Opers and their $q$-deformations provide a geometric way to encode solutions of the Bethe Ansatz equations that arise in quantum integrable systems such as Gaudin models and closed spin chains.

In this talk, we will discuss the extension of this picture to open spin chains, which leads to a class of q-opers satisfying a reflection-invariance condition. For $GL_2$, this construction recovers the known Bethe equations for open XXZ chains with both diagonal and generic boundary conditions. We will also discuss our recent extension of this framework to the higher-rank setting.

We begin with a brief introduction to spin chains and then explain how this class of q-opers can be described using associated QQ-systems and how the nondegenerate solutions of these QQ-systems reproduce the Bethe Ansatz equations for open XXZ spin chains. This is joint work with Peter Koroteev and Myungbo Shim.