Gauge Theory and Boundary Integrability II: Elliptic and Trigonometric Case

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.