BRST Gauge Fixing Research Articles

Quantum instability of gauge theories on κ -Minkowski space

We consider a gauge theory on the 5-d $\kappa$-Minkowski which can be viewed as the noncommutative analog of a $U(1)$ gauge theory. We show that the Hermiticity condition obeyed by the gauge potential $A_\mu$ is necessarily twisted. Performing a BRST gauge-fixing with a Lorentz-type gauge, we carry out a first exploration of the one loop quantum properties of this gauge theory. We find that the gauge-fixed theory gives rise to a non-vanishing tadpole for the time component of the gauge potential, while there is no non-vanishing tadpole 1-point function for the spatial components of $A_\mu$. This signals that the classical vacuum of the theory is not stable against quantum fluctuations. Possible consequences regarding the symmetries of the gauge model and the fate of the tadpole in other gauges of non-covariant type are discussed.

Long, partial-short, and special conformal fields

In the framework of metric-like approach, totally symmetric arbitrary spin bosonic conformal fields propagating in flat space-time are studied. Depending on the values of conformal dimension, spin, and dimension of space-time, we classify all conformal field as long, partial-short, short, and special conformal fields. An ordinary-derivative (second-derivative) Lagrangian formulation for such conformal fields is obtained. The ordinary-derivative Lagrangian formulation is realized by using double-traceless gauge fields, Stueckelberg fields, and auxiliary fields. Gauge-fixed Lagrangian invariant under global BRST transformations is obtained. The gauge-fixed BRST Lagrangian is used for the computation of partition functions for all conformal fields. Using the result for the partition functions, numbers of propagating D.o.F for the conformal fields are also found.

Light-Front Hamiltonian, Path Integral and BRST Formulations of the Chern-Simons-Higgs Theory in the Broken Symmetry Phase

In the present work we study the Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory in two-space one-time dimensions, in the so-called broken symmetry phase of the Higgs potential (where the phase φ(xμ) of the complex matter field Φ(xμ) carries the charge degree of freedom of the complex matter field and is akin to the Goldstone boson) on the light-front (i.e., on the hyperplanes defined by the fixed light-cone time). The theory is seen to possess a set of first-class constraints and the local vector gauge symmetry. The theory being gauge-invariant is quantized under appropriate gauge-fixing conditions. The explicit Hamiltonian and path integral quantization is achieved under the above light-cone gauges. The Heisenberg equations of motion of the system are derived for the physical degrees of freedom of the system. Finally the BRST quantization of the system is achieved under appropriate BRST gauge-fixing, where the BRST symmetry is maintained even under the BRST light-cone gauge-fixing.

A BRST gauge-fixing procedure for Yang–Mills theory on sphere

A gauge-fixing procedure for the Yang–Mills theory on an n-dimensional sphere (or a hypersphere) is discussed in a systematic manner. We claim that Adler's gauge-fixing condition used in massless Euclidean QED on a hypersphere is not conventional because of the presence of an extra free index, and hence is unfavorable for the gauge-fixing procedure based on the BRST invariance principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge condition, which is proved to be equivalent to a generalization of Adler's condition, we apply the BRST gauge-fixing procedure to the Yang–Mills theory on a hypersphere to obtain consistent results. Field equations for the Yang–Mills field and associated fields are derived in manifestly O(n+1) covariant or invariant forms. In the large radius limit, these equations reproduce the corresponding field equations defined on the n-dimensional flat space.

Remarks on gauge fixing and BRST quantization of noncommutative gauge theories

We consider the BRST gauge fixing procedure of the noncommutative Yang-Mills theory and of the gauged U(N) Proca model. An extended Seiberg-Witten map involving ghosts, antighosts and auxiliary fields for non-Abelian gauge theories is studied. We find that the extended map behaves differently for these models. For the Yang-Mills theory in the Lorentz gauge it was not possible to find a map that relates the gauge conditions in the noncommutative and ordinary theories. For the gauged Proca model we found a particular map relating the unitary gauge fixings in both formulations.

Comments on the gauge fixed BRST cohomology and the quantum Noether method

We discuss in detail the relation between the gauge fixed and gauge invariant BRST cohomology. We showed previously that in certain gauges some cohomology classes of the gauge-fixed BRST differential do not correspond to gauge invariant observables. We now show that in addition “accidental” conserved currents may appear. These correspond one-to-one to observables that become trivial in this gauge. We explicitly show how the gauge-fixed BRST cohomology appears in the context of the quantum Noether method.

BRST hamiltonian for bulk quantized gauge theory

By treating the bulk-quantized Yang-Mills theory as a constrained system we obtain a consistent gauge-fixed BRST hamiltonian in the minimal sector. This provides an independent derivation of the 5-d lagrangian bulk action. The ground state is independent of the (anti)ghosts and is interpreted as the solution of the Fokker-Planck equation, thus establishing a direct connection to the Fokker-Planck hamiltonian. The vacuum state correlators are shown to be in agreement with correlators in lagrangian 5-d formulation. It is verified that the complete propagators remain parabolic in one-loop dimensional regularization.

Cohomological analysis of gauge-fixed gauge theories

The relation between the gauge-invariant local BRST cohomology involving the antifields and the gauge-fixed BRST cohomology is clarified. It is shown in particular that the cocycle conditions become equivalent once it is imposed, on the gauge-fixed side, that the BRST cocycles should yield deformations that preserve the nilpotency of the (gauge-fixed) BRST differential. This shows that the restrictions imposed on local counterterms by the quantum Noether condition in the Epstein–Glaser construction of gauge theories are equivalent to the restrictions imposed by BRST invariance on local counterterms in the standard Lagrangian approach.

BRST gauge fixing and the algebra of global supersymmetry

Global supersymmetry (SUSY) in supersymmetric gauge theory is generally broken by gauge fixing. A prescription to extract physical information from such SUSY algebra broken by gauge fixing is analyzed in the path integral framework. If δ SUSY δ BRST Ψ = δ BRST δ SUSY Ψ for a gauge fixing “fermion” Ψ the SUSY charge density is written as a sum of the piece which is naively expected without gauge fixing and a BRST exact piece. If δ SUSY δ SUSY δ BRST = δ BRST δ SUSY δ SUSY Ψ, the equal-time anti-commutator of SUSY charge is written as a sum of a physical piece and a BRST exact piece. We illustrate these properties for N = 1 and N = 2 supersymmetric Yang-Mills theories and for a D = 10 massive superparticle (or “D-particle”) where the κ-symmetry provides extra complications.

BRST formulation of 4-monopoles

A supersymmetric gauge-invariant action is constructed over any four-dimensional Riemannian manifold describing Witten's theory of 4-monopoles. The topological supersymmetric algebra closes off-shell. The multiplets include the auxiliary fields and the Wess - Zumino fields in an unusual way, arising naturally from BRST gauge fixing. A new canonical approach over Riemann manifolds is followed, using a Morse function as a Euclidean time and taking into account the BRST boundary conditions that come from the BFV formulation. This allows a construction of the effective action starting from gauge principles.

On the gauge-fixed BRST cohomology

A crucial property of the standard antifield-BRST cohomology at non-negative ghost number is that any cohomological class is completely determined by its antifield independent part. In particular, a BRST cocycle that vanishes when the antifields are set equal to zero is necessarily exact. This property, which follows from the standard theorems of homological perturbation theory, holds not only in the algebra of local functions, but also in the space of local functionals. The present paper stresses how important it is that the antifields in question be the usual antifields associated with the gauge invariant description. By means of explicit counterexamples drawn from the free Maxwell-Klein-Gordon system, we show that the property does not hold, in the case of local functionals, if one replaces the antifields of the gauge invariant description by new antifields adapted to the gauge fixation. In terms of these new antifields, it is not true that a local functional weakly annihilated by the gauge-fixed BRST generator determines a BRST cocycle; nor that a BRST cocycle which vanishes when the antifields are set equal to zero is necessarily exact.

BRST gauge fixing and regularization

In the presence of consistent regulators, the standard procedure of BRST gauge fixing (or moving from one gauge to another) can require non-trivial modifications. These modifications occur at the quantum level, and gauges exist which are only well-defined when quantum mechanical modifications are correctly taken into account. We illustrate how this phenomenon manifests itself in the solvable case of two-dimensional bosonization in the path integral formalism. As a by-product, we show how to derive smooth bosonization in Batalin-Vilkovisky Lagrangian BRST quantization.

On the consistency of BRST gauge fixing of constrained systems

It is shown that in the context of a mixed Lagrangian-Hamiltonian formulation of gauge theories which result from transformation to collective and fluctuation coordinates it is essential to check the consistency of the chosen BRST gauge fixing with the BRST algebra. Various gauge fixing conditions are considered in the context of a simple model. The prototype model considered here is also shown to possess certain residual BRST and gauge invariances.

Topological conformal gravity in four dimensions

In this paper, we present a new formulation of topological conformal gravity in four dimensions. Such a theory was first considered by Witten as a possible gravitational counterpart of topological Yang-Mills theory, but several problems left it incomplete. The key in our approach is to realise a theory which describes deformations of conformally self-dual gravitational instantons. We first identify the appropriate elliptic complex which does precisely this. By applying the Atiyah-Singer index theorem, we calculate the number of independent deformations of a given gravitational instanton which preserve its self-duality. We then quantise topological conformal gravity by BRST gauge-fixing, and discover how the quantum theory is naturally described by the above complex. Indeed, it is process which closely parallels that of the Yang-Mills theory, and we show how the partition function generates an uncanny gravitational analogue of the first Donaldson invariant.

On the twisted G/H topological models

The twisted G/H models are constructed as twisted supersymmetric gauged WZW models. We analyze the case of G = SU( N), H = SU( N 1) × … × SU( N n ) × U(1) r with rank(G) = rank(H), and discuss possible generalizations. We introduce a non-abelian bosonization of the (1, 0) ghost system in the adjoint of H and in G/H. By computing chiral anomalies in the latter picture we write the quantum action as a decoupled sum of “matter”, gauge and ghost sectors. The action is also derived in the unbosonized version. We invoke a free field parametrization and extract the space of physical states by computing the cohomology of Q, the sum of the BRST gauge-fixing charge and the twisted supersymmetry charge. For a given G we briefly discuss the relation between the various G/H models corresponding to different choices of H. The choice H = G corresponds to the topological G/G theory.

FREE FIELD REPRESENTATION OF THE N=3 SUPERCONFORMAL ALGEBRA FROM HAMILTONIAN REDUCTION OF osp(3|2) AFFINE LIE SUPERALGEBRA

We have constructed the free field representation of the N=3 superconformal algebra with an arbitrary value of the central charge c by using Hamiltonian reduction of the osp (3|2) Lie superalgebra without BRST gauge-fixing procedure.

Conformal integrable field theory from WZNW via quantum Hamiltonian reduction

A new class of gauged WZNW models has been proposed by O'Raifeartaigh. It describes classically a series of conformal integrable field theories including the conformal Toda field theory as a special case. We quantize a constrained WZNW model (the gauged WZNW model with all connections fixed to zero) in the path-integral and BRST gauge-fixing framework and derive the quantum version of the integrable field theories of O'Raifeartaigh by redefining bosonic ghosts for them to be purely local gauge degrees of freedom.

BRST cohomology and BRST gauge fixing

We discuss some general aspects of BRST cohomology. We investigate the BRST gauge-fixing procedure based on the harmonic gauges in the ghost-extended quantum mechanical Hilbert space. We obtain a general classification of states according to their BRST transformation properties. On the basis of these results we find necessary and sufficient conditions for the positivity of the physical state space and establish their equivalence with another set of conditions introduced by Spiegelglas. We apply the method to a system with a non-semisimple, solvable algebra of first-class constraints, which appears in the theory of massless higher spins. We find that both the Dirac and Gupta-Bleuler quantization procedures are contained within the BRST approach, but they correspond to different cohomology classes.

Gauge fixing and renormalization in topological quantum field theory

We show how Witten's topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.

A gauge invariant action in topological quantum field theory

A gauge invariant lagrangian whose BRST gauge fixing corresponds to the theory recently proposed by Witten in the framework of topological quantum field theory is presented. The solutions to the field equations corresponding to this lagrangian consist of self-dual Yang-Mills fields.