BRST Quantization Of Gauge Theories Research Articles

BRST quantization and equivariant cohomology: localization with asymptotic boundaries

We develop BRST quantization of gauge theories with a soft gauge algebra on spaces with asymptotic boundaries. The asymptotic boundary conditions are imposed on background fields, while quantum fluctuations about these fields are described in terms of quantum fields that vanish at the boundary. This leads us to construct a suitable background field formalism that is generally applicable to soft gauge algebras, and therefore to supergravity. We define a nilpotent BRST charge that acts on both the background and the quantum fields, as well as on the background and quantum ghosts.When the background is restricted to be invariant under a residual isometry group, the background ghosts must be restricted accordingly and play the role of the parameters of the background isometries. Requiring in addition that the background ghosts will be BRST invariant as well then converts the BRST algebra into an equivariant one. The background fields and ghosts are then invariant under the equivariant transformations while the quantum fields and ghosts transform under both the equivariant and the background transformations. We demonstrate how this formalism is suitable for carrying out localization calculations in a large class of theories, including supergravity defined on asymptotic backgrounds that admit supersymmetry.

Becchi–Rouet–Stora–Tyutin formalism and zero locus reduction

In the Becchi–Rouet–Stora–Tyutin (BRST) quantization of gauge theories, the zero locus ZQ of the BRST differential Q carries an (anti)bracket whose parity is opposite to that of the fundamental bracket. Observables of the BRST theory are in a 1:1 correspondence with Casimir functions of the bracket on ZQ. For any constrained dynamical system with the phase space N0 and the constraint surface Σ, we prove its equivalence to the constrained system on the BFV-extended phase space with the constraint surface given by ZQ. Reduction to the zero locus of the differential gives rise to relations between bracket operations and differentials arising in different complexes (the Gerstenhaber, Schouten, Berezin–Kirillov, and Sklyanin brackets); the equation ensuring the existence of a nilpotent vector field on the reduced manifold can be the classical Yang–Baxter equation. We also generalize our constructions to the bi-QP manifolds which from the BRST theory viewpoint correspond to the BRST–anti-BRST-symmetric quantization.

BRST-invariant Lagrangian of spontaneously broken gauge theories in a noncommutative geometry.

The quantization of spontaneously broken gauge theories in noncommutative geometry(NCG) has been sought for some time, because quantization is crucial for making the NCG approach a reliable and physically acceptable theory. Lee, Hwang and Ne'eman recently succeeded in realizing the BRST quantization of gauge theories in NCG in the matrix derivative approach proposed by Coquereaux et al. The present author has proposed a characteristic formulation to reconstruct a gauge theory in NCG on the discrete space $M_4\times Z_{_N}$. Since this formulation is a generalization of the differential geometry on the ordinary manifold to that on the discrete manifold, it is more familiar than other approaches. In this paper, we show that within our formulation we can obtain the BRST invariant Lagrangian in the same way as Lee, Hwang and Ne'eman and apply it to the SU(2)$\times$U(1) gauge theory.

BRST quantization of gauge theory in noncommutative geometry: Matrix derivative approach

The Becchi–Rouet–Stora–Tyutin (BRST) quantization of a gauge theory in noncommutative geometry is carried out in the ‘‘matrix derivative’’ approach. BRST/anti-BRST transformation rules are obtained by applying the horizontality condition, in the superconnection formalism. A BRST/anti-BRST invariant quantum action is then constructed, using an adaptation of the method devised by Baulieu and Thierry-Mieg for the Yang–Mills case. The resulting quantum action turns out to be the same as that of a gauge theory in the ’t Hooft gauge with spontaneously broken symmetry. Our result shows that only the even part of the supergroup acts as a gauge symmetry, while the odd part effectively provides a global symmetry. We treat the general formalism first, then work out the SU(2/1) and SU(2/2) cases explicitly.

BRST Quantization of Gauge Theory on M_4×Z_N without Recourse to Noncommutative Geometry

BRST Quantization of Gauge Theory on M_4×Z_N without Recourse to Noncommutative Geometry

BRST invariance and Poincaré-Cartan forms

BRST quantization of gauge theories is considered on the basis of Poincare-Cartan forms in the framework of global variational calculus on fibred manifolds. It is pointed out that the BRST transformation admits a jet-prolonged formulation which is in fact a ≪generalized symmetry≫ of the BRST-invariant Lagrangian (a well-known concept in the framework of integrable systems). We show that the Poincare-Cartan forms are invariant under this extended BRST transformation only modulo contact terms, which vanish on all sections anda fortiori on solutions of field equations.

BRST quantization of gauge theories like SL(2,R) on inner product spaces

Some general formulas are derived for the solutions of a BRST quantization on inner product spaces of finite dimensional bosonic gauge theories invariant under arbitrary Lie groups. A detailed analysis is then performed of SL(2,R) invariant models and some possible geometries of the Lagrange multipliers are derived together with explicit results for a class of SL(2,R) models. Gauge models invariant under a nonunimodular gauge group are also studied in some detail.

BRST quantization of gauge theories in hamiltonian-like gauges

BRST quantization of gauge theories in hamiltonian-like gauges

Extended BRST quantization of gauge theories in the generalized canonical formalism

The rules of canonical quantization of gauge theories are formulated on the basis of the extended BRST symmetry principle. The existence of solutions of the generating equations of the gauge algebra is proved. Equivalence between the extended BRST quantization and the standard method of generalized canonical quantization is established. Ward identities corresponding to invariance of a theory under the extended BRST symmetry are obtained.