Generalized Kontsevich model, topological recursion, and $r$-spin theory

TL;DR

This paper establishes explicit links between the generalized Kontsevich model, topological recursion, and $r$-spin moduli space geometry using polynomial KP integrability and string equations.

Contribution

It provides explicit formulations and proofs of the relationships between these models and extends the method to deformed potentials and related geometric theories.

Findings

Derived explicit formulas for the generalized Kontsevich model with polynomial potentials.

Proved the correspondence between the model, topological recursion, and $r$-spin geometry.

Extended the approach to deformed potentials and their geometric interpretations.

Abstract

By employing polynomial-reduced KP integrability, combined with the string equation, this work establishes explicit relationships between the generalized Kontsevich model, the topological recursion of the spectral curve, and the geometry of moduli spaces of $r$-spin curves. For the generalized Kontsevich model with a polynomial potential, we derive an explicit formulation and provide a proof of these widely expected correspondences. Furthermore, the method is extended to the cases with admissible deformed potentials, where the corresponding geometric theory is a deformed version of $r$-spin theory.