On non-stationary Lam\'e equation from WZW model and spin-1/2 XYZ chain

TL;DR

This paper explores the deep connection between the WZW conformal field theory and the spin-1/2 XYZ chain by comparing their differential equations, revealing a correspondence that enhances understanding of integrable models and conformal blocks.

Contribution

It establishes a novel link between WZW model equations and XYZ chain relations, interpreting Stroganov's odd chain result within the WZW framework.

Findings

Confirmed the correspondence via a trigonometric limit of the XYZ chain.

Eigenstates of the Sutherland model can be derived from deformed conformal blocks.

Mapped the representation space dimension to the number of chain sites.

Abstract

We study the link between WZW model and the spin-1/2 XYZ chain. This is achieved by comparing the second-order differential equations from them. In the former case, the equation is the Ward-Takahashi identity satisfied by one-point toric conformal blocks. In the latter case, it arises from Baxter's TQ relation. We find that the dimension of the representation space w.r.t. the V-valued primary field in these conformal blocks gets mapped to the total number of chain sites. By doing so, Stroganov's "The Importance of being Odd" (cond-mat/0012035) can be consistently understood in terms of WZW model language. We first confirm this correspondence by taking a trigonometric limit of the XYZ chain. That eigenstates of the resultant two-body Sutherland model from Baxter's TQ relation can be obtained by deforming toric conformal blocks supports our proposal.