Higher Deformation Quantization for Kapustin–Witten Theories

We pursue a uniform quantization of all twists of 4-dimensional \(\mathcal N=4\) supersymmetric Yang–Mills theory, using the BV formalism, and we explore consequences for factorization algebras of observables. Our central result is the construction of a one-loop exact quantization on \(\mathbb {R}^4\) for all such twists and for every point in a moduli of vacua. When an action of the group \(\textrm{SO}(4)\) can be defined—for instance, for Kapustin and Witten’s family of twists—the associated framing anomaly vanishes. It follows that the local observables in such theories can be canonically described by a family of framed \(\mathbb E_4\) algebras; this structure allows one to take the factorization homology of observables on any oriented 4-manifold. In this way, each Kapustin–Witten theory yields a fully extended, oriented 4-dimensional topological field theory à la Lurie and Scheimbauer.