scholarly journals Supervariable Approach to the Nilpotent Symmetries for a Toy Model of the Hodge Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
D. Shukla ◽  
T. Bhanja ◽  
R. P. Malik
Keyword(s):  
Brst Symmetry ◽  
Hodge Theory ◽  
Present Theory ◽  
Rigid Rotor ◽  
Geometrical Meaning ◽  
Horizontality Condition ◽  
Symmetry Transformations ◽  
Brst Invariance ◽  
Standard Techniques

We exploit the standard techniques of the supervariable approach to derive the nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for a toy model of the Hodge theory (i.e., a rigid rotor) and provide the geometrical meaning and interpretation to them. Furthermore, wealsoderive the nilpotent (anti-)co-BRST symmetry transformations for this theory within the framework of the above supervariable approach. We capture the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian of our present theory within the framework of augmented supervariable formalism. We also express the (anti-)BRST and (anti-)co-BRST charges in terms of the supervariables (obtained after the application of the (dual-)horizontality conditions and (anti-)BRST and (anti-)co-BRST invariant restrictions) to provide the geometrical interpretations for their nilpotency and anticommutativity properties. The application of the dual-horizontality condition and ensuing proper (i.e., nilpotent and absolutely anticommuting) fermionic (anti-)co-BRST symmetries are completelynovelresults in our present investigation.

scholarly journals Nilpotent charges of a toy model of Hodge theory and an 𝒩 = 2 SUSY quantum mechanical model: (Anti-)chiral supervariable approach

2019 ◽  
Vol 34 (30) ◽  
pp. 1950183
Author(s):  
T. Bhanja ◽  
N. Srinivas ◽  
R. P. Malik
Keyword(s):  
General Formula ◽  
Mechanical Model ◽  
Free Particle ◽  
Brst Symmetry ◽  
Hodge Theory ◽  
Quantum Mechanical ◽  
Rigid Rotor ◽  
Quantum Mechanical Model ◽  
Evolution Parameter ◽  
Symmetry Transformations

We derive the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations for the system of a toy model of Hodge theory (i.e. a rigid rotor) by exploiting the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti-)chiral supervariables that are defined on the appropriately chosen [Formula: see text]-dimensional super-submanifolds of the general [Formula: see text]-dimensional supermanifold on which our system of a one [Formula: see text]-dimensional (1D) toy model of Hodge theory is considered within the framework of the augmented version of the (anti-)chiral supervariable approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism. The general [Formula: see text]-dimensional supermanifold is parametrized by the superspace coordinates [Formula: see text], where [Formula: see text] is the bosonic evolution parameter and [Formula: see text] are the Grassmannian variables which obey the standard fermionic relationships: [Formula: see text], [Formula: see text]. We provide the geometrical interpretations for the symmetry invariance and nilpotency property. Furthermore, in our present endeavor, we establish the property of absolute anticommutativity of the conserved fermionic charges which is a completely novel and surprising observation in our present endeavor where we have considered only the (anti-)chiral supervariables. To corroborate the novelty of the above observation, we apply this ACSA to an [Formula: see text] SUSY quantum mechanical (QM) system of a free particle and show that the [Formula: see text] SUSY conserved and nilpotent charges do not absolutely anticommute.

scholarly journals TOPOLOGICALLY MASSIVE NON-ABELIAN THEORY: SUPERFIELD APPROACH

2011 ◽  
Vol 26 (25) ◽  
pp. 4419-4450 ◽  
Author(s):  
S. KRISHNA ◽  
A. SHUKLA ◽  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Gauge Symmetry ◽  
Brst Symmetry ◽  
Gauge Transformations ◽  
Abelian Theory ◽  
Gauge Symmetries ◽  
Vector Gauge ◽  
Symmetry Transformations ◽  
Brst Invariance ◽  
Lagrangian Densities

We apply the well-established techniques of geometrical superfield approach to Becchi–Rouet–Stora–Tyutin (BRST) formalism in the context of four (3+1)-dimensional (4D) dynamical non-Abelian 2-form gauge theory by exploiting its inherent "scalar" and "vector" gauge symmetry transformations and derive the corresponding off-shell nilpotent and absolutely anticommuting BRST and anti-BRST symmetry transformations. Our approach leads to the derivation of three (anti-)BRST invariant Curci–Ferrari (CF)-type restrictions that are found to be responsible for the absolute anticommutativity of the BRST and anti-BRST symmetry transformations. We derive the coupled Lagrangian densities that respect the (anti-)BRST symmetry transformations corresponding to the "vector" gauge transformations. We also capture the (anti-)BRST invariance of the CF-type restrictions and coupled Lagrangian densities within the framework of our superfield approach. We obtain, furthermore, the off-shell nilpotent (anti-)BRST symmetry transformations when the (anti-)BRST symmetry transformations corresponding to the "scalar" and "vector" gauge symmetries are merged together. These off-shell nilpotent "merged" (anti-)BRST symmetry transformations are, however, found to be non-anticommuting in nature.

scholarly journals Augmented Superfield Approach to Nilpotent Symmetries of the Modified Version of 2D Proca Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-21 ◽  
Author(s):  
A. Shukla ◽  
S. Krishna ◽  
R. P. Malik
Keyword(s):  
Differential Geometry ◽  
Lagrangian Density ◽  
Brst Symmetry ◽  
Physical Realization ◽  
Gauge Invariant ◽  
De Rham ◽  
Symmetry Transformations ◽  
Complete Set ◽  
Brst Invariance ◽  
Augmented Superfield Formalism

We derive the complete set of off-shell nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, and (anti-)co-BRST symmetry transformations forallthe fields of the modified version of two(1+1)-dimensional (2D) Proca theory by exploiting the “augmented” superfield formalism where the (dual-)horizontality conditions and (dual-)gauge invariant restrictions are exploitedtogether. We capture the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian density in the language of superfield approach. We also express the nilpotency and absolute anticommutativity of the (anti-)BRST and (anti-)co-BRST charges within the framework of augmented superfield formalism. This exercise leads to somenovelobservations which have, hitherto, not been pointed out in the literature within the framework of superfield approach to BRST formalism. For the sake of completeness, we also mention, very briefly, a unique bosonic symmetry, the ghost-scale symmetry, and discrete symmetries of the theory and show that the algebra of conserved charges provides a physical realization of the Hodge algebra (satisfied by the de Rham cohomological operators of differential geometry).

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.

scholarly journals Universal Superspace Unitary Operator and Nilpotent (Anti-)Dual-BRST Symmetries: Superfield Formalism

2016 ◽  
Vol 2016 ◽  
pp. 1-17
Author(s):  
T. Bhanja ◽  
N. Srinivas ◽  
R. P. Malik
Keyword(s):  
Gauge Theories ◽  
Unitary Operator ◽  
Group Structure ◽  
Brst Symmetry ◽  
Rigid Rotor ◽  
Unitary Operators ◽  
Bosonic Field ◽  
Hermitian Conjugate ◽  
Symmetry Transformations ◽  
The One

We exploit the key concepts of the augmented version of superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism to derive the superspace (SUSP)dualunitary operator and its Hermitian conjugate and demonstrate their utility in the derivation of the nilpotent and absolutely anticommuting (anti-)dual-BRST symmetry transformations for a set of interesting models of theAbelian1-form gauge theories. These models are the one (0+1)-dimensional (1D) rigid rotor and modified versions of the two (1+1)-dimensional (2D) Proca as well as anomalous gauge theories and 2D model of a self-dual bosonic field theory. We show theuniversalityof the SUSPdualunitary operator and its Hermitian conjugate in the cases ofallthe Abelian models under consideration. These SUSP dual unitary operators, besides maintaining the explicit group structure, provide the alternatives to the dual horizontality condition (DHC) and dual gauge invariant restrictions (DGIRs) of the superfield formalism. The derivations of thedualunitary operators and corresponding (anti-)dual-BRST symmetries are completelynovelresults in our present investigation.

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

2018 ◽  
Vol 33 (04) ◽  
pp. 1850026 ◽  
Author(s):  
B. Chauhan ◽  
S. Kumar ◽  
R. P. Malik
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Scalar Fields ◽  
The Novel ◽  
Complex Scalar ◽  
Symmetry Transformations ◽  
Brst Invariance ◽  
Lagrangian Densities ◽  
Chiral Superfield ◽  
Chiral Superfields

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.

scholarly journals A Specific N=2 Supersymmetric Quantum Mechanical Model: Supervariable Approach

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Aradhya Shukla
Keyword(s):  
Mechanical Model ◽  
Hodge Theory ◽  
Quantum Mechanical ◽  
Geometrical Meaning ◽  
Quantum Mechanical Model ◽  
Symmetry Transformations

By exploiting the supersymmetric invariant restrictions on the chiral and antichiral supervariables, we derive the off-shell nilpotent symmetry transformations for a specific (0 + 1)-dimensional N=2 supersymmetric quantum mechanical model which is considered on a (1, 2)-dimensional supermanifold (parametrized by a bosonic variable t and a pair of Grassmannian variables (θ,θ¯)). We also provide the geometrical meaning to the symmetry transformations. Finally, we show that this specific N=2 SUSY quantum mechanical model is a model for Hodge theory.

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

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
T. Bhanja ◽  
N. Srinivas ◽  
R. P. Malik
Keyword(s):  
Gauge Theories ◽  
Unitary Operator ◽  
Brst Symmetry ◽  
Scalar Fields ◽  
Gauge Field Theory ◽  
Rigid Rotor ◽  
Complex Scalar ◽  
Gauge Invariant ◽  
Hermitian Conjugate ◽  
Symmetry Transformations

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.

scholarly journals COMMENTS ON THE DUAL-BRST SYMMETRY

2011 ◽  
Vol 26 (36) ◽  
pp. 2739-2744 ◽  
Author(s):  
S. KRISHNA ◽  
A. SHUKLA ◽  
R. P. MALIK
Keyword(s):  
Differential Geometry ◽  
Gauge Theories ◽  
Short Note ◽  
Brst Symmetry ◽  
Hodge Theory ◽  
De Rham ◽  
Symmetry Transformations ◽  
Original Motivation

In view of a raging controversy on the topic of dual-Becchi–Rouet–Stora–Tyutin (dual-BRST/co-BRST) and anti-co-BRST symmetry transformations in the context of four (3+1)-dimensional (4D) Abelian two-form and 2D (non-)Abelian one-form gauge theories, we attempt, in our present short note, to settle the dust by taking the help of mathematics of differential geometry, connected with the Hodge theory, which was the original motivation for the nomenclature of "dual-BRST symmetry" in our earlier set of works. It has been claimed, in a recent set of papers, that the co-BRST symmetries are not independent of the BRST symmetries. We show that the BRST and co-BRST symmetries are independent symmetries in the same fashion as the exterior and co-exterior derivatives are independent entities belonging to the set of de Rham cohomological operators of differential geometry.

scholarly journals SUPERFIELD APPROACH TO NILPOTENT SYMMETRIES OF THE FREEDMAN–TOWNSEND MODEL: NOVEL FEATURES

2012 ◽  
Vol 27 (22) ◽  
pp. 1250123 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Present Model ◽  
Brst Symmetry ◽  
The Other ◽  
Gauge Invariant ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Abelian Theory ◽  
Horizontality Condition ◽  
Form Theory ◽  
Symmetry Transformations

We perform the Becchi–Rouet–Stora–Tyutin (BRST) analysis of the Freedman–Townsend (FT) model of topologically massive non-Abelian theory by exploiting its (1-form) Yang–Mills (YM) gauge transformations to show the existence of some novel features that are totally different from the results obtained in such a kind of consideration carried out for the dynamical non-Abelian 2-form theory. We tap here the potential and power of the augmented version of Bonora–Tonin's superfield approach to BRST formalism to derive the full set of off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations where, in addition to the horizontality condition (HC), we are theoretically compelled to exploit the appropriate gauge-invariant restrictions (GIRs) on the (super)fields for the derivation of the appropriate symmetry transformations for all the relevant fields. We compare our key results with that of the other such attempt for the discussion of the present model within the framework of BRST formalism.

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