Lecture notes on topological recursion and geometry

These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of topological recursion: in two dimensional quantum field theories, in cohomological field theories, in the computation of Weil-Petersson volumes of the moduli space of curves; (b) relate them more specifically to Eynard-Orantin topological recursion (revisited from Kontsevich-Soibelman point of view based on quantum Airy structures).Comment: 48 pages, 16 figure

An automorphic generalization of the Hermite-Minkowski theorem

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed.Comment: 30 pages, 1 tabl

Formal multidimensional integrals, stuffed maps, and topological recursion

We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governed by the topological recursion of [Eynard and Orantin, 2007] with initial conditions. In terms of a 1d gas of eigenvalues, this model includes - on top of the squared Vandermonde - multilinear interactions of any order between the eigenvalues. In this problem, the initial data (W10,W20) of the topological recursion is characterized: for W10, by a non-linear, non-local Riemann-Hilbert problem on a discontinuity locus to determine ; for W20, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - W10 being the generating series of disks and W20 that of cylinders. In particular, by substitution one may consider maps whose 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 1/N in the one hermitian matrix model.Comment: 33 pages, 6 figures ; v2, a correction and simplification in the final argument (Section 5

Asymptotic expansion of beta matrix models in the multi-cut regime

We push further our study of the all-order asymptotic expansion in beta matrix models with a confining, offcritical potential, in the regime where the support of the equilibrium measure is a reunion of segments. We first address the case where the filling fractions of those segments are fixed, and show the existence of a 1/N expansion to all orders. Then, we study the asymptotic of the sum over filling fractions, in order to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. We describe the application of our results to study the all-order small dispersion asymptotics of solutions of the Toda chain related to the one hermitian matrix model (beta = 2) as well as orthogonal polynomials outside the bulk.Comment: 59 pages. v4: proof of smooth dependence in filling fraction (Appendix A) corrected, comment on the analogue of the CLT added, typos corrected. v5: Section 7 completely rewritten, interpolation for expansion of partition function is now done by decoupling the cuts, details on comparison to Eynard-Chekhov coefficients added in the introductio

Geometry of Spectral Curves and All Order Dispersive Integrable System

We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between "correlators", the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry