Gauge Theory and Boundary Integrability II: Elliptic and Trigonometric Case

TL;DR

This paper explores boundary integrability in gauge theories via topological-holomorphic Chern-Simons theory on orbifolds, constructing solutions to the boundary Yang-Baxter equation in elliptic and trigonometric cases, including novel $z$-dependent actions.

Contribution

It introduces a new approach to boundary integrability using orbifolded topological gauge theories and constructs explicit semi-classical solutions to the boundary Yang-Baxter equation.

Findings

Constructed semi-classical solutions for boundary Yang-Baxter equation.

Developed novel $z$-dependent $bZ_2$ actions in the trigonometric case.

Matched constructed $K$-matrices with known literature cases.

Abstract

We consider the mixed topological-holomorphic Chern-Simons theory introduced by Costello, Yamazaki and Witten on a $\mathbb{Z}_2$ orbifold. We use this to construct semi-classical solutions of the boundary Yang-Baxter equation in the elliptic and trigonometric cases. A novel feature of the trigonometric case is that the $\mathbb{Z}_2$ action lifts to the gauge bundle in a $z$-dependent way. We construct several examples of $K$-matrices, and check they agree with cases appearing in the literature.