Abstract
Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) approach, we discuss mainly the fermionic (i.e., off-shell nilpotent) (anti-)BRST, (anti-)co-BRST, and some discrete dual symmetries of the appropriate Lagrangian densities for a two (1+1)-dimensional (2D) modified Proca (i.e., a massive Abelian 1-form) theory without any interaction with matter fields. One of the novel observations of our present investigation is the existence of some kinds of restrictions in the case of our present Stückelberg-modified version of the 2D Proca theory which is not like the standard Curci-Ferrari (CF) condition of a non-Abelian 1-form gauge theory. Some kinds of similarities and a few differences between them have been pointed out in our present investigation. To establish the sanctity of the above off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetries, we derive them by using our newly proposed (anti-)chiral superfield formalism where a few specific and appropriate sets of invariant quantities play a decisive role. We express the (anti-)BRST and (anti-)co-BRST conserved charges in terms of the superfields that are obtained after the applications of (anti-)BRST and (anti-)co-BRST invariant restrictions and prove their off-shell nilpotency and absolute anticommutativity properties, too. Finally, we make some comments on (i) the novelty of our restrictions/obstructions and (ii) the physics behind the negative kinetic term associated with the pseudoscalar field of our present theory.
Highlights
One of the simplest gauge theories is the well-known Maxwell U(1) gauge theory which can be generalized to the Proca theory by incorporating a mass term in the Lagrangian density for the bosonic field (thereby rendering the latter field to acquire three degrees of freedom in the physical four (3 + 1)-dimensional (4D) flat Minkowskian spacetime)
The purpose of our present investigation is to concentrate on the two (1 + 1)-dimensional (2D) Stuckelberg-modified version of the Proca theory within the framework of BecchiRouet-Stora-Tyutin (BRST) formalism and show the existence of fermionicBRST andco-BRST symmetry transformations as well as other kinds of discrete and continuous symmetries which provide the physical realizations of the de Rham cohomological operators∗ of differential geometry [7,8,9,10,11]
We have demonstrated the existence of two equivalent Lagrangian densities for the 2D Proca theory which respect the off-shell nilpotent and absolutely anticommutingBRST andco-BRST symmetry transformations
Summary
One of the simplest gauge theories is the well-known Maxwell U(1) gauge theory which can be generalized to the Proca theory by incorporating a mass term in the Lagrangian density for the bosonic field (thereby rendering the latter field to acquire three degrees of freedom in the physical four (3 + 1)-dimensional (4D) flat Minkowskian spacetime). We prove that the massive 2D Abelian 1-form gauge theory (i.e. the Stuckelberg-modified version of the 2D Proca theory) is a field-theoretic example of Hodge theory. We have demonstrated the existence of two equivalent Lagrangian densities for the 2D Proca theory (within the framework of BRST formalism) which respect the off-shell nilpotent and absolutely anticommuting (anti-)BRST and (anti)co-BRST symmetry transformations (separately and independently). We have captured the existence of the new type of restrictions/obstructions within the framework of (anti-)chiral superfield approach to BRST formalism while proving the invariance of the Lagrangian densities under the (anti-)BRST and (anti-) co-BRST symmetry transformations (cf Sec. 6). We focus only on the internal symmetries of our 2D theory and spacetime symmetries of the 2D Minkowskian spacetime manifold do not play any crucial role in our whole discussion
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