Higher algebraic structures in supersymmetric QFT

I'll discuss the algebraic structure of local operators in topologically twisted theories in arbitrary spacetime dimension d. Local operators always form an algebra (associative in any dimension, commutative in d > 1), with a product coming from "collision" of insertion points. However, local operators also turn out to be equipped with second product operation in any dimension, manifesting itself as a Lie bracket of degree 1-d. This bracket is a familiar component of the L-infinity (a.k.a. homotopy Lie algebra) structure of 2d QFT's studied in the 90's, but has remained relatively unexplored in higher dimensions. I'll give some surprisingly simple examples in d=3. Time allowing, I'll also discuss a fundamental relation between the bracket and the Omega background (which turns out to supply its quantization).