About 20 years ago, Beilinson-Drinfeld formulated the categorical geometric Langlands conjecture. About 10 years ago, Kapustin-Witten proposed a physical interpretation for the conjecture. About 4 years ago, Arinkin-Gaitsgory proposed a refined form of the conjecture. We aim to recover the refined conjecture by rigorously constructing the physical theory via derived algebraic geometry. This talk is based on an ongoing joint project with Chris Elliott.