Anti Chiral Superfield Approach to BRST Formalism

Abstract

The Becchi-Rouet-Stora-Tyutin (BRST) formalism [1-4] is a mathematically rich and physically intuitive approach to covariantly quantize the gauge and/or diffeomorphism invariant theories which are endowed with the quantum (anti-)BRST symmetry transformations that are off-shell/on-shell nilpotent of order two and absolutely anticommuting in nature. The on-shell nilpotency property encodes the fermionic (i.e. supersymmetric) nature of the (anti-)BRST symmetry transformations where the equation of motion (EOMs) are invoked for its proof. On the other hand, the off-shell nilpotent (anti-)BRST symmetry transformations are automatically fermionic in nature without any help from the EOMs. This fermionic nature ensures that the (anti-)BRST symmetry transformations change the fermionic fields/variables into bosonic fields/variables and vice-versa. This nilpotency property is also connected with the nilpotency of the exterior derivative of differential geometry [5-10]. The absolute anticommutativity property of the (anti-)BRST symmetry transformations demonstrates their independent identity and linear-independence. In other words, the quantum BRST and anti-BRST symmetry transformations both exist corresponding to a given local classical gauge/diffeomorphism symmetry transformation. In exactly similar fashion, we have two nilpotent (i.e. off-shell as well as on-shell) N = 2 supersymmetric (SUSY) symmetry transformations in the context of N = 2 SUSY quantum mechanical models and/or N = 2 SUSY field theory. However, it has been found that the N = 2 nilpotent SUSY symmetries are not absolutely anticommuting in nature. Thus, there is a distinct difference between the two nilpotent symmetries of the (anti-)BRST symmetries kinds and the other two nilpotent symmetries of the N = 2 SUSY theories (as far as the absolute anticommutativity property is concerned). In fact, the anticommutator of the two N = 2 SUSY transformations generate the spacetime translations of the fields/variables on which it operates unlike the anticommutator of the (anti-)BRST symmetry transformations which turns out to be always zero. In other words, the absolute anticommutativity property of the BRST and anti-BRST symmetry transformations is very sacrosanct.