Elliptic Calogero-Moser system from two dimensional current algebra

TL;DR

The paper derives the elliptic Calogero-Moser system from a two-dimensional current-algebra framework on an elliptic curve through Hamiltonian reduction of the cotangent bundle to the central extension of SL_N currents, augmented by a CP^{N−1} factor. It constructs the Krichever-type Lax operator and shows that the Hamiltonians arise from invariants like tr φ(z,zbar)^2, yielding the elliptic CM Hamiltonian with a Weierstrass wp potential and a quantum shift ν^2→ν(ν−1). It also discusses an elliptic deformation of two-dimensional Yang–Mills theory, proposes a field-theoretic setting for wavefunctions via intertwiners of Verma modules instead of finite-dimensional reps, and ties these ideas to elliptic Sklyanin algebras and potential connections to elliptic Ruijsenaars models. The work reveals a geometric and representation-theoretic origin for elliptic integrable systems and proposes avenues to study spectra, solitons, and related models within this framework.

Abstract

We show that elliptic Calogero-Moser system and its Lax operator found by Krichever can be obtained by Hamiltonian reduction from the integrable Hamiltonian system on the cotangent bundle to the central extension of the algebra of SL(N,C) currents.Elliptic deformation of Yang-Mills theory is presented.