Anti-BRST in the Causal Approach

It is known that the elimination of anomalies in all orders of perturbation theory is an open problem. The constraints given by usual invariance properties and the Wess–Zumino identities are not enough to eliminate the anomalies in the general case of a Yang–Mills theory. So, any new symmetry of the model could restrict further the anomalies and be a solution of the problem. We consider the anti-BRST transform of Ojima in the causal approach and investigate if such new restrictions are obtained. Unfortunately, the result is negative: if we have BRST invariance up to the second order of perturbation theory, we also have anti-BRST invariance up to the same order. Probably, this result is true in all orders of perturbation theory. So, anti-BRST transform gives nothing new, and we have to find other ideas to restrict and eventually eliminate the anomalies for a general Yang–Mills theory.

BRST in the exact renormalization group

We show, explicitly within perturbation theory, that the quantum master equation and the Wilsonian renormalization group flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naïvely break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov–Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov–Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to the choice of the renormalization scheme.

Radiative corrections in 2 + 1 dimensional supergravity

Supergravity in 2 + 1 dimensions has a set of first-class constraints that result in two bosonic and one fermionic gauge invariances. When one uses Faddeev–Popov quantization, these gauge invariances result in four fermionic scalar ghosts and two bosonic Majorana spinor ghosts. The BRST invariance of the effective Lagrangian is found. As an example of a radiative correction, we compute the phase of the one-loop effective action in the presence of a background spin connection, and show that it vanishes. This indicates that unlike a spinor coupled to a gauge field in 2 + 1 dimensions, there is no dynamical generation of a topological mass in this model. An additional example of how a BRST invariant effective action can arise in a gauge theory is provided in Appendix B where the BRST effective action for the classical Palatini action in 1 + 1 dimensions is examined.

(Anti-)chiral supervariable approach to nilpotent and absolutely anticommuting conserved charges of reparametrization invariant theories: A couple of relativistic toy models as examples

We exploit the potential and power of the Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST invariant restrictions on the (anti-)chiral supervariables to derive the proper nilpotent (anti-)BRST symmetries for the reparametrization invariant one (0[Formula: see text]+[Formula: see text]1)-dimensional (1D) toy models of a free relativistic particle as well as a free spinning (i.e. supersymmetric) relativistic particle within the framework of (anti-)chiral supervariable approach to BRST formalism. Despite the (anti-)chiral super expansions of the (anti-)chiral supervariables, we observe that the (anti-)BRST charges, for the above toy models, turn out to be absolutely anticommuting in nature. This is one of the novel observations of our present endeavor. For this proof, we utilize the beauty and strength of Curci–Ferrari (CF)-type restriction in the context of a spinning relativistic particle but no such restriction is required in the case of a free scalar relativistic particle. We have also captured the nilpotency property of the conserved charges as well as the (anti-)BRST invariance of the appropriate Lagrangian(s) of our present toy models within the framework of (anti-)chiral supervariable approach.

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

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.

One-loop polarization operator of the quantum gauge superfield for 𝒩 = 1 SYM regularized by higher derivatives

We consider the general [Formula: see text] supersymmetric gauge theory with matter, regularized by higher covariant derivatives without breaking the BRST invariance, in the massless limit. In the [Formula: see text]-gauge we obtain the (unrenormalized) expression for the two-point Green function of the quantum gauge superfield in the one-loop approximation as a sum of integrals over the loop momentum. The result is presented as a sum of three parts: the first one corresponds to the pure supersymmetric Yang–Mills theory in the Feynman gauge, the second one contains all gauge-dependent terms, and the third one is the contribution of diagrams with a matter loop. For the Feynman gauge and a special choice of the higher derivative regulator in the gauge fixing term, we analytically calculate these integrals in the limit [Formula: see text]. In particular, in addition to the leading logarithmically divergent terms, which are determined by integrals of double total derivatives, we also find the finite constants.

Gauge fixing invariance and anti-BRST symmetry

It is shown that anti-BRST invariance in quantum gauge theories can be considered as the quantized version of the symmetry of classical gauge theories with respect to different gauge fixing mechanisms.

Geometry and Quantum Dynamics

A geometrical derivation of Abelian and non- Abelian gauge theories. The Faddeev–Popov quantisation. BRST invariance and ghost fields. General discussion of BRST symmetry. Application to Yang–Mills theories and general relativity. A brief history of gauge theories.

Axial symmetry, anti-BRST invariance, and modified anomalies

It is shown that, anti-BRST symmetry is the quantized counterpart of local axial symmetry in gauge theories. An extended form of descent equations is worked out, which yields a set of modified consistent anomalies.