Spectral Duality in Integrable Systems from AGT Conjecture

TL;DR

This paper establishes a spectral duality between the N-site Heisenberg spin chain and a reduced gl(N) Gaudin model, connecting gauge theory and conformal field theory through classical and quantum integrable systems.

Contribution

It proves the spectral duality at both classical and quantum levels, linking integrable systems from the AGT conjecture with new insights into their equivalence.

Findings

Classical duality relates Seiberg-Witten differentials and spectral curves via bispectral involution.

Quantum duality shows equivalence of Baxter-Schrodinger equations and quantum spectral curves.

Generalizes spectral self-duality and AHH duality in integrable systems.

Abstract

We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl(N) Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrodinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2x2 and NxN representations of the Toda chain and the famous AHH duality.