scholarly journals GEOMETRICAL ASPECTS OF BRST COHOMOLOGY IN AUGMENTED SUPERFIELD FORMALISM

2004 ◽  
Vol 01 (04) ◽  
pp. 467-492 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Theoretical Model ◽  
Quantum Electrodynamics ◽  
Gauge Theories ◽  
Decomposition Theorem ◽  
Brst Cohomology ◽  
De Rham ◽  
Definition Of ◽  
Field Theoretical Model ◽  
Augmented Superfield Formalism

In the framework of augmented superfield approach, we provide the geometrical origin and interpretation for the nilpotent (anti-)BRST charges, (anti-)co-BRST charges and a non-nilpotent bosonic charge. Together, these local and conserved charges turn out to be responsible for a clear and cogent definition of the Hodge decomposition theorem in the quantum Hilbert space of states. The above charges owe their origin to the de Rham cohomological operators of differential geometry which are found to be at the heart of some of the key concepts associated with the interacting gauge theories. For our present review, we choose the two (1+1)-dimensional (2D) quantum electrodynamics (QED) as a prototype field theoretical model to derive all the nilpotent symmetries for all the fields present in this interacting gauge theory in the framework of augmented superfield formulation and show that this theory is a unique example of an interacting gauge theory which provides a tractable field theoretical model for the Hodge theory.

scholarly journals ONE-FORM ABELIAN GAUGE THEORY AS THE HODGE THEORY

2007 ◽  
Vol 22 (21) ◽  
pp. 3521-3562 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Theoretical Model ◽  
Equations Of Motion ◽  
Hodge Theory ◽  
Laplacian Operator ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
De Rham ◽  
Symmetry Transformations ◽  
Field Theoretical Model

We demonstrate that the two (1+1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical ingredients that are required in the definition of the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, constitute an algebra that is reminiscent of the algebra obeyed by the de Rham cohomological operators. The topological features of the above theory are discussed in terms of the BRST and co-BRST operators. The super-de Rham cohomological operators are exploited in the derivation of the nilpotent (anti-)BRST, (anti-)co-BRST symmetry transformations and the equations of motion for all the fields of the theory, within the framework of the superfield formulation. The derivation of the equations of motion, by exploiting the super-Laplacian operator, is a completely new result in the framework of the superfield approach to BRST formalism. In an Appendix, the interacting 2D Abelian gauge theory (where there is a coupling between the U(1) gauge field and the Dirac fields) is also shown to provide a tractable field theoretical model for the Hodge theory.

scholarly journals NEW LOCAL SYMMETRY FOR QED IN TWO DIMENSIONS

2000 ◽  
Vol 15 (34) ◽  
pp. 2079-2085 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Differential Geometry ◽  
Gauge Theory ◽  
Decomposition Theorem ◽  
Local Symmetry ◽  
Two Dimensions ◽  
De Rham Cohomology ◽  
Brst Cohomology ◽  
Gauge Fixing ◽  
De Rham ◽  
Dimensions Of Space

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.

scholarly journals HAMILTONIAN COHOMOLOGICAL DERIVATION OF FOUR-DIMENSIONAL NONLINEAR GAUGE THEORIES

2002 ◽  
Vol 17 (16) ◽  
pp. 2191-2210 ◽  
Author(s):  
C. BIZDADEA ◽  
E. M. CIOROIANU ◽  
S. O. SALIU
Keyword(s):  
Gauge Theory ◽  
Gauge Theories ◽  
Scalar Fields ◽  
Gauge Algebra ◽  
Brst Cohomology ◽  
Nonlinear Gauge

Consistent couplings among a set of scalar fields, two types of one-forms and a system of two-forms are investigated in the light of the Hamiltonian BRST cohomology, giving a four-dimensional nonlinear gauge theory. The emerging interactions deform the first-class constraints, the Hamiltonian gauge algebra, as well as the reducibility relations.

scholarly journals TOPOLOGICAL FEATURES IN NON-ABELIAN GAUGE THEORY

1999 ◽  
Vol 14 (28) ◽  
pp. 1937-1949 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Equations Of Motion ◽  
Decomposition Theorem ◽  
Laplacian Operator ◽  
Two Dimensional ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Topological Features ◽  
Brst Cohomology ◽  
Topological Field

We discuss the BRST cohomology and exhibit a connection between the Hodge decomposition theorem and the topological properties of a two-dimensional free non-Abelian gauge theory (having no interaction with matter fields). The topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. We obtain two sets of topological invariants with respect to BRST and co-BRST charges on the two-dimensional compact manifold and show that the Lagrangian density of the theory can be expressed as the sum of terms that are BRST and co-BRST invariants. Thus, this theory captures together some of the salient features of both Witten and Schwarz type of topological field theories.

scholarly journals NONCOMMUTATIVE GAUGE THEORIES: MODEL FOR HODGE THEORY

2013 ◽  
Vol 28 (25) ◽  
pp. 1350122 ◽  
Author(s):  
SUDHAKER UPADHYAY ◽  
BHABANI PRASAD MANDAL
Keyword(s):  
Differential Geometry ◽  
Compact Manifold ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Decomposition Theorem ◽  
Hodge Theory ◽  
De Rham ◽  
Symmetry Transformations ◽  
Moyal Plane ◽  
Noncommutative Gauge Theories

The nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST, dual-BRST and anti-dual-BRST symmetry transformations are constructed in the context of noncommutative (NC) 1-form as well as 2-form gauge theories. The corresponding Noether's charges for these symmetries on the Moyal plane are shown to satisfy the same algebra, as by the de Rham cohomological operators of differential geometry. The Hodge decomposition theorem on compact manifold is also studied. We show that noncommutative gauge theories are field theoretic models for Hodge theory.

WEYL FERMION FUNCTIONAL INTEGRAL AND TWO-DIMENSIONAL GAUGE THEORIES

1990 ◽  
Vol 05 (14) ◽  
pp. 2839-2851
Author(s):  
J.L. ALONSO ◽  
J.L. CORTÉS ◽  
E. RIVAS
Keyword(s):  
Gauge Theory ◽  
Gauge Theories ◽  
Functional Integral ◽  
Integral Approach ◽  
Two Dimensional ◽  
Regularization Scheme ◽  
Schwinger Model ◽  
Left Handed ◽  
Definition Of ◽  
Weyl Fermion

In the path integral approach we introduce a general regularization scheme for a Weyl fermionic measure. This allows us to study the functional integral formulation of a two-dimensional U(1) gauge theory with an arbitrary content of left-handed and right-handed fermions. A particular result is that, in contrast with a regularization of the fermionic measure based on a unique Dirac operator, by taking the Dirac fermionic measure as a product of two independent Weyl fermionic measures a consistent and unitary result can be obtained for the Chiral Schwinger Model (CSM) as a byproduct of the arbitrariness in the definition of the fermionic measure.

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

2014 ◽  
Vol 29 (24) ◽  
pp. 1450135 ◽  
Author(s):  
R. Kumar ◽  
S. Krishna ◽  
A. Shukla ◽  
R. P. Malik
Keyword(s):  
Gauge Theory ◽  
Gauge Theories ◽  
Hodge Theory ◽  
Space Time ◽  
Continuous Symmetry ◽  
Gauge Transformations ◽  
Perfect Model ◽  
De Rham ◽  
Symmetry Transformations ◽  
Dimensions Of Space

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.

scholarly journals GAUGE TRANSFORMATIONS, BRST COHOMOLOGY AND WIGNER'S LITTLE GROUP

2004 ◽  
Vol 19 (32) ◽  
pp. 5663-5692 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Degrees Of Freedom ◽  
Gauge Theories ◽  
Internal Symmetry ◽  
Transformation Groups ◽  
Polarization Tensor ◽  
Gauge Transformations ◽  
Brst Cohomology ◽  
Lagrangian Densities ◽  
Translation Subgroup

We discuss the (dual-)gauge transformations and BRST cohomology for the two (1+1)-dimensional (2D) free Abelian one-form and four (3+1)-dimensional (4D) free Abelian two-form gauge theories by exploiting the (co-)BRST symmetries (and their corresponding generators) for the Lagrangian densities of these theories. For the 4D free two-form gauge theory, we show that the changes on the antisymmetric polarization tensor eμν(k) due to (i) the (dual-)gauge transformations corresponding to the internal symmetry group, and (ii) the translation subgroup T(2) of the Wigner's little group, are connected with each other for the specific relationships among the parameters of these transformation groups. In the language of BRST cohomology defined with respect to the conserved and nilpotent (co-)BRST charges, the (dual-)gauge transformed states turn out to be the sum of the original state and the (co-)BRST exact states. We comment on (i) the quasitopological nature of the 4D free two-form gauge theory from the degrees of freedom count on eμν(k), and (ii) the Wigner's little group and the BRST cohomology for the 2D one-form gauge theory vis-à-vis our analysis for the 4D two-form gauge theory.

scholarly journals The BV formalism: Theory and application to a matrix model

2019 ◽  
Vol 31 (10) ◽  
pp. 1950035
Author(s):  
Roberta A. Iseppi
Keyword(s):  
Gauge Theory ◽  
Gauge Symmetry ◽  
General Solution ◽  
Master Equation ◽  
Configuration Space ◽  
Matrix Model ◽  
Gauge Theories ◽  
Brst Cohomology ◽  
Cohomology Complex ◽  
Theory Formula

We review the BV formalism in the context of [Formula: see text]-dimensional gauge theories. For a gauge theory [Formula: see text] with an affine configuration space [Formula: see text], we describe an algorithm to construct a corresponding extended theory [Formula: see text], obtained by introducing ghost and anti-ghost fields, with [Formula: see text] a solution of the classical master equation in [Formula: see text]. This construction is the first step to define the (gauge-fixed) BRST cohomology complex associated to [Formula: see text], which encodes many interesting information on the initial gauge theory [Formula: see text]. The second part of this article is devoted to the application of this method to a matrix model endowed with a [Formula: see text]-gauge symmetry, explicitly determining the corresponding [Formula: see text] and the general solution [Formula: see text] of the classical master equation for the model.

scholarly journals WIGNER'S LITTLE GROUP AND BRST COHOMOLOGY FOR ONE-FORM ABELIAN GAUGE THEORY

2004 ◽  
Vol 19 (16) ◽  
pp. 2721-2737 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Gauge Transformation ◽  
Decisive Role ◽  
Decomposition Theorem ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Gauge Transformations ◽  
Brst Cohomology ◽  
The Relationship ◽  
Translation Subgroup

We discuss the (dual-)gauge transformations for the gauge-fixed Lagrangian density and establish their intimate connection with the translation subgroup T(2) of Wigner's little group for the free one-form Abelian gauge theory in four (3+1)-dimensions (4D) of space–time. Though the relationship between the usual gauge transformation for the Abelian massless gauge field and T(2) subgroup of the little group is quite well known, such a connection between the dual-gauge transformation and the little group is a new observation. The above connections are further elaborated and demonstrated in the framework of Becchi–Rouet–Stora–Tyutin (BRST) cohomology defined in the quantum Hilbert space of states where the Hodge decomposition theorem (HDT) plays a very decisive role.

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