Chain of matrices, loop equations and topological recursion

Random matrices are used in fields as different as the study of multi-orthogonal polynomials or the enumeration of discrete surfaces. Both of them are based on the study of a matrix integral. However, this term can be confusing since the definition of a matrix integral in these two applications is not the same. These two definitions, perturbative and non-perturbative, are discussed in this chapter as well as their relation. The so-called loop equations satisfied by integrals over random matrices coupled in chain is discussed as well as their recursive solution in the perturbative case when the matrices are Hermitean.Comment: 28 pages, 1 figure, contribution to The Oxford Handbook of Random Matrix Theor

Symplectic invariants, Virasoro constraints and Givental decomposition

Following the works of Alexandrov, Mironov and Morozov, we show that the symplectic invariants of \cite{EOinvariants} built from a given spectral curve satisfy a set of Virasoro constraints associated to each pole of the differential form $ydx$ and each zero of $dx$ . We then show that they satisfy the same constraints as the partition function of the Matrix M-theory defined by Alexandrov, Mironov and Morozov. The duality between the different matrix models of this theory is made clear as a special case of dualities between symplectic invariants. Indeed, a symplectic invariant admits two decomposition: as a product of Kontsevich integrals on the one hand, and as a product of 1 hermitian matrix integral on the other hand. These two decompositions can be though of as Givental formulae for the KP tau functions.Comment: 19 page

CFT and topological recursion

We study the quasiclassical expansion associated with a complex curve. In a more specific context this is the 1/N expansion in U(N)-invariant matrix integrals. We compare two approaches, the CFT approach and the topological recursion, and show their equivalence. The CFT approach reformulates the problem in terms of a conformal field theory on a Riemann surface, while the topological recursion is based on a recurrence equation for the observables representing symplectic invariants on the complex curve. The two approaches lead to two different graph expansions, one of which can be obtained as a partial resummation of the other.Comment: Minor correction

Algebraic methods in random matrices and enumerative geometry

We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms, and a sequence of complex numbers Fg . We recall the definition of the invariants Fg, and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several example of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, non-intersecting brownian motions,...Comment: review article, Latex, 139 pages, many figure

Geometric recursion

We propose a general theory to construct functorial assignments $\Sigma \longmapsto \Omega_{\Sigma} \in E(\Sigma)$ for a large class of functors $E$ from a certain category of bordered surfaces to a suitable target category of topological vector spaces. The construction proceeds by successive excisions of homotopy classes of embedded pairs of pants, and thus by induction on the Euler characteristic. We provide sufficient conditions to guarantee the infinite sums appearing in this construction converge. In particular, we can generate mapping class group invariant vectors $\Omega_{\Sigma} \in E(\Sigma)$. The initial data for the recursion encode the cases when $\Sigma$ is a pair of pants or a torus with one boundary, as well as the "recursion kernels" used for glueing. We give this construction the name of Geometric Recursion (GR). As a first application, we demonstrate that our formalism produce a large class of measurable functions on the moduli space of bordered Riemann surfaces. Under certain conditions, the functions produced by the geometric recursion can be integrated with respect to the Weil--Petersson measure on moduli spaces with fixed boundary lengths, and we show that the integrals satisfy a topological recursion (TR) generalizing the one of Eynard and Orantin. We establish a generalization of Mirzakhani--McShane identities, namely that multiplicative statistics of hyperbolic lengths of multicurves can be computed by GR, and thus their integrals satisfy TR. As a corollary, we find an interpretation of the intersection indices of the Chern character of bundles of conformal blocks in terms of the aforementioned statistics. The theory has however a wider scope than functions on Teichm\"uller space, which will be explored in subsequent papers; one expects that many functorial objects in low-dimensional geometry could be constructed by variants of our new geometric recursion.Comment: 97 pages, 21 figures. v2: misprint corrected. v3: revised and abridged version, 66 page

Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture

The BKMP conjecture (2006-2008), proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture had been verified to low genus for several toric CY3folds, and proved to all genus only for C^3. In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion. One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model.Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in 2 steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to K\"ahler radius coincide due to special geometry property implied by the topological recursion.Comment: Pdf Latex, 66 pages+30 pages of appendix, about 30 figures. Revised version: improvement in the presentation of mirror ma

Modular functors, cohomological field theories and topological recursion

Given a topological modular functor $\mathcal{V}$ in the sense of Walker \cite{Walker}, we construct vector bundles over $\bar{\mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction depends on a determination of the logarithm of the eigenvalues of the Dehn twist and central element actions. We show that the intersection of the Chern class with the $\psi$-classes in $\bar{\mathcal{M}}_{g,n}$ is computed by the topological recursion of \cite{EOFg}, for a local spectral curve that we describe. In particular, we show how the Verlinde formula for the dimensions $D_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n}) = \dim \mathcal{V}_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ is retrieved from the topological recursion. We analyze the consequences of our result on two examples: modular functors associated to a finite group $G$ (for which $D_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ enumerates certain $G$-principle bundles over a genus $g$ surface with $n$ boundary conditions specified by $\vec{\lambda}$), and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply-connected Lie group $G$ (for which $\mathcal{V}_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ is the Verlinde bundle).Comment: 50 pages, 2 figures. v2: typos corrected and clarification about the use of ordered pairs of points for glueing. v3: unitarity assumption waived + discussion of families index interpretation of the correlation functions for Wess-Zumino-Witten theorie