Shifted Witten classes and topological recursion

TL;DR

This paper demonstrates that the descendant intersection theory of shifted Witten classes can be computed via topological recursion on spectral curves, leading to a proof of Witten's $r$-spin conjecture relating intersection numbers to the $r$-KdV hierarchy.

Contribution

It introduces a method to derive the $R$-matrix and translation for shifted Witten classes from differential equations, establishing topological recursion for their descendant intersection theory.

Findings

Topological recursion applies to shifted Witten classes on spectral curves.

The $r$-spin intersection theory can be computed by topological recursion on the $r$-Airy spectral curve.

The approach confirms Witten's $r$-spin conjecture linking intersection numbers to the $r$-KdV hierarchy.

Abstract

The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold using the Givental--Teleman reconstruction theorem. We show that the $R$-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Using this, we prove that the descendant intersection theory of the shifted Witten classes satisfies topological recursion on two $1$-parameter families of spectral curves. By taking the limit as the parameter goes to zero for these families of spectral curves, we prove that the descendant intersection theory of the Witten $r$-spin class can be computed by topological recursion on the $r$-Airy spectral curve. We finally show that this proof suffices to deduce Witten's $r$-spin conjecture, already proved by Faber, Shadrin and Zvonkine, which claims that the generating series of $r$-spin intersection numbers is the tau function of the $r$-KdV hierarchy that satisfies the string equation.