Nilpotent charges in an interacting gauge theory and an 𝒩 = 2 SUSY quantum mechanical model: (Anti-)chiral superfield approach

Abstract

We exploit the power and potential of the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism to derive the nilpotent (anti-) BRST symmetry transformations for any arbitrary D-dimensional interacting non-Abelian 1-form gauge theory where there is an SU(N) gauge invariant coupling between the gauge field and the Dirac fields. We derive the conserved and nilpotent (anti-)BRST charges and establish their nilpotency and absolute anticommutativity properties within the framework of ACSA to BRST formalism. The clinching proof of the absolute anticommutativity property of the conserved and nilpotent (anti-)BRST charges is a novel result in view of the fact that we consider, in our present endeavor, only the (anti-)chiral super expansions of the superfields that are defined on the (D, 1)-dimensional super-submanifolds of the general (D, 2)-dimensional supermanifold on which our D-dimensional ordinary interacting non-Abelian 1-form gauge theory is generalized. To corroborate the novelty of the above result, we apply the ACSA to an N = 2 supersymmetric (SUSY) quantum mechanical (QM) model of a harmonic oscillator and show that the nilpotent and conserved N = 2 super charges of this system do not absolutely anticommute.Keywords: (Anti-)chiral superfield approach; interacting non-Abelian 1-form gauge theory with Dirac fields; (anti-)BRST symmetries; (anti-)BRST charges; N = 2 SUSY harmonic oscillator; N = 2 SUSY symmetries; N = 2 SUSY charges, (anti-)chiral supervariable approach; nilpotency property; absolute anticommutativity property * The beauty of the (anti-)chiral superfield/supervariable approach is the observation that we obtain the (anti-)BRST symmetries for all the fields/variables of the theory from the (anti-)BRST (i.e. quantum gauge) invariant restrictions on the (anti-)chiral superfileds/supervariables.

Massive Spinning Relativistic Particle: Revisited under BRST and Supervariable Approaches

“…In other words, the absolute anticommutativity of the (anti-)BRST symmetry transformations and the existence of the coupled (but equivalent) Lagrangians owe their origins to the CF-type restriction which defines a submanifold in the quantum Hilbert space of variables that is defined by the equation: b + b + 2β β = 0. To corroborate the sanctity of our (anti-)BRST symmetry transformations, coupled (but equivalent) Lagrangians, and their invariance(s), we have exploited the theoretical potential and power of ACSA to BRST formalism [20][21][22][23][24] where only the (anti-)chiral supervariables and their suitable expansion(s) along the Grassmannian direction(s) have been considered. One of the novel observations, in this context, has been the proof of absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges within the framework of ACSA to BRST formalism where we have considered only the (anti-)chiral super expansions.…”

“…In the recent set of papers [20][21][22][23][24], we have developed a simpler version of the AVSA where only the (anti-)chiral supervariables/superfields and their appropriate super expansions have been taken into consideration. This superfield approach to BRST formalism has been christened as the (anti-)chiral superfield/supervariable approach (ACSA).…”

Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

“…In other words, it is the consistency conditions of the BRST formalism that lead to the determination of kðτÞ in Equation ( 16) within the ambit of MBTSA. A close look at Equations ( 25)- (27) establishes that a precise determination of QðτÞ in (23) leads to (i) the validity of the absolute anticommutativity (i.e., fs b , s ab gS = 0) of the off-shell nilpotent (anti-)BRST symmetries and (ii) the deduction of the (anti-)BRST invariant (This statement is true only when the whole theory is considered on a submanifold of the Hilbert space of the quantum variables where the CF-type restriction…”

“…Equations ( 32) and ( 33)) besides the phase space variables (x, p x , t, and p t ) whose (anti-)BRST symmetries have already been derived in Section 3 by exploiting the theoretical potential of MBTSA. To achieve the above goal, we exploit the ideas behind ACSA to BRST formalism [25][26][27][28][29]. In this context, first of all, we focus on the derivation of the BRST symmetry transformations 33)).…”

“…This approach has been christened by us as the modified Bonora-Tonin (BT) superfield approach (MBTSA) to BRST formalism. In a recent couple of papers [23,24], we have applied the theoretical beauty of the MBTSA as well as ACSA (i.e., (anti)chiral superfield/supervariable approach) to BRST formalism [25][26][27][28][29] in the context of the 1D diffeomorphism (i.e., reparameterization) invariant theories of the non-SUSY (i.e., scalar) as well as SUSY (i.e., spinning) relativistic free particles.…”