Calogero model revisited, commuting Hamiltonians, Hurwitz numbers

TL;DR

This paper explores advanced Hamiltonians in the Calogero-Sutherland model, revealing their connections to Hurwitz numbers and extending the integrable system framework at a special coupling constant.

Contribution

It introduces a new series of commuting Hamiltonians parametrized by matrices, linking them to Hurwitz numbers and generalizing the Calogero-Sutherland model.

Findings

Identification of Hamiltonians as the center of the enveloping algebra of GL_N(C)

Establishment of a connection between these Hamiltonians and Hurwitz numbers

Extension of the Calogero-Sutherland model to a generalized variant

Abstract

The generalized Mironov-Morozov-Natanson (MMN) equation includes a set of commuting operators, which can be considered as Hamiltonians for the quantum Calogero-Sutherland problem with a special value of the coupling constant (free fermion point). These Hamiltonians can be considered as the center of the enveloping algebra of the group $GL_N(C)$. Another commuting series of Hamiltonians is presented, parametrized by an arbitrary matrix $A\in GL_N$. These Hamiltonians are related to the Hurwitz numbers in the same way as in the case of the MMN equation and generate a generalized variant of the Calogero-Surtheland model.