Gauge Theory and Boundary Integrability

TL;DR

This paper explores the connection between boundary integrable systems and gauge theory, deriving formulas for K-matrices and linking line operators to twisted Yangians, advancing the understanding of boundary conditions in integrable models.

Contribution

It provides a gauge theory framework for boundary integrability, including formulas for K-matrices and the realization of twisted Yangians in this context.

Findings

Derived asymptotic behavior of K-matrices from gauge theory.

Linked line operators on the boundary to twisted Yangian representations.

Realized boundary Yang-Baxter conditions via gauge theory constructions.

Abstract

We study the mixed topological / holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold $(\Sigma\times{\mathbb C})/{\mathbb Z}_2$, obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order $\hbar$ calculation we derive a formula for the the asymptotic behaviour of $K$-matrices associated to rational, quasi-classical $R$-matrices. The ${\mathbb Z}_2$-action on $\Sigma\times {\mathbb C}$ fixes a line $L$, and line operators on $L$ are shown to be labelled by representations of the twisted Yangian. The OPE of such a line operator with a Wilson line in the bulk is shown to give the coproduct of the twisted Yangian. We give the gauge theory realisation of the Sklyanin determinant and related conditions in the $RTT$ presentation of the boundary Yang-Baxter equation.