Formal multidimensional integrals, stuffed maps, and topological recursion
FFormal multidimensional integrals, stuffed maps, andtopological recursion
Ga¨etan Borot MPI f¨ur MathematikVivatsgasse 7, 53111 Bonn, Germany
Abstract
We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governedby the topological recursion of [EO07a] with initial condition. In terms of a d gas of eigenvalues, thismodel includes – on top of the squared Vandermonde – multilinear interactions of any order between theeigenvalues. In this problem, the initial data p ω , ω q of the topological recursion is characterized: for ω ,by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus Γ to determine ; for ω , by arelated but linear, non-local Riemann-Hilbert problem on the discontinuity locus Γ . In combinatorics, thismodel enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology – ω being thegenerating series of disks and ω that of cylinders. In particular, by substitution one may consider mapswhose elementary cells are themselves maps, for which we propose the name ”stuffed maps”. In a sense,our results complete the program of the ”moment method” initiated in the 90s to compute the formal { N in the one hermitian matrix model. It is well-known that the large N expansion of the partition function and correlation functions in a N ˆ N hermitian matrix model with measure:d µ p M q “ d M e ´ NV p M q (1.1)is governed by a topological recursion [ACM92, ACKM93, ACKM95, Eyn04]. This topological recur-sion takes a universal form and it goes far beyond the realm of matrix models. Eynard and Orantinhave defined it axiomatically in the context of algebraic geometry [EO07a], and in this form, it enjoysmany interesting properties (symplectic invariance, special geometry, WDVV equations, . . . ), and hasappeared provably or experimentally in many problems of 2D enumerative geometry: the two her-mitian matrix model [EO08] and the chain of hermitian matrices [CEO06], topological string theoryand Gromov-Witten invariants [BKMP09, BEMS10, EMS09, MP12, NS11, EO12], integrable systems[BE09, BE10, BE11], intersection numbers on the moduli space of curves [EO07b, Eyn11b, Eyn11a],asymptotic of knot invariants [DFM11, BE12, BEM12], . . .In this article, we extend the range of applicability of the topological recursion, by showing itgoverns (in the same universal form) the large N expansion of formal hermitian matrix integrals [email protected] a r X i v : . [ m a t h - ph ] S e p ased on the measure:d µ p M q “ d M exp ´ ÿ k ě h ě p N { t q ´ h ´ k k ! Tr T hk p M p k q , . . . , M p k q k q ¯ , (1.2)where M p k q i “ N b ¨ ¨ ¨ b M b ¨ ¨ ¨ N is a k -th tensor product where M appears in i -th position, and N is the identity matrix. It induces the following measure of eigenvalues of M :d µ p λ , . . . , λ N q “ Vol p U p N qq N ! p π q N N ź i “ d λ i ź ď i ă j ď N p λ i ´ λ j q exp ´ ÿ k ě h ě p N { t q ´ h ´ k k ! N ÿ i ,...,i k “ T gk p λ i , . . . , λ i k q ¯ . (1.3)This is a generalization of the result obtained for arbitrary 2-point interaction (i.e. T hk ” p k, h q ‰ p , q , p , q ) in a recent work with Eynard and Orantin [BEO13]. As we explain in Section 2,the dependence in N of the measure (1.2) is the natural choice in order to have an expansion oftopological nature.We consider in the model (1.2) the partition function: Z “ µ r s “ ˆ d µ p M q , (1.4)and the n -point correlation function: W n p x , . . . , x n q “ µ ” ś nj “ Tr x j ´ M ı c µ r s , (1.5)where the subscript c stands for ”cumulant” expectation value. In the context of formal matrixintegrals, they have by construction a decomposition of the form: Z exp ´ ÿ g ě p N { t q ´ g F g ¯ , (1.6) W n p x , . . . , x n q “ ÿ g ě p N { t q ´ g ´ n W gn p x , . . . , x n q . (1.7)The precise definitions will be given in Section 2. Our main result are Theorems 4.2 and 4.3, fromwhich follows Theorem 5.1, which can be stated informally but with assumptions as follows: Proposition 1.1
If the parameters of T hk are tame (see Definition 4.1), then all W gn p x , . . . , x n q d x ¨ ¨ ¨ d x n can be analytically continued to meromorphic n -forms on C n for thesame Riemann surface C , can be computed by a recursion on g ´ ` n ą , which coincide up to aninitial condition Φ gn p z, z I q with the topological recursion of [EO07a]. The initial data of this recursionis W and W . The tame condition is here the analog of an ”off-criticality” condition in the context of random matrixtheory.
Beyond the effort to develop a complete theory of the topological recursion, let us motivate the studyof models (1.2).It is well-known that formal hermitian matrix integrals with measure (1.1) enumerates maps, i.e.discrete surfaces obtained by polygonal faces with the topology of a disc along their edges. V p x q is2 generating series for the Boltzmann weight of such 2-cells. The large N expansion of the partitionfunction and the correlation functions in these models collect maps of a given topology. Similarly,we show in Section 2 that formal matrix integrals with measure (1.2) enumerate stuffed maps , i.e.maps obtained by gluing 2-cells having the topology of a Riemann surface of genus h with k polygonalboundaries. T hk p x , . . . , x k q is a generating series of such 2-cells. Usual maps carrying self-avoidingloop configurations – the so-called O p n q model, introduced in a special case by [K89] – are equivalentto stuffed maps where the elementary cells may have the topology of a disc (usual faces) or of acylinder (rings of faces carrying the loops). Usual maps with configuration of possibly intersectingloops can also be represented by stuffed maps. Therefore, the result of this article applies to manycombinatorial models studied previously on usual random maps. In a sense, our results completes theprogram of the moment method [ACM92, ACKM93, ACKM95, EK95] initiated in the context of 2dquantum gravity to compute the large N expansion in the one hermitian matrix model (1.1). Ourresult is that the same method, put in the form of the topological recursion [EO07a], applies to allmulti-trace 1 hermitian matrix models.In [GM06], Mari˜no and Garoufalidis claim that, for any closed 3 manifold obtained by filling in aknot K , the U p N q evaluation of the LMO invariants of a 3-manifold can be computed from the U p N q Kontsevich integral of K , which is a formal 1 hermitian matrix model, i.e. of the form (1.2) for certain(in general non-explicit) weights t p X q . This indicates that the large N expansion of those invariantsshould be described by a topological recursion. This will be the matter of a forthcoming work.The convergent version of (2.8) – when it is well-defined – describes a system of N repulsiveparticles with position λ , . . . , λ N , which may have arbitrary k -point interactions. Such integralsfrequently appear in the computation of correlation functions in quantum integrable systems, afterapplying Sklyanin’s quantum separation of variables (see for instance [KKN] in the example of theXXX spin chain and references therein). We first define the formal model 1.2 and the combinatorics of stuffed maps (Section 2), describe theirnester structure in the case of planar maps and analyze some consequences (Section 3). Then, wewrite down the Schwinger-Dyson equations satisfied by the correlation functions (1.5) (Section 4).They are equivalent to functional relations for generating series of stuffed maps, which can be given abijective proof by Tutte’s method. Their analysis (Section 4.2-4.4) shows that W gn have the same typeof monodromies around their discontinuity locus. More precisely, they satisfy a hierarchy of linearloop equations in the terminology of [BEO13] (Theorem 4.2). Then, the Schwinger-Dyson equationscan be recast as quadratic loop equations (Section 4.5, Theorem 4.3), and we can conclude in Section 5using the results of [BEO13] that W gn for 2 g ´ ` n ą p g, n q “ p , q –by the topological recursion (Theorem 5.1).In practice, this reduces the problem of computing the sequence p W gn q n,g to the problem of comput-ing W and W . We show that W is characterized by a scalar non-linear, non-local Riemann-Hilbertproblem with a unknown jump locus Γ (see (3.6)), whereas W is characterized by a related but linear,non-local Riemann-Hilbert problem on Γ (see (3.21)). In general, it seems hopeless to find the solutionfor W p x q and W p x , x q in closed form, but they can easily be obtained recursively as power seriesin the parameters of T hk .The core of our computation is the analysis of the Schwinger-Dyson equation of Section 5 to showlinear and quadratic loop equations (Theorem 4.2 and 4.3), and is relevant both for convergent andformal matrix integrals. It explains why the topological recursion holds in the same universal form in3he class of models (1.2). The other technical details and assumptions are somewhat specific to thecase of formal matrix integrals to which we restrict in this article. In the convergent matrix model,the assumptions and technical steps are of different nature and are more involved, because one needsfirst to justify the existence of a large N expansion for an appropriate topology. In the more simpleconvergent model (1.1), the large N asymptotic expansion were established in the one-cut case in[APS01, BG12], and in the multi-cut case in [BG13] justifying the heuristics of [BDE00, Eyn09] undernatural assumptions on V . The generalization of this approach to the model (1.2) seen as a convergentmatrix model will be addressed in a subsequent work [BGK]. We recall the definition of formal matrix integrals, and describe its underlying combinatorics in termsof stuffed maps. If A is a ring, and t is a collection of variables, A rr t ss is the ring of formal series in t with coefficients in A , whereas A r t s is the polynomial ring of A . Let d M be the Lebesgue measure on the space of N ˆ N hermitian matrices H N , and µ be theGaussian measure: d µ p M q “ d M exp ´ ´ N Tr M t ¯ . (2.1)Let t “ p t h(cid:96) ,...,(cid:96) k q (cid:96),k,h a sequence of formal variables, assumed to be symmetric in (cid:96) , . . . , (cid:96) k . For any k ě h ě
0, we define a formal series depending on variables p “ p p (cid:96) q (cid:96) ě : r T hk p p q “ ÿ (cid:96) ,...,(cid:96) k ě t h(cid:96) ,...,(cid:96) k k ź i “ p (cid:96) i P C rr p ssrr t ss . (2.2)We introduce a exponential generating series: ψ p p q “ exp ´ ÿ k ě h ě p N { t q ´ h ´ k k ! r T hk p p q ¯ P C rr p ssrr t ss . (2.3)Given a matrix M , we will specialize those variables to: p (cid:96) r M s “ Tr M (cid:96) (cid:96) . (2.4)Then, we define the partition function Z and the free energy F as: Z “ µ “ ψ p p r M sqs µ r s P C rr t ss ,F “ ln Z P C rr t ss , (2.5)and the disconnected n -point correlation functions as: W n p x , . . . , x n q “ Z µ ” ψ p p r M sq n ź j “ Tr 1 x j ´ M ı P C rrp x ´ j q j ssrr t ss . (2.6)If I is a set with n elements, we use the notation W n p x I q “ W n pp x i q i P I q . The connected n -pointcorrelators W n p x , . . . , x n q can then be defined as the cumulant expectation values (instead of the4oments) of Tr 1 {p x j ´ M q : W n p x , . . . , x n q “ ÿ J $v ,n w r J s ź i “ W | J i | p x J i q , (2.7)where the sum runs over partitions of v , n w , and r J s denotes the number of subsets in the partition J . Formally, if we disregard the dependence in N that we chose in (2.3), d µ p M q ψ p p r M sq is the mostgeneral measure on the space of N ˆ N hermitian matrices which is invariant under conjugation. Wemay also diagonalize M and consider the measure induced on its eigenvalues λ , . . . , λ N : p d µ ¨ ψ qp λ , . . . , λ N q 9 N ź i “ d λ i ź ď i ă j ď N p λ i ´ λ j q exp ´ ÿ k ě h ě p N { t q ´ h ´ k k ! N ÿ i ,...,i k “ T hk p λ i , . . . , λ i k q ¯ , (2.8)where we have introduced the formal series: T hk p x , . . . , x k q “ ´ δ k, δ h, x t ` ÿ m ,...,m k ě t hm ,...,m k m ¨ ¨ ¨ m k x m ¨ ¨ ¨ x m k k . (2.9) We now introduce the combinatorial model behind (1.2). ‚ An elementary 2-cell of topology p k, h q and perimeters p (cid:96) , . . . , (cid:96) k q is a topological, orientable,connected surface of genus g , with boundaries B i (1 ď i ď k ) endowed with a set V i Ď B i of (cid:96) i ě B i z V i are considered as edges . ‚ A stuffed map of topology p n, g q and perimeters p (cid:96) , . . . , (cid:96) k q is a orientable, connected, discretesurface M of genus g , obtained from n labeled rooted elementary 2-cells with topology of a discand perimeters (cid:96) , . . . , (cid:96) n , and from a finite collection of rooted unlabeled elementary 2-cells,by gluing pairs of edges of opposite orientation. The labeled cells are considered as boundariesof the stuffed map, and the rooting on edges which do not belong to the boundary of M areforgotten after gluing. We denote M g(cid:96) ,...,(cid:96) n this set of stuffed maps. ‚ We say that a elementary 2-cell (or a stuffed map) with boundaries is rooted when a markededge has been chosen on each boundary. By following the cyclic order, the rooting induces alabeling of the edges of the boundaries.For instance, p , q denotes the topology of a disc, p , q denotes the topology of a cylinder, etc. A map – in the usual sense – is a stuffed map made only of elementary 2-cells with topology of a disc.We assign a Boltzmann weight to stuffed maps in the following way: ‚ a weight t per vertex. ‚ a weight t h(cid:96) ,...,(cid:96) k per rooted elementary 2-cell, depending on its topology p k, h q , and on itsperimeters (cid:96) , . . . , (cid:96) k in a symmetric way. 5igure 1: An elementary 2-cell of topology p k “ , h “ q , with perimeters (cid:96) “ (cid:96) “ (cid:96) “ (cid:96) “
3. The corresponding Boltzmann weight is t , , , . ‚ a symmetry factor | Aut M | ´ , where M is a stuffed map in which all constitutive elementary2-cells have been labeled and rooted, thus inducing a labeling for all edges. The identificationof edges is thus represented by a permutation σ which is a product of transposition of the edgelabels. Aut M is the subgroup of permutations of elementary 2-cells labels and rooting, forwhich we get the same stuffed map after identification of the edges according to σ and forgettingall labels which do not decorate the boundary of M .By convention, the stuffed map consisting of only one vertex has 1 boundary of length 0, genus 0, andthus receives a weight 1. Out of a given finite collection of elementary 2-cells, one can only constructa finite number of stuffed maps. This allows the definition: F g “ ÿ M P M g H weight p M q P C rr t ss , (2.10) W gn p x , . . . , x n q “ δ n, δ g, tx (2.11) ` ÿ (cid:96) “p (cid:96) ,...,(cid:96) n q (cid:96) ,...,(cid:96) n ě ” n ź j “ x ´p (cid:96) j ` q j ı´ ÿ M P M g (cid:96) weight p M q ¯ P C rrp x ´ j q j ssrr t ss , where t and N are considered as variables, and t “ p t h (cid:96) q (cid:96) ,h as an infinite sequence of formal variables. Applying the standard techniques invented in [BIPZ78], we quickly review the connection betweenthe combinatorial model of § § µ is a Gaussian measure, Wick’s theorem allows the computation of the coefficients ofthe formal series ln Z , W n and W n defined in (2.5)-(2.6)-(2.7) as sums over Feynman diagrams, whichare fatgraphs. We claim that those fatgraphs are dual to stuffed maps. Indeed, we can representa monomial N ´ h ´ k Tr M (cid:96) ¨ ¨ ¨ Tr M (cid:96) k as a collection of k fatvertices, with (cid:96) i couples of ingoingedge/outgoing edge in cyclic order at the i -th fatvertex. The dual of this collection of fatvertices isa collection of k polygonal faces, with perimeters (cid:96) , . . . , (cid:96) k , which form the boundaries of a single6lementary 2-cell of topology p k, h q . By construction, ψ p p r M sq P C rr t ss defined in (2.3)-(2.4) is thegenerating series of collections of elementary 2-cells, with a weight deduced from § ‚ an extra weight p N { t q χ for each elementary 2-cell with Euler characteristics χ ; ‚ a symmetry factor corresponding 1 { k ! and 1 {p (cid:96) ¨ ¨ ¨ (cid:96) k q corresponding to labeling and rootingthe boundaries of the elementary 2-cells.When we compute the µ expectation value of product of monomials, the Wick theorem mimics thegluing rules of elementary 2-cells along edges of opposite orientations, and each pair of glued edgescomes with a weight t { N . Each vertex in the stuffed map correspond in the dual picture of fatgraphsto a line on which flows a matrix index i P v , N w , and thus receives an extra weight N . Taking intoaccount the symmetry factors, the classical argument of t’Hooft [t’H74] about Euler characteristicscounting implies that the generating series of stuffed maps coincide with the correlation functions inthe model (1.2): F “ ÿ g ě p N { t q ´ g F g , (2.12) W n p x , . . . , x n q “ ÿ g ě p N { t q ´ g ´ n W gn p x , . . . , x n q . (2.13)These equalities holds in C rr t ss (resp. C rrp x ´ j q j ssrr t ss ), meaning that for a given monomial in theformal variables t , only finitely many g ’s contribute to the sum.If all Boltzmann weights t, t h (cid:96) are non-negative, we may also define F g and the coefficients of ś j x ´p (cid:96) j ` q j in W gn as numbers in r , `8s . If the latter happens to be finite for given non-negativevalues t, t (cid:96) , they can also be defined as finite numbers for any real-valued weights t and p t h (cid:96) q so that | t | ď t and |p t h (cid:96) q | ď t h (cid:96) . We first focus on planar stuffed maps M with topology of a disc, i.e. p n, g q “ p , q . All theirconstitutive elementary 2-cells must also be planar ( h “ k boundaries, we end up with k connected components. One of them contains the root edge on theboundary, and is called the exterior , the other ones are tagged interior . The existence of a notion ofexterior and interior implies that planar stuffed maps have a nested structure, that we now describe(see Figure 2-3).The gasket M of M is the map obtained by removing all elementary 2-cells with k ě M of the root edge in M , and filling its holes having perimeter m with new elementary 2-cells with topology of a disc. We obtain in this way a usual map M withtopology of a disc, i.e. made only of elementary 2-cells having the topology of a disc. Some of themwere already 2-cells in M , and the other are called large faces . The gasket M does not contain allinformation about M . It can be retrieved by specifying the configuration in the interior of M Ď M .A hole in M was created by the removal of a planar elementary 2-cell with k ě cement M is planar, distinct holes were created by the removal of distinct 2-cells.The interior of a cement 2-cell can be seen as stuffed maps with topology of a disc, which we call chunks . 7e choose an arbitrary procedure to root the large faces of the gasket: among the points of a largeface γ which are the closest (for graph distance in M ) to the point at the origin of the boundary of M , we choose the one o reached by the leftmost geodesic, and we root γ on the edge with origin o .We also root the corresponding edge on the cement 2-cell filling this large face.Conversely, given a gasket, cement 2-cells rooted on all their boundaries and rooted chunks, we canreconstruct the map M by gluing. The root edge on the chunks and the root edge on the correspondingboundary of a cement 2-cell are identified in this process. This gluing is surjective, and if m i denotethe sequence of perimeters of the chunks, it is actually ś i m i to 1, since we must forget the roots onthe boundaries of the chunks. Let G (cid:96) r t, t s P C rr t ss be the generating series of stuffed maps with topology of a disc and perimeter (cid:96) , and G usual (cid:96) r t, t s P C rr t ss the analog for usual maps, obtained from G (cid:96) r t, t s by setting all t m ,...,m k with k ě G (cid:96) r t, t s “ G usual (cid:96) r t, τ p t, t qs . (3.1)The right-hand side is the generating series for the gasket, which is a usual map whose 2-cells wereeither present in the initial map (weights t ), or are large faces in which we glue a cement planar2-cell with k ě p k ´ q stuffed maps with topology of a disc. We are cautious toadd a symmetry factor to forget the roots on the chunks: τ m p t, t q “ t m ` ÿ k ě p k ´ q ! ÿ m ,...,m k ě t m,m ,...,m k m ¨ ¨ ¨ m k k ź i “ G m i r t, t s“ ÿ k ě p k ´ q ! ÿ m ,...,m k ě t m,m ,...,m k m ¨ ¨ ¨ m k k ź i “ G m i r t, t s (3.2) τ p t, t q represents a sequence of effective face weights allowing to enumerate planar stuffed maps asplanar usual maps. The properties of the generating series of planar usual maps G usual (cid:96) r t, τ s are wellknown, and by (3.1) they can be transferred to the generating series of stuffed maps G (cid:96) r t, t s .We recall the definition of admissible weights [BBG12]. For usual maps, a vertex weight t and asequence of non-negative face weights τ “ p τ , τ , τ , . . . q is admissible is for any (cid:96) ě
1, the generatingseries of pointed rooted maps with topology of a disc t B t G usual (cid:96) r t, τ s is finite. We also say that real-valued t, τ are admissible if | τ | “ p| τ | , | τ | , | τ | , . . . q is admissible. For stuffed maps, we will say thata vertex weight t and a sequence of elementary 2-cells weights t is admissible if the effective faceweights τ p t, t q are admissible. The admissibility condition is not empty: Lemma 3.1
When only a finite number of t m ,...,m k are non-zero and have given values, there exists t c ą so that, for any | t | ă t c , the weights t, t are admissible. Proof.
It correspond to have usual maps with bounded face degree. The existence of t c ą l As a consequence of [BBG12], for stuffed maps, we obtain a planar 1-cut lemma and a functionalrelation: 8igure 2: From top to bottom. First picture : a planar stuffed map with the topology of a disc.The orange arrow denote the root edge. We used different colors for elementary 2-cells of differenttopology. The outer face – peach color – is the marked face. Second picture : the gasket of this stuffedmap. The large faces appear in darker purple.
Lemma 3.2 If t, t is a sequence of admissible weights for planar elementary -cells, then G (cid:96) r t, t s ă 8 for all (cid:96) ě . The formal Laurent series: W p x q “ tx ` ÿ (cid:96) ě G (cid:96) r t, t s x (cid:96) ` (3.3) is the Laurent expansion at of a holomorphic function in C z Γ t, t , where Γ t, t is a segment of thereal line. Besides, W p x q has limits from above and from below on Γ t, t , remains bounded, and ρ p x q “ W p x ´ i0 q ´ W p x ` i0 q π (3.4) assumes positive values at interior points of Γ t, t , and vanishes at the edges l r V p x q of planar elementary 2-cells, whose boundaries are allglued to stuffed maps with topology of a disc, except one boundary which receives a weight x (cid:96) whenit has perimeter (cid:96) . We also include a shift and a sign for convenience: r V p x q “ ´ x t ` ÿ (cid:96) ě ÿ m ,m ,...,m r ě t m ,m ,...,m k m ¨ ¨ ¨ m k x m k ź j “ G m j r t, t s“ ÿ k ě ˛ T k p x, ξ , . . . , ξ k q k ź j “ d ξ j W p ξ j q π . (3.5) Lemma 3.3 If t, t is a sequence of admissible weights for planar elementary -cells, there ex-ists an open disc D t, t centered on and containing the interior of Γ t, t so that the formal series T k p x , . . . , x k q defines a holomorphic function for p x , . . . , x k q P D kt, t , and r V p x q defines a holomor-phic function for x P D t, t . Besides, for any x in the interior of Γ t, t , W p x ` i0 q ` W p x ´ i0 q ` B x r V p x q “ . (3.6) l (3.6) is a non-linear and non-local Riemann-Hilbert problem for W , with unknown discontinuitylocus Γ. We will discuss in § Definition 3.1 If U is an open set of the Riemann sphere, we define M p U q (resp. H p U q ) the spaceof meromorphic (resp. holomorphic) functions on Ω . An open set U Ď p C z Γ which is a neighborhoodof Γ is called an exterior neighborhood of Γ . Let us introduce a generating series of planar elementary 2-cells, in which all but two boundaries areglued to stuffed maps with topology of a disc: r R p x, y q “ ÿ k ě p k ´ q ! ˛ T k p x, y, ξ , . . . , ξ k q k ź j “ d ξ j W p ξ j q π , (3.7) R p x, y q “ ÿ k ě p k ´ q ! ˛ T k p x, y, ξ , . . . , ξ k q k ź j “ d ξ j W p ξ j q π . (3.8)The symmetry factor is the only difference between the two expressions. In order to work with analyticfunctions rather than formal series, we need slightly stronger assumptions. Definition 3.2 ‚ We say that admissible weights t, t are off-critical when B x T p x q is holomorphic in an openneighborhood of Γ t, t . ‚ We say that a sequence p τ m q m is regular when the formal series ÿ m ,...,m r ě τ m ,...,m k m ¨ ¨ ¨ m k x m ¨ ¨ ¨ x m k r P C rr x , . . . , x r ss defines a holomorphic function in D r , where D is an open neighborhood of Γ t, t . We say that admissible weights t, t are completely regular when it is admissible, off-critical, p t m ,...,m k q m is regular for any k ě , and moreover R p x, y q is holomorphic in D , where D isan open neighborhood of Γ t, t . Let t, t be completely regular weights, U be an open exterior neighborhood of Γ, and U be an openneighborhood of Γ. We can define a linear operator r O : H p U q Ñ H p U q by r O φ p x q “ ˛ Γ B x r R p x, ξ q φ p ξ q , O φ p x q “ ˛ Γ B x r R p x, ξ q φ p ξ q . (3.9)Besides, we also define the expressions: S φ p x q “ φ p x ` i0 q ` φ p x ´ i0 q , ∆ φ p x q “ φ p x ` i0 q ´ φ p x ´ i0 q . (3.10)Eqn. 3.6 can be rewritten: for any interior point x of Γ t, t , S W p x q ` r O W p x q ` B x T p x q “ . (3.11)Since the two last terms are holomorphic in a neighborhood of Γ and W p x q remains bounded, wededuce: Lemma 3.4 If t, t are completely regular, W p x q can be decomposed, at α “ a, b the edges of Γ t, t ,as h p x q ` h p x q? x ´ α where h , h are holomorphic in a neighborhood of α . l We start with some preliminaries about analytical continuation. Let Γ “ r a, b s be a segment of R .The domain p C z Γ can be mapped conformally to the exterior of the unit disc D by the Zhukovski map(see Figure 4): x p z q “ a ` b ` a ´ b ´ z ` z ¯ ðñ x p z q “ a ´ b ´ x ´ a ` b ` a p x ´ a qp x ´ b q ¯ . (3.12)The image of the unit circle U by x is r a, b s . We have a holomorphic involution ι p z q “ { z , which has z p a q “ z p b q “ ´ U .From now on, we prefer to work with differential forms rather than functions.Figure 4: Analytic continuation in the z -plane of functions of x via (3.12), ι p z q “ { z . Definition 3.3 If Ω Ď p C is an open set, M p Ω q (resp. H p Ω q ) is the space of meromorphic (resp.holomorphic) -forms in Ω . φ is a holomorphic function in an exterior neighborhood U of Γ, upon multiplication by d x it definesan element ϕ P H p Ω q , where Ω is the exterior neighborhood of U such that x p Ω q “ U . Similarly, if φ is a holomorphic function in a neighborhood U of Γ, it defines an element ϕ P H p Ω q with Ω “ z p U q is an open neighborhood of U stable under ι , and such that ϕ p z q “ ϕ p ι p z qq . We can thus define linearoperators O , r O : H p Ω q Ñ H p Ω q upgrading (3.9) to 1-forms in the z -plane. Besides, if Ω is an openneighborhood of U stable under ι , we may define S , ∆ : M p Ω q Ñ M p Ω q by: S ϕ p z q “ ϕ p z q ` ϕ p ι p z qq , ∆ ϕ p z q “ ϕ p z q ´ ϕ p ι p z qq . (3.13)The restriction of (3.13) to z P U , pulled-back by the map z , agrees with the definition (3.10) in termsof boundary values on Γ. We will apply repeatedly the following principle: Lemma 3.5
Let U be an exterior neighborhood of Γ , and φ P H p U q . Assume that φ has boundaryvalues on the interior of Γ “ r a, b s , that for any α P t a, b u there exists an integer r so that φ p x qp x ´ α q r α { remains bounded when x Ñ α , and that S φ p x q can be analytically continued as a holomorphicfunction in a neighborhood of Γ . Then, ϕ p z q “ φ p x p z qq d x p z q , initially a holomorphic -form in theexterior neighborhood Ω of U such that x p Ω q “ U , can be analytically continued to a meromorphic -form in an open neighborhood Ω of U which is stable under ι . For α “ ˘ , if r α ě , it has a poleof order atmost r α ´ at z “ α . l We assume that t, t are completely regular, and that the generating series of stuffed maps withtopology of a disc W p x q is known. It is considered as a holomorphic function on C z Γ for some segmentΓ Ď R . According to (3.11), S W p x q can be analytically continued as a holomorphic function in aneighborhood of Γ, and thanks to Lemma 3.4, we can apply Lemma 3.5 to define: W p z q “ W p x p z qq d x p z q (3.14)as a meromorphic 1-form in: Ω ε “ (cid:32) z P p C , | z | ą ´ ε ( (3.15)for some ε ą
0. Its only singularity is a simple pole with residue ´ t at z “ 8 , and it satisfies for any z P Ω ε X ι p Ω ε q : SW p z q ` r OW p z q ` d z T p x p z qq “ . (3.16) The operator O will play an important role in the study of higher topologies, let us recall its definitionin the realm of 1-forms: O ϕ p z q “ ÿ k ě p k ´ q ! ˛ U k ´ d z T k p x p z q , x p ζ q , . . . , x p ζ k qq ϕ p ζ q k ź j “ W p ζ j q . (3.17)If Ω is a neighborhood of U stable under ι , we want to study the space of solutions ϕ P M p Ω q of: S ϕ p z q ` O ϕ p z q “ . (3.18)We start with a result of unicity. The unicity result is easy in combinatorics, because the solutionswe will be looking for have by construction power series expansion in t . If we drop the assumption that solutions of (3.18) must have power series expansion in t , the question of unicitycan be addressed under an assumption of strict convexity, see [BEO13, Section 3] in the case where T k ” k ě emma 3.6 Assume Γ is fixed and the weights t, t are completely regular. Let ε ą and consider Ω ε as in (3.15) . The only solution ϕ P H p Ω ε q to the equation: @ z P Ω ε X ι p Ω ε q , S ϕ p z q ` O ϕ p z q “ , (3.19) which has a power series expansion in t , is ϕ ” . The same holds if O is replaced by r O . Proof.
Since O (or r O ) depends linearly on the parameters t m ,...,m k , the leading order ϕ “ ϕ ` O p t q of a power series solution to S ϕ p z q ` O ϕ p z q “ ϕ p z q ` ϕ p ι p z qq “
0, and we remind ι p z q “ { z . By assumption, ϕ is holomorphic in the exterior of the unit disc, and this equationimplies that ϕ is holomorphic in p C z U . Hence, if ϕ is holomorphic in an open neighborhood of U , sois ϕ . Gathering all the information, we see that ϕ is a holomorphic 1-form on the Riemann sphere,thus it vanishes. The same argument shows that ϕ cannot have a non-zero leading order in its powerseries expansion in t , hence it must vanish identically. l Let us comment on the use of this result. Since a power series in a infinite sequence of variable t, t is characterized by its specializations where all but a finite number of variables have been sent to 0,it is enough to study the latter. Lemma 3.1 then tells us that, for any given values for the non-zeroweights, there exists a neighborhood of 0 of values of t so that t, t is admissible, and the solutions wewill be looking for then have a power series expansion in t with non-zero radius of convergence. Sincethere are only a finite number of non-zero weights, they are obviously completely regular in the senseof Definition 3.2. Thus, we do not lose in generality by taking the detour to set t, t to some realadmissible values – which enables us to use the tools of complex analysis – in order to say somethingabout formal series.Although we do not pursue this issue here, it is possible to show that W p x q P C rr x ´ ssr t ss is uniquely determined by the solution of functional equation (3.6) for completely regular weights,together with the requirement that W p x q is holomorphic in C z Γ, is bounded on p C z Γ, and behaveslike t { x when x Ñ 8 . We now turn to the generating series of stuffed maps with topology of a cylinder, which will allowus the representation of any solution of the homogeneous linear equation (3.18). Cylinders can beobtained by marking an extra elementary 2-cell with topology of a disc on a stuffed map with topologyof a cylinder. At the level of generating series, this means: W p x , x q “ ´ ÿ m ě x m ` BB t m ¯ W p x q . (3.20)Applying the differential operator to the functional relation (3.6) yields, for all x in the interior ofΓ t, t and x P C z Γ: W p x ` i0 , x q ` W p x ´ i0 , x q ` O x W p x , x q ` p x ´ x q “ . (3.21)This equation will also be derived from the analysis of Schwinger-Dyson equation in Section 4. Since W p x , x q is symmetric, it satisfies the same equation with respect to x . The subscript of theoperator O indicates on which variable it acts. So, we can apply Lemma 3.5 to W , and define: W p z , z q “ W p x p z q , x p z qq d x p z q d x p z q (3.22)14s a symmetric meromorphic 2-form in p z , z q P Ω ε , and it satisfies: S z W p z , z q ` O z W p z , z q ` d x p z q d x p z qp x p z q ´ x p z qq (3.23)in the domain of analyticity of the left-hand side. We may also define: ω p z , z q “ W p z , z q ` d x p z q d x p z qp x p z q ´ x p z qq , (3.24)which satisfies: S z ω p z , z q ` O z ω p z , z q “ d x p z q d x p z qp x p z q ´ x p z qq . (3.25)A computation done in the proof of Proposition 3.8 in [BEO13] shows that ω p z , z q has its onlysingularities at z “ z , and it is a double pole with leading coefficient 1 and no residues. Let usdefine the local Cauchy kernel : G p z , z q “ ´ ˆ z ω p z , ¨q . (3.26)According to [BEO13, Lemma 2.1], it allows the representation of any solution of the homogeneouslinear equation (3.18) in terms of its singular part only, modulo a holomorphic part. Lemma 3.7
Let Ω be an open neighborhood of U stable under ι , and ϕ P M p Ω q be a solution of S ϕ p z q ` O ϕ p z q “ with a finite number of poles in Ω . Then, r ϕ p z q “ ÿ p P Ω Res z Ñ p ∆ z G p z , z q ϕ p z q (3.27) is such that ϕ p z q ´ r ϕ p z q is holomorphic for z P Ω . We adapt this result to solve (3.18) with a non-zero right-hand-side:
Lemma 3.8
Let Ω be an open neighborhood of U stable under ι , and Ω be the union of Ω and theexterior of the unit disc in p C . Let ψ P H p Ω q . Assume ϕ P M p Ω q satisfies S ϕ p z q` O ϕ p z q` ψ p z q “ forany z P Ω , and has a finite number of poles in Ω . Then, if z lies outside the contour of integrations: r ϕ p z q “ ´ π ˛ U G p z , z q ψ p z q ` ÿ p P Ω Res z Ñ p ∆ z G p z , z q z ϕ p z q (3.28) is such that ϕ p z q ´ r ϕ p z q is holomorphic for z P Ω and satisfies S ϕ p z q ` O ϕ p z q “ for z P Ω . Proof.
It follows from Lemma 3.8 and the fact that φ p z q “ ψ p z q ´ π ¸ U G p z , z q ψ p z q is holomorphicin Ω, and satisfies S φ p z q ` O φ p z q ` ψ p z q “ z P Ω . l And, if we are looking for a solution ϕ which is initially holomorphic in p C z U , and has a powerseries expansion in the parameters t of O , we deduce from Lemma 3.6 that r ϕ p z q “ ϕ p z q . Stuffed maps M of genus g with 1 boundary f can be constructed recursively by Tutte’s decomposition.It consists in removing the root edge of the first boundary, and establishing a bijection between theset of stuffed maps with given topology, and the pieces obtained after the removal. According to theirtopology, several cases can occur: 15 the root edge e was bordered on both sides by f , and its removal disconnects the surface. Weobtain two connected stuffed maps M and M , each having one boundary coming from thesplitting of f , and which are rooted at the edge which was closest to e following f in cyclic order.The handles of M are shared between M and M . ‚ the root edge borders another elementary 2-cell f with k ě K “ v , k w the set of boundaries. Removing the root edge also removes f , and we obtain a stuffed map with r ď k connected components, M , . . . , M r . M i has f i handles and k i ě K i Ď K of | K i | “ k i boundaries of f . One of the boundary in M was incident to f in M . The gluing of M i on f contributed to k i ´ ` f i handles in M , and f itself contributed for h handles. Therefore, we must have h ` ř ri “ p k i ´ ` f i q “ g , which canbe rewritten h ` ` ř ri “ f i ˘ ` k ´ r “ g .In terms of generating series, this bijection implies, for g “ ` W p x q ˘ ` ÿ k ě ˛ k ź j “ d ξ j π B ξ i T k p ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q k ź j “ W p ξ j q “ , (4.1)In this equation, the contour integral is just a way to write the divergent part of a formal Laurentseries: ˛ d ξ π x ´ ξ ´ ÿ m ě m β m ξ m ` ¯ “ ÿ m ě β m x m ` . (4.2)It enforces the matching of perimeters when reconstructing M from its pieces after Tutte’s decompo-sition. In this formal representation, everything happens as if the contour was surrounding and x was closer to than the contour. Similarly, for g ą W g ´ p x, x q ` g ÿ f “ W f p x, x J q W g ´ f p x, x I z J q` ÿ k ě h ě ÿ K $v ,k w f ,...,f r K s ě h `p ř i f i q` k ´r K s“ g ˛ ” k ź j “ d ξ j π ı B ξ T hk p ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q r K s ź i “ W f i | K i | p ξ K i q “ . (4.3)Those relations are equalities between formal series in C rr x ´ ssrr t ss , and (4.3) is still valid for g “ W gn “ g ă g with an arbitrary number n ě δ x ¨ ¨ ¨ δ x n to (4.3), since δ x “ ÿ m ě x m ` BB t m (4.4)amounts to mark an elementary 2-cell with topology of a disc, with the formal variable x coupled toits perimeter. δ x is called the insertion operator . The result is, for any n ě g ě W g ´ n ` p x, x, x I q ` ÿ J Ď I, ď f ď g W f | J |` p x, x J q W g ´ fn ´| J | p x, x I z J q (4.5) ` ÿ i P I B x i ´ W gn ´ p x, x I zt i u q ´ W gn ´ p x I q x ´ x i ¯ ` ÿ k ě h ě ÿ K $v ,k w J Y¨¨¨ Y J r K s “ I ÿ f ,...,f r K s ě h `p ř i f i q` k ´r K s“ g ˛ ” k ź j “ d ξ j π ı B ξ T hk p ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q r K s ź i “ W f i | K i |`| J i | p ξ K i , x J i q “ . W gn can be upgraded to holomorphic functions of x i in some domain of the complex plane, (4.5)will hold in the whole domain of analyticity.We can rewrite those equations in a more compact way by summing over genera with weight p N { t q χ and recalling the definitions (2.12)-(2.13). Introducing: T k p x , . . . , x k q “ ÿ h ě p N { t q ´ h ´ k T hk p x , . . . , x k q , (4.6)we find: W n ` p x, x, x I q ` ÿ J Ď I W | J |` p x, x J q W n ´| J | p x, x I z J q (4.7) ` ÿ i P I B x i ´ W n ´ p x, x I zt i u q ´ W n ´ p x I q x ´ x i ¯ ` ÿ k ě ÿ K $v ,k w J Y¨¨¨ Y J r K s “ I ˛ ” k ź j “ d ξ j π ı B ξ T k p ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q r K s ź i “ W | K i |`| J i | p ξ K i , x J i q “ . The equations (4.7) can also be derived by integration by parts in the matrix integrals described in(2.1), or by expressing the invariance of the matrix integral under infinitesimal change of variables M Ñ M ` εx ´ M . In this context, they are called Schwinger-Dyson equations, and they also hold forconvergent integrals. Definition 4.1
We say that t, t is tame if t, t is completely regular (see Definition 3.2), and if forany m ě , h ě , any partition M $ v , m w , any sequence p f i q ď i ďr M s of nonnegative integers, anyfinite set I , any sequence p J i q ď i ďr M s of pairwise disjoint and maybe empty subsets whose union is I ,the formal series ÿ k ě m ˛ ” k ź j “ d ξ j π ı B x T hk p x, ξ . . . , ξ k q ´ B ξ T hk p ξ , . . . , ξ k q x ´ ξ r M s ź i “ W f i | M i |`| J i | p ξ M i , x J i q k ź j “ m ` W p ξ j q (4.8) which belongs a priori to C rr x, p x ´ i q i P I ssrr t, t ss , is a holomorphic function of x in a neighborhood of Γ t, t and x i in a neighborhood of . Although technical, this condition is similar for usual maps to asking that the model be not critical.It is thus slightly stronger than asking that the coefficients of the generating series considered are finite.This condition allows conveniently the use of analytic functions instead of formal series. If for any h ě
1, the number of boundaries of elementary 2-cells and their perimeter are bounded (i.e. only afinite number of t h(cid:96) ,...,(cid:96) k are non-zero for a given h ). As we already said in § t, t .In this paragraph, we upgrade that the generating series of stuffed maps to analytic functions, andstudy their basic properties. Lemma 4.1
Assume t, t is tame, then W gk p x , . . . , x k q defines a holomorphic function in C z Γ t, t ,which have boundaries values when x i approaches an interior point of Γ t, t , and for any α “ a, b ,there exists an integer r gα,k so that p x i ´ α q r gα,k W gk p x , . . . , x k q remains bounded when x i Ñ α . roof. The statement was established for p n, g q “ p , q in Lemma 3.2. Let p n, g q ‰ p , q , andassume the result is proved for p n , g q such that 2 g ´ ` n ă g ´ ` n . We introduce: P hk p x, ξ ; ξ , . . . , ξ k q “ B x T hk p x, ξ , . . . , ξ k q ´ B ξ T hk p ξ , . . . , ξ k q x ´ ξ . (4.9)For any p n, g q ‰ p , q , we isolate the contribution of W gn in (4.5) and decompose: ` W p x q ` r O W p x q ` B x T p x q ˘ W gn p x, x I q ` W p x q O x W gn p x, x I q (4.10) ´ ÿ k ě ˛ d ξ P k p x, ξ ; ξ , . . . , ξ k qp k ´ q ! W gn p ξ , x I q ” k ź j “ W p ξ j q d ξ j π ı ´ ÿ k ě ˛ d ξ P k p x, ξ ; ξ , . . . , ξ k qp k ´ q ! W gn p ξ , x I q ” k ź j “ j ‰ W p ξ j q d ξ j π ı ` W g ´ n ` p x, x, x I q ` ÿ J Ď I, ď f ď g W f | J |` p x, x J q W g ´ fn ´| J | p x, x I z J q` ÿ i P I B x i ´ W gn ´ p x, x I zt i u q ´ W gn ´ p x I q x ´ x i ¯ ` ÿ k ě h ě ÿ K $v ,k w J Y¨¨¨ Y J r K s “ Iξ “ x ÿ f ,...,f r K s ě h `p ř i f i q` k ´r K s“ g ˛ ” k ź j “ d ξ j π ı B x T hk p x, ξ , . . . , ξ k qp k ´ q ! r K s ź i “ W f i | K i |`| J i | p ξ K i , x J i q´ ÿ k ě h ě ÿ K $v ,k w J Y¨¨¨ Y J r K s “ I ÿ f ,...,f r K s ě h `p ř i f i q` k ´r K s“ g ˛ ” k ź j “ d ξ j π ı P hk p x, ξ ; ξ , . . . , ξ k qp k ´ q ! r K s ź i “ W f i | K i |`| J i | p ξ K i , x J i q “ , where ř means that we excluded all terms containing W gn . We see that (4.10) involves only a finitenumber of terms of the form (4.8), and assuming t, t tame actually justifies the existence of thedecomposition (4.10), and implies that the second, third and last line of (4.10) define holomorphicfunctions of x in a neighborhood of Γ t, t . Then, we can write: W gn p x, x I q “ L gn p x ; x I q W p x q ` O W p x q ` B x T p x q . (4.11)We now come to the key observation. L gn p x ; x I q involve terms which either: ‚ define holomorphic functions in an open neighborhood of Γ t, t . This is the case for O x W gn p x, x I q and the lines involving the P ’s. ‚ or define holomorphic functions in C z Γ t, t , since they involve only W g n with 2 g ´ ` n ă g ´ ` n for which we already have the induction hypothesis.Therefore, W gn p x, x I q upgrades to a holomorphic function in C z Γ t, t , and (4.10) is valid in the wholedomain of analyticity. From the two points above, we infer that L gn p x q behaves as O pp x ´ α q ´ s gn { q for some integer s gn when x Ñ α “ a, b , and has boundary values at any interior point of Γ t, t .Furthermore, 2 W p x q ` r O W p x q ` B x T p x q vanishes like O p? x ´ α q when x Ñ α , and does notvanish elsewhere on Γ t, t . Thus W gn p x, x I q P O ` p x ´ α q ´p s gn ` q{ ˘ when x Ñ α . We thus conclude theproof by induction. l .3 Potentials for higher topologies In this section, we introduce and study generating series called potentials for topology p n, g q : V gn p x ; x , . . . , x n q P C rr x, x ´ , . . . , x ´ n ssrr t ss , (4.12)which will appear in the determination of the monodromy of W gn ’s around their discontinuity locus.The cases p n, g q “ p , q and p , q have a special definition: V p x q “ T p x q , V p x ; x q “ ´ x ´ x . (4.13) T p x q is the potential in the usual sense in random matrix theory, and here in the context of multi-trace matrix models, we may call it ”potential for discs”. For any p n, g q ‰ p , q , p , q , denoting I aset with n ´ p n, g q by: V gn p x ; x I q “ ÿ m ě k ě m ` h ě ÿ M $v ,m w f ,...,f r M s ě h `p ř i f i q` m ´r M s“ gJ Y¨¨¨ Y J r M s “ I ˛ ” k ź j “ d ξ j π ı ˆ ´ m ! T hk p x, ξ , . . . , ξ k ´ qp k ´ ´ m q ! r M s ź i “ W f i | M i |`| J i | p ξ M i , x J i q k ´ ź j “ m ` W p ξ j q ¯ . The ř means that we exclude the term which contains W gn , which is actually equal to O x W gn p x, x I q .Notice that the variables x , . . . , x n play symmetric roles, whereas x plays a special role. Besides,those potentials for 2 g ´ ` n ą T hk of generating series of elementary 2-cellswhich define the model, but also on the generating series of stuffed maps themselves. Yet, the potentialfor topology p n, g q only involves the generating series of stuffed maps W g n with lower topology, i.e.2 g ´ ` n ă g ´ ` n .Combinatorially, V gn p x ; x , . . . , x n q is the generating series of one elementary 2-cell of arbitrarytopology p k, h q , whose first boundary is unrooted and has a perimeter coupled to x , and whose p k ´ q other boundaries are glued to the boundaries of other stuffed maps, so as to form a connected stuffedmap M of genus g with n boundaries, and with the restriction that no stuffed map of topology p n, g q should be used. More precisely, the first boundary of M is the distinguished boundary ofthe elementary 2-cell, while the other boundaries are rooted and their perimeters are coupled to thevariables x , . . . , x n . We may describe M as a stuffed elementary 2-cell of topology p n, g q .An equivalent way to write the sum in (4.14) is: V gn p x ; x I q “ ÿ k ě h ě ÿ K $v ,k w f ,...,f r K s ě h `p ř i f i q` k ´| K |“ gJ Y¨¨¨ Y J r K s “ I ˛ Γ k ´ ” k ź j “ d ξ j π ı T hk p x, ξ , . . . , ξ k qp k ´ q ! r K s ź i “ W f i | K i |`| J i | p ξ K i , x J i q . (4.14)If t, t is tame in the sense of Definition 4.1, one can deduce that (4.14) defines a holomorphic functionof x in a neighborhood of Γ t, t . V gn p x ; x I q can be obtained from V g p x q by successive applications of the insertion operators (4.4) δ x i for i P I , since we have the relation: δ y ` O x W gn p x, x I q ` V gn p x ; x I q ˘ “ O x W gn ` p x, y, x I q ` V gn ` p x ; y, x I q . (4.15)19or later use, we give a formula for p n ` , g ´ q ‰ p , q , p , q : B V g ´ n ` p x, x, x I q (4.16) “ lim y Ñ x B x V g ´ n ` p x, y, x I q“ ÿ k ě h ě ÿ K $v ,k w f ,...,f r K s ě h `p ř i f i q` k ´r K s“ g ´ J Y¨¨¨ Y J r K s “ I ˛ ” k ź j “ d ξ j π ı B x T hk p x, ξ , . . . , ξ k qp k ´ q ! W f | K |`| J |` p x, ξ K , x J q r K s ź i “ W f i | K i |`| J i | p ξ K i , x J i q . It is readily checked from (4.14) by calling 1 the index of the element of the partition K for which thecorresponding subset of I Y t y u in the contains the variable y . W gn ’s We establish the analog of (3.6) for generating series of stuffed maps of higher topologies:
Theorem 4.2
For any x interior to Γ , and any x , . . . , x n P C z Γ , we have: S x W gn p x, x I q ` O x W gn p x, x I q ` B x V gn p x, x I q “ , (4.17) where V gn is the potential for topology p n, g q introduced in (4.14) . As a consequence of Lemma 4.1 and Theorem 4.2, following § n -formin n variables W gn p z , . . . , z n q , holomorphic when z , . . . , z n belong to the exterior of U in C and suchthat: W gn p z , . . . , z n q “ W gn p x p z q , . . . , x p z n qq d x p z q ¨ ¨ ¨ d x p z n q , (4.18)and meromorphic when one of the z i is in a neighborhood of U . Similarly, we have a function of z : V gn p z ; z I q “ V gn p x p z q , x p z I qq ź i P I d x p z i q (4.19)which is holomorphic when z is in a neighborhood of U stable under ι , and such that V gn p ι p z q ; z I q “ V gn p z ; z I q in this neighborhood. Besides, if z I is a set of p n ´ q spectator variablesin the domain of analyticity, and z P Ω ε X ι p Ω ε q for some ε ą
0, (4.17) translates into: S z W gn p z, z I q ` O z W gn p z, z I q ` d z V gn p z ; z I q “ W p z q and W p z , z q , as well as their analytic properties, were already treated in § Proof.
We recall the definitions: S φ p x q “ φ p x ` i0 q ` φ p x ´ i0 q , ∆ φ p x q “ φ p x ` i0 q ´ φ p x ´ i0 q . (4.21)We have the polarization formulas: S p φ ¨ φ qp x q “ ` S φ p x q ¨ S φ p x q ` ∆ φ p x q ¨ ∆ φ p x q ˘ , ∆ p φ ¨ φ qp x q “ ` S φ p x q ¨ ∆ φ p x q ` ∆ φ p x q ¨ S φ p x q ˘ . (4.22)We will compute the discontinuity of the Schwinger-Dyson equations in the form (4.10), and weremind that the terms involving O φ p x q , r O φ p x q and the P ’s are holomorphic in a neighborhood of Γ,20hus have no discontinuity across Γ. For g “
0, there is a huge simplification in the sum over partitions K $ v , k w , since we must have h ` ` ř i f i ˘ ` k ´ r K s “
0, therefore h “ f “ . . . “ f r K s “ r K s “ k , which means that K is the partition consisting of singletons. We will consider the cases p n, g q “ p , q , p , q , p , q which are somewhat special, before explaining the general pattern of theproof, which proceeds by induction on 2 g ´ ` n . It is possible to derive the result for all p n, g q from the result for all p n “ , g q by successive applications of the insertion operator using (4.15) (oneshould not forget to act with δ x on the operator O ). We will take a more direct route, which has itsown pedagogical interest, although it is more cumbersome.For p n, g q “ p , q , the Schwinger-Dyson equation only involves W . Therefore, the ř are empty,and we easily find: ∆ x “` W p x q ˘ ‰ ` ∆ x W p x q ` r O x W p x q ` B x T ˘ “ . (4.23)Using the polarization formula to transform the first term, we infer:∆ x W p x q ` S x W p x q ` r O x W p x q ` B x T p x q ˘ “ . (4.24)Hence, we retrieve the equation (3.6) stating that, on the discontinuity locus (the interior of Γ) of W : S x W p x q ` O x W p x q ` B x T p x q “ . (4.25)By definition V p x q “ T p x q , hence (4.17) for p n, g q “ p , q .For p n, g q “ p , q , the set indexing auxiliary variables is I “ t u , hence in the sum over p J i q ď i ď k in the Schwinger-Dyson equation (4.5), we just have to choose in which J i we put the element 2. Weget first term if we put 2 in J , and p k ´ q equal terms for 2 R J . So, the Schwinger-Dyson equationreads: 2 W p x q W p x, x q ` B x ´ W p x q ´ W p x q x ´ x ¯ (4.26) ` ÿ k ě ˛ Γ k ” k ź j “ d ξ j π ı B ξ T k p ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q W p ξ , x q k ź i “ W p ξ i q (4.27) ` ÿ k ě ˛ Γ k ” k ź j “ d ξ j π ı B ξ T k p ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q W p ξ q W p ξ , x q k ź i “ W p ξ i q “ . (4.28)Then, computing its discontinuity with respect to x and applying the polarization formula for the firstterm, we find: ∆ x W p x q S x W p x, x q ` S x W p x q ∆ x W p x, x q ` p x ´ x q ` ∆ x W p x q O x W p x, x q ` ∆ x W p x, x q ` r O x W p x q ` B x T p x q ˘ “ . We collect the terms:∆ x W p x q ´ S x W p x, x q ` O x W p x, x q ` p x ´ x q ¯ ` ∆ x W p x, x q ` S x W p x q ` r O x W p x q ` B x T p x q ˘ “ , (4.29)and since W satisfies (4.25), we find for any interior point x of Γ: S x W p x, x q ` O x W p x, x q ` p x ´ x q “ . (4.30)21his equation was already derived in § V p x, x q “ ´ x ´ x , we obtain (4.17) for p n, g q “ p , q .We now come to p n, g q “ p , q . The Schwinger-Dyson equation (4.5) reads:2 W p x q W p x q ` W p x, x q` ÿ k ě ˛ Γ k ” k ź j “ d ξ j π ı B ξ T k p ξ , ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q W p ξ q k ź i “ W p ξ i q` ÿ k ě ˛ Γ k ” k ź j “ d ξ j π ı B ξ T k p ξ , ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q W p ξ q W p ξ q k ź i “ W p ξ i q (4.31) ` ÿ k ě ˛ Γ k ” k ź j “ d ξ j π ı B ξ T k p ξ , ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q W p ξ , ξ q k ź i “ W p ξ i q (4.32) ` ÿ k ě ˛ Γ k ” k ź j “ d ξ j π ı B ξ T k p ξ , ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q W p ξ q W p ξ , ξ q k ź i “ W p ξ i q (4.33) ` ÿ k ě ˛ Γ k ” k ź j “ d ξ j π ı B ξ T k p ξ , ξ , . . . , ξ k qp k ´ q ! p x ´ ξ q k ź i “ W p ξ i q “ . (4.34)The discontinuity of the first term can be computed by polarization formula. For the second term,we write similarly: ∆ x ` W p x, x q ˘ “ lim y Ñ x ∆ x S y W p x, y q . (4.35)We find for the discontinuity of (4.31):∆ x W p x q S x W p x q ` S x W p x q ∆ x W p x q ` lim y Ñ x ∆ x S y W p x, y q` ∆ x W p x q ` r O W p x q ` B x T p x q ˘ ` ∆ x W p x q O x W p x q` lim y Ñ x ∆ x O y W p x, y q` ∆ x W p x q ´ ÿ k ě ˛ Γ k ´ ” k ź j “ d ξ j π ı B x T k p x, ξ , . . . , ξ k qp k ´ q ! W p ξ , ξ q k ź i “ W p ξ i q ¯ ` ∆ x W p x q ´ ÿ k ě ˛ Γ k ´ ” k ź j “ d ξ j π ı B x T k p x, ξ , . . . , ξ k qp k ´ q ! k ź i “ W p ξ i q ¯ “ . (4.36)This can be rewritten: lim y Ñ x ∆ y ´ S x W p x, y q ` O x W p x, y q ` p x ´ y q ¯ ` ` S x W p x q ` r O x W p x q ` B x T p x q ˘ ∆ x W p x q (4.37) ` ∆ x W p x q ` S x W p x q ` O x W p x q ` B x V p x q ˘ “ , (4.38)where V p x q is the potential for tori with one boundary introduced in (4.14), namely: V p x q “ ÿ k ě ˛ Γ k ´ ” k ź j “ d ξ j π ı B x T k p x, ξ , . . . , ξ k qp k ´ q ! W p ξ , ξ q k ź i “ W p ξ i q` ÿ k ě ˛ Γ k ´ ” k ź j “ d ξ j π ı B x T k p x, ξ , . . . , ξ k qp k ´ q ! k ź i “ W p ξ i q . (4.39)22n order to obtain (4.36), we have introduced ∆ y ` p x ´ y q ˘ “ W and W , we find at any interior point of Γ: S x W p x q ` O x W p x q ` B x V p x q “ . (4.40)This case was special in the sense that we had to split W p x, x q in lim y Ñ x W p x, y q because of thepole at x “ y in the equation (4.30). This issue is absent for the other values of p n, g q .We now arrive to the general case. Let n ě g ě g ´ ` n ą
0, and p n, g q ‰ p , q . Let us assume that the result (4.17) holds for any W g n such that 2 g ´ ` n ă g ´ ` n .As before, we compute the discontinuity with respect to x of the Schwinger-Dyson equation (4.5). Inthe sum over partitions K $ v , k w , we have to distinguish whether the element of K which contained1 (associated to the variable ξ ), that we call K , is a singleton or not. We denote K the partition of v , k wz K determined by the other elements of K . We then find:∆ x, S x, W g ´ n ` p x, x, x I q ` ∆ x W p x q S x W gn p x, x I q (4.41) ` ÿ J Ď I, ď f ď g p J,f q‰pH , q , p I,g q ∆ x W f | J |` p x, x J q S x W g ´ fn ´| J | p x, x I z J q ` ÿ i P I ∆ x W gn ´ p x, x I zt i u qp x ´ x i q ` ÿ k ě h ě ÿ J Ď I ď f ď g ÿ K $v ,k w J Y¨¨¨ Y J K “ I z J ÿ f ,...,f K ě h `p ř i f i q` k ´pr K s` q“ g ´ f ˛ Γ k ´ ” k ź j “ d ξ j π ı B x T hk p x, ξ , . . . , ξ k qp k ´ q !∆ x W f | J |` p x, x J q r K s ź i “ W f i | K i |`| J i | p ξ K i , x J i q` ÿ k ě h ě ÿ K $v ,k w f ,...,f r K ě h `p ř i f i q` k ´r K s“ gJ Y¨¨¨ Y J r K “ I ˛ Γ k ´ ” k ź j “ d ξ j π ı B x T hk p x, ξ , . . . , ξ k qp k ´ q !∆ x W f | K |`| J |` p x, ξ K , x J q r K s ź i “ W f i | K i |`| J i | p ξ K i , x J i q “ . The indices x, i for the operators ∆ or S in the first line indicate on which of the two variable x theyact. We can collect the terms in three steps: ‚ In the second line, ∆ x W gn ´ p x, x I zt i u q{p x ´ x i q can be included in the term∆ x W gn ´ p x, x I zt i u q S x W p x, x i q arising in the sum over J Ď I . ‚ The prefactor of the terms involving ∆ x W f | J |` p x, x J q in the third/fourth line can be in-cluded in the term ∆ x W f | J |` p x, x J q S x W g ´ fn ´| J | p x, x I z J q of the second line. For p J, f q ‰ p
I, g q ,it produces: a term for which | K i | ` | J i | “ f i “ i , which is equal to∆ x W f | J |` p x, x j q O x W g ´ fn ´| J | p x, x I z J q ; and a term equal to ∆ x W f | J |` p x, x J q V g ´ fn ´| J | p x ; x I z J q , bycomparison with the definition of the potential for higher topologies (4.14). When p J, f q “pH , q , the result is slightly different due to symmetry factors, and we obtain a contribution∆ x W gn p x, x I q ` r O W p x q ` B x T p x q ˘ . ‚ The last two lines are equal to ∆ x V gn p x ; x, x I q . This can be checked by comparing the last twolines with the expression of B V g ´ n ` p x ; x, x I q given in (4.16), and noticing that B x T hk p x, ξ , . . . , ξ k q is by assumption a holomorphic function of x in a neighborhood of Γ.23herefore, we have found: ∆ x W p x q ` S x W gn p x, x I q ` O x W gn p x, x I q ` B x V gn p x ; x I q ˘ (4.42)∆ x W gn p x, x I q ` S x W p x q ` r O x W p x q ` B x T p x q ˘ ` ÿ J Ď I, ď f ď g p J,f q‰pH , q , p I,g q ∆ x W f | J |` p x, x J q ´ S x W g ´ fn ´| J | p x, x I z J q ` O x W g ´ fn ´| J | p x, x I z J q ` δ n ´| J | , δ g ´ f, p x ´ x I z J q ¯ ` ∆ ` S W g ´ n ` p x, x, x I q ` O W g ´ n ` p x, x, x I q ` B V g ´ n ` p x ; x, x I q ˘ “ . where the indices on the operators in the last line indicate on which variable x (the first or the second)they act. By the induction hypothesis, the three last lines vanish: we deduce that for any interiorpoint x of Γ, S x W gn p x, x I q ` O x W gn p x, x I q ` B x V gn p x ; x I q “ , (4.43)which is the desired result. l We define for convenience˘ W gn p z , . . . , z n q “ W gn p z , . . . , z n q ` δ n, δ g, d x p z q d x p z q ` x p z q ´ x p z q ˘ (4.44)The only difference is that now ˘ W p z , z q “ ω p z , z q , thus has a singularity only at z “ z . Theorem 4.3
For any p n, g q ‰ p , q , p , q , the quadratic differential form in z : Q gn p z ; z I q “ ˘ W g ´ n ` p z, ι p z q , z I q ` ÿ J Ď I ď f ď g ˘ W f | J |` p z, z J q ˘ W g ´ fn ´| J | p ι p z q , z J q (4.45) has double zeroes at z “ ˘ , i.e. x p z q P t a, b u . The content of this theorem is that, although W gn can have poles of high order at z “ ˘
1, thecombination Q gn p z ; z I q does not. Proof.
To arrive to (4.45), we have going to recast the Schwinger-Dyson equation (4.10) using thesame decomposition of the sum over partitions K $ v , k w which led to (4.42). We find: r Q gn p z ; z I q ` W g ´ n ` p z, z, z I q ` O z, W g ´ n ` p z, z, z I q ` p ´ δ n, δ g, q d V g ´ n ` p z ; z, z I q (4.46) ` ` W p z q ` r O z W p z q ` d z V p z q ˘ W gn p z, z I q` W p z q ` O z W gn p z, z I q ` d z V gn p z ; z I q ˘ ` ÿ J Ď I, ď f ď g p J,f q‰pH , q , p I,g q W f | J |` p z, z J q ` W g ´ fn ´| J | p z, z I z J q ` O z W g ´ fn ´| J | p z, z I z J q ` d z V g ´ fn ´| J | p z ; z I z J q ˘ “ r Q gn p z ; z I q “ ´ d x p z q d z i ´ W gn ´ p z I q d x p z i q ` x p z q ´ x p z i q ˘ ¯ (4.47) ´ ÿ k ě h ě ÿ K $v ,k w J Y¨¨¨ Y J r K s “ I ÿ f ,...,f r K s ě h `p ř i f i q` k ´r K s“ g ˛ U k P hk p z, ζ ; ζ , . . . , ζ k qp k ´ q ! ` x p z q ´ x p ζ qq r K s ź i “ W f i | K i |`| J i | p ζ K i , z J i q . ř i P I d x p z q d z i ` W gn ´ p z I qp x p z q´ x p z i qq ˘ to the Schwinger-Dyson equations was included in theterm V appearing in the sum of the fourth line. We have introduced the differential form version of(4.9), and: P hk p z, ζ ; ζ , . . . , ζ k q “ d x p ζ q d z T hk p x p z q , x p ζ q , . . . , x p ζ k qq ´ d x p z q d ζ T hk p x p ζ q , x p ζ q , . . . , x p ζ k qq x p z q ´ x p ζ q , (4.48)and in (4.47), the variables ζ i are integrated over the unit circle. We already observe that r Q gn p z ; z I q hasa double zero at z “ t˘ u , since it is a holomorphic function in a neighborhood of z “ ˘ ` d x p z q ˘ . We also recognize in (4.46) combinations which can be represented using: W g n p ι p z q , z J q “ ´ W g n p z, z J q ´ O z W g n p z, z J q ´ d z V g n p z, z J q . (4.49)If we rewrite the equality (4.46) in terms of ω g n p z, z J q and ω g n p ι p z q , z J q , we conclude after some algebrathat Q gn p z ; z I q “ r Q gn p z ; z I q . l Assuming that t, t are tame, we are going to show that the generating series of stuffed maps W gn (in the x variables) or W gn (in the z variables), are given up to a shift – which is essential – by thetopological recursion of [EO07a] applied to the initial data: ω p z q “ W p x p z qq d x p z q , (5.1)together with the Bergman kernel: ω p z , z q “ ´ W p x p z q , x p z qq ` ` x p z q ´ x p z q ˘ ¯ d x p z q d x p z q , (5.2)and local involution given by ι p z q “ { z (this is Theorem 5.1 below).For this purpose, we remind the definition of the local Cauchy kernel: G p z , z q “ ´ ˆ z ω p z , ¨q , (5.3)and introduce the recursion kernel : K p z , z q “ ´
12 ∆ z G p z , z q ∆ z ω p z q “ ´ ´ zι p z q ω p z , ¨q ω p z q ´ ω p ι p z qq . (5.4)For 2 g ´ ` n ą
0, we introduce the meromorphic forms: ω gn p z , . . . , z n q “ W gn p z , . . . , z n q d x p z q ¨ ¨ ¨ d x p z n q (5.5)and from Theorem 4.2, we have the inhomogeneous linear equations: S z ω gn p z, z I q ` O z ω gn p z, z I q ` d z V gn p z, z I q “ ω gn p z, z I q is meromorphic in a neighborhood of U , and haspoles only at z “ ˘
1. Therefore, and since we are working in the realm of formal series in t, t , we canapply Lemma 3.8 and Lemma 3.6 to represent, for p n, g q ‰ p , q , p , q : ω gn p z, z I q “ Φ gn p z ; z I q ` Res z Ñ˘ ∆ z G p z , z q z ω gn p z, z I q (5.7)Φ gn p z ; z I q “ π ˛ z P U G p z , z q d z V gn p z ; z I q “ π ˛ z P U ω p z , z q V gn p z ; z I q . (5.8)25he expression (5.8) is valid when z is outside U , and can be analytically continued inside U . Weremind that Φ gn p z, z I q is holomorphic in a neighborhood of U . Then, we decompose Q gn defined in(4.45) as: Q gn p z ; z I q “ S z ω p z q S z ω gn p z, z I q `
12 ∆ z ω p z q ∆ z ω gn p z, z I q ` E gn p z ; z I q (5.9) E gn p z ; z I q “ ω g ´ n ` p z, ι p z q , z I q ` ÿ J Ď I, ď f ď g p J,f q‰pH , q , p I,g q ω f | J |` p z, z J q ω g ´ fn ´| J | p ι p z q , z I z J q . (5.10)The first term in (5.9) has a double zero at z “ ˘
1. So does Q gn p z ; z I q according to Theorem 4.3.Therefore, if we plug the expression for ∆ z W gn p z, z I q in terms of Q gn p z, z I q , we find that the only termcontributing to the residue in (5.7) is E gn p z, ι p z q , z I q . So, we have proved: Theorem 5.1 If t, t is tame, we have the recursion relation, for any p n, g q ‰ p , q , p , q : ω gn p z , z I q “ Φ gn p z ; z I q` Res z Ñ K p z , z q ” ω g ´ n ` p z, ι p z q , z I q` ÿ J Ď I, ď f ď g p J,h q‰pH , q , p I,g q ω f | J |` p z, z J q ω g ´ fn ´| J | p ι p z q , z I z J q ı . (5.11)This is a topological recursion, since the right-hand side involves only ω g n with 2 g ´ ` n ă g ´ ` n .The form of the recursion is universal, it only depends on the model through the initial condition ω and ω , and the monodromy operator ι . Evaluating ω g n p z , z J q at z “ ι p z q is done by Theorem 4.2,which led to the expression (5.6) for the monodromy. ´ boundary For p n, g q “ p , q , (5.11) becomes: ω p z q “ Φ p z q ` Res z Ñ˘ K p z , z q ω p z, ι p z qq , (5.12)and (5.8) gives:Φ p z q “ π p k ´ q ! ÿ k ě ˆ U k T k p x p ζ q , . . . , x p ζ k qq ω p z , ζ q ω p ζ , ζ q k ź j “ ω p ζ j q` π p k ´ q ! ÿ k ě ˆ U k T k p x p ζ q , . . . , x p ζ k qq ω p z , ζ q k ź j “ ω p ζ j q . (5.13) boundaries For p n, g q “ p , q , we compute from (5.11): ω p z , z , z q “ Φ p z ; z , z q ` Res z Ñ˘ K p z , z q ´ ω p z, z q ω p ι p z q , z q ` ω p z, z q ω p ι p z q , z q ¯ “ Res z Ñ˘ ω p z, z q ω p z, z q ω p z, z q x p z q d y p z q , (5.14)where we have defined the function y which is the analytic continuation of ∆ x W p x q in the z -plane,and has simple zeroes at z “ ˘
1. The integrand in (5.14) has a simple pole at z “ ˘ x p z q in the denominator. Hence, the residue can be evaluated: ω p , z q ω p , z q ω p , z q x p q y p q ` ω p´ , z q ω p´ , z q ω p´ , z q x p´ q y p´ q , (5.15)26here α means that we divide the 1-form by d z and evaluate the function obtained in this way at z “ α . Besides, from (5.8), we have:Φ p z , z , z q “ π ÿ k ě p k ´ q ! ˛ U k T k p x p ζ q , . . . , x p ζ k qq ω p ζ , z q ω p ζ , z q ω p ζ , z q k ź j “ ω p ζ j q . (5.16)We observe that both ω p z , z , z q and Φ p z , z , z q are symmetric in their 3 variables, although thisis not obvious of the definition.We leave to a future investigation the study of the symmetry properties of ω gn p z , . . . , z n q andΦ gn p z , . . . , z n q . The generating series of connected closed stuffed maps of genus g is denoted F g (see (2.12)). It ischaracterized by its derivatives with respect to the parameters t of the model: B F g B t hm ,...,m k “ p´ q k Res x Ñ8 ¨ ¨ ¨ Res x n Ñ8 ” k ź i “ x m i i d x i ı´ ÿ K $v ,k w f ,...,f r K s ě h `p ř i f i q` k ´r K s“ g W f i | K i | p x K i q ¯ . (5.17)The residue just picks up the coefficient of x ´p m ` q ¨ ¨ ¨ x ´p m k ` q k in the Laurent expansion at ofthe integrand. We leave to a future investigation the simultaneous integration of (5.17) to get a closedformula for F g in terms of W g n ’s. For usual maps, this step was performed systematically in [Che06],but the problem here seems more complicated since the evaluation of ω gn p z , . . . q at z “ ι p z q involvesthe operator O in (3.17) and thus depends explicitly on t . In the terminology of [BEO13], Theorem 4.2 means that ω ‚‚ defined by (5.1)-(5.2)-(5.8) satisfy linearloop equations, which are here solvable thanks to Lemma 3.6 because we work in the realm of formalseries in t, t . Theorem 4.3 then established that ω ‚‚ satisfies quadratic loop equation. The recursionformula (5.11) is then shown in [BEO13, Proposition 2.7] to be a consequence of those two properties, § W gn p x , . . . , x n q the generating series of maps (resp. the coefficients in a large N expansion of the n -point correlationfunctions) and the ω gn satisfying the usual topological recursion of [EO07a], was: ω gn p z , . . . , z n q “ W gn p x p z q , . . . , x p z n qq ` δ n, d x p z q d x p z q ` x p z q ´ x p z q ˘ ¯ . (5.18)It included a shift only for the unstable topologies p n, g q “ p , q , p , q . Here, for stuffed maps (or forthe multi-trace hermitian matrix model), there is a shift between the residue formula and ω gn for any p n, g q , and this shift is given by Φ gn (see (5.8)), in terms of the potentials for topology p n, g q discussedin § gn as a way to include an ”initial condition” for unstable topologiesin the topological recursion. 27 cknowledgments I thank B. Eynard, E. Guitter and N. Orantin for asking questions which led to this project, theorganizers of the Journ´ees Cartes in June 2013 at the IPhT CEA Saclay where it was initiated, aswell as S. Garoufalidis, I.K. Kostov and S. Shadrin. This work is supported by the Max-Planck-Gesellschaft.
A Two matrix model realization of stuffing
Consider two N ˆ N hermitian matrices with formal measure:d µ p M , M q 9 d M d M det p ´ α M b M q ´ γ exp ` ´ N Tr V p M q ´ N Tr V p M q ˘ . (A.1)It induces on M the distribution:d µ p M q 9 d M exp ` ´ N Tr V p M q ˘ ˆ H N d M exp ` ´ N Tr V p M q ˘ det p ´ αM b M q ´ γ (A.2) d M exp ` ´ N Tr V p M q ˘ ˆ H N d M exp ´ ´ N Tr V p M q ` γ ÿ (cid:96) ě α (cid:96) (cid:96) Tr M (cid:96) Tr M (cid:96) ¯ d M exp ` ´ N Tr V p M q ` ÿ k ě γ k k ! ÿ (cid:96) ,...,(cid:96) k ě ˇ T (cid:96) ,...,(cid:96) k (cid:96) ¨ ¨ ¨ (cid:96) k k ź i “ Tr M (cid:96) i ¯ , (A.3)where: ˇ T (cid:96) ,...,(cid:96) k “ α (cid:96) `¨¨¨` (cid:96) k A Tr M (cid:96) ¨ ¨ ¨ Tr M (cid:96) k D M ,c , (A.4)and by definition: x f p M qy M “ ´ H N d M exp ` ´ N Tr V p M q ˘ f p M q ´ H N d M exp ` ´ N Tr V p M q ˘ , (A.5)and the subscript c stands for ”cumulant”. In other words, the marginal distribution of M inthe model (A.1) is of the form (1.2), where T (cid:96) ,...,(cid:96) k are by definition the coefficients of the k -pointcorrelators ˇ W k of the matrix M for the measure defined in (A.5):d x ¨ ¨ ¨ d x k ˇ T k p x , . . . , x k q “ ÿ (cid:96) ,...,(cid:96) k ě ˇ T (cid:96) ,...,(cid:96) k k ź i “ x (cid:96) i ´ d x i (A.6) “ ˇ W k ` {p αx q , . . . , {p αx k q ˘ d ` ´ {p α x q ˘ ¨ ¨ ¨ d ` ´ {p α x k qq , ˇ W k p ξ , . . . , ξ k q “ A k ź j “ Tr 1 ξ j ´ M E M ,c . (A.7)The fatgraphs underlying the formal model (A.1) are dual to usual maps with two types of faces(associated to M or to M ), and the particular coupling between M and M ensures that we cancollect faces of the same type in clusters which are actually usual maps made of faces of type M only.Therefore, a map appearing in the combinatorial description behind (A.1) can be seen as a stuffedmap (in the sense of § M , whose elementary 2-cells are themselves usual maps(of arbitrary topology) made of faces of type M . This justifies the name of ”stuffing”.28 eferences [ACKM93] J. Ambjørn, L.O. Chekhov, C. Kristjansen, and Yu. Makeenko, Matrix model calculationsbeyond the spherical limit , Nucl. Phys. B (1993), 127–172, hep-th/9302014 .[ACKM95] ,
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