Cohomological representations of quantum tau functions

TL;DR

This paper explores cohomological representations of quantum tau functions linked to quantum DR hierarchies, providing new formulas and equations for their correlators, especially for the topological solution.

Contribution

It introduces two cohomological representations for quantum tau function correlators and establishes fundamental equations and vanishing results.

Findings

Two cohomological representations involving $A$-classes and $\,Omega$-classes.

String and dilaton equations are proven for these tau functions.

Certain correlator vanishings are demonstrated.

Abstract

In 2016, Buryak and Rossi introduced the quantum Double Ramification (DR) hierarchies which associate a quantum integrable hierarchy to any Cohomological Field Theory (CohFT). Shortly after, they introduced, in collaboration with Dubrovin and Gu\'er\'e, the quantum tau functions of these hierarchies. In this work, we study quantum tau functions associated to a specific solution called the topological solution. We provide two cohomological representations for the correlators of these tau functions. The first representation involves an analog in the quantum setting of the $A$-class of the DR-DZ equivalence. The second representation, valid for CohFT of low degree, involves the so-called $\Omega$-classes. Furthermore, we establish the string and dilaton equations for these tau functions, and present certain vanishing of their correlators.