Deformation of Batalin-Vilkovisky Structures
Noriaki Ikeda
TL;DR
This work develops a general AKSZ-BV framework for deforming BV structures on graded bundles across arbitrary dimensions, revealing that $n=2$ deformations yield Lie algebroids and $n=3$ deformations yield Courant algebroids, with higher $n$ suggesting $n$-algebroids yet to be fully characterized. By analyzing both Abelian BF and Chern-Simons-with-BF topological theories, it connects BV master equations to concrete algebroid axioms and constructs nonlinear gauge theories via BRST cohomology. The quantum version demonstrates that in $n=2$ the deformation quantization emerges as Kontsevich’s star product on Poisson manifolds, while higher-dimensional quantum deformations remain an open area. Overall, the paper links BV deformation theory with generalized geometry and higher algebroid structures, offering a unifying perspective on geometry, topological field theories, and quantization.
Abstract
A Batalin-Vilkovisky formalism is most general framework to construct consistent quantum field theories. Its mathematical structure is called {\it a Batalin-Vilkovisky structure}. First we explain rather mathematical setting of a Batalin-Vilkovisky formalism. Next, we consider deformation theory of a Batalin-Vilkovisky structure. Especially, we consider deformation of topological sigma models in any dimension, which is closely related to deformation theories in mathematics, including deformation from commutative geometry to noncommutative geometry. We obtain a series of new nontrivial topological sigma models and we find these models have the Batalin-Vilkovisky structures based on a series of new algebroids.