Topological BF Theory Research Articles

BRST symmetries for the tangent gauge group

For any principal bundle P, one can consider the subspace of the space of connections on its tangent bundle TP given by the tangent bundle TA of the space of connections A on P. The tangent gauge group acts freely on TA. Appropriate BRST operators are introduced for quantum field theories that include as fields elements of TA, as well as tangent vectors to the space of curvatures. As the simplest application, the BRST symmetry of the so-called BF-Yang–Mills theory is described and the relevant gauge fixing conditions are analyzed. A brief account on the topological BF theories is also included and the relevant Batalin–Vilkovisky operator is described.

Cohomology and renormalization of BFYM theory in three dimensions

The first-order formalism for the 3D Yang-Mills theory is considered and two different formulations are introduced, in which the gauge theory appears to be a deformation of the topological BF theory. We perform the quantization and the algebraic analysis of the renormalization of both the models, which are found to be anomaly free. We discuss also their stability against radiative corrections, giving the full structure of possible counterterms, requiring an involved matricial renormalization of fields and sources. Both models are then proved to be equivalent to the Yang-Mills theory at the renormalized level.

Renormalization Transformations of the 4-D BFYM Theory

We study the most general renormalization transformations for the first-order formulation of the Yang–Mills theory. We analyze, in particular, the trivial sector of the BRST cohomology of two possible formulations of the model: the standard one and the extended one. The latter is a promising starting point for the interpretation of the Yang–Mills theory as a deformation of the topological BF theory. This work is a necessary preliminary step towards any perturbative calculation, and completes some recently obtained results.

BRST and Anti-BRST Symmetry in Topological BF Theories

We realize the quantization of arbitrary-dimensional non-Abelian BF theories from the point of view of both the BRST symmetry and anti-BRST symmetry. The method relies on the possibility of enlarging the system of fields occurring in such theories by auxiliary fields introduced geometrically in terms of a superspace formalism. The off-shell nilpotency of the BRST and anti-BRST transformations of these fields is automatically guaranteed, thanks to the structure equations and the Bianchi identities. From this follows a direct construction of the BRST invariant full quantum action which leads, after the elimination of the auxiliary fields, to the same quantum action obtained in the context of the Batalin–Vilkovisky formalism. We also discuss how the constructed BRST invariant quantum action is anti-BRST invariant. In addition, we show that the non-Abelian BF theory in any space–time dimension possesses a vector supersymmetry of ghost number +1.

Black holes in the gauge theoretic formulation of dilatonic gravity

We show that two-dimensional topological BF theories coupled to particles carrying non-Abelian charge admit a new coupling involving the Lagrange multiplier field. When applied to the gauge theoretic formulation of dilatonic gravity it gives rise to a source term for the gravitational field. We show that the system admits black hole solutions.

ℵ 0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO( N) symmetry. We call these ℵ 0 (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ 0 supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ 0 globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ℵ 0 supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.

Topological BF theories in 3 and 4 dimensions

In this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2-knots (in 4 dimensions). The vacuum expectation values of such observables give a wide range of invariants. Here we consider mainly the 3 dimensional case, where these invariants include Alexander polynomials, HOMFLY polynomials and Kontsevich integrals.

IS THE CONDENSATION OF STRINGS THE ORIGIN OF EINSTEIN GRAVITY?

We propose a mechanism to generate the metric in which the Kalb-Ramond symmetry existing in the topological BF theory is broken through the condensation of the string fields which are so introduced as to couple the antisymmetric tensor fields B, invariantly under the Kalb-Ramond symmetry. In the chiral decomposition of the local Lorentz group, the non-Abelian B fields need to be generalized to the string fields. The mechanism of the condensation is discussed, in view of the confinement problem and the polymer physics.

BRST quantization of non-abelian BF topological theories

The off-shell nilpotent BRST charge and the BRST invariant effective action for non-abelian BF topological theories over D-dimensional manifolds are explicitly constructed. These theories have the feature of being reducible with exactly D - 3 stages of reducibility. The adequate extended phase space including the different levels of ghosts for ghosts is explicitly obtained. Using the structure of the resulting BRST charge we show that for topological BF theories the semi-classical approximation completely describes the quantum theory. The independence of the partition function on the metric also follows from our explicit construction in a straightforward way.

Topological selection of trajectories of a particle in a BF field theory

Topological BF theories are analysed by considering a test massive particle subject to the field on an external given background. The equations of motion are solved completely. It is shown that the topological selection of trajectories is implemented by a topological equation whose set of solutions is related to the intersection invariant theory of p and 1-chains on a ( p+1)-dimensional manifold.

On the BRST charge for topological BF theories

Topological quantum field theories are discussed using the generalized canonical approach to constrained systems. We give the general structure of the canonical effective action of linearly dependent first class constrained systems. When this construction is applied to metric independent classical actions, one is able to ensure - under some explicitly given boundary conditions - metric independence of the associated partition function. The quantum analysis of the non-abelian BF theories in four dimensions is performed on these lines; their off-shell nilpotent BRST charge is constructed and a new direct proof of the topological nature of the associated quantum theories is presented. We do also comment on some properties of the BF actions in higher dimensions.

Renormalization and finiteness of topological BF theories

We show that the BF theory in any space-time dimension, when quantized in a certain linear covariant gauge, possesses a vector supersymmetry. The generator of the latter together with those of the BRS transformations and of the translations form the basis of a superalgebra of the Wess-Zumino type. We give a general classification of all possible anomalies and invariant counterterms. Their absence, which amounts to ultraviolet finiteness, follows from purely algebraic arguments in the lower-dimensional cases.

Infrared and ultraviolet finiteness of topological BF theory in two dimensions

The two-dimensional topological BF model is considered in the Landau gauge in the framework of perturbation theory. Due to the singular behaviour of the ghost propagator at long distances, a mass term to the ghost fields is introduced as an infrared regulator. Relying on the supersymmetric algebraic structure of the resulting massive theory, the authors study the infrared and ultraviolet renormalizability of the model, with the outcome that it is perturbatively finite.