Igor Batalin

Generalized Poisson sigma models

Abstract A general master action in terms of superfields is given which generates generalized Poisson sigma models by means of a natural ghost number prescription. The simplest representation is the sigma model considered by Cattaneo and Felder. For Dirac brackets considerably more general models are generated.

Completely anticanonical form of Sp(2)-symmetric Lagrangian quantization

Abstract The Sp(2)-symmetric Lagrangian quantization scheme is represented in a completely anticanonical form. Antifields are assigned to all field variables including former “parametric” ones πAa. The antibrackets (F,G)a as well as the operators Δa and Va are extended to include the new anticanonical pairs πAa, Φ A . A new version of the gauge fixing mechanism i the Lagrangian effective action is proposed. The corresponding functional integral is shown to be gauge independent.

Superfield algorithms for topological field theories

A superfield algorithm for master actions of a class of gauge field theories including topological ones in arbitrary dimensions is presented generalizing a previous treatment in two dimensions. General forms for master actions in superspace are given, and possible theories are determined by means of a ghost number prescription and the master equations. The resulting master actions determine the original actions together with their gauge invariances. Generalized Poisson sigma models in arbitrary dimensions are constructed by means of this algorithm, and other applications in low dimensions are given including the Chern-Simon model.

Solving general gauge theories on inner product spaces

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form |ph>=e^{[Q, psi]} |ph>_0 where Q is the nilpotent BRST operator, psi a hermitian fermionic gauge fixing operator, and |ph>_0 BRST invariant states determined by a hermitian set of BRST doublets in involution. |ph>_0 does not belong to an inner product space although |ph> does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for |ph>_0 and the corresponding psi. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that |ph>_0 are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set |ph>_0 are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.

General quantum antibrackets

The recently introduced quantum antibracket is further generalized such that the odd operator Q can be arbitrary. We give exact formulas for quantum antibrackets of arbitrary higher orders and for their generalized Jacobi identities. We review applications of the quantum antibrackets to the BV and BFV-BRST quantizations and include some new aspects.

General triplectic quantization

Abstract The general structure of the Sp(2)-covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism, the so-called triplectic quantization, as presented in our previous paper with A.M. Semikhatov is further generalized and clarified. We present new unified expressions for the generating operators which are more invariant and which yield a natural realization of the operator Va and provide for a geometrical explanation for its presence. ThisVa operator provides then for an invariant definition of a degenerate Poisson bracket on the triplectic manifold being non-degenerate on a naturally defined submanifold. We also define inverses to non-degenerate antitriplectic metrics and give a natural generalization of the conventional calculus of exterior differential forms which, e.g., explains the properties of these inverses. Finally we define and give a consistent treatment of second class hyperconstraints.

DUALITIES BETWEEN POISSON BRACKETS AND ANTIBRACKETS

Recently it has been shown that antibrackets may be expressed in terms of Poisson brackets and vice versa for commuting functions in the original bracket. Here we also introduce generalized brackets involving higher antibrackets or higher Poisson brackets where the latter are of a new type. We give generating functions for these brackets for functions in arbitrary involutions in the original bracket. We also give master equations for generalized Maurer–Cartan equations. The presentation is completely symmetric with respect to Poisson brackets and antibrackets.

SUPERFIELD FORMULATION OF THE PHASE SPACE PATH INTEGRAL

Abstract We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.

A systematic study of finite BRST-BFV transformations in generalized Hamiltonian formalism

We study systematically finite BRST-BFV transformations in the generalized Hamiltonian formalism. We present explicitly their Jacobians and the form of a solution to the compensation equation determining the functional field dependence of finite Fermionic parameters, necessary to generate an arbitrary finite change of gauge-fixing functions in the path integral.

GAUGE THEORY OF SECOND-CLASS CONSTRAINTS WITHOUT EXTRA VARIABLES

We show that any theory with second-class constraints may be cast into a gauge theory if one makes use of solutions of the constraints expressed in terms of the coordinates of the original phase space. We perform a Lagrangian path integral quantization of the resulting gauge theory and show that the natural measure follows from a superfield formulation.