Extended $r$-spin theory in all genera and the discrete KdV hierarchy

TL;DR

This paper extends r-spin theory to all genera with a new family of classes, linking them to the discrete KdV hierarchy and introducing F-CohFTs that generalize cohomological field theories.

Contribution

It constructs a new family of cohomology classes generalizing Witten's r-spin classes, forming F-CohFTs, and connects these to the discrete KdV hierarchy.

Findings

The new classes generalize r-spin classes to all genera.

Partition function solves the discrete KdV hierarchy.

The classes form an F-CohFT, not a traditional CohFT.

Abstract

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus $0$ it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.