Using technique of wheeled props we establish a correspondence between the
homotopy theory of unimodular Lie 1-bialgebras and the famous
Batalin-Vilkovisky formalism. Solutions of the so called quantum master
equation satisfying certain boundary conditions are proven to be in 1-1
correspondence with representations of a wheeled dg prop which, on the one
hand, is isomorphic to the cobar construction of the prop of unimodular Lie
1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop
of unimodular Poisson structures. These results allow us to apply properadic
methods for computing formulae for a homotopy transfer of a unimodular Lie
1-bialgebra structure on an arbitrary complex to the associated quantum master
function on its cohomology. It is proven that in the category of quantum BV
manifolds associated with the homotopy theory of unimodular Lie 1-bialgebras
quasi-isomorphisms are equivalence relations.
It is shown that Losev-Mnev's BF theory for unimodular Lie algebras can be
naturally extended to the case of unimodular Lie 1-bialgebras (and, eventually,
to the case of unimodular Poisson structures). Using a finite-dimensional
version of the Batalin-Vilkovisky quantization formalism it is rigorously
proven that the Feynman integrals computing the effective action of this new BF
theory describe precisely homotopy transfer formulae obtained within the
wheeled properadic approach to the quantum master equation. Quantum corrections
(which are present in our BF model to all orders of the Planck constant)
correspond precisely to what are often called "higher Massey products" in the
homological algebra.Comment: 42 pages. The journal versio
