Spectral Duality Between Heisenberg Chain and Gaudin Model

TL;DR

The paper establishes a comprehensive higher-rank extension of the spectral duality between the GL_k Heisenberg spin chain and a special reduced gl_N Gaudin model, demonstrating both classical and quantum equivalence. It constructs an explicit Poisson map via Dirac reduction and the Adams–Harnad–Hurubise duality, linking the reduced Gaudin phase space to the Heisenberg chain’s phase space and matching their spectral data. In the quantum regime, it shows that the Baxter equations of the two systems coincide under a precise ordering, yielding exact wavefunction correspondences: ψ_XXX(z) = ψ_Gaudin(z) exp[(1/(Nħ))∫^z b_ħ dz]. Framed within the AGT correspondence, these results illuminate how gauge-theory–inspired integrable models map onto conformal/Gaudin-type systems, highlighting the role of spectral curves, SW differentials, and bispectrality in enabling dual descriptions with practical implications for exact quantization and monodromy data.

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.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4