Abelian 2-form Gauge Theory Research Articles

A 3D field-theoretic example for Hodge theory

We focus on the continuous symmetry transformations for the three (2 + 1)-dimensional (3D) system of a combination of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We establish that this combined system is a tractable field-theoretic model of Hodge theory. The symmetry operators of our present system provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. Our present investigation is important in the sense that, for the first time, we are able to establish an odd dimensional (i.e., D = 3) field-theoretic system to be an example for Hodge theory (besides earlier works on a few interesting (0 + 1)-dimensional (1D) toy models as well as a set of well-known SUSY quantum mechanical systems of physical interest). For the sake of brevity, we have purposely not taken into account the 3D Chern-Simon term for the Abelian 1-form gauge field in our theory which allows the mass as well as the gauge invariance to co-exist together.

Modified Stückelberg Formalism: Free Massive Abelian 2-Form Theory in 4D

We demonstrate that the celebrated Stückelberg formalism is modified in the case of a massive four (3 + 1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism where there is the existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual Stückelberg-modified massive Abelian 2-form gauge theory. The modification in the Stückelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the Stückelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick of using the equations of motion to remove the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent.

Massive 4D Abelian 2-form theory: Nilpotent symmetries from the (anti-)chiral superfield approach

The off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries are obtained by using the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism for the four [Formula: see text]-dimensional (4D) Stückelberg-modified massive Abelian 2-form gauge theory. We perform exactly similar kind of exercise for the derivation of the off-shell nilpotent (anti-)co-BRST symmetry transformations, too. In the above derivations, the symmetry invariant restrictions on the superfields play very important and decisive roles. To prove the sanctity of the above nilpotent symmetries, we generalize our 4D ordinary theory (defined on the 4D flat Minkowskian space–time manifold) to its counterparts [Formula: see text]-dimensional (anti-)chiral super-submanifolds of the [Formula: see text]-dimensional supermanifold which is parametrized by the superspace coordinates [Formula: see text] where [Formula: see text] [Formula: see text] are the bosonic coordinates and a pair of Grassmannian variables [Formula: see text] are fermionic: [Formula: see text], [Formula: see text] in nature. One of the novel observations of our present endeavor is the derivation of the Curci–Ferrari (CF)-type restrictions from the requirement of the symmetry invariance of the coupled (but equivalent) Lagrangian densities of our theory within the framework of ACSA to BRST formalism. We also exploit the standard techniques of ACSA to capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST as well as the conserved (anti-)co-BRST charges. In a subtle manner, the proof of the absolute anticommutativity of the above conserved charges also implies the existence of the appropriate CF-type restrictions on our theory. This proof is also a novel result.

A massive field-theoretic model for Hodge theory

Within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism, we show that the four (3+1)-dimensional (4D) massive Abelian 2-form gauge theory (without any interaction with matter fields) is a model for the Hodge theory because its discrete and continuous symmetry transformations (and their corresponding Noether conserved charges) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. For this purpose, we incorporate the pseudo-scalar and axial-vector fields which appear in the theory with negative kinetic terms (but with proper definition of mass). The negative kinetic terms, for the above fields, are essential so that our theory could respect the discrete symmetry transformations which provide the physical realizations of the Hodge duality operation in the domain of differential geometry. Thus, our present endeavour, not only provides the physical realizations of all the mathematical ingredients connected with the de Rham cohomological operators of differential geometry, it also sheds light on the existence and emergence of fields with negative kinetic terms. We discuss the implications and relevance of the latter fields in the context of current models of dark matter and dark energy as well as the bouncing models of Universe.

Abelian 1-Form Gauge Theory in Very Special Relativity

We study an Abelian 1-form gauge theory in the context of very special relativity. The very special relativity regards subgroup of Lorentz group as fundamental symmetry of the nature than the full Lorentz group. The rotational symmetry of the Lorentz symmetry breaks down by chosing a privileged direction in spacetime. We discuss the BRST invariant and anti-BRST invariant superspace formulations for such BV actions. The VSR modified BRST invariant BV action corresponding to the Abelian 1-form gauge theory can be written manifestly covariant manner in a superspace with one Grassmann coordinate. Moreover, we can write down the the VSR modified extended BRST and extended anti-BRST invariant BV actions in a superspace with two Grassmann coordinates. The results thus obtained is in complete agreement with the Lorentz invariant theory.

Nilpotent symmetries of a 4[formula omitted] model of Hodge theory: Augmented (anti-)chiral superfield formalism

We derive the continuous nilpotent symmetries of the four (3+1)-dimensional (4D) model of the Hodge theory (i.e. 4D Abelian 2-form gauge theory) by exploiting the beauty and strength of the symmetry invariant restrictions on the (anti-)chiral superfields. The above off-shell nilpotent symmetries are the Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-)co-BRST transformations which turn up beautifully due to the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral superfields that are defined on the (4, 1)-dimensional (anti-)chiral super-submanifolds of the general (4, 2)-dimensional supermanifold on which our ordinary 4D theory is generalized. The latter supermanifold is characterized by the superspace coordinates ZM=(xμ,θ,θ̄) where xμ(μ=0,1,2,3) are the bosonic coordinates and a pair of Grassmannian variables θ and θ̄ are fermionic in nature as they obey the standard relationships: θ2=θ̄2=0,θθ̄+θ̄θ=0). The derivation of the proper (anti-)co-BRST symmetries and proof of the absolute anticommutativity property of the conserved (anti-)BRST and (anti-) co-BRST charges are novel results of our present investigation (where only the (anti-)chiral superfields and their super-expansions have been taken into account).

(Anti-)chiral superfield approach to interacting Abelian 1-form gauge theories: Nilpotent and absolutely anticommuting charges

We derive the off-shell nilpotent (fermionic) (anti-)BRST symmetry transformations by exploiting the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism for the interacting Abelian 1-form gauge theories where there is a coupling between the U(1) Abelian 1-form gauge field and Dirac as well as complex scalar fields. We exploit the (anti-)BRST invariant restrictions on the (anti-)chiral superfields to derive the fermionic symmetries of our present D-dimensional Abelian 1-form gauge theories. The novel observation of our present investigation is the derivation of the absolute anticommutativity of the nilpotent (anti-)BRST charges despite the fact that our ordinary D-dimensional theories are generalized onto the (D,[Formula: see text]1)-dimensional (anti-) chiral super-submanifolds (of the general (D,[Formula: see text]2)-dimensional supermanifold) where only the (anti-)chiral super expansions of the (anti-)chiral superfields have been taken into account. We also discuss the nilpotency of the (anti-)BRST charges and (anti-)BRST invariance of the Lagrangian densities of our present theories within the framework of ACSA to BRST formalism.

Basic Brackets of a 2D Model for the Hodge Theory Without its Canonical Conjugate Momenta

We deduce the canonical brackets for a two (1 + 1)-dimensional (2D) free Abelian 1-form gauge theory by exploiting the beauty and strength of the continuous symmetries of a Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density that respects, in totality, six continuous symmetries. These symmetries entail upon this model to become a field theoretic example of Hodge theory. Taken together, these symmetries enforce the existence of exactly the same canonical brackets amongst the creation and annihilation operators that are found to exist within the standard canonical quantization scheme. These creation and annihilation operators appear in the normal mode expansion of the basic fields of this theory. In other words, we provide an alternative to the canonical method of quantization for our present model of Hodge theory where the continuous internal symmetries play a decisive role. We conjecture that our method of quantization is valid for a class of field theories that are tractable physical examples for the Hodge theory. This statement is true in any arbitrary dimension of spacetime.

Universal Superspace Unitary Operator for Some Interesting Abelian Models: Superfield Approach

Within the framework of augmented version of superfield formalism, we derive the superspace unitary operator and show its usefulness in the derivation of Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for a set of interesting models for the Abelian 1-form gauge theories. These models are (i) a one (0+1)-dimensional (1D) toy model of a rigid rotor, (ii) the two (1+1)-dimensional (2D) modified versions of the Proca and anomalous Abelian 1-form gauge theories, and (iii) the 2D self-dual bosonic gauge field theory. We provide, in some sense, the alternatives to the horizontality condition (HC) and the gauge invariant restrictions (GIRs) in the language of the above superspace (SUSP) unitary operator. One of the key observations of our present endeavor is the result that the SUSP unitary operator and its Hermitian conjugate are found to be thesameforallthe Abelian models under consideration (including the 4DinteractingAbelian 1-form gauge theories with Dirac and complex scalar fields which have been discussed earlier). Thus, we establish theuniversalityof the SUSP operator for the above Abelian theories.

Superspace unitary operator in QED with Dirac and complex scalar fields: Superfield approach

We exploit the strength of the superspace (SUSP) unitary operator to obtain the results of the application of the horizontality condition (HC) within the framework of the augmented version of the superfield formalism that is applied to the interacting systems of Abelian 1-form gauge theories where the U(1) Abelian 1-form gauge field couples to the Dirac and complex scalar fields in the physical four (3 + 1)-dimensions of spacetime. These interacting theories are generalized onto a (4, 2)-dimensional supermanifold that is parametrized by the four -dimensional (4D) spacetime variables and a pair of Grassmannian variables. To derive the (anti-)BRST symmetries for the matter fields, we impose the gauge-invariant restrictions (GIRs) on the superfields defined on the (4, 2)-dimensional supermanifold. We discuss various outcomes that emerge out from our knowledge of the SUSP unitary operator and its Hermitian conjugate. The latter operator is derived without imposing any operation of Hermitian conjugation on the parameters and fields of our theory from outside. This is an interesting observation in our present investigation.

Abelian p-form (p = 1, 2, 3) gauge theories as the field theoretic models for the Hodge theory

Taking the simple examples of an Abelian 1-form gauge theory in two (1+1)-dimensions, a 2-form gauge theory in four (3+1)-dimensions and a 3-form gauge theory in six (5+1)-dimensions of space–time, we establish that such gauge theories respect, in addition to the gauge symmetry transformations that are generated by the first-class constraints of the theory, additional continuous symmetry transformations. We christen the latter symmetry transformations as the dual-gauge transformations. We generalize the above gauge and dual-gauge transformations to obtain the proper (anti-)BRST and (anti-)dual-BRST transformations for the Abelian 3-form gauge theory within the framework of BRST formalism. We concisely mention such symmetries for the 2D free Abelian 1-form and 4D free Abelian 2-form gauge theories and briefly discuss their topological aspects in our present endeavor. We conjecture that any arbitrary Abelian p-form gauge theory would respect the above cited additional symmetry in D = 2p(p = 1, 2, 3, …) dimensions of space–time. By exploiting the above inputs, we establish that the Abelian 3-form gauge theory, in six (5+1)-dimensions of space–time, is a perfect model for the Hodge theory whose discrete and continuous symmetry transformations provide the physical realizations of all aspects of the de Rham cohomological operators of differential geometry. As far as the physical utility of the above nilpotent symmetries is concerned, we demonstrate that the 2D Abelian 1-form gauge theory is a perfect model of a new class of topological theory and 4D Abelian 2-form as well as 6D Abelian 3-form gauge theories are the field theoretic models for the quasi-topological field theory.

BRST symmetry and Darboux transformations in Abelian 2-form gauge theory

We analyse the constraints of an Abelian 2-form gauge theory using the Faddeev-Jackiw symplectic formalism. Further, this theory is treated as a constrained system in the context of the Batalin-Fradkin-Vilkovisky formalism to retrieve the BRST symmetry. Using the fields decompositions the effective action for the Abelian 2-form gauge theory is written in terms of a diagonalized uncanonical part and the BRST exact one. The nilpotent BRST and contracting homotopy σ closed transformations with field redefinitions are shown as the Darboux transformations used in the Faddeev-Jackiw formalism.

Abelian 3-form gauge theory: Superfield approach

We discuss a D-dimensional Abelian 3-form gauge theory within the framework of Bonora-Tonin's superfield formalism and derive the off-shell nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for this theory. To pay our homage to Victor I. Ogievetsky (1928-1996), who was one of the inventors of Abelian 2-form (antisymmetric tensor) gauge field, we go a step further and discuss the above D-dimensional Abelian 3-form gauge theory within the framework of BRST formalism and establish that the existence of the (anti-)BRST invariant Curci-Ferrari (CF) type of restrictions is the hallmark of any arbitrary p-form gauge theory (discussed within the framework of BRST formalism).

Dual-BRST symmetry: 6D Abelian 3-form gauge theory

Within the framework of the Becchi–Rouet–Stora–Tyutin (BRST) formalism, we demonstrate the existence of the novel off-shell nilpotent (anti-)dual-BRST symmetries in the context of a six (5+1)-dimensional (6D) free Abelian 3-form gauge theory. Under these local and continuous symmetry transformations, the total gauge-fixing term of the Lagrangian density remains invariant. This observation should be contrasted with the off-shell nilpotent (anti-)BRST symmetry transformations, under which, the total kinetic term of the theory remains invariant. The anticommutator of the above nilpotent (anti-)BRST and (anti-)dual-BRST transformations leads to the derivation of a bosonic symmetry in the theory. There exists a discrete symmetry transformation in the theory which provides a thread of connection between the nilpotent (anti-)BRST and (anti-)dual-BRST transformations. This theory is endowed with a ghost-scale symmetry, too. We discuss the algebra of these symmetry transformations and show that the structure of the algebra is reminiscent of the algebra of de Rham cohomological operators of differential geometry.

BRST analysis of topologically massive gauge theory: novel observations

A dynamical non-Abelian 2-form gauge theory (with B∧F term) is endowed with the “scalar” and “vector” gauge symmetry transformations. In our present endeavor, we exploit the latter gauge symmetry transformations and perform the Becchi–Rouet–Stora–Tyutin (BRST) analysis of the four (3+1)-dimensional (4D) topologically massive non-Abelian 2-form gauge theory. We demonstrate the existence of some novel features that have, hitherto, not been observed in the context of BRST approach to 4D (non-)Abelian 1-form as well as Abelian 2-form and 3-form gauge theories. We comment on the differences between the novel features that emerge in the BRST analysis of the “scalar” and “vector” gauge symmetries.

Notoph gauge theory: Superfield formalism

We derive absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for the 4D free Abelian 2-form gauge theory by exploiting the superfield approach to BRST formalism. The antisymmetric tensor gauge field of the above theory was christened as the “notoph” (i.e. the opposite of “photon”) gauge field by Ogievetsky and Palubarinov way back in 1966–67. We briefly outline the problems involved in obtaining the absolute anticonimutativity of the (anti-) BRST transformations and their resolution within the framework of geometrical superfield approach to BRST formalism. One of the highlights of our results is the emergence of a Curci-Ferrari type of restriction in the context of 4D Abelian 2-form (notoph) gauge theory which renders the nilpotent (anti-) BRST symmetries of the theory to be absolutely anticommutative in nature.

VARIOUS FORMS OF BRST SYMMETRY IN ABELIAN 2-FORM GAUGE THEORY

We derive the various forms of BRST symmetry using Batalin–Fradkin–Vilkovisky approach in the case of Abelian 2-form gauge theory. We show that the so-called dual BRST symmetry is not an independent symmetry but the generalization of BRST symmetry obtained from the canonical transformation in the bosonic and ghost sector. We further obtain the new forms of both BRST and dual-BRST symmetry by making a general transformation in the Lagrange multipliers of the bosonic and ghost sector of the theory.

GENERALIZED BRST TRANSFORMATION IN ABELIAN RANK-2 ANTISYMMETRIC TENSOR FIELD THEORY

We generalize the BRST transformations in Abelian rank-2 tensor field theory by allowing the parameter to be finite and field-dependent and show that such transformations play crucial role in studying the Abelian 2-form gauge theory in noncovariant gauges. The generating functionals in different effective theories are connected through these generalized BRST transformations with different choice of finite parameter. We further consider the generalization of the anti-BRST transformations and show that even anti-BRST transformations with the finite field dependent parameter relate the generating functional for different effective theories. Such generalized BRST/anti-BRST transformations are useful in connecting generating functionals corresponding to different effective theories. Field/antifield formulation of this Abelian 2-form gauge theory is also studied in the context of generalized BRST transformations. We show that generalized BRST transformation connects the generating functional corresponding to the different solutions of quantum master equation in field/antifield formulation.

A NOTE ON THE (ANTI-)BRST INVARIANT LAGRANGIAN DENSITIES FOR THE FREE ABELIAN 2-FORM GAUGE THEORY

We show that the previously known off-shell nilpotent [Formula: see text] and absolutely anticommuting (sb sab + sab sb = 0) Becchi–Rouet–Stora–Tyutin (BRST) transformations (sb) and anti-BRST transformations (sab) are the symmetry transformations of the appropriate Lagrangian densities of a four (3+1)-dimensional (4D) free Abelian 2-form gauge theory which do not explicitly incorporate a very specific constrained field condition through a Lagrange multiplier 4D vector field. The above condition, which is the analogue of the Curci–Ferrari restriction of the non-Abelian 1-form gauge theory, emerges from the Euler–Lagrange equations of motion of our present theory and ensures the absolute anticommutativity of the transformations s(a)b. Thus, the coupled Lagrangian densities, proposed in our present investigation, are aesthetically more appealing and more economical.

ABSOLUTE ANTICOMMUTATIVITY OF THE NILPOTENT SYMMETRIES IN THE HAMILTONIAN FORMALISM: FREE ABELIAN 2-FORM GAUGE THEORY

The celebrated Curci–Ferrari (CF) type of restrictions are invoked to obtain the off-shell nilpotent and absolutely anticommuting (anti-)BRST as well as (anti-)co-BRST symmetry transformations in the context of the Lagrangian description of the physical four (3+1)-dimensional (4D) free Abelian 2-form gauge theory. We show that the above CF-type conditions, which turn out to be the secondary constraints of the theory, remain invariant with respect to the time-evolution of the above 2-form gauge system in the Hamiltonian formulation. This time-evolution invariance (i) physically ensures the linear independence of the BRST versus anti-BRST as well as co-BRST versus anti-co-BRST symmetry transformations, and (ii) provides a logical reason behind the imposition of the above CF-type restrictions in the proof of the absolute anticommutativity of the off-shell nilpotent (anti-)BRST as well as (anti-)co-BRST symmetry transformations.