Quasi-algebraic quantization for the B-twist Langlands TQFT

This is the first part of a program to construct hyperholomorphic families of boundary conditions for the Kapustin–Witten B-twist of the Langlands QFT, otherwise known as $\mathit{\text{(BBB)-branes}}$. We define the category of quasi-algebraic sheaves over the Deligne moduli stack, which serves as an analog of the twistor space of Hitchin’s moduli stack. This allow us to construct a representation of a simplified version of the Moore–Tachikawa category which is motivated by the relative Langlands program in the sense of Ben-Zvi–Sakellaridis–Venkatesh.