1/160 as a special value, which we recognise as α (4) 3,c , one of the multicritical couplings (2.13). At (α 2 , α 3 ) = (−1/8, 1/160) we further have that ∆ 1 = 0, while D 4 goes as (1 + 8960α 4 ) 3 revealing the third multicritical value α 4 = −1/8960. Also, at (α 2 , α 3 ) = (−1/8, 1/160) we observe that Q in (B.4) goes as (1 + 8960α 4 ) 1/2 , and p in (B.2) goes as (1 + 8960α 4 ). Expanding away from the multicritical point reveals distinct non-analytic behaviour in the solutions of N 4 (α) = 0. For instance, fixing (α 2 , α 3 ) = (−1/8, 1/160) and deviating slightly away from α 4 = −1/8960, we uncover a fourth root non-analyticity.

C Non-analyticities: normalisation condition
In this appendix we prove the following.

D Non-analyticities: action
In this appendix we prove the following.
Claim. Along where we used i1), i2), i3) (D.14) and c2) (D. 15) and H m is the m th harmonic number. Additionally (D.7) contains a coupling andx dependent part. Using (D.5) we obtain along (D.1)  Only the sums in the first line of (D.9) and the last sum could contribute to the leading nonanalyticity in (D.2). We treat the sums independently. For the second sum in the second line we use that (D.15) vanishes for < m and the first non-vanishing term arises for = m proportional to (2m−r )/(m−r ) . Since our leading non-analyticity grows us O( m/(m−r ) ), r = 1, . . . , m − 2 the former is subleading with respect to the non-analyticity we are after. For the first sum of the second line we show that for s ∈ H Thex dependent term therefore exactly cancels the 4 th term in (D.4).
We are left to show for k ≥ 1 −12 (−1) k Γ(k + 1) (1 − 3n) . (E.11) The last product we write in terms of stirling numbers s 1 (n, k) for the first kind. The Stirling numbers s 1 (n, k) enumerate (−1) n−k times the number of partitions of the symmetric group S n with exactly k cycles. By definition, they are also the coefficients of the falling factorial (x) n ≡ x(x − 1)(x − 2) · · · (x − n + 1) = n k=0 s 1 (n, k)x k . (E.12) Applying this to the product in (E.11) with x = 1/3 we find where in going to the last line we substituted n → k − n + 1. We now define L k ≡ k n=0 3 n s 1 (k + 1, k + 1 − n) , R k ≡ k! k n=0 3 n (−1) n 3 k n + k n n k − n . (E.14) We show that both expressions satisfy We start with L k . Using the recursion relation for the Stirling numbers s 1 (k + 1, n + 1) = s 1 (k, n) − ks 1 (k, n + 1) , (E. 16) and s 1 (k, 0) = 0 we find L k ≡ k n=0 3 n s 1 (k + 1, k + 1 − n) = We now show the same recursion equation for R k . We have Since L 0 = R 0 and L 1 = R 1 so (E.15) completes the proof.