Supereigenvalue models and topological recursion

Abstract

We show that the Eynard-Orantin topological recursion, in conjunction with simple auxiliary equations, can be used to calculate all correlation functions of supereigenvalue models. Hermitian Matrix Models and Topological RecursionIn this section we introduce formal Hermitian matrix models, and review the connections between Virasoro constraints, loop equations and topological recursion. A detailed discussion can be found in [24]. Formal Hermitian Matrix ModelsThe question of convergence of matrix integrals is a rather complex one. However, for many applications of matrix models in physics and enumerative geometry, convergence is not really necessary. More precisely, in this context -even though it is not always explicitly mentioned -we are often interested in so-called formal matrix models, rather than convergent matrix models. The difference between the two is well explained in [24].In this paper we focus on formal matrix models. An important consequence of formal matrix models is that the quantities of interest, such as the partition function, the free energy and correlation functions, all possess a well-defined 1/N expansion. Let us now define formal matrix models.

Topological recursion in the Ramond sector

“…genus correlation functions, and most importantly, no correlation functions have a pole at the irregular ramification point due to a supersymmetric correction. Hence, the recursive formalism evidently differs from the Eynard-Orantin topological recursion unlike what is shown in [10] for the NS sector. We summarize the results of this paper in Theorem 5.2.…”

“…Even such restricted models give rise to interesting results as outlined in section 1. Discussions and computations in the paper closely follow the presentation of [10,22].…”

“…We are working in a more abstract framework that potentially unifies both the NS sector and Ramond sector into one package. However, since such abstraction is beyond the scope of this paper, we leave more discussion to a paper in progress [11]. We also note that it is an open question whether the new recursive formalism shown in this paper (or perhaps the more abstract one in [11]) has interesting applications beyond the study of supereigenvalue models.…”

“…(2.15) [34] proved (see also appendix A of [10]) that the free energy of those in the NS sector depends on Grassmann couplings ξ N S l+ 1 2 only up to quadratic order. We now prove that this feature also holds in the Ramond sector:…”

“…For the NS sector, [10] showed that all correlation functions can be recursively computed by the Eynard-Orantin topological recursion in conjunction with an auxiliary Grassmann-valued polynomial equation, which can be thought of as the supersymmetric partner of the algebraic curve for the Eynard-Orantin topological recursion. This is because of great simplification thanks to Becker's formula [7,34] that gives an explicit relation to Hermitian matrix models.…”