Twisted homotopy algebras: spontaneous symmetry breaking, anomalies, localisation, and supersymmetric twists

Abstract Classical background fields are a foundational technique in quantum field theory, playing a central role in developments such as the Higgs mechanism. Independently, supersymmetric twisting à la Witten has emerged as a key tool underlying phenomena such as supersymmetric localisation. Although these constructions are traditionally treated as distinct, they arise on an equal footing within the homotopy-algebraic approach to quantum field theory. In this work, we formalise this connection by interpreting both supersymmetric twisting and classical backgrounds as instances of twisting curved quantum L ∞ -superalgebras. Using the language of homotopy algebras and the Batalin–Vilkovisky formalism, we provide a unified algebraic framework that encompasses topological/holomorphic twists, spontaneous symmetry breaking, computation of anomalies, and supersymmetric localisation à la Festuccia–Seiberg. We examine a variety of applications and examples illustrating this perspective in supersymmetric and general quantum fields theories alike. As a byproduct, we introduce a notion of twisting for quantum L ∞ -algebras and a homotopy-algebraic reformulation of the one-particle-irreducible effective action.