Harmonic locus and Calogero-Moser spaces

TL;DR

This paper explores the harmonic locus of monodromy-free Schrödinger operators with rational potentials, linking it to Calogero-Moser spaces, and provides new insights into the inverse Wronskian problem and fixed point character computations.

Contribution

It establishes a novel identification of the harmonic locus with Calogero-Moser spaces fixed by a symplectic action, and solves the inverse Wronskian problem for the multiplicity-free case.

Findings

Identified the harmonic locus with a subset of Calogero-Moser space fixed by $\

Solved the inverse Wronskian problem for the multiplicity-free locus.

Computed characters of the $\

Abstract

We study the harmonic locus consisting of the monodromy-free Schr\"odinger operators with rational potential and quadratic growth at infinity. It is known after Oblomkov that it can be identified with the set of all partitions via the Wronskian map for Hermite polynomials. We show that the harmonic locus can also be identified with the subset of the Calogero--Moser space introduced by Wilson, which is fixed by the symplectic action of $\mathbb C^\times.$ As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by describing the partition in terms of the spectrum of the corresponding Moser matrix. We also compute the characters of the $\mathbb C^\times$-action at the fixed points, proving, in particular, a conjecture of Conti and Masoero. In the Appendix written by N. Nekrasov there is an alternative proof of this result, based on the space of instantons and ADHM construction.