Tautological relations and integrable systems

TL;DR

The paper introduces conjectural tautological relations in the cohomology of moduli spaces of curves, linking them to integrable hierarchies and proving special cases using advanced geometric methods.

Contribution

It proposes new conjectural relations in tautological cohomology that imply properties of integrable hierarchies and proves these relations in specific cases.

Findings

Relations imply properties of DZ and DR hierarchies

Proved relations for n=1 and arbitrary g

Proved relations for g=0 and arbitrary n

Abstract

We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Gu\'er\'e and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case $n=1$ and arbitrary $g$ using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hern\'andez Iglesias. We also prove all the above mentioned relations in the case $g=0$ and arbitrary $n$.