Higher Deformation Quantization for Kapustin-Witten Theories

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 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 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. Contents 1. Introduction 2.1. Classical BV Formalism 2.2. Quantum BV formalism 2.3. Equivariant BV formalism 3. Twists of 4d N = 4 Super Yang-Mills Theory 3.1. Summary of Twists 3.2. Anomaly Vanishing 4. Perturbation Theory Around a Vacuum 4.1. Moduli of Vacua 4.2. Families of Classical Field Theories over [g * /G] 4.3. Families of Quantum Field Theories over [g * /G]

Snowmass White Paper: The Quest to Define QFT

“…Therefore, the likely status of the FA approach is that it provides an alternative viewpoint, and technically quite a different approach to constructing concrete models. Some papers that use this framework include [473][474][475][476][477][478][479][480][481][482][483][484][485][486][487][488], also [489][490][491][492]. It is also suggested by [493] that this approach has close relations to [494].…”

Superconformal Algebras and Holomorphic Field Theories

“…Our main tool will be a spectral sequence converging to the cohomology of Obs(g, V ), similar to the one considered in [26]. This filtration also appears in [19] where it is used in the context of 4d N = 4 supersymmetric Yang-Mills theory. The key property of this spectral sequence is that the first page computes the cohomology of the observables where we turn off the interactions which are of cubic order and higher.…”