Meromorphic differentials, twisted DR cycles and quantum integrable hierarchies

TL;DR

This paper introduces twisted double ramification hierarchies derived from intersection theory of meromorphic differentials, establishing their integrability and revealing deep connections to classical integrable systems like the KdV hierarchy.

Contribution

It defines new twisted hierarchies based on intersection theory, proves their integrability and tau symmetry, and uncovers their relation to the KdV hierarchy and intersection numbers.

Findings

Twisted hierarchies are integrable and tau symmetric.

The trivial cohomological field theory yields the KdV hierarchy.

New identities relate intersection numbers of tautological classes.

Abstract

We define twisted versions of the classical and quantum double ramification hierarchy construction based on intersection theory of the strata of meromorphic differentials in the moduli space of stable curves and $k$-twisted double ramification cycles for $k=1$, respectively, we prove their integrability and tau symmetry and study their connection. We apply the construction to the case of the trivial cohomological field theory to find it produces the KdV hierarchy, although its relation to the untwisted case is nontrivial. The key role of the KdV hierarchy in controlling the intersection theory of several natural tautological classes translates this relation into a series of remarkable identities between intersection numbers involving psi-classes, Hodge classes, Norbury's theta class and the strata of meromorphic differentials.