Topological recursion in the Ramond sector

Abstract

We investigate supereigenvalue models in the Ramond sector and their recursive structure. We prove that the free energy truncates at quadratic order in Grassmann coupling constants, and consider super loop equations of the models with the assumption that the 1/N expansion makes sense. Subject to this assumption, we obtain the associated genus-zero algebraic curve with two ramification points (one regular and the other irregular) and also the supersymmetric partner polynomial equation. Starting with these polynomial equations, we present a recursive formalism that computes all the correlation functions of these models. Somewhat surprisingly, correlation functions obtained from the new recursion formalism have no poles at the irregular ramification point due to a supersymmetric correction-the new recursion may lead us to a further development of supersymmetric generalizations of the Eynard-Orantin topological recursion.

Correlators in the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector

“…Let us collect the coefficients of t l 1 and t l 2 in (2.12) and set to zero, respectively, we obtainC 17) and the recursive relations…”

“…The supereigenvalue models have attracted considerable attention. They can be regarded as supersymmetric generalizations of matrix models [1]- [17]. Many of the important features of matrix models, such as the Virasoro constraints, the loop equations, the genus expansions and the moment descriptions have their supersymmetric counterparts in the supereigenvalue models.…”

Airy Structures for Semisimple Lie Algebras

“…It is thus conceivable that results concerning Airy structures (and their supersymmetric generalizations [19], related to supereigenvalue models and the corresponding topological recursion [20][21][22][23][24]) may find applications in some of the subjects listed above.…”

Super Quantum Airy Structures

“…However, such relations are obscure, even though corresponding supersymmetric structures in matrix models are known. Indeed, supersymmetric generalizations of matrix models, referred to supereigenvalue models, have been introduced and discussed some time ago [3,8], and also more recently [20,[26][27][28]61]. By construction, loop equations for such supereigenvalue models can be rewritten in the form of super-Virasoro constraints.…”

“…This generalizes the reformulation of loop equations in terms of Virasoro constraints in the non-supersymmetric case, and thus one might hope that super-Virasoro constraints for supereigenvalue models lead immediately to supersymmetric topological recursion. However, in [ 20 , 61 ] it was shown that such a generalization is not automatic: one can indeed write down a recursive system that determines the partition function of a supereigenvalue model, but it is augmented by an auxiliary equation, which does not have a simple interpretation. In a sense, this makes the super quantum Airy structures that we introduce here even more interesting, and revealing their meaning in the context of supereigenvalue models is an important task.…”