Quantum KdV hierarchy and quasimodular forms

TL;DR

This paper extends the connection between the spectrum of quantum integrable hierarchies and quasimodular forms from the dispersionless KdV to the full quantum KdV and ILW hierarchies, providing a general criterion for quasimodularity.

Contribution

It introduces a broad criterion for quasimodularity and extends the relation to quantum hierarchies associated with the Double Ramification cycle.

Findings

Spectrum of quantum KdV related to quasimodular forms

Extension of quasimodularity to quantum ILW hierarchy

General effective criterion for quasimodularity

Abstract

Dubrovin has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg-de Vries (KdV) hierarchy is given by shifted symmetric functions; the latter are related by the Bloch-Okounkov Theorem to quasimodular forms on the full modular group. We extend the relation to quasimodular forms to the full quantum KdV hierarchy (and to the more general quantum Intermediate Long Wave hierarchy). These quantum integrable hierarchies have been defined by Buryak and Rossi in terms of the Double Ramification cycle in the moduli space of curves. The main tool and conceptual contribution of the paper is a general effective criterion for quasimodularity.