The Geometry of the Master Equation and Topological Quantum Field Theory

TL;DR

This paper explores the geometry of $QP$-manifolds within the Batalin-Vilkovisky formalism, linking them to topological quantum field theories like sigma-models and Chern-Simons theory, and provides classification results.

Contribution

It offers a geometric framework for $QP$-manifolds, constructs new examples, and demonstrates their application to topological quantum field theories.

Findings

Classification theorem for $QP$-manifolds under certain conditions

Construction of action functionals for 2D topological sigma-models

Derivation of Chern-Simons theory as a sigma-model with a Lie algebra target

Abstract

In Batalin-Vilkovisky formalism a classical mechanical system is specified by means of a solution to the {\sl classical master equation}. Geometrically such a solution can be considered as a $QP$-manifold, i.e. a super\m equipped with an odd vector field $Q$ obeying $\{Q,Q\}=0$ and with $Q$-invariant odd symplectic structure. We study geometry of $QP$-manifolds. In particular, we describe some construction of $QP$-manifolds and prove a classification theorem (under certain conditions). We apply these geometric constructions to obtain in natural way the action functionals of two-dimensional topological sigma-models and to show that the Chern-Simons theory in BV-formalism arises as a sigma-model with target space $\Pi {\cal G}$. (Here ${\cal G}$ stands for a Lie algebra and $\Pi$ denotes parity inversion.)