Higher Deformation Quantization for Kapustin-Witten Theories

TL;DR

This paper develops a uniform one-loop exact quantization for all twists of 4D N=4 supersymmetric Yang-Mills theories, enabling the construction of extended topological field theories via factorization algebras.

Contribution

It introduces a novel, uniform quantization method for all twists of 4D N=4 SYM, including the Kapustin-Witten family, and constructs associated extended topological field theories.

Findings

Quantization is one-loop exact on b5^4 for all twists.

Framing anomaly vanishes when SO(4) symmetry acts, as in Kapustin-Witten twists.

Local observables form framed  algebras, enabling factorization homology on 4-manifolds.

Abstract

We pursue a uniform quantization of all twists of 4-dimensional 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 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 \`a la Lurie and Scheimbauer.