Spectral Duality Between Heisenberg Chain and Gaudin Model

TL;DR

This paper extends the spectral duality between integrable systems, specifically linking higher spin Heisenberg chains with reduced Gaudin models, by establishing explicit classical Poisson maps and dualities.

Contribution

It generalizes the spectral duality to higher spin chains and constructs explicit Poisson maps between the models.

Findings

Proves N-site GL(k) Heisenberg chain is dual to a special reduced k+2-point gl(N) Gaudin model.

Constructs explicit classical Poisson maps via Dirac reduction and AHH duality.

Extends spectral duality framework inspired by the AGT conjecture.

Abstract

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL(k) Heisenberg chain is dual to the special reduced k+2-points gl(N) Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.