Quantum $${L_\infty}$$ L ∞ Algebras and the Homological Perturbation Lemma

Abstract

Quantum $${L_\infty}$$ algebras are a generalization of $${L_\infty}$$ algebras with a scalar product and with operations corresponding to higher genus graphs. We construct a minimal model of a given quantum $${L_\infty}$$ algebra via the homological perturbation lemma and show that it’s given by a Feynman diagram expansion, computing the effective action in the finite-dimensional Batalin–Vilkovisky formalism. We also construct a homotopy between the original and this effective quantum $${L_\infty}$$ algebra.