Communications in Number Theory and Physics

Volume 18 (2024)

Number 2

Quantum KdV hierarchy and quasimodular forms

Pages: 405 – 439

DOI: https://dx.doi.org/10.4310/CNTP.2024.v18.n2.a4

Authors

Jan-Willem M. van Ittersum (Max-Planck-Institut für Mathematik, Bonn, Germany; and Department of Mathematics and Computer Science, University of Cologne, Germany)

Giulio Ruzza (Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Portugal)

Abstract

Dubrovin $\href{https://doi.org/10.1007/s00023-015-0449-2}{[10]}$ 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 $\href{https://doi.org/10.1007/JHEP07(2014)141}{[1]}$ 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 $\href{https://doi.org/10.1007/s11005-015-0814-6}{[6]}$ 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.

Keywords

partitions, modular forms, double ramification cycles, quantum integrable hierarchies

2010 Mathematics Subject Classification

Primary 11F11, 37K10. Secondary 05A17, 14H70.

Full Text (PDF format)

G.R. acknowledges support from the Fonds de la Recherche Scientifique-FNRS under EOS project O013018F and from the FCT Grant 2022.07810.CEECIND.

Received 14 October 2022

Accepted 7 April 2024

Published 15 July 2024