The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves

TL;DR

This paper constructs a quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves, revealing a modular, explicit hierarchy involving fermionic fields, based on advanced intersection number calculations.

Contribution

It introduces the first explicit quantum integrable hierarchy from a cohomological field theory with fermionic fields, using novel intersection number results.

Findings

Derived a closed, modular expression for the hierarchy

First explicit example involving fermionic fields in such hierarchies

Connected intersection theory with quantum integrable systems

Abstract

We construct the quantum double ramification hierarchy associated with the Gromov-Witten theory of elliptic curves. We use results of Oberdieck and Pixton on the intersection numbers of the double ramification cycle, the Gromov-Witten classes of the elliptic curve and the Hodge class $\lambda_{g-1}$ together with vanishing results for $\lambda_{g-2}$ to produce a closed, modular expression for the resulting integrable hierarchy. It is the first explicit nontrivial example of a quantum integrable hierarchy from a cohomological field theory containing fermionic fields, which correspond to the odd classes in the cohomology of the elliptic curve.