# The Geometry of the Master Equation and Topological Quantum Field Theory

@article{Alexandrov1997TheGO,
title={The Geometry of the Master Equation and Topological Quantum Field Theory},
author={M. Alexandrov and M. Kontsevich and A. Schwarz and O. Zaboronsky},
journal={International Journal of Modern Physics A},
year={1997},
volume={12},
pages={1405-1429}
}
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… Expand
551 Citations

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#### References

SHOWING 1-10 OF 22 REFERENCES
Topological sigma models
A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surfaceΣ to an arbitrary almost complex manifoldM. It possesses a fermionic BRST-like symmetry,Expand
Chern-Simons perturbation theory. II
• Mathematics, Physics
• 1994
We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, theExpand
Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation
The complete quantum theory of covariant closed strings is constructed in detail. The nonpolynomial action is defined by elementary vertices satisfying recursion relations that give rise toExpand
Chern-Simons Perturbation Theory
• Physics, Mathematics
• 1991
We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, theExpand
Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space
This is an expanded and updated version of a talk given at the Conference on Topics in Geometry and Physics at the University of Southern California, November 6, 1992. It is a survey talk, aimed atExpand
Semiclassical approximation in Batalin-Vilkovisky formalism
The geometry of supermanifolds provided with aQ-structure (i.e. with an odd vector fieldQ satisfying {Q, Q}=0), aP-structure (odd symplectic structure) and anS-structure (volume element) or withExpand
Quantum field theory and the Jones polynomial
It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the JonesExpand
Mirror Manifolds And Topological Field Theory
These notes are devoted to explaining aspects of the mirror manifold problem that can be naturally understood from the point of view of topological field theory. Basically this involves studying theExpand
Geometry of Batalin-Vilkovisky quantization
The geometry ofP-manifolds (odd symplectic manifolds) andSP-manifolds (P-manifolds provided with a volume element) is studied. A complete classification of these manifolds is given. ThisExpand
CLOSED STRING FIELD THEORY
Abstract A gauge invariant cubic action describing bosonic closed string field theory is constructed. The gauge symmetries include local spacetime diffeomorphisms. The conventional closed stringExpand