Quasi-algebraic quantization for the B-twist Langlands TQFT

Abstract

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 \textit{(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.