Quantum $L_{\infty }$ 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.

v2: 27 pages, fixed typos and the section 4.4; v3: published version - shortened and removed the appendix on relationship between quantum master actions and brackets

References in corpus (5)

• On the perturbation lemma, and deformations
• Deformation theory and rational homotopy type
• Renormalisation and the Batalin-Vilkovisky formalism
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• Discrete BF theory

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