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, 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 Lie algebras in any QFT, called L_\infty-algebras. These algebras 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 discuss how this formalism enables the systematic computation of BPS operators in supersymmetric QFTs (even if they are not "fully protected"), and describe the "holomorphic confinement" of N=1 SYM. Time permitting, I will prove a non-renormalization theorem for holomorphic-topological QFTs with more than one topological direction. Based on arXiv:2403.13049.