Quantum Geometric Langlands Categories from N = 4 Super Yang-Mills Theory

We describe the family of supersymmetric twists of $\mathcal N = 4$ super Yang--Mills theory using derived algebraic geometry, starting from holomorphic Chern--Simons theory on $\mathcal N = 4$ super twistor space. By considering an ansatz for categorical geometric quantization of the family of further twists of a fixed holomorphic twist, we give a quantum field-theoretic synthesis of the categories of twisted D-modules occuring in the quantum geometric Langlands correspondence.