Rooted trees with level structures, $\Omega$-classes and double ramification cycles

TL;DR

This paper establishes new relations in the tautological ring of the moduli space of curves involving rooted trees, $ abla$-classes, and double ramification structures, resolving a recent conjecture and connecting to integrable hierarchies.

Contribution

It proves a new system of relations in the tautological ring involving rooted trees with level structures, $ abla$-classes, and double ramification cycles, confirming a recent conjecture.

Findings

Resolved a conjecture on relations in the tautological ring.

Connected double ramification structures with integrable hierarchies.

Established new relations involving $ abla$-classes and rooted trees.

Abstract

We prove a new system of relations in the tautological ring of the moduli space of curves involving stable rooted trees with level structure decorated by the top Chern class of the Hodge bundle and $\Omega$-classes and double ramification structures. In particular, this resolves a recent conjecture on these relations as well as connects with one of the two sides of the recently established DR/DZ equivalence between the integrable hierarchies constructions of Buryak and of Dubrovin--Zhang.