9412228

8/14/2019 9412228

1/196

a r X i v : h e p - t h / 9 4 1 2 2 2 8 v 1 2 8 D e c 1 9 9 4

CCNY-HEP-94/03KUL-TF-94/12

UB-ECM-PF 94/15UTTG-11-94

hep-th/9412228May 1994

Antibracket, Antieldsand Gauge-Theory Quantization

Joaquim Gomis 1, Jordi Pars 2 and Stuart Samuel 3

Theory Group, Department of PhysicsThe University of Texas at Austin

RLM 5208, Austin, Texasand

Departament dEstructura i Constituents de la Materia Facultat de Fsica, Universitat de Barcelona

Diagonal 647, E-08028 Barcelona

Catalonia

Instituut voor Theoretische Fysica Katholieke Universiteit Leuven

Celestijnenlaan 200D B-3001 Leuven, Belgium

Department of PhysicsCity College of New York

138th St and Convent AvenueNew York, New York 10031 U.S.A.

1 Permanent address: Dept. dEstructura i Constituents de la Materia, U. Barcelona.E-mail: [email protected] Wetenschappelijk Medewerker, I. I. K. W., Belgium.E-mail: jordi=paris%tf%[email protected] E-mail: [email protected]
http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228v1http://arxiv.org/abs/hep-th/9412228http://arxiv.org/abs/hep-th/9412228http://arxiv.org/abs/hep-th/9412228v1

8/14/2019 9412228

2/196

Abstract

The antibracket formalism for gauge theories, at both the classical andquantum level, is reviewed. Gauge transformations and the associatedgauge structure are analyzed in detail. The basic concepts involved inthe antibracket formalism are elucidated. Gauge-xing, quantum effects,and anomalies within the eld-antield formalism are developed. Theconcepts, issues and constructions are illustrated using eight gauge-theorymodels.

8/14/2019 9412228

3/196

Contents

1 Introduction 2

2 Structure of the Set of Gauge Transformations 102.1 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Irreducible and Reducible Gauge Theories . . . . . . . . . . . . . . . 142.3 Trivial Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . 172.4 The Gauge Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Examples of Gauge Theories 243.1 The Spinless Relativistic Particle . . . . . . . . . . . . . . . . . . . . 24

3.2 Yang-Mills Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Topological Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . 293.4 The Antisymmetric Tensor Field Theory . . . . . . . . . . . . . . . . 313.5 Abelian p-Form Theories . . . . . . . . . . . . . . . . . . . . . . . . . 333.6 Open Bosonic String Field Theory . . . . . . . . . . . . . . . . . . . . 343.7 The Massless Relativistic Spinning Particle . . . . . . . . . . . . . . . 383.8 The First-Quantized Bosonic String . . . . . . . . . . . . . . . . . . . 44

4 The Field-Antield Formalism 484.1 Fields and Antields . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 The Antibracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Classical Master Equation and Boundary Conditions . . . . . . . . . 514.4 The Proper Solution and the Gauge Algebra . . . . . . . . . . . . . . 544.5 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 564.6 The Classical BRST Symmetry . . . . . . . . . . . . . . . . . . . . . 57

5 Examples of Proper Solutions 595.1 The Spinless Relativistic Particle . . . . . . . . . . . . . . . . . . . . 595.2 Yang-Mills Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3 Topological Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . 625.4 The Antisymmetric Tensor Field Theory . . . . . . . . . . . . . . . . 625.5 Abelian p-Form Theories . . . . . . . . . . . . . . . . . . . . . . . . . 625.6 Open String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 635.7 The Massless Relativistic Spinning Particle . . . . . . . . . . . . . . . 665.8 The First-Quantized Bosonic String . . . . . . . . . . . . . . . . . . . 66

8/14/2019 9412228

4/196

6 The Gauge-Fixing Fermion 686.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2 Gauge-Fixing Auxiliary Fields . . . . . . . . . . . . . . . . . . . . . . 716.3 Delta-Function Gauge-Fixing Procedure . . . . . . . . . . . . . . . . 746.4 Other Gauge-Fixing Procedures . . . . . . . . . . . . . . . . . . . . . 836.5 Gauge-Fixed Classical BRST Symmetry . . . . . . . . . . . . . . . . 856.6 The Gauge-Fixed Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Gauge-Fixing Examples 897.1 The Spinless Relativistic Particle . . . . . . . . . . . . . . . . . . . . 897.2 Yang-Mills Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.3 Topological Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . 917.4 The Antisymmetric Tensor Field Theory . . . . . . . . . . . . . . . . 937.5 Open String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 947.6 The Massless Relativistic Spinning Particle . . . . . . . . . . . . . . . 967.7 The First-Quantized Bosonic String . . . . . . . . . . . . . . . . . . . 97

8 Quantum Effects and Anomalies 1018.1 Quantum-BRST Transformation and Its Cohomology . . . . . . . . . 1018.2 Satisfying the Quantum Master Equation . . . . . . . . . . . . . . . . 1048.3 Remarks on Renormalization . . . . . . . . . . . . . . . . . . . . . . . 105

8.4 The Effective Action and the Zinn-Justin Equation . . . . . . . . . . 1068.5 Quantum Master Equation Violations: Generalities . . . . . . . . . . 1108.6 Canonical Transformations and the Quantum

Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.7 The Anomaly at the One-Loop Level . . . . . . . . . . . . . . . . . . 114

9 Sample Anomaly Calculations 1199.1 Computation for the Spinless Relativistic Particle . . . . . . . . . . . 1199.2 The Abelian Chiral Schwinger Model . . . . . . . . . . . . . . . . . . 1239.3 Anomaly in the Open Bosonic String . . . . . . . . . . . . . . . . . . 126

10 Brief Discussion of Other Topics 13410.1 Applications to Global Symmetries . . . . . . . . . . . . . . . . . . . 13410.2 A Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . 13510.3 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13710.4 Cohomological Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 13810.5 Equivalence with the Hamiltonian BFV Formalism . . . . . . . . . . 14510.6 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8/14/2019 9412228

5/196

10.7 The Antibracket Formalism in General Coordinates . . . . . . . . . . 15310.8 The D=26 Closed Bosonic String Field Theory . . . . . . . . . . . . . 15510.9 Extended Antibracket Formalism for

Anomalous Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . 159

A Appendix: Right and Left Derivatives 162

B Appendix: The Regularity Condition 164

C Appendix: Anomaly Trace Computations 166

References 171

8/14/2019 9412228

6/196

This work is dedicated to Joseph and Marie,to Pilar,

and to the memory of Pere and Francesca.

8/14/2019 9412228

7/196

J. Gomis, J. Pars and S. Samuel Antibracket, Antields and . . . 2

1 Introduction

The known fundamental interactions of nature are all governed by gauge theo-ries. The presence of a gauge symmetry indicates that a theory is formulated in aredundant way, in which certain local degrees of freedom do not enter the dynamics.Conversely, when there are degrees of freedom, which do not enter the lagrangian, atheory possesses local invariances. Although one can in principle eliminate the gaugedegrees of freedom, there are reasons for not doing so. These reasons include manifestcovariance, locality of interactions, and calculational convenience.

The rst example of a gauge theory was electrodynamics. Electric and magneticforces are generated via the exchange of photons. Being particles of spin 1, photons

involve a vector eld, A

. However, not all four components of the electromagneticpotential A enter dynamically. Two degrees of freedom correspond to the two pos-sible physical polarizations of the photon. The longitudinal degree of freedom playsa role in interactions via virtual exchanges of photons. The remaining gauge degreeof freedom does not enter the theory. Consequently, electromagnetism is describedby a gauge theory. When it was realized that the weak interactions could be uniedwith electromagnetism in an SU (2) U (1) gauge theory [129, 266, 213] and thatthis theory is renormalizable [ 243, 244], the importance of non-abelian gauge theo-ries [276] grew enormously. The strong interactions are also governed by an SU(3)non-abelian gauge theory. The fourth fundamental force is gravity. It is based on

Einsteins general theory of relativity and uses general coordinate invariance. Whenformulated in terms of a metric or any other convenient elds, gravity also possessesgauge symmetries.

The quantization of gauge theories is not always straightforward. In the abeliancase, relevant for electromagnetism, the procedure is well understood. In contrast,quantization of a non-abelian theory and its renormalization is more complicated.Quantization generally involves the introduction of ghost elds. Typically, a gauge-xing procedure is used to render dynamical all degrees of freedom. Ghost eldsare used to compensate for the effects of the gauge degrees of freedom [101], so thatunitarity is preserved. In electrodynamics in the linear gauges, ghosts decouple andcan be ignored. In non-abelian gauge theories, convenient gauges generically involveinteracting ghosts. A major step in understanding these issues was the Faddeev-Popov quantization procedure [ 98, 83], which relied heavily on the functional-integralapproach to quantization [ 102, 1, 165]. From this viewpoint, the presence of ghostelds is understood as a measure effect. In dividing out the volume of gaugetransformations in function space, a Jacobian measure factor arises. This factor isproduced naturally by introducing quadratic terms in the lagrangian for ghosts andthen integrating them out. It was realized at a later stage that the gauge-xed action

8/14/2019 9412228

8/196


retains a nilpotent, odd, global symmetry involving transformations of both elds andghosts. This Becchi-Rouet-Stora-Tyutin (BRST) symmetry [ 36, 254] is what remainsof the original gauge invariance. In fact, for closed theories, the transformation lawfor the original elds is like a gauge transformation with gauge parameters replacedby ghost elds. In general, this produces nonlinear transformation laws. The relationsamong correlation functions derived from BRST symmetry involve the insertions of the BRST variation of elds. These facts require the use of composite operators andit is convenient to introduce sources for these transformations. The Ward identities[265] associated with the BRST invariance treated in this way are the Slavnov-Tayloridentities [233, 241]. The Slavnov-Taylor identities and BRST symmetry have playedan important role in quantization, renormalization, unitarity, and other aspects of

gauge theories.Ghosts elds have been useful throughout the development of covariant gauge-

eld-theory quantization [ 181, 182, 184, 205, 3]. It is desirable to have a formulationof gauge theories that introduces them from the outset and that automatically in-corporates BRST symmetry [ 32]. The eld-antield formulation has these features[36, 277, 24, 25, 26, 27]. It relies on BRST symmetry as fundamental principle anduses sources to deal with it [36, 254, 277]. It encompasses previous ideas and develop-ments for quantizing gauge systems and extends them to more complicated situations(open algebras, reducible systems, etc.) [ 113, 114, 172, 238, 81]. In 1975, J. Zinn-Justin, in his study of the renormalization of Yang-Mills theories [ 277], introducedthe above-mentioned sources for BRST transformations and a symplectic structure( , ) (actually denoted by him) in the space of elds and sources, He expressed theSlavnov-Taylor identities in the compact form ( , ) = 0, where , the generatingfunctional of the one-particle-irreducible diagrams, is known as the effective action(see also [187]). These ideas were developed further by B. L. Voronov and I. V. Tyutinin [263, 264] and by I. A. Batalin and G. A. Vilkovisky in refs.[ 24, 25, 26, 27, 28]. Theseauthors generalized the role of ( , ) and of the sources for BRST transformations andcalled them the antibracket and antields respectively. Due to their contributions,this quantization procedure is often referred to as the Batalin-Vilkovisky formalism.

The antibracket formalism gained popularity among string theorists, when it wasapplied to the open bosonic string eld theory [ 56, 246]. It has also proven quiteuseful for the closed string eld theory and for topological eld theories. Only withinthe last few years has it been applied to more general aspects of quantum eld theory.

In some sense, the BRST approach, which was driven, in part, by renormalizationconsiderations, and the eld-antield formalism, which was motivated by classicalconsiderations such as gauge structure, are not so different. When sources are in-troduced for BRST transformations, the BRST approach resembles the eld-antield

8/14/2019 9412228

9/196


one. Antields, then, have a simple intepretation: They are the sources for BRSTtransformations. In this sense, the eld-antield formalism is a general method fordealing with gauge theories within the context of standard eld theory.

The general structure of the antibracket formalism is as follows. One introducesan antield for each eld and ghost, thereby doubling the total number of originalelds. The antibracket ( , ) is an odd non-degenerate symplectic form on the spaceof elds and antields. The original classical action S 0 is extended to a new actionS , in an essentially unique way, to arrive at a theory with manifest BRST symmetry.One equation, the master equation ( S, S ) = 0, reproduces in a compact way thegauge structure of the original theory governed by S 0. Although the master equationresembles the Zinn-Justin equation, the content of the two is different since S is a

functional of quantum elds and antields and is a functional of classical elds.The antibracket formalism currently appears to be the most powerful method for

quantizing a gauge theory. Beyond tree level, order h terms usually need to be addedto the action, thereby leading to a quantum action W . These counterterms are ex-pected to render nite loop contributions, after a suitable regularization procedurehas been introduced. The master equation must be appropriately generalized to theso-called quantum master equation. It involves a potentially singular operator .The regularization procedure and counterterms should also render and its actionon W well-dened. Violations of the quantum master equation are equivalent togauge anomalies [251]. To calculate correlation functions and scattering amplitudesin perturbation theory, a gauge-xing procedure is selected. This procedure elimi-nates antields in terms of functionals of elds. When appropriately implemented,propagators exist, and the usual Feynman graph methods can be used. In addition,for the study of symmetry properties, renormalization and anomalies, a modied ver-sion of the gauge-xing procedure is available which keeps antields. In short, theantibracket formalism has manifest gauge invariance or BRST symmetry, providesthe extra elds needed for covariant quantization, permits a perturbative expansionof the quantum theory, and allows the study of quantum corrections to the symmetrystructure of the theory.

The eld-antield formalism can treat systems that cannot be handled by Faddeev-Popov functional integration approach. This is particularly clear for theories in whichquartic ghost interactions arise [ 172, 81]. Faddeev-Popov quantization leads to anaction bilinear in ghost elds, and fails for the case of open algebras. An openalgebra occurs when the commutator of two gauge transformations produces a termproportional to the equations of motion and not just another gauge transformation[81, 27]. In other words, the gauge algebra closes only on-shell. Such algebras occurin gravity [110] and supergravity [114, 172, 81, 258] theories. The ordinary Faddeev-

8/14/2019 9412228

10/196


Popov procedure also does not work for reducible theories. In reducible theories, thegauge generators are all not independent [ 67, 171, 6, 37, 228, 250, 260, 80, 242]. Somemodications of the procedure have been developed by introducing ghosts for ghosts[228, 178]. However, these modications [228, 148, 178, 116] do not work for thegeneral reducible theory. Even for Yang-Mills theories, the Faddeev-Popov procedurecan fail, if one considers exotic gauge-xing procedures for which extraghosts appear[172, 199, 200]. The eld-antield formalism is sufficiently general to encompasspreviously known lagrangian approaches to the quantization of gauge theories.

Perhaps the most attractive feature of the eld-antield formalism is its imita-tion of a hamiltonian Poisson structure in a covariant way. In some instances, thehamiltonian approach to quantization has the advantage of being manifestly unitary.

However, it is necessarily non-covariant since the time variable is treated in a mannerdifferent from the space variables. In addition, the gauge invariances usually must bexed at the outset. In compensation for this, one needs to impose constraints on theHilbert space of states. In the eld-antield approach, the antibracket plays the roleof the Poisson bracket. As a consequence, hamiltonian concepts, such as canonicaltransformations, can be formulated and used [ 262, 263, 264, 27, 105, 251]. At thesame time, manifest covariance and BRST invariance are maintained. Since the an-tibracket formalism proceeds via the functional integral, the powerful techniques of functional integration are available.

A non-trivial aspect of the eld-antield approach is the construction of the quan-tum action W . When loop effects are ignored, W S provides the solution to themaster equation. A straightforward but not necessarily simple procedure is availablefor obtaining S given the classical action S 0 and its gauge invariances for a nite-reducible system. When quantum effects are incorporated, W must satisfy the moresingular quantum master equation. However, there is currently no known method thatguarantees the construction of W . The problem is that the eld-antield formalismdoes not automatically provide the functional integration measure. These issues arelinked with those associated with unitarity, renormalization, quantum gauge invari-ance, and anomalies. Because these aspects of gauge theories are inherently difficult,

it is not surprising that the eld-antield formalism does not provide a simple solu-tion.Another, less serious weakness, is that the antibracket formalism involves quite

a bit of mathematical machinery. Sometimes, a gauge theory is expressed in a formwhich is more complicated than necessary. This can make computations somewhatmore difficult.

The organization of this article is as follows. Sect. 2 discusses gauge structure.Some notation is presented during the process of introducing gauge transformations.

8/14/2019 9412228

11/196


The distinction between irreducible and reducible gauge theories is made. The latterinvolve a redundant set of gauge invariances so that there are relations among thegauge generators. As a result, there exists gauge invariances for gauge invariances,and ghosts for ghosts. A theory is Lth-stage reducible if there are gauge invariancesfor the gauge invariances for the gauge invariances, etc., L-fold times. The generalform of the gauge structure for a rst-stage reducible case is determined. In Sect.3, specic gauge theories are presented to illustrate the concepts of Sect. 2. Thespinless relativistic particle, non-abelian Yang-Mills theories, topological Yang-Millstheory, the antisymmetric tensor eld, free abelian p-form theories, open bosonicstring eld theory, the massless relativistic spinning particle, and the rst-quantizedbosonic string are treated. The spinless relativistic particle of Sect. 3.1 is also used to

exemplify notation. The massless relativistic spinning particle provides an example of a simple supergravity theory, namely a theory with supersymmetric gauge invariances.This system is used to illustrate the construction of supersymmetric and supergravitytheories. A review of the construction of general-coordinate-invariant theories is givenin the subsection on the rst-quantized open bosonic string. These mini-reviewsshould be useful to the reader who is new to these subjects.

The key concepts of the eld-antield formalism are elucidated in Sect. 4. An-tields are introduced and the antibracket is dened. The latter is used to denecanonical transformations. They can be quite helpful in simplifying computations.Next, the classical master equation ( S, S ) = 0 is presented. When appropriate bound-ary conditions are imposed, it reproduces, in a compact way, the gauge structure of Sect. 2. A suitable action S satisfying the master equation is called a proper solution.Given the gauge-structure tensors of a rst-stage reducible theory, Sect. 4.4 presentsthe generic proper solution. The last part of Sect. 4 denes and discusses the classicalBRST symmetry. Examples of proper solutions are provided in Sect. 5 for the gaugeeld theories presented in Sect. 3.

Sect. 6 begins the passage from the classical to the quantum aspects of theeld-antield formalism. The gauge-xing procedure is discussed. The gauge-xingfermion is a key concept. It is used as a means of eliminating antields in terms

of functions of elds. The result is an action that is suitable for use in the pathintegral. Only in this context and in performing standard perturbative computationsare antields eliminated. It is shown that results are independent of the choice of , if the quantum action W satises the quantum master equation. To implementgauge-xing, more elds and their antields must be introduced. How this worksfor irreducible and rst-stage reducible theories is treated rst. Then, for referencepurposes, the general Lth-stage reducible case is considered. Delta-function typegauge-xing is treated in Sect. 6.3. Again, irreducible and rst-stage reducible cases

8/14/2019 9412228

12/196


are presented rst. Again, for reference purposes, the general Lth-stage reduciblecase is treated. Gauge-xing by a gaussian averaging process is discussed in Sect.6.4. After gauge-xing, a classical gauge-xed BRST symmetry can be dened. SeeSect. 6.5. The freedom to perform canonical transformations permits one to work inany appropriate eld basis. This freedom can be quite useful. Concepts tend to havedifferent interpretations in different bases. One basis, associated with and calledthe gauge-xed basis, is the last topic of Sect. 6. Examples of gauge-xing proceduresare provided in Sect. 7. With the exception of the free p-form theory, the theories arethe ones considered in Sects. 3 and 5.

Quantum effects and possible gauge anomalies are analyzed in Sect. 8. Thekey concepts are quantum-BRST transformations and the quantum master equa-

tion. Techniques for assisting in nding solutions to the quantum master equationare provided in Sects. 8.2, 8.5 and 8.6. The generating functional for one-particle-irreducible diagrams is generalized to the eld-antield case in Sect. 8.4. This allowsone to treat the quantum system in a manner similar to the classical system. TheZinn-Justin equation is shown to be equivalent to the quantum master equation.When unavoidable violations of the latter occur, the gauge theory is anomalous. SeeSect. 8.5. Explicit formulas at the one-loop level are given. In Sect. 9, sample anomalycalculations are presented. It is shown that the spinless relativistic particle does nothave an anomaly. In Sect. 9.2, the eld-antield treatment of the two-dimensionalchiral Schwinger model is presented. Violations of the quantum master equation areobtained. This is expected since the theory is anomalous. A similar computationis performed for the open bosonic string. For D = 26, the theory is anomalous, asexpected. Some of the details of the calculations are relegated to Appendix C.

Section 10 briey presents several additional topics. The application of the eld-antield formalism to global symmetries is presented. A review is given of the geo-metric interpretation of E. Witten [ 273]. The next topic is the role of locality. Thissomewhat technical issue is important for renormalizability and for cohomologicalaspects. A summary of cohomological methods is given. Next, the relation betweenthe hamiltonian and antibracket approaches is discussed. The question of unitarity

is the subject of Sect. 10.6. One place where the eld-antield formalism has playedan essential role is in the D = 26 closed bosonic string eld theory. This example israther complicated and not suitable for pedagogical purposes. Nevertheless, generalaspects of the antibracket formalism for the closed string eld theory are discussed.Finally, an overview is given of how to handle anomalous systems using an extendedset of elds and antields.

Appendix A reviews the mathematical aspects of left and right derivatives, inte-gration by parts, and chain rules for differentiation. Appendix B discusses in more

8/14/2019 9412228

13/196


detail the regularity condition, which is a technical requirement of the antibracketformalism.

At every stage of development of the formalism, there exists some type of BRSToperator. In the space of elds and antields before quantization, a classical nilpo-tent BRST transformation B is dened by using the action S and the antibracket:B F = ( F, S ). From B , a gauge-xed version B is obtained by imposing the condi-tions on antields provided by the gauge-xing fermion . At the quantum level, aquantum version B of B emerges. In the context of the effective action formulation,a transformation B cq , acting on classical elds, can be dened by using in lieu of S .Several subsections are devoted to the BRST operator, its properties and its utility.

The existence of a BRST symmetry is crucial to the development. Observables are

those functionals which are BRST invariant and cannot be expressed as the BRSTvariation of something else. In other words, observables correspond to the elementsof the BRST cohomology. The nilpotency of B and B are respectively equivalentto the classical and quantum master equations. The traditional treatment of gaugetheories using BRST invariance is reviewed in [32]. For this reason, we do not discussBRST quantization in detail.

The antibracket formalism is rather versatile in that one can use any set of elds(and antields) related to the original elds (and antields) by a canonical transfor-mation. However, under such a change, the meaning of certain concepts change. Forexample, the gauge structure, as determined by the master equation, has a differentinterpretation in the gauge-xed basis than in the original basis. Most of this reviewuses the second viewpoint. The treatment in the gauge-xed basis is handled in Sects.6.6 and 8.4.

The material in each section strives to fulll one of three purposes. A key purposeis to present computations that lead to understanding and insight. Sections 2, 4, 6.1,6.5, 6.6, 8 and 10 are mainly of this character. The second purpose is pedagogical.This Introduction falls into this category in that it gives a quick overview of theformalism and the important concepts. Sections 3, 5, 7, and 9 analyze specially chosengauge theories which allow the reader to understand the eld-antield formalism in

a concrete manner. Finally, some material is included for technical completeness.Sections 6.26.4 present methods for gauge-xing the generic gauge theory. Parts of sections 2.2, 2.4 and 8.7 are also for reference purposes. Probably the reader shouldnot initially try to read these sections in detail. Many sections serve a dual role.

A few new results on the antibracket formalism are presented in this review.They are included because they provide insight for the reader. We have tried tohave a minimum overlap with other reviews. In particular, cohomological aspects arecovered in [36, 89, 57, 32, 93, 8, 152, 157, 253] and more-detailed aspects of anomalies

8/14/2019 9412228

14/196


are treated in [ 253]. Pedagogical treatments are given in references [ 157, 253]. Incertain places, material from reference [ 206] has been used.

This review focuses on the key points and concepts of antibracket formalism.There is some emphasis on applications to string theory. Our format is to rstpresent the material abstractly and then to supply examples. The reader who isnew to this subject and mainly interested in learning may wish to reverse this order.Exercises can be generated by verifying the abstract results in each of the samplegauge theories of Sects. 3, 5, 7, and 9. Other systems, which have been treatedby eld-antield quantization and may be of use to the reader, are the free spin 52eld [26], the spinning string [136], the 10-dimensional Brink-Schwarz superparticleand superstring [ 123, 139, 173, 190, 211, 43, 232, 44, 227]1, chiral gravity [77], W 3

gravity [161, 45, 74, 162, 72, 257], general topological eld theories [66, 185, 50, 49,127, 164, 191, 158, 79], the supersymmetric Wess-Zumino model [ 33, 159] and chiralgauge theories in four-dimensions [251]. The antibracket formalism has found variousinterpretations in mathematics [ 125, 126, 127, 209, 189, 198, 218, 219, 237]. Someother recent relevant work can be found in [ 261, 21, 22, 47, 147]. The referencing inthis review is thorough but not complete. A restriction has been made to only citeworks directly relevant to the issues addressed in each section. Multiple referencesare done rst chronologically and then alphabetically. The titles of references areprovided to give the reader a better indication of the content of each work.

We work in Minkowski space throughout this article. Functional integrals aredened by analytic continuation using Wick rotation. This is illustrated in the com-putations of Appendix C. We use to denote the at-space metric with the signa-ture convention ( 1, 1, 1, . . . , 1). Flat-space indices are raised and lowered with thismetric. The epsilon tensor 0 1 2 ... d 1 is determined by the requirement that it beantisymmetric in all indices and that 012...d 1 = 1, where d1 and d are respectivelythe dimension of space and space-time. We often use square brackets to indicate afunctional of elds and antields to avoid confusion with the antibracket, i.e., S [, ]in lieu of S (, ).

1 See [227] for additional references.

8/14/2019 9412228

15/196


2 Structure of the Set of Gauge Transformations

The most familiar example of a gauge structure is the one associated with anon-abelian Yang-Mills theory [ 276], namely a Lie group. The commutator of twoLie-algebra generators produces a Lie-algebra generator. When a basis is used, thiscommutator algebra is determined by the structure constants of the Lie group. Forexample, for the Lie algebra su (2) there are three generators and the structure con-stant is the anti-symmetric tensor on three indices . A commutator algebra, asdetermined by a set of abstract structure constants, does not necessarily lead to a Liealgebra. The Jacobi identity, which expresses the associativity of the algebra, mustbe satised [258].

Sometimes, in more complicated eld theories, the transformation rules involveeld-dependent structure constants. Such cases are sometimes referred to as softalgebras [17, 236]. In such a situation, the determination of the gauge algebra is morecomplicated than in the Yang-Mills case. The Jacobi identity must be appropriatelygeneralized [17, 24, 84]. Furthermore, new structure tensors beyond commutatorstructure constants may appear and new identities need to be satised.

In other types of theories, the generators of gauge transformations are not inde-pendent. This occurs when there is a gauge invariance for gauge transformations.One says the system is reducible. A simple example is a theory constructed usinga three-form F which is expressed in terms of a two-form B by applying the exte-

rior derivative F = dB. The gauge invariances are given by the transformation ruleB = dA for any one-form A. The theory is invariant under such transformationsbecause the lagrangian is a functional of F and F is invariant: F = dB = ddA = 0.However, the gauge invariances are not all independent since modifying A by A = dfor some zero-form leads to no change in the transformation for B . When A = d,B = dA = dd = 0. The structure of a gauge theory is more complicated than theYang-Mills case when there are gauge invariances for gauge transformations.

Another complication occurs when the commutator of two gauge transformationsproduces a term that vanishes on-shell, i.e., when the equations of motion are used.When equations of motion appear in the gauge algebra, how should one proceed?

In this section we discuss the above-mentioned complications for a generic gaugetheory. The questions are (i) what are the relevant gauge-structure tensors and (ii)what equations do they need to satisfy. The answers to these questions lead us to thegauge structure of a theory.

This section constitutes a somewhat technical but necessary prelude. A readermight want to consult the examples in Sect. 3. The more interesting development of the eld-antield formalism begins in Sect. 4.

8/14/2019 9412228

16/196


2.1 Gauge Transformations

This subsection introduces the notions of a gauge theory and a gauge transfor-mation. It also denes notation. The antibracket approach employs an elaboratemathematical formalism. Hence, one should try to become quickly familiar withnotation and conventions.

Consider a system whose dynamics is governed by a classical action S 0[], whichdepends on n different elds i(x), i = 1 , , n . The index i can label space-timeindices , of tensor elds, the spinor indices of fermion elds, and/or an indexdistinguishing different types of generic elds. At the classical level, the elds arefunctions of space-time. In the quantum system, they are promoted to operators. Inthis section, we treat the classical case only.

Let (i) = i denote the statistical parity of i . Each i is either a commuting eld( i = 0) or an anticommuting eld ( i = 1). One has i(x) j (y) = ( 1) i j j (y)i(x).

Let us assume that the action is invariant under a set of m0 (m0 n) non-trivialgauge transformations, which, when written in innitesimal form, read

i(x) = R i () (x) , where = 1 or 2 . . . or m0 . (2.1)

Here, (x) are innitesimal gauge parameters, that is, arbitrary functions of thespace-time variable x, and R i are the generators of gauge transformations. These gen-erators are operators that act on the gauge parameters. In kernel form, ( R i () ) (x)can be represented as dyR

i (x, y) (y).It is convenient to adopt the following compact notation [ 82, 83]. Unless otherwise

stated, the appearance of a discrete index also indicates the presence of a space-time variable. We then use a generalized summation convention in which a repeateddiscrete index implies not only a sum over that index but also an integration over thecorresponding space-time variable. As a simple example, consider the multiplicationof two matrices g and h, written with explicit matrix indices. In compact notation,

f A B = gAC hC B (2.2)

becomes not only a matrix product in index space but also in function space. Eq.( 2.2)represents

f AB (x, y) =C dz gAC (x, z ) hC B (z, y) (2.3)

in conventional notation. In other words, the index A in Eq.(2.2) stands for A andx in Eq.(2.3). Likewise, B and C in Eq.(2.2) represent {B, y}and {C, z}. Thegeneralized summation convention for C in compact notation yields a sum over thediscrete index C and an integration over z in conventional notation in Eq.( 2.3). The

8/14/2019 9412228

17/196


indices A, B and C in compact notation implicitly represent space-time variables x,y, z, etc., and explicitly can be eld indices i,j,k, etc., gauge index ,,, etc., orany other discrete index in the formalism.

With this convention, the transformation laws

i(x) = dyR i (x, y) (y) (2.4)

can be written succinctly asi = R i

. (2.5)

The index in Eq.(2.5) corresponds to the indices y and in Eq.(2.4). The index iin Eq.(2.5) corresponds to the indices x and i in Eq.(2.4). The compact notation is

illustrated in the example of Sect. 3.1. Although this notation might seem confusing atrst, it is used extensively in the antibracket formalism. In the next few paragraphs,we present equations in both notations.

Each gauge parameter is either commuting, ( ) = 0, or is anti-commuting, = 1. The former case corresponds to an ordinary symmetry whilethe latter is a supersymmetry. The statistical parity of R i , (R i ), is determined fromEq.( 2.1): (R i ) = ( i + ) (mod 2).

Let S 0,i (, x) denote the variation of the action with respect to i(x):

S 0,i (, x)

r S 0[]

i (x), (2.6)

where the subscript r indicates that the derivative is to be taken from the right (seeAppendix A). Henceforth, when a subscript index i, j , etc., appears after a commait denotes the right derivative with respect to the corresponding eld i , j , etc.. Incompact notation, we write Eq.( 2.6) as S 0,i = r S 0 i where the index i here stands forboth x and i in Eq.(2.6).

The statement that the action is invariant under the gauge transformation inEq.( 2.1) means that the Noether identities

dxn

i=1S 0,i (x) R

i (x, y) = 0 (2.7)

hold, or equivalently, in compact notation

S 0,i R i = 0 . (2.8)

Eq.( 2.8) is derived by varying S 0 with respect to right variations of the i givenby Eq.( 2.1). When using right derivatives, the variation S 0 of S 0, or of any otherobject, is given by S 0 = S 0,i i . If one were to use left derivatives, the variation

8/14/2019 9412228

18/196


of S 0 would read S 0 = i l S 0 i . Eq.(2.7) is sometimes zero because the integrand isa total derivative. We assume that surface terms can be dropped in such integrals this is indeed the case when Eq.( 2.7) is applied to gauge parameters that fall off sufficiently fast at spatial and temporal innity. The Noether identities in Eq.( 2.8)are the key equations of this subsection and can even be thought of as the denitionof when a theory is invariant under a gauge transformation of the form in Eq.( 2.1).

To commence perturbation theory, one searches for solutions to the classical equa-tions of motion, S 0,i (, x) = 0, and then expands about these solutions. We assumethere exists at least one such stationary point 0 = { j0}so that

S 0,i |0 = 0 . (2.9)Equation ( 2.9) denes a surface in function space, which is innite dimensionalwhen gauge symmetries are present.

As a consequence of the Noether identities, the equations of motion are not in-dependent. Furthermore, new saddle point solutions can be obtained by performinggauge transformations on any particular solution. These new solutions should not beregarded as representing new physics however elds related by local gauge trans-formations are considered equivalent.

The Noether identities also imply that propagators do not exist. By differentiatingthe identities from the left with respect j , one obtains

l i

S 0,j R j = l r S 0

i jR j + S 0,j

lR j i

(1) i j = 0 , l r S 0 i j

R j0

= 0 , (2.10)

i.e., the hessian l r S 0 i j of S 0 is degenerate at any point on the stationary surface. The R i are on-shell null vectors of this hessian. Since propagators involve theinverse of this hessian, propagators do not exist for certain combinations of elds.This means that the standard loop expansion cannot be straightforwardly applied. A

method is required to overcome this problem.Technically speaking, to study the structure of the set of gauge transformationsit is necessary to assume certain regularity conditions on the space for which theequations of motion S 0,i = 0 hold. The interested reader can nd these conditionsin Appendix B. A key consequence of the regularity conditions is that if a functionF () of the elds vanishes on-shell, that is, when the equations of motion areimplemented, then F must be a linear combination of the equations of motion, i.e.,

F ()| = 0 F () = S 0,i i() , (2.11)

8/14/2019 9412228

19/196


where | indicates the restriction to the surface where the equations of motion hold[81, 28, 106, 103, 105]. Eq.(2.11) can be thought of as a completeness relation for theequations of motion. We shall make use of Eq.( 2.11) frequently.

Throughout Sect. 2, we assume that the gauge generators are xed once and forall. One could take linear combinations of the generators to form a new set. Thiswould change the gauge-structure tensors presented below. This non-uniqueness isnot essential and is discussed in Sect. 4.5.

To see explicit examples of the abstract formalism that follows, one may want toglance from time to time at the examples of Sect. 3.

2.2 Irreducible and Reducible Gauge Theories

It is important to know any dependences among the gauge generators. Only withthis knowledge is possible to determine the independent degrees of freedom. Thepurpose of this subsection is to analyze this issue in more detail for the generic case.

The simplest gauge theories, for which all gauge transformations are independent,are called irreducible. When dependences exist, the theory is reducible. In reduciblegauge theories, there is a kind of gauge invariance for gauge transformations or whatone might call level-one gauge invariances. If the level-one gauge transformationsare independent, then the theory is called rst-stage reducible . This may not happen.Then, there are level-two gauge invariances, i.e., gauge invariances for the level-one

gauge invariances and so on. This leads to the concept of an L-th stage reducibletheory . In what follows we let ms denote the number of gauge generators at the s-thstage regardless of whether they are independent.

Let us dene more precisely the above concepts. Assume that all gauge invariancesof a theory are known and that the regularity condition described in Appendix B issatised. Then, the most general solution to the Noether identities ( 2.8) is a gaugetransformation, up to terms proportional to the equations of motion:

S 0,i i = 0 i = R i0 0 0 + S 0,j T ji , (2.12)

where T ij must satisfy the graded symmetry property

T ij = (1) i j T ji . (2.13)The R i0 0 are the gauge generators in Eq.( 2.1). For notational convenience, we haveappended a subscript 0 on the gauge generator and the gauge index . This subscriptindicates the level of the gauge transformation. The second term S 0,j T ji in Eq.(2.12)is known as a trivial gauge transformation. Such transformations are discussed inthe next subsection. It is easily checked that the action is invariant under such

8/14/2019 9412228

20/196


transformations due to the trivial commuting or anticommuting properties of the S 0,j .The rst term R i

0 0 0 in Eq.(2.12) is similar to a non-trivial gauge transformation

of the form of Eq.(2.1) with 0 = 0 . The key assumption in Eq.( 2.12) is that theset of functionals R i0 0 exhausts on-shell the relations among the equations of motion,namely the Noether identities. In other words, the gauge generators are on-shell acomplete set. This is essentially equivalent to the regularity condition.

If the functionals R i0 0 are independent on-shell then the theory is irreducible. Insuch a case,

rank R i0 0 = m0 , (2.14)

where m0 is the number of gauge transformations. The rank of the hessian

rank l r S 0 i j = n rank R i (2.15)

is n m0. Dene the net number of degrees of freedom ndof to be the number of eldsthat enter dynamically in S 0, regardless of whether they propagate. 2 Then for anirreducible theory ndof is n m0 since there are m0 gauge degrees of freedom. Notethat ndof matches the rank of the hessian in Eq.( 2.15).

If, however, there are dependences among the gauge generators, and the rankof the generators is less than their number, rank R i0 0 < m 0, then the theory isreducible. If m0 m1 of the generators are independent on-shell, then there are m1relations among them and there exist m1 functionals R

01 1 such that

R i0 0 R 01 1 = S 0,j V

ji1 1 , 1 = 1 , . . . , m 1 ,

(R 01 1 ) = 0 + 1 (mod 2) , (2.16)

for some V ji1 1 , satisfying V ij

1 1 = (1) i j V ji1 1 . Here, 1 is the statistical parityof the level-one gauge parameter. The R 01 1 are the on-shell null vectors for R i0 0since R i0 0 R

01 1

= 0. The presence of V ji1 1 in Eq.(2.16) is a way of extendingthis statement off-shell. Here and elsewhere, when a combination of eld equationsappears on the right-hand side of an equation, it indicates the off-shell extension of an

on-shell statement; such an extension can be performed using the regularity postulateof Appendix B. Note that, if = R1 1 1 for any 1 , then i in Eq.(2.5) is zero on-

shell, so that no gauge transformation is produced. In Eq.( 2.16) it is assumed thatthe reducibility of the R i0 0 is completely contained in R

01 1 , i.e., R

01 1 also constitute

a complete set

R i0 0 0 = S 0,j M ji0 0 = R 01 1

1 + S 0,j T j 00 , (2.17)2 In electromagnetism, n dof = 3, but there are only two propagating degrees of freedom corre-

sponding to the two physical polarizations.

8/14/2019 9412228

21/196


for some 1 and some T j 00 .If the functionals R 0

1 1are independent on-shell

rank R 01 1 = m1 ,

then the theory is called rst-stage reducible . One also has rank R i0 0 = m0 m1and the net number of degrees of freedom in the theory is n m0 + m1. Since trueand gauge degrees of freedom have been determined,

rank l r S 0 i j

= n m0 + m1 .

If the functionals R 01 1 are not all independent on-shell, relations exist among them

and the theory is second-or-higher-stage reducible. Then, the on-shell null vectors of R 01 1 and higher R-type tensors must be found.

One continues the above construction until it terminates. A theory is L-th stagereducible [26] if there exist functionals

R s 1s s , s = 1 , . . . , m s , s = 0 , . . . , L , (2.18)

such that R i0 0 satises Eq.( 2.8), i.e., S 0,i Ri0 0 = 0, and such that, at each stage, the

R s 1s s constitute a complete set, i.e.,

R s 1s s s = S 0,j M j s 1s

s = R ss+1 , s +1 s +1 + S 0,j T j ss ,

R s 2s 1, s 1 R s 1s s = S 0,i V

i s 2s s , s = 1 , . . . , L ,

rank R s 1s s =L

t= s(1)t s m t , s = 0 , . . . , L , (2.19)

where we have dened R 10 0 R i0 0 and 1 i. The R s 1s s are the on-shell nullvectors for R s 2s 1 s 1 . The statistical parity of R s

1s s is s 1 + s (mod 2), where

s is the statistical parity of the s-level gauge transformation associated with theindex

s. Finally,

ndof = rank l r S 0 i j

= n L

s=0(1)s ms (2.20)

is the net number of degrees of freedom.

8/14/2019 9412228

22/196


2.3 Trivial Gauge Transformations

As mentioned in the last subsection, trivial gauge transformations exist. Sincethey are proportional to the equations of motion they do not lead to conservationlaws. This subsection discusses their role in the gauge algebra.

Given that the nite invertible gauge transformations satisfy the group axioms,their innitesimal counterparts necessarily form an algebra. Besides the usual gaugetransformations ( 2.1), there are the trivial transformations, dened as

i = S 0,j ji , ji = (1) i j ij , (2.21)where ji are arbitrary functions. It is easily demonstrated that, as a consequence

of the symmetry properties of ji

, the transformations in Eq.( 2.21) leave the actioninvariant. In studying the structure of the gauge transformations, it is necessary totake into consideration the presence of such transformations.

To determine their effect on the gauge algebra, consider the commutator of atrivial transformation with any other transformation. Calling the latter r i = r i ,one has

[ , r ]i = r i,k S 0,j jk S 0,j ji,k r k S 0,jk r k ji .

Given that r is a symmetry transformation of S 0, it follows by differentiation by j

that

S 0,k rk

= 0 S 0,jk rk

+ S 0,k rk,j = 0 ,

so that the commutator becomes

[ , r ]i = S 0,j r j,k ki (1) i j r i,k kj ji,k r k = S 0,j ji ,

from which one concludes that the commutator of a trivial transformation with anyother transformation is a trivial transformation. Hence, the trivial transformationsare a normal subgroup H of the full group of gauge transformations, G.

The trivial gauge transformations are of no physical signicance: They neitherlead to conserved currents nor do they prevent the development of a perturbative

expansion about a stationary point. They are simply a consequence of having morethan one degree of freedom. On these grounds, it would seem sensible to dispense withthem and restrict oneself to the quotient G = G/H . However, this is only possible incertain cases. In general, the commutator of two non-trivial gauge transformationsproduces trivial gauge transformations. Furthermore, for reasons of convenience,particularly when it is desirable to have manifest covariance or preserve locality, onesometimes wants to include trivial transformations. Hence, the full group G is usedfor studying the gauge structure of the theory.

8/14/2019 9412228

23/196


2.4 The Gauge Structure

In this section we restrict ourselves to the simpler cases of irreducible and rst-stage-reducible gauge theories. To avoid cumbersome notation, we use R i for R i0 0 ,Z a for R

01 1 , and in Eq.( 2.16) we use V jia for V

ji1 1 , so that the indices 0 and 1

respectively correspond to and a.The general strategy in obtaining the gauge structure is as follows [ 81]. The rst

gauge-structure tensors are the gauge generators themselves, and the rst gauge-structure equations are the Noether identities ( 2.8). One computes commutators,commutators of commutators, etc., of gauge transformations. Graded symmetrizationproduces identity equations for the structure tensors that must be satised. Genericsolutions are obtained by exploiting the consequences of the regularity conditions,namely, completeness. In using completeness, additional gauge-structure tensors ap-pear. They enter in higher-order symmetrized commutator identity equations. Theprocess is continued until it terminates.

Although this section provides some insight, it is somewhat technical so that thereader may wish to skip it at rst. If one is only interested in the irreducible case,one should read to Eq.( 2.36). For reasons of space, many details of the algebra areomitted. As an exercise, the reader can provide the missing steps.

Consider the commutator of two gauge transformations of the type in Eq.( 2.1).On one hand, a direct computation leads to

[1, 2]i = R i,j R j (1) R i,j R j 1 2 .

On the other hand, since this commutator is also a gauge symmetry of the action itsatises the Noether identity so that, factoring out the gauge parameters 1 and 2 ,one may write

S 0,i R i,j R j (1) R i,j R j = 0 .

Taking into account Eq.( 2.12) the above equation implies the following importantrelation among the generators

Ri,j R

j (1)

Ri,j R

j = R

i T

S 0,j E

ji , (2.22)

for some gauge-structure tensors T and E ji . This equation denes T

and E

ji .

Restoring the dependence on the gauge parameters 1 and 2 , the last two equationsimply

[1, 2]i R i T 1 2 S 0,j E ji 1 2 , (2.23)where T are known as the structure constants of the gauge algebra. The wordsstructure constants are in quotes because in general the T depend on the elds of

8/14/2019 9412228

24/196


the theory and are not constant. The possible presence of the E ji term is due tothe fact that the commutator of two gauge transformations may give rise to trivialgauge transformations [ 81, 24, 27].

The gauge algebra generated by the R i is said to be open if E ij = 0, whereas the

algebra is said to be closed if E ij = 0. Moreover, Eq.( 2.22) denes a Lie algebra if the algebra is closed, E ij = 0, and the T

do not depend on the elds i .

The gauge-structure tensors have the following symmetry properties under theinterchange of indices

E ij = (1) i j E ji = (1) E ij ,T =

(

1) T . (2.24)

In other words, E ij is graded-antisymmetric both in lower indices and in upper indicesand T is graded-antisymmetric in lower indices. The statistical parity of structuretensors is determined by the sum of the parities of the tensor indices, so that (R i ) =( + i) (mod 2), T = ( + + ) (mod 2), and E

ij = ( i + j + + )

(mod 2).The next step determines the restrictions imposed by the Jacobi identity. In

general, it leads to new gauge-structure tensors and equations [ 172, 258, 84, 28]. Theidentity

cyclic over 1 , 2, 3

[1, [2, 3]] = 0 ,

produces the following relations among the tensors R, T and E

cyclic over 1 , 2, 3R iA

S 0,j B ji 1 2 3 = 0 , (2.25)

where we have dened

3A T ,k Rk T T +(1) ( + ) T ,k Rk T T + ( 1) ( + ) T ,k Rk T T , (2.26)

and

3B ji E ji,k Rk E ji T (1) i R j,k E ki + ( 1) j ( i + )R i,k E kj

+ ( 1) ( + ) RHS of above line with

(2.27)

+( 1) ( + ) RHS of rst line with

.

8/14/2019 9412228

25/196

8/14/2019 9412228

26/196


and to replace gauge parameters by ghosts, as is done in the BRST formalism [ 32].The ghost elds obey the same boundary conditions as gauge parameters. The ghostscan be used as a compact way of writing the gauge-structure equations. However, inorder to do this, the symmetry properties of T , E

ij , D

j , M

kji , etc., need to be

correctly incorporated. Note that these tensors are graded anti-symmetric in lower-index gauge indices , , etc., whereas the ghosts satisfy C C = ( 1)( +1)( +1) C C .If one is given a graded anti-symmetric tensor T 1 2 3 4 ... , then a way to make it intoa graded symmetric tensor with symmetry factors 1 + 1, 2 + 1, etc., associatedwith indices 1, 2, etc., is to multiply by a factor of (1) i for every other index iin T 1 2 3 4 ... . In other words, one replaces T 1 2 3 4 ... by (1) 2 + 4 + ...T 1 2 3 4 ... .Using this device, one arrives at a compact way of writing the Noether identity ( 2.8),

the gauge commutator relation ( 2.22), as well as Eqs.(2.28) and (2.30) which arisefrom the Jacobi identity:

S 0,i R iC = 0 , (2.33)2R i,j R

j R i T + S 0,j E ji (1) C C = 0 , (2.34)

A S 0,j D j (1) C C C = 0 , (2.35)B ji + ( 1) i R jD i (1) j ( i + )R iD j + S 0,k M kji (1) C C C = 0 ,

(2.36)where A and B

ji are dened in Eqs.( 2.26) and (2.27). The graded-anticommuting

nature of the ghosts automatically produces the appropriate graded-cyclic sums.Equations ( 2.33) through ( 2.36) are key equations for an irreducible algebra.Now let us consider a rst-stage reducible gauge theory. In this case, the existence

of non-trivial relations among the generators in Eq.( 2.16) leads to the appearance of new tensor quantities.

For rst-stage reducible theories there are on-shell null vectors for the generatorsR i . Let Z a denote these null vectors. In Eq.( 2.16), the Z a are called R1a when = 0and a = 1. The null vectors are independent on-shell. Their presence modies thesolutions of the Jacobi identities in Eqs.( 2.35) and (2.36) as well as higher-commutatorstructure equations. In addition there are new structure equations. One of these isEq.( 2.16) itself:

R i Z b = S 0,j V

jib . (2.37)

Another is derived as follows. Take relation ( 2.22) and multiply it by Z a to obtain

R i,j R j (1) R i,j R j R i T + S 0,j E ji Z a = 0 .

Use Eq.(2.37) to express R j Z a as a term proportional to equations of motion. Alsodo the same with R i,j R j Z a and make use of the Noether identity in Eq.( 2.8). After

8/14/2019 9412228

27/196


a little algebra, one nds that the previous equation can be written in the form

R i (1) a Z a,j R j T Z a = S 0,j M jia ,for some quantity M jia . Terms proportional to the equations of motion have beencollected into M jia . Using the completeness of the null vectors Z a , the general solutionto this equation is

(1) a Z a,j R j T Z a = Z d Ada S 0,j G j a . (2.38)Eq.( 2.38) is a new gauge-structure equation for the rst-stage reducible case. Twonew structure tensors Ada and G

j a arise.

The null vectors also lead to modications of the solution of the Jacobi identity.Eq.( 2.25) still holds but its solution is different. Instead of Eq.( 2.35), one obtains

A + Z c F

c S 0,j D j (1) C C C = 0 , (2.39)

where we have made use of the completeness of the null vectors Z a . In this equationA stands for the combination of terms in Eq.( 2.26).

Multiplying Eq.( 2.39) by R i and using the Jacobi identity result of Eq.( 2.25) leadto a modication of Eq.( 2.29) involving B ij . The new result reads

S 0,j B ji (1)

j ( i + )

RiD

j + V

jic F

c (1)

C

C

C

= 0 ,when written using ghosts. The general solution is

B ji + ( 1) i R jD i (1) j ( i + )R iD j +V jic F

c + S 0,k M

kji (1) C C C = 0 , (2.40)

where B ji is given in Eq.(2.27).By taking more and more commutators of gauge transformations, more structure

functions and equations appear, some of which involve graded symmetrizations in therst-stage gauge indices a, b, etc.. As in the irreducible case, it is useful to introduceghosts a to automatically incorporate graded symmetrization. Equations ( 2.37) and(2.38) can then be written as

R i Z a S 0,j V jia a = 0 , (2.41)

and(1) a Z a,j R j T Z a + Z d Ada + S 0,j G j a a C = 0 . (2.42)

8/14/2019 9412228

28/196


To summarize, key equations for rst-stage reducible theories are Eqs.( 2.33), (2.34)and (2.39) (2.42). Besides the null vectors Z

a, the new structure tensors are V ji

a,

Ada , G j a , F a as well as higher-level tensors.

Needless to say, for a higher-order reducible theory the number of quantities andequations increases considerably. The complexity of the formalism makes the studyof the gauge structure at higher levels quite complicated. A more sensible approach isto have a generating functional whose expansion in terms of auxiliary elds producesthe generic gauge-structure tensors. In addition, it is desirable to have a simple singleequation which, when expanded in terms of auxiliary elds, generates the entire setof gauge-structure equations. The eld-antield method [ 24, 25, 26] provides sucha formalism. The generating functional for structure tensors is a generalized action

subject to certain boundary conditions and the classical master equation containsall the gauge-structure equations. Before presenting the abstract machinery, it is of pedagogical value to consider some examples of the formalism of this section.

8/14/2019 9412228

29/196


3 Examples of Gauge Theories

This section presents eight gauge theories, which will be used in Sect. 5 to illustratethe antibracket formalism. The theories are (1) the spinless relativistic particle, (2)Yang-Mills theories, (3) four-dimensional topological Yang-Mills theories, (4) the four-dimensional antisymmetric tensor eld, (5) abelian p-form theories, (6) the openbosonic string eld theory, (7) the massless relativistic spinning particle, and (8) therst-quantized bosonic string. Models (1), (2), (7) and (8) are closed and irreducible.Models (3) and (4) are rst-stage reducible. Model (5) is p-stage reducible and model(6) is an innitely reducible open system. For each theory, the classical action S 0 andits gauge symmetries are rst presented. Then, the non-zero gauge-structure tensors

are obtained. The determination of the structure tensors is the rst computationalstep in the antibracket formalism. The results in this section are used in Sect. 5 toobtain proper solutions S .

In the rst subsection on the spinless relativistic particle, we illustrate the compactnotation of Sect. 2.1. Models (1) and (8) are respectively one and two-dimensionalgravity theories. Model (8), the rst-quantized bosonic string, is used to explain theconstruction of general-coordinate-invariant theories, i.e., gravities. In the subsection3.7 on the massless relativistic spinning particle, we provide a mini-review of super-symmetry and supergravity. A brief introduction to string eld theory is given inSect. 3.6.

As exercises for the reader, we suggest the following three computations. (i)Verify that the gauge transformations leave S 0 invariant. (ii) Given S 0 and its gaugesymmetries, obtain the results presented for the gauge-structure tensors. (iii) VerifyEqs.(2.33)(2.36) for the irreducible theories, and verify Eqs.( 2.33), (2.34) and (2.39) (2.42) for the rst-stage reducible theories.

3.1 The Spinless Relativistic Particle

One of the simplest examples of a model with a gauge invariance is the freerelativistic particle. It actually corresponds to a 0+1 dimensional gravity theory withscalar elds. The supersymmetric generalization of the spinless relativistic particle ispresented in Sect. 3.7.

Let us use this system to illustrate the formalism in Sect. 2. The degrees of freedom are a particle coordinate x and an einbein e both of which are functions of a single proper time variable . The action is given by

S 0[x , e] = d 12 x xe m2e , i = ( x , e) , (3.1)

8/14/2019 9412228

30/196


where a dot over a variable indicates a derivative with respect to proper time. Thevariations of the action with respect to the elds, Eq.( 2.6), are

S 0, = d

d xe

, S 0,e =12

x2

e2 m2 , (3.2)where S 0,e is the variation of the action with respect to eld e; in other words, we alsouse e as a eld index for the einbein e. If the equation of motion S 0,e = 0 is used tosolve for e, and this solution is substituted into the action in Eq.( 3.1), one nds thatthe action becomes the familiar one: S 0 = d mx2. Classically, this actionand the one in Eq.( 3.1) are equivalent.

The innitesimal gauge transformations for this system can be written as

x =x

e , e = . (3.3)

It is straightforward to verify that Eq.( 3.3) is a symmetry of Eq.( 3.1). The Noetheridentity in Eq.( 2.8) reads

d xe dd xe 12 x2e2 + m2 dd ( ) = 0 , (3.4)which is veried using integration by parts.

The transformations laws in Eq.( 3.3) in the form of Eq.(2.5) are

R =x

e , Re =

dd

. (3.5)

Eq.( 3.5) says that R is the operator that is multiplication by x

e and Re is dd . In

kernel form using compact index notation, they are

R =x( )

e ( ) , Re =

dd

( ) .Recall that in using compact notation the index of R i in Eq.(2.5) represents not

only a discrete index labelling the different gauge transformations but also a space-time index. Since there is only one type of gauge transformation the discrete indextakes on only one value, which we drop for convenience. Hence the index of R i isreplaced by the space-time variable . In this subsection, we use the Greek letters ,, and to denote proper time variables. Likewise the index i on R i in Eq.(2.5)represents not only a eld index or e but also a proper time variable .

The algebra of the gauge transformations is simply

[1, 2] = [(1) , (2)] = 0 ,

8/14/2019 9412228

31/196


where (1) indicates a gauge transformation with parameter 1. These abelian gaugetransformations ( 3.3) are related to the standard reparametrization transformations

R x = x , R e =d

d (e) (3.6)

through the following redenition of the gauge parameter

e .The algebra of reparametrization transformations reads

[R (1), R (2)] = R (12) , (3.7)

with the parameter 12 given by12 = 12 12 . (3.8)

Eqs.(3.7) and (3.8) correspond to the usual diffeomorphism algebra. Since the com-mutator of two gauge transformations is a gauge transformation, the algebra is closedand E ji in Eq.(2.23) is zero. This example illustrates the effect of eld-dependentredenitions of the gauge parameters or, equivalently, of the gauge generators: Anabelian algebra can be transformed into a non-abelian one. The converse of this alsoholds. One can transform any given non-abelian algebra into an abelian algebra usingeld-dependent redenitions, a result known as the abelianization theorem [ 27]. Thefact that this process can spoil the locality of the transformations is one of the reasonsfor using the non-abelian version.

It may appear unusual that a single family of gauge transformations producesnon-abelian commutation relations. This is due to the local non-commutativity of reparametrization transformations that arises from the time derivatives in Eq.( 3.6).Indeed, when 1 and 2 have non-overlapping support, i.e., 1( ) = 0 where 2( ) = 0and vice-versa, 12 = 0.

It is instructive to see how a non-zero structure constant T for the diffeomor-phism algebra arises using compact notation. In what follows, gauge indices, , ,

etc. are replaced by proper time variable , , , . From Eq.( 3.6) one sees that thetransformation operators R for reparametrizations are

R = x ( ) ( ) ,

Re = e ( )d

d ( ) + e ( ) ( ) =

dd

[e ( ) ( )] .For the x degrees of freedom, a straightforward computation yields

R , = ( )d

d ( ) ,

8/14/2019 9412228

32/196


R , R =

d ( ) dd ( ) x () ( )= x ( ) ( ) ( ) + x ( ) ( )

dd

( ) .Antisymmetrizing in and and comparing with Eq.( 2.22) one nds

T = ( )d

d ( ) ( )

dd

( ) , (3.9)which is in agreement with Eqs.( 3.7) and (3.8). For e, straightforward computationproduces

Re ,e = ( )d

d ( ) + ( )d

d ( ) =d

d [ ( ) ( )] ,Re ,e R

e = e ( )

dd

( )d

d ( ) + e ( ) ( )

dd

( ) +2e ( ) ( )

dd

( ) + e ( ) ( )d2

d 2 ( ) + e ( ) ( ) ( ) .

Antisymmetrizing in and and using Eq.( 2.23), one nds T is again given byEq.( 3.9).

Although compact notation is useful to represent the formalism of gauge theoriesin full generality, it is cumbersome for specic theories, especially for those in whichmore natural notation has already been established. In the examples that follow,we do not explicitly display equations in compact form but use more conventionalnotation.

3.2 Yang-Mills Theories

Yang-Mills theories [276] are perhaps the most familiar gauge theories. For eachLie algebra Gthere is different theory. The fundamental elds are gauge potentials Aawhere there is an index a for each generator T a of G. In a matrix representation, thegenerators are antihermitian matrices which are conventionally normalized so thatT r (T a T b) = 12 ab . The generators satisfy

[T a , T b] = f ab cT c , (3.10)

where f ab c are the structure constants of G. They are real and antisymmetric in lowerindices f ab c = f ba c and they must satisfy the Jacobi identity

f ab ef ec d + f ca ef ebd + f bcef ea d = 0 . (3.11)

8/14/2019 9412228

33/196


The Yang-Mills action is

S 0[Aa] = 14 ddx F a (x)F

a (x) = 12 d

dx T r [F (x)F (x)] , (3.12)

where d is the dimension of space-time, F (x) F a (x)T a , and where the eldstrengths F a are

F a (x) Aa (x) Aa(x) f bca Ab(x)Ac (x) . (3.13)The equations of motion, gauge transformations and gauge algebra are

(D F )a D a bF b = 0 , (3.14)Aa = ( D)

a D abb , (3.15)[(1), (2)]Ac = (12)A

c = D

cd f ab

db1a2 , (3.16)

so that c12 = f abcb1a2. The covariant derivatives D ab and Da

b in the adjointrepresentation are

D ab = a

b f cba Ac ,Da b = a b + f ca bAc , (3.17)

where D ab is applied to elds b with an upper index b and Dab is applied to elds

b with a lower index b. One has

ddxa D abb = ddx (Dba a ) b . (3.18)The operator R i in Eq.(2.5) corresponds to D a b. The covariant derivative satises

[D , D ]a b = f cba F c ,[D , D ]a

b = f ca bF c . (3.19)

In using compact notation, the spatial dependence as well as index dependence of

tensors needs to be specied. For local theories, the spatial dependence is proportionalto delta functions or a nite number of derivatives acting on delta functions. Whenthe spatial-temporal part of a tensor structure is a delta function, it is proportionalto the identity operator in x-space when regarded as an operator. In such cases, it isconvenient to drop explicitly such identity operators.

Eq.( 3.16) is in the form of Eq.(2.23) with E ji = 0 and T cab = f abc where two

identity operators or delta functions are implicit. One concludes that this exampleconstitutes a closed, irreducible gauge algebra.

8/14/2019 9412228

34/196


It is useful to verify the key equations for an irreducible closed algebra given inEqs.(2.33)-(2.36). The generator of gauge transformations is the covariant derivativein Eq.(3.17). Using Eqs.(3.14) and (3.15), the Noether identity in Eq.( 2.33) reads

ddx (D F )b (D C)b = 0 .To verify this equation, integrate by parts, use the antisymmetry of F in and ,use Eq.(3.19), and then make use of the antisymmetry of f cda in c and d:

ddx (D F )b (D C)b = ddx (D D F )b Cb =

12 d

d

x ([D

, D

]F )b Cb

= 12 d

d

x f bda

F a F

d C

b

= 0 .A straightforward computation of the 2 R i,j R

j C C term in the commutator alge-

bra equation of Eq.( 2.34) produces 2f bac Cb Ad f de bCe Ca which, after a lit-tle algebra that makes use of Eq.( 3.11), leads to D c d f ab dCbCa . Using this for2R i,j R

j C C in Eq.(2.34), one concludes, as expected, that T cab = f ab c and E ji = 0.

When T cab = f abc is used, the gauge-structure Jacobi equation in Eq.( 2.35) leads to

3Adabc = ( LHS of Eq.(3.11)) so that Adabc = 0. Finally, the other consequence of the Jacobi identity, namely Eq.( 2.36), produces the tautology 0 = 0. All terms arezero because the tensors B ji , D i and M

kji are all zero. Higher-level equations

(which were not displayed in Sect. 2.4) are automatically satised because higher-leveltensors are identically zero.

3.3 Topological Yang-Mills Theory

In four-dimensions, the action for topological Yang-Mills theory [ 272, 35] is pro-portional to the Pontrjagin index

S 0 =14 d4x F a F a , (3.20)

where the dual eld strength F a is given by F a = 12 F a , and where 0123 = 1.

The interest in this system is its connection [ 272] to Donaldson theory [90].This action is invariant under the gauge transformations in Eq.( 3.15) because the

lagrangian is constructed as a group invariant of the eld strength. In addition, sincethe theory is topological, the transformation

Aa = a , (3.21)

8/14/2019 9412228

35/196


leaves the action invariant, as a short calculation veries. The two gauge transforma-tions form a closed algebra since

[(1, 1), (2, 2)]A = (12 , 12)A ,

c12 f ab cb1a2 , c12 f ab c b1a2 + b1a2 . (3.22)However, the gauge generators are obviously off-shell linearly dependent since the

transformations include ordinary gauge transformations when = D. Sincethe gauge transformations are not all independent, one has a reducible gauge theoryand the coefficients Z a in Eq.(2.37) and Aba introduced in Eq.( 2.38) are non-zero. Of course, the theory can be made irreducible by eliminating ordinary gauge transforma-

tions from the set of all transformations. However, for other theories it is not so easyto reduce the full set to an irreducible subset without spoiling locality or relativisticcovariance and often it is convenient to formulate the theory as a reducible system.

Let us use topological Yang-Mills theory to illustrate the gauge-structure formal-ism of Sect. 2.4. The eld index i corresponds to both a gauge index and a vectorLorentz index since the eld is Aa : i a. There are two types of gauge transfor-mations so that the gauge index of Sect. 2.4 corresponds to the group index b in thecase of an ordinary gauge transformation or to the pair c in the case of a topologicalgauge transformation: (b,c). The generator of ordinary gauge transformationsis the covariant derivative in Eq.( 3.17) and the generator of topological transforma-tions is a delta function:

Rab = Da

b , Rab = ab . (3.23)

The null vectors, denoted as Z b in Sect. 2.4, are

Z ab = ab , Z

ab = D a b . (3.24)

The number of null vectors is equal to the number of gauge generators. It is easilyveried that Eq.( 2.41) holds since Rbc Z ca a + Rbc Z c a a = D bcca a + bc (D c a a ) =0. This computation implies that V

ji

a = 0 in Eq.( 2.41).The non-zero structure constants are

T cab = f abc , T c a b = T

c a b =

f ab

c , (3.25)

whereas T ca b = T ca b = T c ab = T ca b = T ca b = 0. As in the example of Sect.3.2, A is zero since 3Adabc = f ae df bce + f bedf ca e + f ce df ab e = 0. One also ndsAd a b c = Ad a b c = Ad a b c = Adabc = 0.

8/14/2019 9412228

36/196


Eq.( 2.39) holds because A = F c = D j = 0. Eq.( 2.40) holds because B

ji

and M kji

are also zero. Finally, Eq.( 2.42) is valid as long as G j a

= 0 and

Acab = f ab c , Aca b = 0 . (3.26)In verifying Eq.(2.42) it is useful to note that Z c a,b = f ab c, and Z ca,j = 0 bothwhen j = b and when j = b. For Eq.( 2.42), there are four cases to verify: (i) = cand = b, (ii) = c and = b, (iii) = c and = b and (iv) = c and = b.Case (i) gives f bdcda cdAdab = 0. Case (ii) is automatically zero because eachterm in Eq.( 2.42) is zero. Case (iii), after a little algebra, results in the expressionf ba c aCb + f ab c aCb + f ad cf bed + f bdcf ea d + f ed cf ab d aCbAe which is zerobecause of the antisymmetry of f ab c in ab and the Jacobi identity for the Lie algebrastructure constants in Eq.( 3.11). For case (iv), one nds f ab c f ba c = 0. Inshort, the important non-zero gauge structure tensors are given in Eqs.( 3.23) (3.26).Eqs.(2.33), (2.34) and (2.39) (2.42) are all satised.

In a topological theory, the number of local degrees of freedom is zero. One ndsthat ndof is 4N (for Aa ) minus 4N (for a) minus N (for a ) plus N (for the nullvectors in Eq.( 3.24)). Hence, the net number of local degrees of freedom is zero.

3.4 The Antisymmetric Tensor Field Theory

Another example of rst-stage reducible theory is the antisymmetric tensor gaugetheory. Consider a tensor eld B a in four dimensions satisfying B a = B a whosedynamics is described by the action [ 112, 242]

S 0 Aa, Ba = d4x 12AaAa 12B a F a , (3.27)

where Aa is an auxiliary vector eld. The eld strength F a is given in terms of Aa as in the Yang-Mills case via Eq.( 3.13). The action is invariant under the gaugetransformations

B a = Da

bb , Aa = 0 . (3.28)

The covariant derivative D a b is given in Eq.(3.17). The equations of motion derivedfrom Eq.(3.27) are

r S 0B a

= F a = 0 , r S 0Aa

= Aa + Da

bB b = 0 . (3.29)

In spite of the presence of Lie-algebra structure constants f ab c, the model has anabelian gauge algebra, i.e., T = 0, due to the fact that the vector eld, whichappears in the covariant derivative, does not transform.

8/14/2019 9412228

37/196


The gauge transformations ( 3.28) have an on-shell null vector. Indeed, taking

b = D b cc , (3.30)

one nds using Eq.(3.19) that

B a = (D D )a

bb =

12

f cba F c b ,

which vanishes on-shell, since F c = 0 when the equations of motion ( 3.29) are used.Since the null vectors are independent, this theory is on-shell rst-stage reducible.

It is instructive to determine the gauge-structure tensors and verify Eqs.( 2.33),(2.34) and (2.39) (2.42). The eld index i in Sect. 2 corresponds to a in the case

of the antisymmetric eld B a and corresponds to a in the case of Aa . The gaugeindex of Sect. 2 corresponds to b which are the indices of b . The null index a of Sect. 2 is a Lie algebra generator index. The generator of gauge transformations R i is

Ra b = D ab , R

ab = 0 , (3.31)

where the second equation holds because Aa does not transform and is the atspace-time metric. The null vectors Z a are the covariant derivative operators

Z ba = Db

a . (3.32)

Using Eq.(2.41), one nds Rc b Z ba = 12 f ab cF b so that

V b c a = bdf ad c ,V jia = 0 , if j = b or i = b , (3.33)

that is, V jia = 0 if i or j corresponds to the eld index of Ab . The derived quantitiesA and B

ij are zero. Other gauge-structure tensors also vanish:

T = E ji = D

j = F

c = M

kji = A

da = 0 . (3.34)

The gauge-structure equations are all satised. Eq.( 2.33) holds because the actionis invariant under gauge transformations, as is easily checked. Eq.( 2.41) is satised. Infact, it was used above in Eq.( 3.33) to obtain V jia . Eqs.(2.39) and (2.40) are satisedbecause all the tensors entering these equations are zero. Each term in Eq.( 2.34) iszero: The rst term R i,j R

j is zero because R i,j = 0 when j = a , i.e., when j is

a eld index of B a , and R j = 0 when j = a, i.e., when j is a eld index of Aa .The other terms vanish because the structure tensors vanish. Likewise each term inEq.( 2.42) is zero: The rst term Z ,j R

j is zero because Z

,j like R i,j is zero when

j = a .

8/14/2019 9412228

38/196


Let N be the number of generators of G, i.e., the dimension of the Lie algebra

G. The number of degrees of freedom ndof is 4N for Aa plus 6N for B a

minus

the number of gauge transformations 4 N plus the number of null vectors N , so thatEq.( 2.20) reads ndof = 7 N .

3.5 Abelian p-Form Theories

It is not hard to nd an example of an L-th stage reducible theory. Let A be a p-form and dene F to be its eld strength: F = dA where d is the exterior derivative.For p + 1 less than the dimension d of spacetime, an action for this theory is

S 0 = 12 F F , (3.35)where is the dual star operation that takes a q-form into a d q form and is the

wedge product. On basis q-forms, it is dened by

(dx1 dx2 . . . dxq ) =

1(n p)!

1 2 ... d q +1 q +1 q+2 q+2 . . . d d dx q +1 dx q +2 . . . dx d ,

where is the at space-time metric and 1 2 ... d is the antisymmetric tensorsymbol. The case p = 1 corresponds to abelian Yang-Mills theory. Using dd = 0, onesees that the action is invariant under the gauge transformation

A = d p 1 , (3.36)

where p 1 is a p 1 form. This gauge transformation has its own gauge invariance.In fact, there is a tower of gauge invariances for gauge invariances given by

p 1 = d p 2 ,...

1

= d0

,

where q is a q-form. Hence, the theory is p 1 stage reducible. The number of degrees of freedom is

ndof =d p

d p1

+d

p2 . . .(1) p

d1

+( 1) pd0

=d 1

p. (3.37)

Note that dq is the dimension of the space of q-forms in d dimensional space-time.

8/14/2019 9412228

39/196


The gauge generators at the s-th stage, R s 1s s , correspond to the exterior deriva-tive d acting on the space of ( p

1

s) forms, d( p 1 s):

R0 d( p 1) ,R1 d( p 2) ,...

R p 1 d(0) .Because dd = 0, Eq.( 2.19) holds off-shell and all V i s 2s s are zero. With the excep-

tion of the gauge generators, all gauge-structure tensors are zero.If n( p+ 1) = d = dimension of space time, a topological term can be added to the

action S 0 = F F . . . F n terms

. (3.38)

For n = 2 one can consider the quantization of S 0 alone if d = 2 ( p + 1). This wouldbe another example of a topological theory.

Abelian p-form theories provide a good background for covariant open string eldtheory, a non-abelian generalization of p-form theory which is innite-stage reducible.

3.6 Open Bosonic String Field Theory

The covariant d = 26 open string eld theory was obtained by E. Witten [ 270].It resembles a Chern-Simons theory. The fundamental object is a string eld A.Although one can proceed without a detailed understanding of A, A can be expandedas a series in rst-quantized string states whose coefficients are ordinary particle elds.Each member of this innite tower of states corresponds to a particular vibrationalmode of the string. In this manner, string theory is able to incorporate collectivelymany particles. For example, the open bosonic string possesses a tachyonic scalar, amassless vector eld, and numerous massive states of all possible spins. For reviews onopen bosonic string eld theory, see refs.[215, 168, 247]. Also useful is the discussion

in Sect. 7.7, where the rst quantization of the bosonic string is treated.Covariant open string eld theory can be formulated axiomatically [ 270, 271].Fields are classied according to their string ghost number. If the string ghost numberof B is g(B) = p then we say that B is a string p-form in a generalized sense. Theingredients of abstract string theory are a derivation Q, a star operation whichcombines pairs of elds to produce a new eld, and an integration operation whichyields a complex number B for each integration over a string eld B. These objectssatisfy ve axioms:

8/14/2019 9412228

40/196


(1) The nilpotency of Q: QQ = 0.

(2) Absence of surface terms in integration: QA = 0. This axiom isequivalent to an integration-by-parts rule.(3) Graded distributive property of Q across :Q (A B) = QA B + ( 1)g(A) A QB .(4) Associativity of the star product: ( A B) C = A (B C ).

(5) Graded commutativity of the star product under the integral:

A B = ( 1)g(A)g(B )

B A.

The ghost number of a star product of elds is the sum of their ghost numbers:g(A B) = g(A)+ g(B). The derivation Q increases the ghost number by 1: g(QA) =g(A) + 1. In some circles, including refs.[270, 271], the ghost number is shifted by

3/ 2 so that A has ghost number 1/ 2 instead of 1.The axioms are satised for non-abelian Chern-Simons theory in three dimensions.

The eld is a non-abelian vector potential which is converted into a Lie-algebra-valued 1-form by multiplying by dx : Aa b Aabdx As(T s )a bdx , where the T sare a set of matrix generators for the Lie group. The derivation Q is the exteriorderivative d. The star product is the wedge product and a matrix multiplication:(A B)a b = Aa c B cb. Integration is an integral over a three-dimensional manifoldM without boundary and a trace over the Lie-algebra indices: A M Aa a . Axiom(1) is satised because dd = 0. Axiom (2) holds because M has no boundary. Axiom(3) is satised because the exterior derivative d is graded distr

Date post:	31-May-2018
Category:	Documents
Upload:	mlmilleratmit
View:	224 times
Download:	0 times

9412228

Documents