Topological Field Theories and Geometry of Batalin-Vilkovisky Algebras

TL;DR

This paper explores how deformations of topological field theories, analyzed via the Batalin-Vilkovisky formalism, induce rich geometric structures like Courant algebroids and Poisson structures on target spaces.

Contribution

It demonstrates that deformations of BF theories in various dimensions produce specific geometric and algebraic structures, linking topological field theories with advanced geometry.

Findings

Deformations of 3D Chern-Simons-BF theory induce Courant algebroid structures.

2D deformed BF theories induce Poisson structures.

n-dimensional BF theories induce Batalin-Vilkovisky algebra structures.

Abstract

The algebraic and geometric structures of deformations are analyzed concerning topological field theories of Schwarz type by means of the Batalin-Vilkovisky formalism. Deformations of the Chern-Simons-BF theory in three dimensions induces the Courant algebroid structure on the target space as a sigma model. Deformations of BF theories in $n$ dimensions are also analyzed. Two dimensional deformed BF theory induces the Poisson structure and three dimensional deformed BF theory induces the Courant algebroid structure on the target space as a sigma model. The deformations of BF theories in $n$ dimensions induce the structures of Batalin-Vilkovisky algebras on the target space.