Liouville reflection operator, affine Yangian and Bethe ansatz

TL;DR

This paper explores the integrable structure of conformal field theory using Liouville reflection operators and affine Yangian, deriving Bethe ansatz equations for spectra related to the Intermediate Long Wave hierarchy.

Contribution

It establishes a connection between Liouville reflection operators, the affine Yangian of gl(1), and Bethe ansatz equations within conformal field theory.

Findings

Constructed commuting transfer matrices for the ILW hierarchy.

Derived Bethe ansatz equations for the spectra.

Linked Liouville reflection operators with affine Yangian structures.

Abstract

In these notes we study integrable structure of conformal field theory by means of Liouville reflection operator/Maulik-Okounkov $R$-matrix. We discuss the relation between $RLL$ and current realization of the affine Yangian of $\mathfrak{gl}(1)$. We construct the family of commuting transfer matrices related to the Intermediate Long Wave hierarchy and derive Bethe ansatz equations for their spectra discovered by Nekrasov and Okounkov and independently by one of the authors. Our derivation mostly follows the one by Feigin, Jimbo, Miwa and Mukhin, but is adapted to the conformal case.