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

Quantum $L_{\mathrm{\infty }}$ algebras are a generalization of $L_{\mathrm{\infty }}$ algebras with a scalar product and with operations corresponding to higher genus graphs. We construct a minimal model of a given quantum $L_{\mathrm{\infty }}$ 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 $L_{\mathrm{\infty }}$ algebra.