BRST Symmetry Research Articles

Holomorphic vector field and topological sigma model on ℂP1 worldsheet

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold [Formula: see text] with potential terms induced from a Killing vector on [Formula: see text].3. One of the well-known fixed-point theorems is the Bott residue formula9 which represents the intersection number of Chern classes of holomorphic vector bundles on a Kähler manifold [Formula: see text] as the sum of contributions from fixed point sets of a holomorphic vector field [Formula: see text] on [Formula: see text]. In this paper, we derive the Bott residue formula by using the topological sigma model (A-model) that describes dynamics of maps from [Formula: see text] to [Formula: see text], with potential terms induced from the vector field [Formula: see text]. Our strategy is to restrict phase space of path integral to maps homotopic to constant maps. As an effect of adding a potential term to the topological sigma model, we are forced to modify the BRST symmetry of the original topological sigma model. Our potential term and BRST symmetry are closely related to the idea used in the paper by Beasley and Witten2 where potential terms induced from holomorphic section of a holomorphic vector bundle and corresponding supersymmetry are considered.

Higher-Genus Contributions to an Induced Supergravity Action

The higher-genus contributions to a supergravity action induced by the evaluation of the superstring path integral are considered. Three extra terms are found in the BRST transformation that would arise in the supergravity action. The supersymmetric BRST contour integral over the ideal boundary of an in.nite-genus surface is found to be vanishing if the harmonic measure is zero. The e.ect of a conformal transformation is demonstrated to be consistent with the vanishing of the BRST superspace contour integral. Since the commutator of BRST and conformal Since the BRST symmetry and conformal transformations transofrmations is proportional to the .rst current in four-dimensional gravity, and the vacuum is invariant under both transformations, conditions on the worldsheet .elds result from the dynamics in the embedding space. A connection between conformal models on the two-dimensional worldsheet and the quantum action of the gravitational theory in four dimensions is established through the ghost sector. The number of degrees of freedom in the gravity theory must be reduced to achieve this result. The introduction of new variables related to the embedding of two-dimensional surfaces in three-geometries and then fourmanifolds is then required to satisfy the condition of quantum consistency characteristic of conformal .eld theories.

Higgs potential from Weyl conformal gravity

We consider Weyl’s conformal gravity coupled to a complex matter field in Weyl geometry. It is shown that a Higgs potential naturally arises from a [Formula: see text] term in moving from the Jordan frame to the Einstein frame. A massless Nambu–Goldstone boson, which stems from spontaneous symmetry breakdown of the Weyl gauge invariance, is absorbed into the Weyl gauge field, thereby the gauge field becoming massive. We present a model where the gravitational interaction generates a Higgs potential whose form is a perfect square. Finally, we show that a theory in the Jordan frame is gauge-equivalent to the corresponding theory in the Einstein frame via the BRST formalism.

Matter-free higher spin gravities in 3D: Partially-massless fields and general structure

We study the problem of interacting theories with (partially)-massless and conformal higher spin fields without matter in three dimensions. A new class of theories that have partially-massless fields is found, which significantly extends the well-known class of purely massless theories. More generally, it is proved that the complete theory has to have a form of the flatness condition for a connection of a Lie algebra, which, provided there is a non-degenerate invariant bilinear form, can be derived from the Chern-Simons action. We also point out the existence of higher spin theories without the dynamical graviton in the spectrum. As an application of a more general statement that the frame-like formulation can be systematically constructed starting from the metric one by employing a combination of the local BRST cohomology technique and the parent formulation approach, we also obtain an explicit uplift of any given metric-like vertex to its frame-like counterpart. This procedure is valid for general gauge theories while in the case of higher spin fields in d-dimensional Minkowski space one can even use as a starting point metric-like vertices in the transverse-traceless gauge. In particular, this gives the fully off-shell lift for transverse-traceless vertices.

U(1) gauged model of FJ-type chiral boson based on Batalin–Fradkin–Vilkovisky formalism

The BRST quantization of the U(1) gauged model of FJ-type chiral boson for [Formula: see text] and [Formula: see text] are performed using the Batalin–Fradkin–Vilkovisky formalism. BFV formalism converts the second-class algebra into an effective first-class algebra with the help of auxiliary fields. Explicit expressions of the BRST charge, the involutive Hamiltonian, and the preserving BRST symmetry action are given and the full quantization has been carried through. For [Formula: see text], this Hamiltonian gives the gauge invariant Lagrangian including the well-known Wess–Zumino term, while for [Formula: see text] the corresponding Lagrangian has the additional new type of the Wess–Zumino term. The spectra in both cases have been analysed and the Wess–Zumino actions in terms of auxiliary fields are identified.

Massive Spinning Relativistic Particle: Revisited under BRST and Supervariable Approaches

We discuss the continuous and infinitesimal gauge, supergauge, reparameterization, nilpotent Becchi-Rouet-Stora-Tyutin (BRST), and anti-BRST symmetries and derive corresponding nilpotent charges for the one 0+1-dimensional (1D) massive model of a spinning relativistic particle. We exploit the theoretical potential and power of the BRST and supervariable approaches to derive the (anti-)BRST symmetries and coupled (but equivalent) Lagrangians for this system. In particular, we capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of the newly proposed (anti-)chiral supervariable approach (ACSA) to BRST formalism where only the (anti-)chiral supervariables (and their suitable super expansions) are taken into account along the Grassmannian direction(s). One of the novel observations of our present investigation is the derivation of the Curci-Ferrari- (CF-) type restriction by the requirement of the absolute anticommutativity of the (anti-)BRST charges in the ordinary space. We obtain the same restriction within the framework of ACSA to BRST formalism by (i) the symmetry invariance of the coupled Lagrangians and (ii) the proof of the absolute anticommutativity of the conserved and nilpotent (anti-)BRST charges. The observation of the anticommutativity property of the (anti-)BRST charges is a novel result in view of the fact that we have taken into account only the (anti-)chiral super expansions.

Restricted Weyl symmetry

We discuss the physics of a restricted Weyl symmetry in a curved space-time where a gauge parameter $\Omega(x)$ of Weyl transformation satisfies a constraint $\Box \Omega = 0$. First, we present a model of QED where we have a restricted gauge symmetry in the sense that a $U(1)$ gauge parameter $\theta(x)$ obeys a similar constraint $\Box \theta = 0$ in a flat Minkowski space-time. Next, it is precisely shown that a global scale symmetry must be spontaneously broken at the quantum level. Finally, we discuss the origin of the restricted Weyl symmetry and show that its symmetry can be derived from a full Weyl symmetry by taking a gauge condition $R = 0$ in the BRST formalism.

Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors

The full two-loop amplitudes for five massless states in Type II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The α′ → 0 limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type II supergravity. Investigations of the α′ expansion of the Type II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.

Infinitesimal Gribov copies in gauge-fixed topological Yang-Mills theories

We study the Gribov problem in four-dimensional topological Yang-Mills theories following the Baulieu-Singer approach in the (anti-)self-dual Landau gauges. This is a gauge-fixed approach that allows to recover the topological spectrum, as first constructed by Witten, by means of an equivariant (or constrained) BRST cohomology. As standard gauge-fixed Yang-Mills theories suffer from the gauge copy (Gribov) ambiguity, one might wonder if and how this has repercussions for this analysis. The resolution of the small (infinitesimal) gauge copies, in general, affects the dynamics of the underlying theory. In particular, treating the Gribov problem for the standard Landau gauge condition in non-topological Yang-Mills theories strongly affects the dynamics of the theory in the infrared. In the current paper, although the theory is investigated with the same gauge condition, the effects of the copies turn out to be completely different. In other words: in both cases, the copies are there, but the effects are very different. As suggested by the tree-level exactness of the topological model in this gauge choice, the Gribov copies are shown to be inoffensive at the quantum level. To be more precise, following Gribov, we discuss the path integral restriction to the Gribov horizon. The associated gap equation, which fixes the so-called Gribov parameter, is however shown to only possess a trivial solution, making the restriction obsolete. We relate this to the absence of radiative corrections in both gauge and ghost sectors. We give further evidence by employing the renormalization group which shows that, for this kind of topological model, the gap equation indeed forbids the introduction of a massive Gribov parameter.

A worldline theory for supergravity

The mathcal{N} = 4 supersymmetric spinning particle admits several consistent quantizations, related to the gauging of different subgroups of the SO(4) R-symmetry on the worldline. We construct the background independent BRST quantization for all of these choices which are shown to reproduce either the massless NS-NS spectrum of the string, or Einstein theory with or without the antisymmetric tensor field and/or dilaton corresponding to different restrictions. Quantum consistency of the worldline implies equations of motion for the background which, in addition to the admissible string backgrounds, admit Einstein manifolds with or whithout a cosmological constant. The vertex operators for the Kalb-Ramond, graviton and dilaton fields are obtained from the linear variations of the BRST charge. They produce the physical states by action on the diffeomorphism ghost states.

BRST Reduction of Quantum Algebras with $$^*$$-Involutions

In this paper we investigate the compatibility of the BRST reduction procedure with the Hermiticity of star products. First, we introduce the generalized notion of abstract BRST algebras with corresponding involutions. In this setting we define adjoint BRST differentials and as a consequence one gets new BRST quotients. Passing to the quantum BRST setting we show that for compact Lie groups the new quantum BRST quotient and the quantum BRST cohomology are isomorphic in zero degree implying that reduction is compatible with Hermiticity.

BRST cohomology of timelike Liouville theory

We compute the Hermitian sector of the relative BRST cohomology of the spacelike and timelike Liouville theories with generic real central charge cL in each case, coupled to a spacelike Coulomb gas and a generic transverse CFT. This paper is a companion of [1], and its main goal is to completely characterize the cohomology of the timelike theory with cL≤ 1 which was defined there. We also apply our formulas to revisit the BRST cohomology of the spacelike Liouville theory with cL> 1, which includes generalized minimal gravity. We prove a no-ghost theorem for the Hermitian sector in the timelike theory and for some spacelike models.

Landau-Khalatnikov-Fradkin transformations, Nielsen identities, their equivalence, and implications for QCD

The Landau-Khalatnikov-Fradkin transformations (LKFTs) represent an important tool for probing the gauge dependence of the correlation functions within the class of linear covariant gauges. Recently these transformations have been derived from first principles in the context of non-Abelian gauge theory (QCD) introducing a gauge invariant transverse gauge field expressible as an infinite power series in a Stueckelberg field. In this work we explicitly calculate the transformation for the gluon propagator, reproducing its dependence on the gauge parameter at the one loop level and elucidating the role of the extra fields involved in this theoretical framework. Later on, employing a unifying scheme based upon the BRST symmetry and a resulting generalized Slavnov-Taylor identity, we establish the equivalence between the LKFTs and the Nielsen identities which are also known to connect results in different gauges.

Superspace formulation in the absence of full anti-BRST symmetry

There are two known approaches to construct the six-dimensional superspace formulation of a gauge theory where the additional two dimensions are Grassmannian. One of the approaches (the second approach in this paper) requires both BRST and anti-BRST symmetries for the superspace formulation. In this paper, we consider an effective theory in a recently introduced quadratic gauge which has two pairs of ghosts. The theory does not have a full anti-BRST symmetry and only has full BRST symmetry. We show that it is possible to partially formulate this effective theory in the superspace using the second approach even in the absence of full anti-BRST symmetry due to the peculiar structure of the theory. We encounter a couple of technical difficulties which further brings useful insights.

BRST Cohomologies of Mixed and Second Class Constraints

The cohomological resolution of mixed constraints is constructed and shown to give Gupta–Bleuler space of physical states. The differential space and so-called anomalous BRST complex is constructed in detail. Special structures associated with anomaly are demonstrated and proved to be of important significance. Finally, the formalism is applied to a spinorial system with second class constraints. In this case, the Laplace operators are proved to define the (irreducibility) equations for fields carrying arbitrary (high) spin.

Lorentz symmetry breaking in supersymmetric quantum electrodynamics

Lorentz symmetry is one of the fundamental symmetries of nature; however, it can be broken by several proposals such as quantum gravity effects, low energy approximations in string theory and dark matter. In this paper, Lorentz symmetry is broken in supersymmetric quantum electrodynamics using aether superspace formalism without breaking any supersymmetry. To break the Lorentz symmetry in three-dimensional quantum electrodynamics, we must use the [Formula: see text] aether superspace. A new constant vector field is introduced and used to deform the deformed generator of supersymmetry. This formalism is required to fix the unphysical degrees of freedom that arise from the quantum gauge transformation required to quantize this theory. By using Yokoyama’s gaugeon formalism, it is possible to study these gaugeon transformations. As a result of the quantum gauge transformation, the supersymmetric algebra gets modified and the theory is invariant under BRST symmetry. These results could aid in the construction of the Gravity’s Rainbow theory and in the study of superconformal field theory. Furthermore, it is demonstrated that different gauges in this deformed supersymmetric quantum electrodynamics can be related to each other using the gaugeon formalism.

Ghostbusters: unitarity and causality of non-equilibrium effective field theories

: For a non-equilibrium physical system defined along a closed time path (CTP), a key constraint is the so-called largest time equation, which is a consequence of unitarity and implies causality. In this paper, we present a simple proof that if the propagators of a non-equilibrium effective action have the proper pole structure, the largest time equation is obeyed to all loop orders. Ghost fields and BRST symmetry are not needed. In particular, the arguments for the proof can also be used to show that if ghost fields are introduced, their contributions vanish.

Faddeev-Popov Ghost and BRST Symmetry in Yang-Mills Theory

Ghost fields arise from the quantization of the gauge field with constraints (gauge fixing) through the path integral method. By substituting a form of identity, an effective propagator will be obtained from the gauge field with constraints and this is called the Faddeev-Popov method. The Grassmann odd properties of the ghost field cause the gauge transformation parameter to be Grassmann odd, so a BRST transformation is defined. Ghost field emergence with Grassmann odd properties can also be obtained through the least action principle with gauge transformation, and thus the relations between the BRST transformation parameters and the ghost field is obtained.

Background Independence in Gauge Theories

Classical field theory is insensitive to the split of the field into a background configuration and a dynamical perturbation. In gauge theories, the situation is complicated by the fact that a covariant (w.r.t. the background field) gauge fixing breaks this split independence of the action. Nevertheless, background independence is preserved on the observables, as defined via the BRST formalism, since the violation term is BRST exact. In quantized gauge theories, however, BRST exactness of the violation term is not sufficient to guarantee background independence, due to potential anomalies. We define background-independent observables in a geometrical formulation as flat sections of the observable algebra bundle over the manifold of background configurations, with respect to a flat connection which implements background variations. A theory is then called background independent if such a flat (Fedosov) connection exists. We analyze the obstructions to preserve background independence at the quantum level for pure Yang–Mills theory and for perturbative gravity. We find that in the former case, all potential obstructions can be removed by finite renormalization. In the latter case, as a consequence of power-counting non-renormalizability, there are infinitely many non-trivial potential obstructions to background independence. We leave open the question whether these obstructions actually occur.

Chiral Schwinger model with Faddeevian anomaly and its BRST quantization

We consider chiral Schwinger model with Faddeevian anomaly, and carry out the quantization of both the gauge-invariant and non-invariant version of this model has been. Theoretical spectra of this model have been determined both in the Lagrangian and Hamiltonian formulation and a necessary correlation between these two are made. BRST quantization using BFV formalism has been executed which shows spontaneous appearance of Wess–Zumino term during the process of quantization. The gauge invariant version of this model in the extended phase space is found to map onto the physical phase space with the appropriate gauge fixing condition.