Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

TL;DR

This paper develops the theory of finite type modules over the quantum toroidal gl(1) algebra, introduces a q-character framework, and applies these results to diagonalize transfer matrices and derive Bethe Ansatz equations.

Contribution

It introduces the concept of finite type modules for the quantum toroidal gl(1) algebra and connects their q-characters to transfer matrix eigenvalues and Bethe Ansatz solutions.

Findings

Finite type modules have a finite number of eigenvalues for the Cartan current.

Transfer matrices for these modules are polynomials satisfying a TQ relation.

Bethe Ansatz equations are derived for the eigenvalues of the transfer matrices.

Abstract

We study highest weight representations of the Borel subalgebra of the quantum toroidal gl(1) algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of `finite type' modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current \psi^+(z) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogous to those of the six vertex model. In our setting T_{V,W}(u;p) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal gl(1) with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules $V$ the corresponding transfer matrices, Q(u;p) and T(u;p), are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q(u;p). Then we show that the eigenvalues of T_{V,W}(u;p) are given by an appropriate substitution of eigenvalues of Q(u;p) into the q-character of V.