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 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 Superfield approach to nilpotent symmetries in 3D Jackiw–Pi model of massive non-Abelian theory

2014 ◽  
Vol 92 (9) ◽  
pp. 1033-1042 ◽  
Author(s):  
S. Gupta ◽  
R. Kumar ◽  
R.P. Malik
Keyword(s):  
Gauge Theory ◽  
Brst Symmetry ◽  
Point Of View ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
Abelian Gauge Theory ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Symmetry Transformations ◽  
Augmented Superfield Formalism

In the available literature, only the Becchi–Rouet–Stora–Tyutin (BRST) symmetries are known for the Jackiw–Pi model of the three (2 + 1)-dimensional (3D) massive non-Abelian gauge theory. We derive the off-shell nilpotent [Formula: see text] and absolutely anticommuting (sbsab + sabsb = 0) (anti-)BRST transformations s(a)b corresponding to the usual Yang–Mills gauge transformations of this model by exploiting the “augmented” superfield formalism where the horizontality condition and gauge invariant restrictions blend together in a meaningful manner. There is a non-Yang–Mills (NYM) symmetry in this theory, too. However, we do not touch the NYM symmetry in our present endeavor. This superfield formalism leads to the derivation of an (anti-)BRST invariant Curci–Ferrari restriction, which plays a key role in the proof of absolute anticommutativity of s(a)b. The derivation of the proper anti-BRST symmetry transformations is important from the point of view of geometrical objects called gerbes. A novel feature of our present investigation is the derivation of the (anti-)BRST transformations for the auxiliary field ρ from our superfield formalism, which is neither generated by the (anti-)BRST charges nor obtained from the requirements of nilpotency and (or) absolute anticommutativity of the (anti-)BRST symmetries for our present 3D non-Abelian 1-form gauge theory.

scholarly journals Nilpotent (anti-)BRST and (anti-)co-BRST symmetries in 2D non-Abelian gauge theory: Some novel observations

2021 ◽  
Author(s):  
S. Kumar ◽  
B. K. Kureel ◽  
R. P. Malik
Keyword(s):  
Differential Geometry ◽  
Gauge Theory ◽  
Brst Symmetry ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Conserved Charges ◽  
Symmetry Operators ◽  
Symmetry Transformations ◽  
Continuous Symmetries ◽  
Lagrangian Densities

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-)co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two (1[Formula: see text]+[Formula: see text]1)-dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel.

scholarly journals Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds

2018 ◽  
Vol 30 (05) ◽  
pp. 1850012 ◽  
Author(s):  
C. I. Lazaroiu ◽  
C. S. Shahbazi
Keyword(s):  
Gauge Theory ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Global Solutions ◽  
Mathematical Object ◽  
Mathematical Structure ◽  
Quantization Condition ◽  
Local Analysis ◽  
Abelian Gauge ◽  
Abelian Gauge Theory

We give the global mathematical formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields on a Lorentzian four-manifold, for the generalized situation when the duality structure of the Abelian gauge theory is described by a flat symplectic vector bundle [Formula: see text] defined over the scalar manifold [Formula: see text]. The construction uses a taming of [Formula: see text], which we find to be the correct mathematical object globally encoding the inverse gauge couplings and theta angles of the “twisted” Abelian gauge theory in a manner that makes no use of duality frames. We show that global solutions of the equations of motion of such models give classical locally geometric U-folds. We also describe the groups of duality transformations and scalar-electromagnetic symmetries arising in such models, which involve lifting isometries of [Formula: see text] to the bundle [Formula: see text] and hence differ from expectations based on local analysis. The appropriate version of the Dirac quantization condition involves a discrete local system defined over [Formula: see text] and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows, in particular, that a generalization of part of the mathematical structure familiar from [Formula: see text] supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetry.

scholarly journals NILPOTENT SYMMETRIES FOR MATTER FIELDS IN NON-ABELIAN GAUGE THEORY: AUGMENTED SUPERFIELD FORMALISM

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4899-4915 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
General Scheme ◽  
Brst Symmetry ◽  
Grassmann Algebra ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Symmetry Transformations ◽  
Matter Fields ◽  
Augmented Superfield Formalism ◽  
Standing Problem

In the framework of superfield approach to Becchi–Rouet–Stora–Tyutin (BRST) formalism, the derivation of the BRST and anti-BRST nilpotent symmetry transformations for the matter fields, present in any arbitrary interacting gauge theory, has been a long-standing problem. In our present investigation, the local, covariant, continuous and off-shell nilpotent (anti-)BRST symmetry transformations for the Dirac fields [Formula: see text] are derived in the framework of the augmented superfield formulation where the four (3 + 1)-dimensional (4D) interacting non-Abelian gauge theory is considered on the six (4 + 2)-dimensional supermanifold parametrized by the four even space–time coordinates xμ and a couple of odd elements (θ and [Formula: see text]) of the Grassmann algebra. The requirement of the invariance of the matter (super)currents and the horizontality condition on the (super)manifolds leads to the derivation of the nilpotent symmetries for the matter fields as well as the gauge and the (anti)ghost fields of the theory in the general scheme of augmented superfield formalism.

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 DUAL BRST SYMMETRY FOR QED

2001 ◽  
Vol 16 (08) ◽  
pp. 477-488 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Brst Symmetry ◽  
Discrete Symmetry ◽  
Two Dimensions ◽  
Abelian Gauge ◽  
Dirac Fields ◽  
Gauge Fixing ◽  
De Rham ◽  
Symmetry Transformations ◽  
Dimensions Of Space ◽  
Field Theoretical Model

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.

scholarly journals BRST COHOMOLOGY AND HODGE DECOMPOSITION THEOREM IN ABELIAN GAUGE THEORY

2000 ◽  
Vol 15 (11) ◽  
pp. 1685-1705 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Single Photon ◽  
Equations Of Motion ◽  
Decomposition Theorem ◽  
Hodge Decomposition ◽  
Laplacian Operator ◽  
Two Dimensional ◽  
Abelian Gauge ◽  
Brst Charge ◽  
Brst Cohomology

We discuss the Becchi–Rouet–Stora–Tyutin (BRST) cohomology and Hodge decomposition theorem for the two-dimensional free U(1) gauge theory. In addition to the usual BRST charge, we derive a local, conserved and nilpotent co(dual)-BRST charge under which the gauge-fixing term remains invariant. We express the Hodge decomposition theorem in terms of these charges and the Laplacian operator. We take a single photon state in the quantum Hilbert space and demonstrate the notion of gauge invariance, no-(anti)ghost theorem, transversality of photon and establish the topological nature of this theory by exploiting the concepts of BRST cohomology and Hodge decomposition theorem. In fact, the topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. On the two-dimensional compact manifold, we derive two sets of topological invariants with respect to the conserved and nilpotent BRST- and co-BRST charges and express the Lagrangian density of the theory as the sum of terms that are BRST- and co-BRST invariants. Mathematically, this theory captures together some of the key features of both Witten- and Schwarz-type of topological field theories.

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.

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