Extended systems of Baxter Q-functions and fused flags I: simply-laced case

Abstract

The spectrum of integrable models is often encoded in terms of commuting functions of a spectral parameter that satisfy functional relations. We propose to describe this commutative algebra in a covariant way by means of the extended Q-system that comprise Q-vectors in each of the fundamental representations of the (Langlands dual of) the underlying symmetry algebra. These Q-vectors turn out to parameterise a collection of complete flags which are fused with one another in a particular way. We show that the fused flag is gauge equivalent to a finite-difference oper, explicit equivalence depends on (an arbitrary choice of) a Coxeter element.The paper considers the case of simple Lie algebras with a simply-laced Dynkin diagram. For the A r series, the construction coincides with already known results in the literature. We apply the proposed formalism to the case of the D r series and the exceptional algebras E r , r = 6, 7, 8. In particular, we solve Hirota bilinear equations in terms of Q-functions and give the explicit character solution of the extended Q-system in the D r case. We also show how to build up the extended Q-system of D r type starting either from vectors, by a procedure similar to the A r scenario which however constructs a fused flag of isotropic spaces, or from pure spinors, via fused Fierz relations.Finally, for the case of rational, trigonometric, and elliptic spin chains, we propose an explicit ansatz for the analytic structure of Q-functions of the extended Q-system. We conjecture that the extended Q-system constrained in such a way is always in bijection with actual Bethe algebra of commuting transfer matrices of these models and moreover can be used to show that the Bethe algebra has a simple joint spectrum.

Bethe Algebra using Pure Spinors

“…Section IV studies gl r -invariant ways to compute transfer matrices. Some of the relations in Sections II-IV are also present in [8] (modulo adjustment of conventions) and we thus provide their group-theoretical interpretation from the perspective of [5]. Section V is a proof of concept: we demonstrate how to use the proposed technology to explicitly compute the spectrum of rational spin chains, the results are available in the attached Mathematica notebook.…”

“…In [5], H. Shu and the authors described the Bethe algebra for any simply-laced g in terms of an extended Q-system. This approach offers full covariance: if an integrable model is based on (the quantum deformation of the current or the loop algebra of) g then Q-functions generating the Bethe algebra are Plücker coordinates of the fused flag corresponding to the Langlands dual L ĝ.…”

“…For so 2r models, candidates for analogs of Q a can be spotted in the work [8] by G. Ferrando, R. Frassek, and V. Kazakov who considered Q-system on the Weyl orbit (we use the terminology of [5]) and proposed determinant-type formulae for transfer matrices in symmetric powers of the vector representation. These formulae feature only r Q-functions and, by simple inspection, are gl r -invariant, where gl r ⊂ so 2r corresponds to the standard embedding U(r) ⊂ SO(2r) (we do not need to specify real forms and hence select gl r for notation).…”

“…We build on the findings of [5] and [8] to offer and investigate the explicit gl r -covariant description of so 2r Bethe algebra. The key observation is the following: Because Qfunctions are Plücker coordinates, those of them who define maximal isotropic subspaces are components of pure spinors.…”

“…The paper is organised as follows: in Section II we summarise the relevant findings of [5] about the extended so 2r Q-system, with a slight update of notations. Section III describes gl r parameterisation of the Q-system leading to formulation of spinor Q-system which is selfcontained for the most interesting examples, and this is the main conceptual message of the article.…”