Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model

TL;DR

The paper establishes remarkable identities for the elliptic Calogero-Sutherland (eCS) model by deriving them from a second-quantized, finite-temperature quantum field theory of anyons on a circle. It introduces a theta-function kernel $F_{N,M}({\bf x};{\bf y})$ built from pairwise theta factors and proves two families of operator identities involving $H_{\lambda,N}({\bf x})$, $H_{\lambda,M}({\bf y})$, and a beta-derivative term, with explicit constants $c_{N,M}$ and $\tilde{c}_{N,M}$. A dual set $\tilde{F}_{N,M}$ with mixed statistics ($\lambda$ and $1/\lambda$) is also shown, along with total momentum constraints. In addition to the quantum-field-theoretic derivation, the paper provides elementary direct proofs of these identities and discusses special cases (e.g., $M=0$ and $N=M$) and potential connections to Q-operators and conformal Ward identities on the torus. The results offer a generating-function perspective on eCS identities and suggest avenues for elliptic perturbation theory and links to CFT frameworks.

Abstract

We present further remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers $N$ and $M$, and a particular function of $N+M$ variables arising as anyon correlation function of $N$ particles and $M$ anti-particles. In addition to identities obtained from anyons with the same statistics parameter $λ$, we also obtain ``dual'' relations involving ``mixed'' correlation functions of anyons with two different statistics parameters $λ$ and $1/λ$. We also give alternative, elementary proofs of these identities by direct computations.