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

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

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

Loop equations for Gromov-Witten invariants of P1\mathbb{P}^1

We show that non-stationary Gromov-Witten invariants of $\mathbb{P}^1$ can be extracted from open periods of the Eynard-Orantin topological recursion correlators $\omega_{g,n}$ whose Laurent series expansion at $\infty$ compute the stationary invariants. To do so, we overcome the technical difficulties to global loop equations for the spectral $x(z) = z + 1/z$ and $y(z) = \ln z$ from the local loop equations satisfied by the $\omega_{g,n}$, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov-Witten theory of $\mathbb{P}^1$.Comment: 27 pages, 1 figur

Tracy-Widom GUE law and symplectic invariants

We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curve \Sigma_{TW} describing the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This explains directly how the Tracy-Widow law F_{GUE}, governing the distribution of the maximal eigenvalue in hermitian random matrices, can also be recovered from symplectic invariants.Comment: pdfLatex, 36 pages, 1 figure. v2: typos corrected, re-sectioning, a reference adde

Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

We consider the Gopakumar-Ooguri-Vafa correspondence, relating ${\rm U}(N)$ Chern-Simons theory at large $N$ to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients $\mathbb{S}^{\Gamma} = \Gamma\backslash\mathbb{S}^3$ of the three-sphere by the free action of a finite isometry group. Guided by string theory dualities, we propose a large $N$ dual description in terms of both A- and B-twisted topological strings on (in general non-toric) local Calabi-Yau threefolds. The target space of the B-model theory is obtained from the spectral curve of Toda-type integrable systems constructed on the double Bruhat cells of the simply-laced group identified by the ADE label of $\Gamma$. Its mirror A-model theory is realized as the local Gromov-Witten theory of suitable ALE fibrations on $\mathbb{P}^1$, generalizing the results known for lens spaces. We propose an explicit construction of the family of target manifolds relevant for the correspondence, which we verify through a large $N$ analysis of the matrix model that expresses the contribution of the trivial flat connection to the Chern-Simons partition function. Mathematically, our results put forward an identification between the $1/N$ expansion of the $\mathrm{sl}_{N + 1}$ LMO invariant of $\mathbb{S}^\Gamma$ and a suitably restricted Gromov-Witten/Donaldson-Thomas partition function on the A-model dual Calabi-Yau. This $1/N$ expansion, as well as that of suitable generating series of perturbative quantum invariants of fiber knots in $\mathbb{S}^\Gamma$, is computed by the Eynard-Orantin topological recursion.Comment: 65 page

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

Simple maps, Hurwitz numbers, and Topological Recursion

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation $x \leftrightarrow y$ on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of $x$ and $y$. We propose an argument to prove this statement conditionally to a mild version of symplectic invariance for the $1$-hermitian matrix model, which is believed to be true but has not been proved yet. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters $(\lambda_1,\ldots,\lambda_n)$ are strictly monotone double Hurwitz numbers with ramifications $\lambda$ above $\infty$ and $(2,\ldots,2)$ above $0$. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.Comment: 66 pages, 7 figure