R. P. Malik

New topological field theories in two dimensions

It is shown that two (1 + 1)-dimensional (2D) free Abelian and self-interacting non-Abelian gauge theories (without any interaction with matter fields) belong to a new class of topological field theories (TFTs). These new theories capture together some of the key features of Witten and Schwarz types of TFT because they are endowed with symmetries that are reminiscent of the Schwarz-type theories but their Lagrangian density has the appearance of the Witten-type theories. The topological invariants for these theories are computed on a 2D compact manifold and their recursion relations are obtained. These new theories are shown to provide a class of tractable field theoretical models for the Hodge theory in two dimensions of flat (Minkowski) spacetime where there are no propagating degrees of freedom associated with the 2D gauge boson.

DUAL BRST SYMMETRY FOR QED

We show the existence of a co(dual)-BRST symmetry for the usual BRST invariant Lagrangian density of an Abelian gauge theory in two dimensions of space–time where a U(1) gauge field is coupled to the Noether conserved current (constructed by the Dirac fields). Under this new symmetry, it is the gauge-fixing term that remains invariant and the symmetry transformations on the Dirac fields are analogous to the chiral transformations. This interacting theory is shown to provide a tractable field theoretical model for the Hodge theory. The Hodge dual operation is shown to correspond to a discrete symmetry in the theory and the extended BRST algebra for the generators of the underlying symmetries turns out to be reminiscent of the algebra obeyed by the de Rham cohomology operators of differential geometry.

Hodge decomposition theorem for Abelian two form gauge theory

We show that the Becchi-Rouet-Stora-Tyutin (BRST)/anti-BRST invariant (3+1)-dimensional two-form gauge theory has further nilpotent symmetries (dual BRST/anti-dual BRST) that leave the gauge fixing term invariant. The generator for the dual BRST symmetry is analogous to the co-exterior derivative of differential geometry. There exists a bosonic symmetry which keeps the ghost terms invariant and it turns out to be the analogue of the Laplacian operator. The Hodge duality operation is shown to correspond to a discrete symmetry in the theory. The generators of all these continuous symmetries are shown to obey the algebra of the de Rham cohomology operators of differential geometry. We derive the extended BRST algebra constituted by six conserved charges and discuss the Hodge decomposition theorem in the quantum Hilbert space of states.

A field-theoretic model for Hodge theory

We demonstrate that the four-dimensional (4D) ((3+1)-dimensional) free Abelian 2-form gauge theory presents a tractable field-theoretical model for the Hodge theory where the well-defined symmetry transformations correspond to the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, obey an algebra that is reminiscent of the algebra obeyed by the cohomological operators. The discrete symmetry transformation of the theory represents the realization of the Hodge duality operation that exists in the relationship between the exterior and co-exterior derivatives of differential geometry. Thus, we provide the realizations of all the mathematical quantities, associated with the de Rham cohomological operators, in the language of the symmetries of the present 4D free Abelian 2-form gauge theory.

Abelian 2-form gauge theory: special features

It is shown that the four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theory provides an example of (i) a class of field theoretical models for the Hodge theory, and (ii) a possible candidate for the quasi-topological field theory (q-TFT). Despite many striking similarities with some of the key topological features of the two (1 + 1)-dimensional (2D) free Abelian (and self-interacting non-Abelian) gauge theories, it turns out that the 4D free Abelian 2-form gauge theory is not an exact TFT. To corroborate this conclusion, some of the key issues are discussed. In particular, it is shown that the (anti-)BRST and (anti-)co-BRST invariant quantities of the 4D 2-form Abelian gauge theory obey recursion relations that are reminiscent of the exact TFTs but the Lagrangian density of this theory is not found to be able to be expressed as the sum of (anti-)BRST and (anti-)co-BRST exact quantities as is the case with the topological 2D free Abelian (and self-interacting non-Abelian) gauge theories.

NEW LOCAL SYMMETRY FOR QED IN TWO DIMENSIONS

A new local, covariant and nilpotent symmetry is shown to exist for the interacting BRST invariant U(1) gauge theory in two dimensions of space–time. Under this new symmetry, it is the gauge-fixing term that remains invariant and the corresponding transformations on the Dirac fields turn out to be the analogue of chiral transformations. The extended BRST algebra is derived for the generators of all the underlying symmetries, present in the theory. This algebra turns out to be the analogue of the algebra obeyed by the de Rham cohomology operators of differential geometry. Possible interpretations and implications of this symmetry are pointed out in the context of BRST cohomology and Hodge decomposition theorem.

Unique nilpotent symmetry transformations for matter fields in QED: augmented superfield formalism

We derive the off-shell nilpotent (anti-)BRST symmetry transformations for the interacting U(1) gauge theory of quantum electrodynamics (QED) in the framework of the augmented superfield approach to the BRST formalism. In addition to the horizontality condition, we invoke another gauge invariant condition on the six (4,2)-dimensional supermanifold to obtain the exact and unique nilpotent symmetry transformations for all the basic fields present in the (anti-)BRST invariant Lagrangian density of the physical four (3+1)-dimensional QED. The above supermanifold is parametrized by four even space–time variables

An alternative to the horizontality condition in the superfield approach to BRST symmetries

x^{\mu}

Abelian 2-form gauge theory: superfield formalism

(with μ=0,1,2,3) and two odd variables (θ and

BRST, anti-BRST and gerbes

\bar{\theta}