The Geometry of the Master Equation and Topological Quantum Field Theory

  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},
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
Batalin–Vilkovisky quantization and the algebraic index
Abstract Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship betweenExpand
AKSZ-type Topological Quantum Field Theories and Rational Homotopy Theory
We reformulate and motivate AKSZ-type topological field theories in pedestrian terms, explaining how they arise as the most general Schwartz-type topological actions subject to a simple constraint,Expand
One-dimensional Chern–Simons theory and the $\hat{A}$ genus
We construct a Chern‐Simons gauge theory for dg Lie and L‐infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin‐Vilkovisky formalism and Costello’sExpand
Flat family of QFTs and quantization of d-algebras
Exploiting the path integral approach al la Batalin and Vilkovisky, we show that any anomaly-free Quantum Field Theory (QFT) comes with a family parametrized by certain moduli space M, which tangentExpand
Doubled Formalism, Complexification and Topological Sigma-Models
We study a generalization of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) formulation of the A- and B-models which involves a doubling of coordinates, and can be understood as aExpand
Geometry of Localized Effective Theory, Exact Semi-classical Approximation and Algebraic Index
In this paper we propose a general framework to study the quantum geometry of $\sigma$-models when they are effectively localized to small quantum fluctuations around constant maps. Such effectiveExpand
BRST and Topological Gauge Theories
In this thesis, the BRST-quantization of gauge theories is discussed. A detailed analysis of the BRST-quantization on inner product spaces is performed for a class of abelian models, includingExpand
Noncommutative supergeometry, duality and deformations
Following Lett. Math. Phys. 50 (1999) 309, we introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector fieldExpand
Topological Field Theories induced by twisted R-Poisson structure in any dimension
We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions ≥ 2 whose target space has a geometrical structure that suitably generalizes Poisson or twistedExpand
Frobenius-Chern-Simons gauge theory
Given a set of differential forms on an odd-dimensional noncommutative manifold valued in an internal associative algebra H, we show that the most general cubic covariant Hamiltonian action, withoutExpand


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
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
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
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