BV pushforward as a quasi-isomorphism

Abstract

Given a BV theory on a space of fields split into two subspaces ("infrared" and "ultraviolet"), one has the BV pushforward map $P_{\ast }$, sending observables to observables of the effective theory on the infrared space. This note proves that $P_{\ast }$ is a quasi-isomorphism of BV complexes, by realizing it as a part of a strong deformation retraction constructed using the homological perturbation lemma. Two proofs are given: (i) comparing Feynman diagrams for $P_{\ast }$ with "cable diagrams" arising from homological perturbation theory and (ii) using topological quantum mechanics. This construction gives a formula for the quasi-inverse $i_{\mathrm{int}}$ of $P_{\ast }$ - the map lifting observables of the effective theory to the full theory. The topological quantum mechanics perspective - and its realization as an AKSZ theory - allows one to write $i_{\mathrm{int}}$ as a path integral (realizing cable diagrams for $i_{\mathrm{int}}$ as Feynman diagrams) and to study its classical limit.

62 pages, 15 figures. v.2: references added