# 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 A Alexandrov and Maxim Kontsevich and Albert S. Schwarz and Oleg Zaboronsky},
journal={International Journal of Modern Physics A},
year={1997},
volume={12},
pages={1405-1429}
}
• Published 2 February 1995
• Mathematics
• International Journal of Modern Physics A
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…
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## References

SHOWING 1-10 OF 18 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,

### Chern-Simons perturbation theory. II

• 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, the

### Chern-Simons Perturbation Theory

• Physics
• 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, the

### Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space

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### Semiclassical approximation in Batalin-Vilkovisky formalism

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### Quantum field theory and the Jones polynomial

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### Mirror Manifolds And Topological Field Theory

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### 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. This