Mathematical Physics and Algebraic Geometry

Speaker: Justin Kulp (Simons Center for Geometry and Physics)

Title: Twisted Tools for (Untwisted) Quantum Field Theory

Abstract: One of the most important properties of QFTs is that they can be deformed by “turning on interactions.” Essentially every observable can be viewed as coupling the theory to some external system. Famously, adding interactions (generically) breaks scale invariance, leading to familiar ideas of EFTs and RG flows in the space of QFTs. An underappreciated fact is that one can actually consider flows generated by any transformation, not just the usual scale transformations.

In my talk, I will discuss a flow in the space of QFTs coming from (an analogue of) BRST symmetry. The beta-function for this “BRST-flow” controls deformations of the QFT and is highly mathematically constrained, endowing the space of interactions with an L∞ algebra structure. The structure constants/brackets of the L∞ algebra are highly computable (requiring only a first course in QFT to compute) and contain familiar information such as anomalies and Operator Product Expansion coefficients. I will prove a non-renormalization theorem for holomorphic-topological QFTs with more than one topological direction, which can be thought of as a generalization of a formality theorem of Kontsevich. Time permitting, I will discuss how this formalism enables the systematic computation of minimal BPS operators in supersymmetric QFTs and describe the “holomorphic confinement” of N=1 SYM.  Based on arXiv:2403.13049.