BRST Transformations Research Articles

Local superfield Lagrangian BRST quantization

A θ-local formulation of superfield Lagrangian quantization in non-Abelian hypergauges is proposed on the basis of an extension of general reducible gauge theories to special superfield models with a Grassmann parameter θ. We solve the problem of describing the quantum action and the gauge algebra of an L-stage-reducible superfield model in terms of a BRST charge for a formal dynamical system with first-class constraints of (L+1)-stage reducibility. Starting from θ-local functions of the quantum and gauge-fixing actions, with an essential use of Darboux coordinates on the antisymplectic manifold, we construct a superfield generating functionals of Green’s functions, including the effective action. We present two superfield forms of BRST transformations, considered as θ-shifts along vector fields defined by Hamiltonian-like systems constructed in terms of the quantum and gauge-fixing actions and an arbitrary θ-local boson function, as well as in terms of corresponding fermion functionals, through Poisson brackets with opposite Grassmann parities. The gauge independence of the S-matrix is proved. The Ward identities are derived. Connection is established with the BV method, the multilevel Batalin-Tyutin formalism, as well as with the superfield quantization scheme of Lavrov, Moshin, and Reshetnyak, extended to the case of general coordinates.

Comments on BRST quantization of strings

The BRST quantization of strings is revisited and the derivation of the path integral measure for scattering amplitudes is streamlined. Gauge invariances due to zero modes in the ghost sector are taken into account by using the Batalin-Vilkovisky formalism. This involves promoting the moduli of Riemann surfaces to quantum mechanical variables on which BRST transformations act. The familiar ghost and antighost zero mode insertions are recovered upon integrating out auxiliary fields. In contrast to the usual treatment, the gauge-fixed action including all zero mode insertions is BRST invariant. Possible anomalous contributions to BRST Ward identities due to boundaries of moduli space are reproduced in a novel way. Two models are discussed explicitly: bosonic string theory and topological gravity coupled to the topological A-model.

DISCRETIZED BRST TRANSFORMATIONS AND QUANTUM MECHANICAL POTENTIAL OF A FREE PARTICLE ON SD

We evaluated a quantum mechanical potential of a free particle on D-dimensional sphere. This system has the second-class constraints. We change the second-class constraints into first-class ones with new canonical variables. A BRST transformation is induced by the first-class constraints and is discretized. A discretized BRST invariant path integral is considered and the quantum mechanical potential is evaluated as R/12.

Characteristic classes of gauge systems

We define and study invariants which can be uniformly constructed for any gauge system. By a gauge system we understand an (anti-)Poisson supermanifold provided with an odd Hamiltonian self-commuting vector field called a homological vector field. This definition encompasses all the cases usually included into the notion of a gauge theory in physics as well as some other similar (but different) structures like Lie or Courant algebroids. For Lagrangian gauge theories or Hamiltonian first class constrained systems, the homological vector field is identified with the classical BRST transformation operator. We define characteristic classes of a gauge system as universal cohomology classes of the homological vector field, which are uniformly constructed in terms of this vector field itself. Not striving to exhaustively classify all the characteristic classes in this work, we compute those invariants which are built up in terms of the first derivatives of the homological vector field. We also consider the cohomological operations in the space of all the characteristic classes. In particular, we show that the (anti-)Poisson bracket becomes trivial when applied to the space of all the characteristic classes, instead the latter space can be endowed with another Lie bracket operation. Making use of this Lie bracket one can generate new characteristic classes involving higher derivatives of the homological vector field. The simplest characteristic classes are illustrated by the examples relating them to anomalies in the traditional BV or BFV-BRST theory and to characteristic classes of (singular) foliations.

The effective action for superfield Lagrangian quantization in reducible hypergauges

The rules of local superfield Lagrangian quantization in reducible non-Abelian hypergauge functions are formulated for an arbitrary gauge theory. The generating functionals of standard and vertex Green’s functions which depend on the Grassmann variable η via super(anti)fields and sources are constructed. The difference between the local quantum and the gauge fixing action determines an almost Hamiltonian system such that translations with respect to η along the solutions of this system define the superfield BRST transformations. The Ward identities are derived and the gauge independence of the S-matrix is proved.

Unitarity, Becchi-Rouet-Stora-Tyutin symmetry, and Ward identities in orbifold gauge theories

We discuss the use of BRST symmetry and the resulting Ward identities as consistency checks for orbifold gauge theories in an arbitrary number of dimensions. We demonstrate that both the usual orbifold symmetry breaking and the recently proposed Higgsless symmetry breaking are consistent with the nilpotency of the BRST transformation. The corresponding Ward identities for four-point functions of the theory engender relations among the coupling constants that are equivalent to the sum rules from tree level unitarity. We present the complete set of these sum rules also for inelastic scattering and discuss applications to six-dimensional models and to incomplete matter multiplets on orbifold fixed points.

BRST Symmetry and the Scale Transformation of Ghost Fields in Gauge Theory on NC Spacetime

BRST symmetry is very important to quantize gauge theory and to construct a covariant canonical formulation. It may also be important for gauge theory on noncommutative (NC) spacetime. The BRST symmetry of U(N) gauge theory on NC spacetime has recently been studied by Soroush, who manipulated the component fields of the gauge and fermion fields. We define the BRST transformations of the representation fields of gauge and fermion fields and show its nilpotency in a transparent way. The scale symmetry of ghost fields is also investigated. BRST charge and FP ghost charge are derived and shown to be the generators of the respective symmetries. As a byproduct, it is shown that Noether’s theorem is recovered on NC spacetime by redefining the current.

Nilpotent symmetries for QED in superfield formalism

In the framework of superfield approach, we derive the local, covariant, continuous and nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations on the U(1) gauge field (Aμ) and the (anti-)ghost fields ((C̄)C) of the Lagrangian density of the two (1+1)-dimensional QED by exploiting the (dual-)horizontality conditions defined on the four (2+2)-dimensional supermanifold. The long-standing problem of the derivation of the above symmetry transformations for the matter (Dirac) fields (ψ̄,ψ) in the framework of superfield formulation is resolved by a new set of restrictions on the (2+2)-dimensional supermanifold. These new physically interesting restrictions on the supermanifold owe their origin to the invariance of conserved currents of the theory. The geometrical interpretation for all the above transformations is provided in the framework of superfield formalism.

More on ghost condensation in Yang-Mills theory: BCS versus Overhauser effect and the breakdown of the Nakanishi-Ojima annex SL(2,Bbb R) symmetry

We analyze the ghost condensates , and in Yang-Mills theory in the Curci-Ferrari gauge. By combining the local composite operator formalism with the algebraic renormalization technique, we are able to give a simultaneous discussion of , and , which can be seen as playing the role of the BCS, respectively Overhauser effect in ordinary superconductivity. The Curci-Ferrari gauge exhibits a global continuous symmetry generated by the Nakanishi-Ojima (NO) algebra. This algebra includes, next to the (anti-)BRST transformation, a SL(2,R) subalgebra. We discuss the dynamical symmetry breaking of the NO algebra through these ghost condensates. Particular attention is paid to the Landau gauge, a special case of the Curci-Ferrari gauge.

Non-perturbative BRST invariance and what it might be good for

We construct a local, gauge-fixed, lattice Yang-Mills theory with an exact BRST invariance, and with the same perturbative expansion as the standard Yang-Mills theory. The ghost sector, and some of its BRST transformation rules, are modified to get around Neuberger's theorem. A special term is introduced in the action to regularize the Gribov horizons, and the limit where the regulator is removed is discussed. We conclude with a few comments on what might be the physical significance of this theory. We speculate that there may exist new strong-interaction phases apart from the anticipated confinement phase. (For additional technical details see ref. [1].)

On new forms of the BRST transformations

We show how to derive systematically new forms of the BRST transformations for a generic gauge fixed action. They arise after a symmetry of the gauge fixed action is found in the sector involving the Lagrange multiplier and its canonical momentum.

Chern-Simons in the Seiberg-Witten map for non-commutative Abelian gauge theories in 4D

A cohomological BRST characterization of the Seiberg-Witten (SW) map is given. We prove that the coefficients of the SW map can be identified with elements of the cohomology of the BRST operator modulo a total derivative. As an example, it will be illustrated how the first coefficients of the SW map can be written in terms of the Chern-Simons three form. This suggests a deep topological and geometrical origin of the SW map. The existence of the map for both Abelian and non-Abelian case is discussed. By using a recursive argument and the associativity of the $\star$-product, we shall be able to prove that the Wess-Zumino consistency condition for non-commutative BRST transformations is fulfilled. The recipe of obtaining an explicit solution by use of the homotopy operator is briefly reviewed in the Abelian case.

Algebraic Aspects of the Background Field Method

The background field method allows the evaluation of the effective action by exploiting the (background) gauge invariance, which in general yields Ward identities, i.e., linear relations among the vertex functions. In the present approach an extra gauge fixing term is introduced right at the beginning in the action and it is chosen in such a way that BRST invariance is preserved. The background effective action is considered and it is shown to satisfy both the Slavnov–Taylor (ST) identities and the Ward identities. This allows the proof of the background equivalence theorem with the standard techniques. In particular we consider a BRST doublet where the background field enters with a non-zero BRST transformation. The rationale behind the introduction of an extra gauge fixing term is that of removing the singularity of the Legendre transform of the background effective action, thus allowing the construction of the connected amplitudes generating functional Wbg. By using the relevant ST identities we show that the functional Wbg gives the same physical amplitudes as the original one we started with. Moreover we show that Wbg cannot in general be derived from a classical action by the Gell–Mann–Low formula. As a final point of the paper we show that the BRST doublet generated from the background field does not modify the anomaly of the original underlying gauge theory. The proof is algebraic and makes no use of arguments based on power-counting.

Test of the absence of kinetic terms around the tachyon vacuum in cubic string field theory

It has been conjectured that the bosonic open string theory around the non-perturbative tachyon vacuum has no open string dynamics at all. We explore, in the cubic open string field theory with level truncation approximation, the possibility that this conjecture is realized by the absence of kinetic terms of the string field fluctuations. We study the kinetic terms with two and four derivatives for the lower level scalar modes as well as their BRST transformation properties. The behavior of the coefficients of the kinetic terms in the neighborhood of the non-perturbative vacuum supports our expectation that the BRST invariant scalar component lacks its kinetic term.

Topologically massive non-abelian BF models in arbitrary space–time dimensions

This work extends to the D-dimensional space–time the topological mass generation mechanism of the non-abelian BF model in four dimensions. In order to construct the gauge invariant non-abelian kinetic terms for a ( D-2)-form B and a 1-form A, we introduce an auxiliary ( D-3)-form V. Furthermore, we obtain a complete set of BRST and anti-BRST transformation rules of the fields using the so-called horizontality condition, and construct a BRST/anti-BRST invariant quantum action for the model in D-dimensional space–time.

Higher order anomaly consistency conditions: renormalization and non-locality

We study the Slavnov–Taylor Identities (STI) breaking terms, up to the second order in perturbation theory. We investigate which requirements are needed for the first order Wess–Zumino consistency condition to hold true at the next order in perturbation theory. We find that: (a) If the cohomologically trivial contributions to the first order ST breaking terms are not removed by a suitable choice of the first order normalization conditions, the Wess–Zumino consistency condition is modified at the second order. (b) Moreover, if one fails to remove the cohomologically trivial part of the first order STI breaking local functional, the second order anomaly actually turns out to be a non-local functional of the fields and the external sources. By using the Wess–Zumino consistency condition and the Quantum Action Principle, we show that the cohomological analysis of the first order STI breaking terms can actually be performed also in a model (the massive Abelian Higgs–Kibble one) where the BRST transformations are not nilpotent.

BRST algebra on quantum bundles

A quantum analogue of the BRST algebra is given. The method is based on the construction of a differential algebra of generalized quantum forms carrying a bigraduation. This is realized over the base quantum space of a quantum vector bundle associated to a quantum principal bundle. Using this approach, we introduce the quantum gauge, ghost and matter fields via connections and sections. Imposing constraints on the curvatures leads to the quantum BRST transformations of these fields.

Quantum gauge symmetry from finite field dependent BRST transformations

Using the technique of finite field dependent BRST transformations we show that the classical massive Yang–Mills theory and the pure Yang–Mills theory whose gauge symmetry is broken by a gauge fixing term are identical from the view point of quantum gauge symmetry. The explicit infinitesimal transformations which leave the massive Yang–Mills theory BRST invariant are given.

Instanton calculus, topological field theories and N = 2 super Yang-Mills theories

The results obtained by Seiberg and Witten for the low-energy Wilsonian effective actions of N=2 supersymmetric theories with gauge group SU(2) are in agreement with instanton computations carried out for winding numbers one and two. This suggests that the instanton saddle point saturates the non-perturbative contribution to the functional integral. A natural framework in which corrections to this approximation are absent is given by the topological field theory built out of the N=2 Super Yang-Mills theory. After extending the standard construction of the Topological Yang-Mills theory to encompass the case of a non-vanishing vacuum expectation value for the scalar field, a BRST transformation is defined (as a supersymmetry plus a gauge variation), which on the instanton moduli space is the exterior derivative. The topological field theory approach makes the so-called "constrained instanton" configurations and the instanton measure arise in a natural way. As a consequence, instanton-dominated Green's functions in N=2 Super Yang-Mills can be equivalently computed either using the constrained instanton method or making reference to the topological twisted version of the theory. We explicitly compute the instanton measure and the contribution to $u=<\Tr \phi^2>$ for winding numbers one and two. We then show that each non-perturbative contribution to the N=2 low-energy effective action can be written as the integral of a total derivative of a function of the instanton moduli. Only instanton configurations of zero conformal size contribute to this result. Finally, the 8k-dimensional instanton moduli space is built using the hyperkahler quotient procedure, which clarifies the geometrical meaning of our approach.

Direct algebraic restoration of Slavnov-Taylor identities in the Abelian Higgs-Kibble model

A purely algebraic method is devised in order to recover Slavnov-Taylor identities (STI), broken by intermediate renormalization. The counterterms are evaluated order by order in terms of finite amplitudes computed at zero external momenta. The evaluation of the breaking terms of the STI is avoided and their validity is imposed directly on the vertex functional. The method is applied to the Abelian Higgs-Kibble model. We show that, since there are no anomalies, the imposition of the STI turns out to be equivalent to the solution of a linear problem. The presence of several invariants for the linearized ST operator S 0 implies that there are many possible solutions, corresponding to different normalization conditions. Moreover, we find more equations than unknowns (over-determined problem). This leads us to the consideration of consistency conditions, that must be obeyed if the restoration of STI is possible. It is suggested that the choice of the basis of counterterms (normalization conditions) and of the linearly independent equations (consistency conditions) can be of great help in actual calculations. An explicit mass term for the gauge field is introduced, in order to study the consistency conditions on general ground and in particular in the case where BRST transformations are not nilpotent.