Transfer matrices of rational spin chains via novel BGG-type resolutions

TL;DR

This paper develops BGG-type formulas for transfer matrices of certain finite-dimensional irreducible representations of classical Lie algebras, expressing them via infinite-dimensional modules and factorizing into Baxter Q-operators, with geometric insights from flag varieties.

Contribution

Introduces new BGG-type resolutions for finite-dimensional modules and expresses transfer matrices in terms of infinite-dimensional modules, linking representation theory and integrable models.

Findings

Derived BGG-type formulas for transfer matrices.

Expressed transfer matrices as products of Baxter Q-operators.

Connected geometric representation theory with integrable systems.

Abstract

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.