The geometry of the master equation and topological quantum field theory. (English) Zbl 1073.81655

Alexandrov, M.; Schwarz, A.; Zaboronsky, O.; Kontsevich, M.

Int. J. Mod. Phys. A 12, No. 7, 1405-1429 (1997).

Summary: In Batalin-Vilkovisky formalism, a classical mechanical system is specified by means of a solution to the classical master equation. Geometrically, such a solution can be considered as a \(QP\)-manifold, i.e. a supermanifold 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 a 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 \(\Pi {\mathcal G}\). Here stands \(G\) for a Lie algebra and \(\Pi\) denotes parity inversion.

MSC:

81T70	Quantization in field theory; cohomological methods
58D29	Moduli problems for topological structures
81T45	Topological field theories in quantum mechanics