Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum KdV hierarchy and shifted symmetric functions

Jan-Willem van Ittersum, Giulio Ruzza

TL;DR

This work analyzes the spectral problem for the quantum double ramification (DR) hierarchy, a quantum deformation of the KdV hierarchy, by exploiting the boson-fermion correspondence and fermionic quartic expressions. It proves that the first-order dispersion correction to quantum KdV eigenvalues is a shifted symmetric function in the underlying partition, explicitly written in terms of the $Q_k(\lambda)$; it also provides an explicit first-order eigenvector correction whose Schur expansion is supported only on partitions at minimal Hamming distance. As an application, the authors evaluate certain double Hodge integrals on moduli spaces of curves and derive a border-strip–type description of eigenvector corrections. The results illuminate the connection between moduli-space intersection theory, shifted symmetric functions, and quantum integrable systems, and they reveal a quasimodular structure in the associated generating functions.

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.

Quantum KdV hierarchy and shifted symmetric functions

TL;DR

This work analyzes the spectral problem for the quantum double ramification (DR) hierarchy, a quantum deformation of the KdV hierarchy, by exploiting the boson-fermion correspondence and fermionic quartic expressions. It proves that the first-order dispersion correction to quantum KdV eigenvalues is a shifted symmetric function in the underlying partition, explicitly written in terms of the ; it also provides an explicit first-order eigenvector correction whose Schur expansion is supported only on partitions at minimal Hamming distance. As an application, the authors evaluate certain double Hodge integrals on moduli spaces of curves and derive a border-strip–type description of eigenvector corrections. The results illuminate the connection between moduli-space intersection theory, shifted symmetric functions, and quantum integrable systems, and they reveal a quasimodular structure in the associated generating functions.

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.
Paper Structure (22 sections, 21 theorems, 147 equations)

This paper contains 22 sections, 21 theorems, 147 equations.

Table of Contents

  1. Introduction and results
  2. Generating functions of double ramification intersection numbers
  3. Spectral problem
  4. Results
  5. Further comments and conjectures
  6. Quantum double ramification hierarchies
  7. Recursion
  8. Reduction to c=0
  9. Differential polynomials formalism
  10. Partitions and the boson-fermion correspondence
  11. Partitions
  12. Infinite wedge formalism
  13. Boson-fermion correspondence
  14. First order correction to the eigenvalues
  15. Quadratic and quartic fermionic operators
  16. ...and 7 more sections

Key Result

Theorem 1.6

For all $k\geq 0$ we have where the shifted symmetric function $Q_k$ is defined by Eq. eq:Qk.

Theorems & Definitions (52)

  • Remark 1.1
  • Example 1.2
  • Remark 1.3: Quantum KdV
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Conjecture 1.9
  • Remark 1.10
  • ...and 42 more