Twisted Tools for (Untwisted) Quantum Field Theory

One of the most important properties of quantum field theories is that they can be deformed by "turning on interactions." Indeed, essentially every observable can be viewed as coupling the theory to some external system. Famously, such interactions will (generically) break 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, generating a "homotopical" generalization of a Lie algebras in any QFT. These "homotopical" Lie brackets are highly computable (requiring only a first course in QFT to compute) and contain familiar information such as anomalies and Operator Product Expansion coefficients. Time permitting, I will discuss how these brackets are even more strongly constrained in holomorphic/topological theories, and prove the absence of perturbative corrections to HT theories with more than one topological direction. Based on arXiv:2403.13049.