Twisted Homotopy Algebras: Supersymmetric Twists, Spontaneous Symmetry Breaking, Anomalies and Localisation

TL;DR

This paper develops a unified homotopy-algebraic framework for supersymmetric quantum field theory, connecting twisting, classical backgrounds, symmetry breaking, anomalies, and localisation through quantum $L_$-superalgebras.

Contribution

It formalizes the connection between twisting and classical backgrounds as instances of twisting curved quantum $L_$-superalgebras within a homotopy-algebraic approach.

Findings

Unified algebraic framework for twists and backgrounds

Introduction of twisting for quantum $L_$-algebras

Homotopy reformulation of effective actions

Abstract

Twisting and classical background fields are two foundational techniques in supersymmetric quantum field theory, central to developments ranging from the Higgs mechanism to topological twisting and supersymmetric localisation. While traditionally treated as distinct procedures, they appear on an equal footing in the homotopy-algebraic approach to quantum field theory. In this work, we formalise this connection by interpreting both twisting and the introduction of classical backgrounds as instances of twisting curved quantum $L_\infty$-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 \`a la Festuccia--Seiberg. As a byproduct, we introduce a notion of twisting for quantum $L_\infty$-algebras and a homotopy-algebraic reformulation of the one-particle-irreducible effective action.