The planar limit of $$ \mathcal{N} $$ = 2 chiral correlators

We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of $$ \mathcal{N} $$ N = 2 SQCD on S4, to all orders in the ’t Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ℝ4. Specifically, we compute all the terms with a single value of the ζ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ℝ4.

Genus expansion of matrix models and ћ expansion of KP hierarchy

We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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 (higher order) free cumulants established by Collins et al. [22], and implement the symplectic transformation $$x \leftrightarrow y$$ x ↔ 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 the symplectic invariance for the 1-hermitian matrix model, which is believed to be true but has not been proved yet. Our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion. 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 $$(\lambda _1,\ldots ,\lambda _n)$$ ( λ 1 , … , λ n ) are strictly monotone double Hurwitz numbers with ramifications $$\lambda $$ λ above $$\infty $$ ∞ and $$(2,\ldots ,2)$$ ( 2 , … , 2 ) above 0. Combining with a recent result of Dubrovin et al. [24], this implies an ELSV-like formula for these Hurwitz numbers.

The sine-law gap probability, Painlevé 5, and asymptotic expansion by the topological recursion

The goal of this paper is to rederive the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled Hermitian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ-function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any Hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N-5).