Baxter's Relations and Spectra of Quantum Integrable Models

TL;DR

This paper develops generalized Baxter's relations for transfer-matrices in quantum affine algebras, interprets them in the Grothendieck ring, and proves a conjecture relating spectra to Baxter polynomials, advancing understanding of quantum integrable systems.

Contribution

It constructs and proves generalized Baxter's relations for transfer-matrices in quantum affine algebras and relates spectra to Baxter polynomials, confirming a longstanding conjecture.

Findings

Transfer-matrices associated with prefundamental representations have polynomial eigenvalues.

Spectra of transfer-matrices are expressed in terms of generalized Baxter polynomials.

Proves a conjecture relating spectra to Baxter polynomials for all untwisted quantum affine algebras.

Abstract

Generalized Baxter's relations on the transfer-matrices (also known as Baxter's TQ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the category O introduced by Jimbo and the second author in arXiv:1104.1891 involving infinite-dimensional representations constructed in arXiv:1104.1891, which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Combining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 (arXiv:math/9810055). We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras.