Quantum KdV hierarchy and shifted symmetric functions

TL;DR

This paper investigates the spectral properties of the quantum KdV hierarchy, revealing that eigenvalues relate to shifted symmetric functions and providing explicit formulas for eigenvectors, with applications to Hodge integrals.

Contribution

It establishes a connection between eigenvalues of the quantum KdV hierarchy and shifted symmetric functions, using boson-fermion correspondence, and derives explicit eigenvector formulas.

Findings

Eigenvalues are given by shifted symmetric functions to first order in dispersion.

Provides a closed-form evaluation of certain double Hodge integrals.

Explicit formula for eigenvectors with Schur expansion supported on minimal Hamming distance partitions.

Abstract

We study spectral properties of the quantum Korteweg-de Vries hierarchy defined by Buryak and Rossi. We prove that eigenvalues to first order in the dispersion parameter are given by shifted symmetric functions. The proof is based on the boson-fermion correspondence and an analysis of quartic expressions in fermions. As an application, we obtain a closed evaluation of certain double Hodge integrals on the moduli spaces of curves. Finally, we provide an explicit formula for the eigenvectors to first order in the dispersion parameter. In particular, we show that its Schur expansion is supported on partitions for which the Hamming distance is minimal.