Minimal models of quantum homotopy Lie algebras via the BV-formalism

Abstract

Using the BV-formalism of mathematical physics an explicit construction for the minimal model of a quantum L∞-algebra is given as a formal super integral. The approach taken herein to these formal integrals is axiomatic, and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix. Additionally, there exists a canonical differential graded Lie algebra morphism mapping formal functions on homology to formal functions on the whole space. An inverse L∞-algebra morphism to this differential graded Lie algebra morphism is constructed as a formal super integral.

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

“…In section 5, we describe the relation of this paper to the mentioned works [2,5,8,11,24] in more detail.…”

“…However, since ω is non-degenerate, this means that ∆A + 1 2 {A, A} is an odd constant, which can only be 0. We finish by introducing the weight grading of Braun and Maunder [5].…”

“…We will begin by introducing homotopies of quantum L ∞ algebras, following [5,8]. Then, we will see that the perturbation lemma directly gives a homotopy between the original and the effective action.…”

“…Braun and Maunder [5] define the path integral explicitly and use it to compute the effective action. They then prove that the effective action again solves a quantum master equation (and hence defines a quantum L ∞ algebra).…”

“…It is a simple consequence of the properties of the path integral that the resulting W again solves the quantum master equation on F (H). This approach has already been used in a similar context, either by directly defining W as a diagram expansion [2,8,11,24] or by defining the path integral [5].…”