Random Finite Noncommutative Geometries and Topological Recursion

In this paper we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples ${(\mathcal{A}, \mathcal{H}, D , \gamma , J) \,}$, called random matrix geometries of type ${(1,0) \,}$, with a fixed fermion space ${(\mathcal{A}, \mathcal{H}, \gamma , J) \,}$, and a distribution of the form ${e^{- \mathcal{S} (D)} {\mathop{}\!\mathrm{d}} D}$ over the moduli space of Dirac operators. The action functional ${\mathcal{S} (D)}$ is considered to be a sum of terms of the form ${\prod_{i=1}^s \mathrm{Tr} \left( {D^{n_i}} \right)}$ for arbitrary ${s \geqslant 1 \,}$. The Schwinger-Dyson equations satisfied by the connected correlators ${W_n}$ of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients ${W_{g,n}}$ of the large $N$ expansion of ${W_n}$'s enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve ${\left( {\Sigma , \omega_{0,1} , \omega_{0,2}} \right)}$ of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential ${\omega_{0,2}}$ in terms of the formal parameters of the model.


Introduction and basics
In metric noncommutative geometry, the formalism of spectral triples [15] encodes, in the commutative case, the data of a Riemannian metric on a spin manifold in terms of the Dirac operator.More precisely, by Connes' reconstruction theorem [18] one knows that a spin Riemannian manifold can be fully constructed if we are given a commutative spectral triple satisfying some natural conditions like reality, and regularity.A simple manifestation of this fact is a distance formula [14] according to which one can recover the Riemannian metric from the interaction between the Dirac operator and the algebra of smooth functions on a spin manifold through their actions on the Hilbert space of L 2 -spinors.This naturally leads to the view that spectral triples in general, without commutativity assumption, can be regarded as noncommutative spin Riemannian manifolds.
The data of a real spectral triple (A, H, D, γ, J) consists of a * -algebra A together with a *representation π : A → L(H) on a Hilbert space H, a self-adjoint Dirac operator D, a Z/2-grading γ, and an anti-linear isometry J acting on H.The above-mentioned operators should satisfy certain (anti)-commutation relations and technical functional analytic conditions (see [15], [12] and [19] for the detailed axiomatic definition of a real spectral triple).
A spectral triple (A, H, D) is called finite if the Hilbert space H is finite dimensional, i.e.
H ∼ = C n .The data (A, H, γ, J) corresponding to a finite real spectral triple is referred to as a fermion space.Given a fixed fermion space (A, H, γ, J), the moduli space of Dirac operators of the finite real spectral triple (A, H, D, γ, J) consists of all possible self-adjoint operators D (up to unitary equivalence) which satisfy the axiomatic definition of a real spectral triple [27].It is considered as the space of all possible geometries, that is Riemannian metrics, over the noncommutative space (A, H, γ, J).
The theory of spectral triples has been used in constructing geometric models of matter coupled to gravity, using an action functional, called the spectral action, which is given in terms of the spectrum of the Dirac operator (see [16], [11], [17] and [12]; see also [24] for a recent work in the spirit of matrix models).
This paper is about a second application of the idea of spectral triples by creating a connection with the recently emerged theory of topological recursion [21].Roughly speaking, if we understand quantization of gravity as a path integral over the space of metrics, it is natural to consider models of Euclidean quantum gravity over a finite noncommutative space in which one integrates over the moduli space of Dirac operators for a fixed fermion space.Given a fermion space (A, H, γ, J), the distribution over the moduli space of Dirac operators is considered to be of the form e −S(D) dD , where the action functional S(D) is defined in terms of the spectrum of the Dirac operator D.
The investigation of the relation between models of quantum gravity on a certain class of finite noncommutative spaces and (anti)-Hermitian matrix ensembles started in the work of Barrett and Glaser ( [3], cf. also [2]), although largely through numerical simulations.In this paper we consider a much larger class of models and show that an analytic approach to analyzing these models is possible, using techniques of topological recursion and blobbed topological recursion pioneered by Eynard, Orantin [21,22], Chekhov [13], and Borot [4,6,9].
In the following, we recall the definition of a particular type of finite real spectral triples whose Dirac operators are classified in terms of (anti)-Hermitian matrices in [2].Denote the real Clifford algebra associated to the vector space R n and the pseudo-Euclidean metric η of signature (p, q) , given by by Cℓ p,q . 1 Consider the complexification Cℓ n := Cℓ p,q ⊗ R C of Cℓ p,q .Let {e i } n i=1 be the standard oriented orthonormal basis, i.e. η(e i , e j ) = ±δ ij , for R n .The chirality operator Γ is given by where s ≡ q − p (mod 8) , 0 ⩽ s < 8 .We denote by V p,q the unique (up to unitary equivalence) irreducible complex Cℓ p,q -module, where, for n = p + q odd, the chirality operator Γ acts trivially on V p,q . 23We refer to γ i = ρ(e i ) , i = 1, • • • , n , i.e. the Clifford multiplication by e i 's, as the gamma matrices.There exist a Hermitian inner product ⟨•, •⟩ on V p,q such that the gamma matrices act unitarily with respect to it, i.e. ⟨γ i u, γ i v⟩ = ⟨u, v⟩ , i = 1, • • • , n [26]. 4onsider the Hilbert space (V p,q , ⟨•, •⟩).Let C : V p,q → V p,q be a real structure of KO-dimension s ≡ q − p (mod 8) (see, e.g., [27]) on V p,q such that (Cℓ n , V p,q , Γ, C) (1.4) satisfies all the axioms of a fermion space. 5efinition 1.1.A matrix geometry of type (p, q) is a finite real spectral triple (A, H, D, γ, J), where the corresponding fermion space is given by: The Dirac operators of type (p, q) matrix geometries are expressed in term of gamma matrices γ i , and commutators or anti-commutators with given Hermitian matrices H and anti-Hermitian matrices L (see [2] and [3]).For a recent survey of interactions between fuzzy spectral triples and random matrix theory initiated in this paper we recommend [25].
2 Random matrix geometries of type (1, 0) In this section we describe a model for Euclidean quantum gravity on finite noncommutative spaces corresponding to the random matrix geometries of type (1, 0) .The Dirac operator of type (1, 0) 2 Let ρ : Cℓp,q → Hom C (V, V ) be an irreducible complex unitary representation of Cℓp,q .It can be shown [26] that: • If n = p + q is even, then the representation ρ is unique up to unitary equivalence; Both possibilities can occur, and the corresponding representations are inequivalent.
matrix geometries is given by [3]: where H N denotes the space of N × N Hermitian matrices.The Dirac operator D acts on the Hilbert space H = M N (C) in the following way The moduli space of Dirac operators is isomorphic to the space of Hermitian matrices H N .A distribution of the form is considered over H N , where is the canonical Lebesgue measure on H N .Let us describe the action functional S(D) .Let Suggested by Connes' spectral action, we define the action functional S(D) of the model by where and In the definition of the action functional, t is a fixed parameter ("temperature"), the (α n , αn I ) n,n I are formal parameters, and, for each s, the summation over n I ∈ N s ↑ is a finite sum. 6Let (2.9) For large N , the term S stable (D) can be considered as higher order terms in ℏ-expansion of the action functional S(D) . Using we get In addition, we have where I = {1, • • • , s} .By substituting (2.11) and (2.12) into (2.6), we get the expression for the action functional S(D) in terms of the spectrum of the Hermitian matrix H ∈ H N .

Topological expansion of the action functional
In the following, we rewrite the action functional S(D) , in a succinct form, as a summation over a finite set of (properly defined) equivalence classes of surfaces.
We start by recalling the notion of a surface with polygonal boundary.A compact, connected, oriented surface C is referred to have n polygonal boundary components of perimeters with a cellular decomposition into ℓ i 0-cells and ℓ i 1-cells.We refer to the 1-cells in each connected component of ∂C as the sides of that polygon.We define an equivalence relation between surfaces with polygonal boundaries in the following way: Definition 3.1.Two compact, connected, oriented surfaces C 1 , C 2 with polygonal boundaries are considered equivalent if there exists an orientation-preserving diffeomorphism F : C 1 → C 2 which restricts to a cellular homeomorphism f : ∂C 1 → ∂C 2 whose inverse is also a cellular map.
The set C of equivalence classes of compact, connected, oriented surfaces with polygonal bound-aries is in bijective correspondence with the set where g denotes the genus of the corresponding closed surface.Inspired by [4], we refer to the combinatorial data (g; ⃗ ℓ ), and its corresponding equivalence class [C] ∈ C of surfaces with polygonal boundaries, as the elementary 2-cell of type (g; ⃗ ℓ ).
We isolate the free part of the action functional and denote it by Consider the following two sets: We rewrite S unstable (D) in the following from where t 1,1 = −1/t , and for (ℓ For each 1 ⩽ s ⩽ g , and 0 ⩽ m ⩽ s , let where two (s + m)-tuples are considered equivalent if there exists a permutation σ ∈ S s+m which maps one to the other.Consider the set We rewrite S stable (D) in the following form where, for each ⃗ ℓ ∈ L s,m , t = t s−m α⃗ ℓ , and α⃗ ℓ is a finite linear combination of the formal parameters αn I 's with integral coefficients.
Consider the following two sets of elementary 2-cells: We identify the set with the corresponding set of equivalence classes [C] of surfaces with polygonal boundaries.We assign a Boltzmann weight, equal to t , to each elementary , ⃗ ℓ ∈ N n ↑ , let where P ℓ (H) is defined by (3.3).We rewrite (3.6) and (3.9), respectively, in the following from where β 0 (∂C) denotes the zeroth Betti number, i.e. the number of connected components, of the boundary ∂C of a surface C .Thus, we get: Proposition 3.1.The action functional S(D) for the random matrix geometries of type (1, 0) , given by (2.6), can be decomposed in the following form where and C is given by (3.12). 4 The corresponding 1-Hermitian matrix model From now on, we consider the multi-trace 1-Hermitian matrix model corresponding to the random matrix geometries of type (1, 0) with the distribution dρ = e −S(D) dD , in the sense of formal matrix integrals.In other words, we treat the term S int (H) in (3.16) as a perturbation of S 0 (H) .
Consider the normalized Gaussian measure over H N with total mass one.Here Denote by t the sequence of Boltzmann weights t in S int (H) .We consider as a formal power series in t , i.e. an exponential generating function.The partition function Z N of the model is defined by where the second integral is understood in the sense that we expand Φ(H) as a power series in t , and interchange the integration with the summation.
of the model are defined as the joint moments of i.e.
where (x j 1 N − H) −1 denotes the resolvent of H . 7 The connected n-point correlators n ⩾ 1 , of the model are defined as the joint cumulants of X j 's, i.e.

Topological expansion of the correlators
where W g,n (x 1 , • • • , x n ) is defined, in the following, by (4.17). 8  Proof.We use the Wick's Theorem and the techniques of [10] to relate the formal matrix integrals in our model to the combinatorics of stuffed maps [4], [9].Considering (4.6) and (4.7), the computation To compute (4.11) using Wick's Theorem, we represent each term of the form , in (4.11), by a marked face of perimeter ℓ j with Boltzmann weight x −(ℓ j +1) j , 7 Strictly speaking, in the context of formal matrix integrals, one works with the formal series 8 One should not misinterpret (4.10) as the asymptotic expansion of the connected correlators as N → ∞ (see [20], [8]).j = 1, • • • , n .9Also, we represent each term of the form in (4.11), by a surface C i of Euler characteristic χ(C i ) and Boltzmann weight t  and we have Since, for a connected closed stuffed map M = (S, G) of genus g with n marked faces, we get (4.10),where stuffed maps with n marked faces of perimeters ℓ j , j = 1, • • • , n .By (4.17), the generating se- ] of the stuffed maps, corresponding to our model, of genus g with n polygonal boundaries of perimeters

Large-N spectral distribution
Using (4.18), we see that the generating series Q 0;ℓ , ℓ ∈ N , of the rooted planar stuffed maps with topology of a disk and perimeter ℓ, is given by where t denotes the sequence of Boltzmann weights t If the Boltzmann weights t (or, equivalently, the formal parameters given values, then there exists a critical temperature t c > 0 such that, for any |t| < t c , we have . From now on, we restrict ourselves to the case 0 < t < t c , where t c is specified according to each set of given values to α n 's.Hence, we have the following one-cut Lemma [4], [5]: For given values to the Boltzmann weights t , and 0 < t < t c , the series is the Laurent expansion at x = ∞ of a holomorphic function, denoted by W 0,1 (x) , on C\Γ , where , t .The limits lim ϵ→0 + W 0,1 (s ± iϵ) , ∀s ∈ Γ o exist, and the jump discontinuity assumes positive values on the interior Γ o of the discontinuity locus Γ, and vanishes at ∂Γ.
Consider the measure µ = φ(s) ds on R , where ds denotes the Lebesgue measure, and φ(s) is given by (4.21).By the Sokhotski-Plemelj Theorem, the function In addition, consider the empirical spectral distribution (empirical measure) on R , where {λ i } N i=1 , λ i ∈ R , denotes the eigenvalues of the random Hermitian matrix H ∈ H N corresponding to the random matrix geometries of type (1, 0) with the distribution dρ = e −S(D) dD . 12y (4.6), we have Motivated by (4.10), we assume that lim Therefore, considering (4.22) and (4.24), the expected distribution of the eigenvalues {λ i } N i=1 is given by up to terms with exponential decay, as N → ∞ . 13We refer to the measure µ = φ(s) ds as the large-N spectral distribution.In addition, from now on, we assume that the sequence of Boltzmann weights t , and the parameter t , are tame, in the sense of [4], Definition 4.1.Hence, each a priori defined as the generating series of stuffed maps, upgrades to a holomorphic function on (C\Γ) n which has a jump discontinuity when one of x i 's crosses Γ .

Schwinger-Dyson equations
Our main tool for analyzing the W g,n (x 1 , • • • , x n )'s is an infinite system of equations, called the Schwinger-Dyson equations (SDEs), satisfied by the n-point correlators of the model.In the matrix model framework, they were introduced by Migdal [28], and referred to as the loop equations.There are several versions of SDEs for matrix models (and some other closely related models in statistical physics, e.g., the β-ensembles) in the literature (see, e.g., [20], [7]).However, the root of all of them is the invariance of the integral of a top degree differential form under a 1-parameter family of orientation-preserving diffeomorphisms on a manifold.
To put the above-mentioned differential geometric fact in a precise form, consider an oriented connected Riemannian n-manifold M with the Riemannian volume form ω .Let V be a smooth vector field on M with a local flow ϕ t : M → M .Consider a smooth function Ψ : using Cartan's magic formula and Stokes' theorem, we get In (5.2), the exterior derivative, the interior product by V , and the divergence of V are denoted by d , ι V , and div(V ) , respectively.
The Schwinger-Dyson equations for the multi-trace 1-Hermitian matrix models are derived in [4], using Tutte's decomposition applied to the stuffed maps.In this section, we give a proof of them based on the above-mentioned differential geometric fact. 14onsider the action of the unitary group U where For each 1 ⩽ k ⩽ 2g , let (5.7) Because of the symmetry of (μ N ) k , we can replace Tk (s in (5.8).We refer to the symmetric polynomials actions.We will need the following technical lemma on derivatives of T k in the proof of SDEs: Lemma 5.1.For each fixed element λ of E = {λ i } N i=1 , we have where (5.11) and, for each 2 ⩽ r ⩽ k , let Λr = Since T k is a symmetric polynomial, we have and k (λ I ) , ∀λ I ∈ Λr .

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We rewrite the measure dρ = e −S(D) dD over H N in the following from dρ = e −S(D) dD = exp By the Weyl integration formula, the measure dρ induces a measure dρ , given by on the space R N of eigenvalues of Hermitian matrices.In (5.19), ∆(λ) denotes the Vandermonde determinant, i.e.
and c N = 2 . 15Let where Γ ⊂ R is a strict ϵ-enlargement of the support Γ of the large-N spectral distribution µ , i.e.
Γ ⊂ Γo , and Γ\Γ has small Lebesgue measure.We assume that if we replace dρ with dϱ = 1 Ω dρ (5.22) in the definition of the partition function and the correlators, then they get modified by terms of exponential decay as N → ∞ .
where Proof.Let {τ j } n−1 j=1 be a sequence of parameters.Consider the random variables X given by (5.24) Let P ⃗ τ be the probability measure on R N defined by where f ∈ C 0 (R N ) , and dϱ . (5.26) We denote the joint cumulants (resp.moments) of {X j } r j=1 with respect to the probability measure (5.27) Let Ω be the compact subset of R N given by (5.21).Consider the smooth vector field where ⃗ e i , i = 1, • • • , N , denote the standard constant unit vector fields on R N .One gets the rank n Schwinger-Dyson equation for the connected correlators , by considering the invariance of under the flow of the vector field V .
We rewrite Z ⃗ τ N in the following form where Ψ(λ) = 3 m=1 ψ m (λ) is given by and (5.33)By (5.2), the invariance of Z ⃗ τ N under the flow of V is equivalent to up to the boundary term. 16sing the Cauchy integral formula, the term dψ 1 (V )/ψ 1 can be expressed in the following way: By interchanging the integration on R N with the contour integral, we get (5.36) Considering Lemma 5.1, we have where the integration is a k-times iterated contour integral along C Γ .Thus, using (4.14), we have (5.38) By similar steps, we get and (5.40) By substituting (5.36), (5.38), (5.39), and (5.40) into (5.34),we get where Considering the definition of joint cumulant, for given finite subsets (5.42) Therefore, by taking the derivative of each term of (5.41), we get (5.23).

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The system of Schwinger-Dyson equations is not closed in the sense that, for each n ⩾ 1 , the rank n SDE gives an expression for However, we will see that they are "asymptotically" closed as N → ∞ , and we can solve them to find the coefficients W g,n (x 1 , • • • , x n ) of the large N expansion of the correlators.
For each 1 ⩽ k ⩽ 2g , and h ⩾ 0 , let where g(C) denotes the genus of a surface C .The k-point interactions T k (s 1 , • • • , s k ) of the model can be rewritten in the following form where the summation includes finitely many terms.Considering (5.44) and (4.10), for each n ⩾ 1 , g ⩾ 0 , the rank n Schwinger-Dyson equation to order N 3−2g−n gives: Before continuing, we introduce the following notations which are used in this article.Let V ⊂ X be an open subset of a Riemann surface X.We denote by O, O * , M , Ω, and Q, the sheaves on X defined by

Spectral curve
We start by analyzing the rank one Schwinger-Dyson equation to leading order in N , i.e. the Equation (5.45) for n = 1 and g = 0 , given by Recall that where V(ξ) is given by (2.7), and We fix a simply-connected open neighborhood U ⊂ C such that Γ ⊂ U .Consider the following integral operator, called the master operator [4], with the kernel We rewrite (6.1) in the following form where Since W 0,1 (x) ∈ O(C\Γ) , using (4.21) and (4.20), we have where m ℓ = s ℓ φ(s) ds , ℓ ⩾ 1 , (6.9) denotes the moments of the large-N spectral distribution µ = φ(s) ds , and Q 0;ℓ is given by (4.19).
Thus, the polynomial Q(ξ) can be expressed in terms of m ℓ 's in the following way: Considering W 0,1 (ξ) = O(1/ξ) as ξ → ∞ , we rewrite the contour integral in (6.6) as: where denotes the noncommutative derivative, aka "finite difference quotient", of Q(ξ) .The polynomial has the following expression in terms of m ℓ 's: where the polynomials Q(x) and P (x) are given by (6.10) and (6. of U \Γ under the Joukowski map x has two connected components V e ⊂ C\D and V i ⊂ D , whose common boundary is the unit circle T (see Figure 1).The exterior neighborhood V e is mapped to the interior neighborhood V i under ι : z → 1/z .By Proposition 6.1 and Lemma 4.2, the function W 0,1 (x) ∈ O(U \Γ) can be expressed in the following way where M (x) ∈ O * (U ) , and Γ = [a, b] ⊂ R is the support of the large-N spectral distribution µ .

Since
considering (6.17), the function W 0,1 (x(z)) has an analytic continuation to Consider the open neighborhood Υ = D\V i of the point z = 0 .The Riemann surface which is homeomorphic to a disk, is called the spectral curve of the model.Let be the restriction of the Joukowski map to Σ .The map x| V : V ⊂ Σ → U is a two-sheeted ramified covering map with the ramification points and the branch points x = a , x = b .In addition, the spectral curve Σ is equipped with a local biholomorphic involution Using the Schwinger-Dyson equations (5.45) recursively, it can be shown [20] that each W g,n (x, x I ) for fixed (x i ) i∈I ∈ C\Γ , initially defined as a holomorphic function on C\Γ , has a meromorphic continuation W g,n (x(z 1 ), x I ) ∈ M (Σ) to Σ .By doing the same process for the other arguments x i , Let K Σ → Σ be the canonical line bundle, i.e. the holomorphic cotangent bundle, on the spectral curve Σ. Denote by π i : Σ n → Σ the projection map onto the i-th component.Let be the n-times external tensor product of K Σ , where π * i K Σ denotes the pullback of K Σ under π i .The sections of the holomorphic line bundle K ⊠n Σ → Σ n are referred to as the differentials of degree n over Σ n .A differential of degree n is called symmetric if it is invariant under the natural action of the symmetric group S n on the line bundle K ⊠n Σ → Σ n .In the theory of (blobbed) topological recursion [21,6,4], one constructs meromorphic symmetric differentials of degree n from the meromorphic functions W g,n (x(z 1 ), , where dx(z i ) denotes the pullback π * i (dx) of the 1-form dx under π i : Σ n → Σ .In (6.24), the bidifferential, i.e. differential of degree two, is the pullback of the fundamental symmetric bidifferential of the second kind with biresidue 1 over Before continuing, we introduce several operators in the following which are used in our investigation of the differentials ω g,n (z 1 , • • • , z n ) .Denote by P + and P − the orthogonal idempotents corresponding to the involution ι : V → V , given by (1 + ι * ) , and respectively, where ι * denotes the pullback under ι .Let P± = 2 P ± .( The domain of the operators P ± can be M (V ) or Q(V ) , depending on the context which they are used.For a fixed ϵ > 0 , consider the closed counter-clockwise-oriented contour , and Bires|∆ B = 1 , where ∆ ⊂ X 2 denotes the diagonal divisor.
• For each p ∈ X, the 1-form B(p, •) has vanishing ai-periods, where {ai, bi} g i=1 is the symplectic basis of H1(X, Z) specifying the Torelli making of X.
By considering (7.2) as x 1 → s ± iϵ , s ∈ Γ o , it can be shown [4] that Using the identity theorem for holomorphic functions, and the Riemann's removable singularities theorem, we deduce from (7.4) that for all z ∈ V ⊂ Σ .Considering (7.5), we get the following equation for the bidifferential where B0 (z, ζ) is given by (6.25).
Consider the function where In the remaining part of this section, we consider the following set of assumptions, and derive an explicit expression for F ζ (z) .To get an explicit expression for the meromorphic function F ζ (z) , z ∈ CP 1 , we find the principal part of the germ of F ζ (z) at its poles on CP 1 , and then analyze the corresponding Mittag-Leffler problem.More precisely, we consider the following long exact sequence of Čech cohomology groups It is well known that Therefore, given a section f ∈ H 0 (CP 1 , M /O) , there exists a meromorphic function f ∈ H 0 (CP 1 , M ) , unique up to a constant, whose local singular behavior is given by f .
The meromorphic function F ζ (z) does not have any poles in CP 1 \D because the simple zeroes of x ′ (z) at the ramification points R cancels out the simple poles of W 0,2 (x(z), x(ζ)) .We use (7.6) to analyze the possible poles of F ζ (z) in D .Rewrite the function T 0,2 (ξ, η) , given by (6.3), in the following form where each ν k,m is a linear combination of the Boltzmann weights t where, for each 1 (7.15) We recall the following basic fact from the theory of projective connections on Riemann surfaces (see, e.g., [31]).
Let v = v − p , and ŵ = w − p .By expanding E f (v, w) as a series in v and ŵ , one gets as v, ŵ → 0 , where is called the Schwarzian derivative of f , and H(v, ŵ) is a sum of terms in v and ŵ of strictly positive degree.
Considering (7.15), the function F ζ (z) has a pole of order two at z = ι(ζ) .Using (7.17), we have as z → ι(ζ) .Thus, the principal part of the germ of In addition, by (7.15), the function F ζ (z) has a pole at z = 0 .Since W 0,2 (x(z), x(ζ)) has a zero of order 2 at z = ∞ , the term F ζ (ι(z))/z 2 , in (7.15), is regular at z = 0 .Therefore, the principal part of the germ of F ζ (z) at z = 0 is the same as the principal part of the Laurent polynomial and is given by where By the Mittag-Leffler theorem, we have leads to a linear equation of the following form satisfied by the x i 's: where the coefficients C ij depend only on b , and t We assume that the given values to the Boltzmann weights t Hence, we get the explicit expression of has only a pole of order 2 at ∆ , and its singular behavior is given by where the orthogonal idempotents P ± : F → F± are given by (6.27).
Proof.We rewrite the operator K in the following form Since K = K ι * , we have (K ± K)P ∓ = 0 .(8.12)

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In the remaining part of this section, we try to explain some elementary aspects of the blobbed topological recursion formula, which will be discussed in the next section, in a simpler setup.Consider the operator where the operators P+ and Ô are given by (6.28) and (6.30), respectively.Let P ⊂ V be a fixed finite set of points in V ⊂ Σ such that P ∩ γ = ∅ .Suppose we are given a set of germs s i ∈ Q p i /Ω p i , at the points p i ∈ P , i = 1, • • • , |P | , and an ι-invariant holomorphic 1-form ψ ∈ Ω inv (V ) .Denote by E ⊂ H 0 (Σ, K Σ ) the subspace of holomorphic 1-forms η on Σ whose restriction to V ⊂ Σ satisfies the homogeneous equation

.14)
Let A ⊂ Q(Σ) be the set of meromorphic 1-forms ϕ on Σ whose restriction to V ⊂ Σ satisfies the inhomogeneous equation and their singularities satisfy

.17)
The set A ⊂ Q(Σ) is an affine space over the vector space E .In the following, we investigate the space A .
The operator K maps the space A to the meromorphic 1-form ϕ 0 ∈ Q(Σ) , given by where each si ∈ Q p i is a representative of s i ∈ Q p i /Ω p i , ∀p i ∈ P .Using (8.15), we get Thus, by Lemma 8.1, we have Proof.Considering (7.6) and (7.37), since Ω inv (V ) ⊂ ker( Ô) , we have By decomposing the operator K 0 in the same way as in Lemma 8.1, using (8.19), we get On the other hand, we have . Thus, we get (8.28).
be the image of A under the operator 1 − K .Considering Proposition 8.2, the space A h consists of holomorphic 1-forms η on Σ whose restriction to V ⊂ Σ satisfies Therefore, A h is an affine subspace of H 0 (Σ, K Σ ) over the vector space E .
Lemma 8.3.The holomorphic 1-form is an element of the affine subspace A h ⊂ H 0 (Σ, K Σ ) , 21 where the boundary ∂Σ of the spectral curve Σ is assumed to have the induced orientation from Σ, i.e. it is clock-wise oriented.

Blobbed topological recursion formula
In this section we show that all the stable coefficients W g,n (x 1 , • • • , x n ) of the large N expansion of the correlators of our model can be computed recursively using the blobbed topological recursion formula [4].In the following, without further explicit mention, a couple (g, n) is assumed to be stable, i.e. (g, n) ̸ = (0, 1), (0, 2) .
Let f (x) ∈ O(C\Γ) be a holomorphic function on C\Γ with a jump discontinuity on Γ.We denote by δf (s) and σf (s) the functions given by δf (s) = lim where δ s,2 (resp.σ s,1 ) means that the operator δ (resp.σ) is acting on the second (resp.first) argument.In (9.3), the functions V g,n (x; x I ) , called the potentials for higher topologies [4], 22 are given by V g,n (x; x I ) = δ g,2 δ n,1 T 2,1 (x) Considering (6.17), the zeroes of the 1-from P− ω 0,1 (z) in V ⊂ Σ occur only at the ramification points R , and their order is exactly two.Since the quadratic differential Qg,n (z; z I ) has double zeroes at R , the 1-form 1 P− ω 0,1 (z) Qg,n (z; z I ) (9.13) is holomorphic on V .Therefore, the singularities of ω g,n (z, z I ) in V is the same as the 1-form 1 P− ω 0,1 (z) E g,n (z, ι(z); z I ) .

Proposition 4 . 1 .
The connected n-point correlators W n (x 1 , • • • , x n ) of the random matrix geometries of type (1, 0) with the distribution dρ = e −S(D) dD have a large N expansion of topological type, given by The orientation on each C i induces an orientation on ∂C i .Let Ξ be the collection of all surfaces representing the terms in(4.11)  in the above-mentioned way.Consider a pairing σ on the set of sides of the connected components of the boundary of surfaces in Ξ .We glue the surfaces in Ξ along the sides of their boundary, according to σ, such that the gluing map reverses the orientation.The resulting stuffed map M = (S, G) consists of an oriented, not necessarily connected surface S , and a graph G embedded into S .10We denote by S the surface which one gets by deleting the marked faces from S .It can be shown that each vertex (resp.edge) of G contributes a weight N (resp.t/N )[10].In addition, each unmarked connected component U of S\G contributes a weight (N/t) χ(U) .Hence, the exponent of N , in the total contribution corresponding to M = (S, G) , equals χ( S) . 11n addition to the pre-mentioned Boltzmann weights assigned to the 2-cells of M , we assign a Boltzmann weight equal to t to each vertex of G .The total Boltzmann weight of the isomorphism class [M ] of a stuffed map M , denoted by Bw([M ]), is defined to be the product of all Boltzmann weights assigned to the cells in M divided by the order |Aut(M )| of the automorphism group of M .The contribution of the isomorphism class [M ] of a Boltzmann-weighted not necessarily connected

. 13 )
Let M g,n (C) be the set of isomorphism classes of the Boltzmann-weighted connected closed stuffed maps M = (S, G) of genus g with n marked faces, such that the equivalence class of each unmarked connected component of S\G (in the sense of Definition 3.1) is in C .Considering (4.13), N on H N by conjugation, i.e. u • H := uHu −1 for u ∈ U N and H ∈ H N .Since the action functional S(D) : H N → R is invariant under the prementioned action of U N on H N , we can rewrite S(D) as a function of the eigenvalues {λ i } N i=1 , λ i ∈ R , of the random Hermitian matrix H ∈ H N .Let (μ N ) k := μN × • • • × μN k-times (5.3) be the product measure on R k corresponding to the unnormalized empirical measure μN = N i=1 δ λ i (5.4) on R .For each elementary 2-cell [C] ∈ C of type (g; ⃗ ℓ ) , ⃗ ℓ ∈ N k ↑ , we rewrite T [C] (H) , defined by (3.13), in the following from

Theorem 5 . 2 .
For any x, (x i ) i∈I ∈ C\ Γ, the rank n Schwinger-Dyson equation for the connected correlators of the model (up to the boundary term) is given by and C Γ is a closed counter-clockwise-oriented contour in an ϵ-tubular neighborhood of Γ which encloses Γ .
. Consider the Riemannian manifold R N with the Euclidean metric, and the Riemannian volume form dλ := N i=1 dλ i .

. 14 )Proposition 6 . 1 .Figure 1 :
Figure 1: Illustration of the Joukowski map in the case where the boundary of U is an ellipse with the foci x = a and x = b .The open neighborhoods U , V e , and V i are colored green, brown and yellow, respectively.

16 )
14), respectively.The coefficients of Q(x) and P (x) depend on the Boltzmann weights t (0) ⃗ ℓ , and the moments m ℓ of the large-N spectral distribution µ .Let the interval Γ = [a, b] ⊂ R , and the open neighborhood U ⊂ C , Γ ⊂ U , be the same as the above-mentioned ones.We recall the Joukowski map x : C\{0} → C given by Denote by T and D the unit circle and the open unit disk in C , respectively.The preimage x −1 (U \Γ)

Hypothesis 7. 1 .
(i) For each fixed ζ ∈ C\D , the function F ζ (z) has a meromorphic continuation to the whole CP 1 ; (ii) The support Γ of the large-N spectral distribution µ of the model is of the form Γ = [−b, b] ⊂ R . 19 Let D ⊂ C be an open neighborhood of a point p ∈ C .Let f be a biholomorphism on D .Consider the holomorphic function E f (v, w) on D × D given by

. 24 )
Since the function F ζ (z) has a zero of order two at z = ∞ , the constant function c(ζ) should be equal to zero.To get an explicit expression for F ζ (z) in terms of z, ζ , it suffices to find b k (ζ)'s, as a function of ζ , explicitly.Note that each b k (ζ) is a linear combination of the Fourier coefficients a n (ζ) = 1 2πi γ F ζ (z) z n+1 dz , n ∈ Z , (7.25) of the restriction of F ζ (z) to γ .Considering the degree of the Laurent polynomial Q 2 (z, ζ), it suffices to find only x i = a −i−1 (ζ) , 1 ⩽ i ⩽ d − 1 .For each 1 ⩽ i ⩽ d − 1 , the identity equivalently, the formal parameters α n , 3 ⩽ n ⩽ d) of the model are such that the above-mentioned matrix (C ij ) is invertible.

■Lemma 8 . 4 .
.34) for each z ∈ V ⊂ Σ .Therefore, we get T η 0 = ψ .(8.35)In general, the subspace E ⊂ H 0 (Σ, K Σ ) is non-trivial, and its dimension depends on the given values to the Boltzmann weights t (0) ⃗ ℓ (or, equivalently, the formal parameters α n , 3 ⩽ n ⩽ d) of the model.Proof.For simplicity, we give a proof in the case where the support Γ of the large-N spectral distribution µ of the model is of the form Γ = [−b, b] ⊂ R .Consider a holomorphic 1-form η = f (z) dz ∈ E .We have