Becchi-Rouet-Stora-Tyutin Transformations Research Articles

Quantization of Weyl invariant unimodular gravity with antisymmetric ghost fields

The enforcement of the unimodularity condition in a gravity theory by means of a Lagrange multiplier leads, in general, to inconsistencies upon quantization. This is so, in particular, when the classic linear splitting of the metric between the background and quantum fields is used. To avoid the need of introducing such a Lagrange multiplier while using the classic linear splitting, we carry out the quantization of unimodular gravity with extra Weyl symmetry by using Becchi–Rouet–Stora–Tyutin (BRST) techniques. Here, two gauge symmetries are to be gauge-fixed: transverse diffeomorphisms and Weyl transformations. We perform the gauge-fixing of the transverse diffeomorphism invariance by using BRST transformations that involve antisymmetric ghost fields. We show that these BRST transformations are compatible with the BRST transformations needed to gauge-fix the Weyl symmetry, so that they can be combined in a set of transformations generated by a single BRST operator. Newton’s law of gravitation is derived within the BRST formalism we put forward as well as the Slavnov–Taylor equation.

FLPR model: Nilpotent (anti-)co-BRST symmetries

We demonstrate the existence of a set of novel off-shell nilpotent and absolutely anticommuting continuous symmetry transformations, within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism, which are over and above the usual off-shell nilpotent and absolutely anticommuting (anti-)BRST transformations that are respected by the quantum version of the classical first-order Lagrangian for the Friedberg-Lee-Pang-Ren (FLPR) model that describes the motion of a single non-relativistic particle of unit mass (moving under the influence of a general rotationally invariant spatial two-dimensional potential). We christen these novel set of fermionic transformations as the (anti-)co-BRST transformations because the gauge-fixing terms remain invariant under them. We derive the conserved and off-shell nilpotent (anti-)BRST and (anti-)co-BRST charges and comment on the physicality criteria with respect to them where we establish the presence of the operator forms of the first-class constraints (of the original classical gauge theory) at the quantum level which is consistent with the Dirac-quantization conditions.

Modified Stückelberg Formalism: Free Massive Abelian 2-Form Theory in 4D

We demonstrate that the celebrated Stückelberg formalism is modified in the case of a massive four (3 + 1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism where there is the existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual Stückelberg-modified massive Abelian 2-form gauge theory. The modification in the Stückelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the Stückelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick of using the equations of motion to remove the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent.

Nilpotent symmetries of a modified massive Abelian 3-form theory: Augmented superfield approach

We derive the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations for any arbitrary D-dimensional Stückelberg-modified massive Abelian 3-form theory within the framework of augmented version of superfield approach (AVSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism where, in addition to the horizontality condition (HC), we exploit the theoretical strength of the gauge invariant restriction (GIR) to deduce the proper transformations for the gauge, associated (anti-)ghost fields, auxiliary fields, Stückelberg compensating field, etc. In fact, it is an elegant and delicate combination of HC and GIR (within the ambit of AVSA) that is crucial for all our discussions and derivations. One of the highlights of our present endeavour is the deduction of a new set of (anti-)BRST invariant Curci-Ferrari (CF)-type restrictions which are not found in the massless version of our present theory where only the HC plays an important role in the derivations of all the (anti-)BRST transformations and a very specific set of CF-type restrictions. The alternative ways of the derivation of the full set of the latter, from various theoretical considerations, are also interesting results of our present investigation.

Superfield Approaches to a Model of Bosonic String: Curci-Ferrari-Type Restrictions

Exploiting the theoretical potential of the modified Bonora-Tonin superfield approach (MBTSA) as well as the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete set of off-shell nilpotent (anti-)BRST symmetry transformations corresponding to the classical two- 1 + 1 dimensional (2D) diffeomorphism symmetry transformations on the worldsheet (that is traced out by the motion of a model of bosonic string). Only the BRST symmetry transformations for this model have been discussed in the earlier literature. We derive the (anti-)BRST invariant Curci-Ferrari- (CF-) type restrictions (using MBTSA) which turn out to be the root cause behind the absolute anticommutativity of the above (anti-)BRST symmetry transformations. We capture the symmetry invariance of the (anti-)BRST invariant Lagrangian densities within the ambit of ACSA. The derivation of the proper anti-BRST transformations (corresponding to the already-known BRST transformations) and the (anti-)BRST invariant CF-type restrictions are the novel results in our present endeavor.

Nilpotent Symmetries of a Model of 2D Diffeomorphism Invariant Theory: BRST Approach

Within the framework of the Becchi-Rouet-Stora-Tyutin (BRST) formalism, we discuss the full set of proper BRST and anti-BRST transformations for a 2D diffeomorphism invariant theory which is described by the Lagrangian density of a standard bosonic string. The above (anti-)BRST transformations are off-shell nilpotent and absolutely anticommuting. The latter property is valid on a submanifold of the space of quantum fields where the 2D version of the universal (anti-)BRST invariant Curci-Ferrari (CF) type of restrictions is satisfied. We derive the precise forms of the BRST and anti-BRST invariant Lagrangian densities as well as the exact expressions for the conserved (anti-)BRST and ghost charges. The lucid derivation of the proper anti-BRST symmetry transformations and the emergence of the CF-type restrictions are completely novel results for our present bosonic string which has already been discussed earlier in literature where only the BRST symmetry transformations have been pointed out. We briefly mention the derivation of the CF-type restrictions from the modified version of the Bonora-Tonin superfield approach, too.

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

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 [Formula: see text]-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.

Hamiltonian BRST formalism for gauge fields on black hole spacetimes

We investigate the Becchi-Rouet-Stora-Tyutin (BRST) formalism for gauge theories on spherically symmetric black hole spacetimes, with or without a cosmological constant ($\Lambda\geq0$). This is illustrated through the example of scalar electrodynamics. We first demonstrate that the horizons contribute additional surface terms to the Gauss law constraint of the theory when gauge transformations are not required to vanish on the horizons. We then consider the BRST invariant path integral including these surface terms following the Hamiltonian BRST formalism. We fix a radiation-like gauge which involves null components of the electromagnetic field at the horizons. We find that the presence of the surface terms in the constraint forces the ghost and gauge fixing actions to include additional surface integrals at the horizons. The null combination of the gauge fields at the horizons is shown to modify the ghost number charge of the theory through additional terms at the horizons. We also demonstrate how one can construct a gauge-fixing fermion which generates its own nilpotent symmetry transformations, called co-BRST transformations, that leave the theory invariant. The BRST and co-BRST transformations are further used to identify dressed (gauge invariant) fields of theory, whose dressings are affected by the presence of Killing horizons. We conclude with a discussion of potential applications of our results in soft limits and thermal field theories on black hole backgrounds.

Nilpotent Symmetries in Jackiw-Pi Model: Augmented Superfield Approach

We derive the complete set of off-shell nilpotent (s^2_{(a)b} = 0) and absolutely anticommuting (s_b s_{ab} + s_{ab} s_b = 0) Becchi-Rouet-Stora-Tyutin (BRST) (s_b) as well as anti-BRST symmetry transformations (s_{ab}) corresponding to the combined Yang-Mills and non-Yang-Mills symmetries of the (2 + 1)-dimensional Jackiw-Pi model within the framework of augmented superfield formalism. The absolute anticommutativity of the (anti-)BRST symmetries is ensured by the existence of two sets of Curci-Ferrari (CF) type of conditions which emerge naturally in this formalism. The presence of CF conditions enables us to derive the coupled but equivalent Lagrangian densities. We also capture the (anti-)BRST invariance of the coupled Lagrangian densities in the superfield formalism. The derivation of the (anti-)BRST transformations of the auxiliary field \rho is one of the key findings which can neither be generated by the nilpotent (anti-)BRST charges nor by the requirements of the nilpotency and/or absolute anticommutativity of the (anti-)BRST transformations. Finally, we provide a bird's-eye view on the role of auxiliary field for various massive models and point out few striking similarities and some glaring differences among them.

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

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.

Self-dual chiral boson: augmented superfield approach

We exploit the standard tools and techniques of the augmented version of Bonora-Tonin (BT) superfield formalism to derive the off-shell nilpotent and absolutely anticommuting (anti-)BRST and (anti-)co-BRST symmetry transformations for the Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density of a self-dual bosonic system. In the derivation of the full set of the above transformations, we invoke the (dual-)horizontality conditions, (anti-)BRST and (anti-)co-BRST invariant restrictions on the superfields that are defined on the (2, 2)-dimensional supermanifold. The latter is parameterized by the bosonic variable x^\mu\,(\mu = 0,\, 1) and a pair of Grassmanian variables \theta and \bar\theta (with \theta^2 = \bar\theta^2 = 0 and \theta\bar\theta + \bar\theta\theta = 0). The dynamics of this system is such that, instead of the full (2, 2) dimensional superspace coordinates (x^\mu, \theta, \bar\theta), we require only the specific (1, 2)-dimensional super-subspace variables (t, \theta, \bar\theta) for its description. This is a novel observation in the context of superfield approach to BRST formalism. The application of the dual-horizontality condition, in the derivation of a set of proper (anti-)co-BRST symmetries, is also one of the new ingredients of our present endeavor where we have exploited the augmented version of superfield formalism which is geometrically very intuitive.

Dual-BRST symmetry: 6D Abelian 3-form gauge theory

Within the framework of the Becchi–Rouet–Stora–Tyutin (BRST) formalism, we demonstrate the existence of the novel off-shell nilpotent (anti-)dual-BRST symmetries in the context of a six (5+1)-dimensional (6D) free Abelian 3-form gauge theory. Under these local and continuous symmetry transformations, the total gauge-fixing term of the Lagrangian density remains invariant. This observation should be contrasted with the off-shell nilpotent (anti-)BRST symmetry transformations, under which, the total kinetic term of the theory remains invariant. The anticommutator of the above nilpotent (anti-)BRST and (anti-)dual-BRST transformations leads to the derivation of a bosonic symmetry in the theory. There exists a discrete symmetry transformation in the theory which provides a thread of connection between the nilpotent (anti-)BRST and (anti-)dual-BRST transformations. This theory is endowed with a ghost-scale symmetry, too. We discuss the algebra of these symmetry transformations and show that the structure of the algebra is reminiscent of the algebra of de Rham cohomological operators of differential geometry.

TOPOLOGICALLY MASSIVE NON-ABELIAN THEORY: SUPERFIELD APPROACH

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.

Notoph gauge theory: Superfield formalism

We derive absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for the 4D free Abelian 2-form gauge theory by exploiting the superfield approach to BRST formalism. The antisymmetric tensor gauge field of the above theory was christened as the “notoph” (i.e. the opposite of “photon”) gauge field by Ogievetsky and Palubarinov way back in 1966–67. We briefly outline the problems involved in obtaining the absolute anticonimutativity of the (anti-) BRST transformations and their resolution within the framework of geometrical superfield approach to BRST formalism. One of the highlights of our results is the emergence of a Curci-Ferrari type of restriction in the context of 4D Abelian 2-form (notoph) gauge theory which renders the nilpotent (anti-) BRST symmetries of the theory to be absolutely anticommutative in nature.

Off-shell nilpotent finite Becchi–Rouet–Stora–Tyutin (BRST)/anti-BRST transformations

We develop the off-shell nilpotent finite field dependent Becchi–Rouet–Stora–Tyutin (BRST) transformations and show that for different choices of the finite field dependent parameter these transformations connect the generating functionals corresponding to different effective theories. We also construct both on-shell and off-shell finite field dependent anti-BRST transformations for Yang–Mills theories and show that these transformations play the similar role in connecting different generating functionals of different effective theories. Analogous to the finite field dependent BRST transformations, the nontrivial Jacobians of the path integral measure which arise due to the finite field dependent anti-BRST transformations are responsible for the new results. We consider several explicit examples in each case to demonstrate the results.

BRST, anti-BRST and their geometry

We continue the comparison between the field theoretical and geometrical approaches to the gauge field theories of various types, by deriving their Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST transformation properties and comparing them with the geometrical properties of the bundles and gerbes. In particular, we provide the geometrical interpretation of the so-called Curci–Ferrari conditions that are invoked for the absolute anticommutativity of the BRST and anti-BRST symmetry transformations in the context of non-Abelian one-form gauge theories as well as the Abelian gauge theory that incorporates a two-form gauge field. We also carry out the explicit construction of the three-form gauge fields and compare it with the geometry of 2-gerbes.

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

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 xμand 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.

Quantized gauge-affine gravity in the superfiber bundle approach

The quantization of gauge-affine gravity within the superfiber bundle formalism is proposed. By introducing an even pseudotensorial 1-superform over a principal superfibre bundle with superconnection, we obtain the geometrical Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST transformations of the fields occurring in such a theory. Reducing the four-dimensional general affine group double-covering to the Poincare group double-covering we also find the BRST and anti-BRST transformations of the fields present in Einstein's gravity. Furthermore, we give a prescription leading to the construction of both BRST-invariant gauge-fixing action for gauge-affine gravity and Einstein's gravity.

Two-form gauge fields in the Becchi–Rouet–Stora–Tyutin superspace

We consider a geometrical description for tensor gauge fields. Based on this geometrical treatment, we develop the theory involving one- and two-form gauge fields by means of the Becchi–Rouet–Stora–Tyutin (BRST) superfield formalism. This permits us to directly obtain invariant Lagrangians for both BRST and anti-BRST transformations and we shall see that all the ingredients of the formalism (ghosts, ghost-for-ghosts and all the auxiliary fields) naturally occur. We introduce collective fields to construct the field–antifield quantum action in a generic gauge. We deal with both Abelian and non-Abelian cases. In this last case, the BRST superspace formulation sheds more light on this still open problem.

Consistent Fujikawa regularization of the spinning string in superspace from Pauli-Villars regularization.

We use Pauli-Villars (PV) regularization of the spinning string in superspace to identify those Fujikawa regulators which produce consistent anomalies. We use a general background gauge, and obtain, after elimination of the nonpropagating fields and using a super-Einstein and Lorentz-invariant mass term, a diagonal Fujikawa regulator with only super-Weyl anomalies. The PV extended action has extra twisted'' Becchi-Rouet-Stora-Tyutin (BRST) symmetries; dropping these, the BRST transformations of the PV fields turn into off-shell superconformal transformations plus field equations. Thus, the transformations of superconformal field theory are just on-shell BRST transformations.