Equivariant symplectic geometry of gauge fixing in Yang–Mills theory

Equivariant symplectic geometry of gauge fixing in Yang–Mills theoryLevent Akant Citation: Journal of Mathematical Physics 49, 033512 (2008); doi: 10.1063/1.2897049 View online: http://dx.doi.org/10.1063/1.2897049 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/49/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Yang–Mills equations of motion for the Higgs sector of SU(3)-equivariant quiver gauge theories J. Math. Phys. 51, 072302 (2010); 10.1063/1.3429582 Large N behavior of two dimensional supersymmetric Yang-Mills quantum mechanics J. Math. Phys. 48, 012302 (2007); 10.1063/1.2408399 Gravity and YangMills Fields: Geometrical Approaches AIP Conf. Proc. 751, 64 (2005); 10.1063/1.1891531 Gauge theories of Yang–Mills vector fields coupled to antisymmetric tensor fields J. Math. Phys. 44, 1006 (2003); 10.1063/1.1528271 Quantization of a particle in a background Yang–Mills field J. Math. Phys. 39, 867 (1998); 10.1063/1.532357

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/578858469/x01/AIP-PT/CiSE_JMPArticleDL_121714/Awareness_LibraryF.jpg/47344656396c504a5a37344142416b75?x

http://scitation.aip.org/search?value1=Levent+Akant&option1=author

http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov

http://dx.doi.org/10.1063/1.2897049

http://scitation.aip.org/content/aip/journal/jmp/49/3?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jmp/51/7/10.1063/1.3429582?ver=pdfcov


http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.1891531?ver=pdfcov



Equivariant symplectic geometry of gauge fixingin Yang–Mills theory

Levent Akant1,a�

1Feza Gursey Institute, Emek Mahallesi, Rasathane Yolu No. 68, Cengelkoy,Istanbul 346 84, Turkey

�Received 23 October 2007; accepted 15 February 2008; published online 20 March 2008�

The Faddeev–Popov gauge fixing in Yang–Mills theory is interpreted as equivariantlocalization. It is shown that the Faddeev–Popov procedure amounts to a construc-tion of a symplectic manifold with a Hamiltonian group action. The BRST coho-mology is shown to be equivalent to the equivariant cohomology based on thissymplectic manifold with Hamiltonian group action. The ghost operator is inter-preted as a �pre�symplectic form and the gauge condition as the moment mapcorresponding to the Hamiltonian group action. This results in the identification ofthe gauge fixing action as a closed equivariant form, the sum of an equivariantsymplectic form, and a certain closed equivariant 4-form, which ensures conver-gence. An almost complex structure compatible with the symplectic form is con-structed. The equivariant localization principle is used to localize the path integralsonto the gauge slice. The Gribov problem is also discussed in the context of equi-variant localization principle. As a simple illustration of the methods developed inthe paper, the partition function of N=2 supersymmetric quantum mechanics iscalculated by equivariant localization. © 2008 American Institute of Physics.�DOI: 10.1063/1.2897049�

I. INTRODUCTION

There are certain field theories with BRST-like symmetries for which the path integrals local-ize on small subsets of the space of all field configurations. Yang–Mills theory on a Riemannsurface,1 Chern–Simons theory on a Seifert manifold,2 and the so-called G /G models3 provideexamples where this phenomenon takes place. It is well known that the reason for this localizationis that the BRST-like symmetry and the field content of these models form a differential complexfor equivariant cohomology and that the action functionals are closed forms in that differentialcomplex. Localization follows from basic properties of integration of equivariant forms.1,2,4 Theaim of this paper is to interpret the Faddeev–Popov gauge fixing procedure5,6 �see also Refs. 7 and8� as equivariant localization. Such a reformulation improves our understanding of the BRSTsymmetry9 and completes its geometric interpretation10,11 by identifying the antighost and theLautrup–Nakanishi auxiliary field12 as geometric objects that arise in the Cartan model of equi-variant cohomology. The role of Gribov horizons13 in equivariant localization will also be dis-cussed. A discussion in the framework of Batalin–Vilkovisky14 formalism will not be attempted atthis stage �see Refs. 15 and 16�. This will be the subject matter of a future work. The relationbetween BRST cohomology and equivariant cohomology is well known in topological gaugetheories.17,18 Here, we will consider the physical gauge theories.

The equivariant cohomology can be thought of as the generalization of the de Rham theory tomanifolds with group actions, where the usual exterior derivative is replaced by the so-calledCartan derivative. If a compact group G acts freely on a manifold M, then the correspondingequivariant cohomology is equivalent to the de Rham cohomology of the quotient manifold M /G.

a�Electronic mail: [email protected].

JOURNAL OF MATHEMATICAL PHYSICS 49, 033512 �2008�

49, 033512-10022-2488/2008/49�3�/033512/29/$23.00 © 2008 American Institute of Physics


130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

http://dx.doi.org/10.1063/1.2897049

http://dx.doi.org/10.1063/1.2897049

http://dx.doi.org/10.1063/1.2897049

If the action is not free, then the quotient space M /G is not a smooth manifold and one cannotdefine the de Rham cohomology of M /G. Although the de Rham cohomology of the quotientspace is not well defined when the group action is not free, the equivariant cohomology still makessense �it is just not equivalent to de Rham cohomology of a finite dimensional smoothmanifold�.19,20 It is in this respect that the equivariant cohomology is to be thought of as thegeneralization of de Rham theory. In the present paper, we will assume the manifold M to be asymplectic manifold and the group action to be symplectic �canonical� and Hamiltonian.21 A veryinteresting property of closed equivariant forms on such manifolds is that their integrals localizeon the critical points of a certain function, which is related to the moment map of the group action.Equivariant localization principle for Abelian group actions was discovered by Duistermaat andHeckman,22 while the non-Abelian case was developed by Witten in Ref. 1, and Jeffrey andKirwan in Ref. 4 �for a short review see also Ref. 23�. Some useful reviews on equivariantcohomology are Refs. 24–26.

In this paper, we will work with linear covariant gauges for which the gauge fixing actionreads

Sgf =� ddx tr�1

2�b2 − ib� · A − c̄� · �c� , �1�

where ��=��+ad�A�� is the covariant derivative in the adjoint representation. Our analysis seemsto be valid for more general gauge fixing conditions as well.

We will see that the gauge fixing procedure involves the construction of a symplectic manifoldwith a symplectic and Hamiltonian group action. It is precisely this geometric construction thatwill allow us to see that the nonminimal BRST complex �which, on top of the gauge field and theghost, contains also the antighost and the Lautrup–Nakanishi auxiliary field� is, in fact, a Cartanmodel of equivariant cohomology. More precisely, the symplectic manifold in question will be avector bundle over a gauge orbit and the group action will be given by fiber translations. Ghostswill be interpreted as fundamental 1-forms corresponding to the action of the gauge group10 �seealso Refs. 27 and 28�, whereas the antighosts will be identified with fundamental 1-forms of fibertranslations. The equivariant cohomology based on the fiber translations will be constructed. Thiscohomology will have the same generators as the BRST complex and, moreover, the action of theCartan derivative on the generators will be shown to be the same as the action of the BRSToperator. The only difference between the two cohomologies will be the gradings assigned to thegenerators. Although our group of fiber translations is Abelian, we will use the language andmethods of Witten’s non-Abelian localization principle, which are more natural for the problem.The ghost operator will be interpreted as a �pre�symplectic form on our vector bundle and thegauge condition as the moment map corresponding to fiber translations. In fact, we will see thatthe gauge fixing action is an equivariant form; it will be the sum of the so-called equivariantsymplectic form and an equivariant 4-form.1 In order to apply equivariant localization principle,we will need the presymplectic form �ghost operator� to be nondegenerate. It is well known thatthe ghost operator is degenerate on the so-called Gribov horizons.13 These horizons must beexcluded from the region of integration since they cause overcounting of gauge fields in the pathintegral. In the standard treatment,13 one usually restricts the path integral to the interior of the firstGribov horizon and thus avoids all Gribov horizons. So, we will see that the problem encounteredin the standard treatment and the one encountered in our analysis based on the use of equivariantlocalization principle are the same: the degeneracy of the ghost operator. The solution to theproblem will be the same: we will a priori restrict the region of integration to the interior of thefirst Gribov horizon. However, a consistent use of equivariant localization principle in calculatingintegrals over a region R �in our case the interior of the first Gribov horizon� with �R�� requires

1In this respect, the gauge fixing action will have the same equivariant structure as the action of Yang–Mills theory on aRiemann surface in first order formalism for which the non-Abelian localization principle was developed in the first placeby Witten in Ref. 1.

033512-2 Levent Akant J. Math. Phys. 49, 033512 �2008�


130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

not only the nondegeneracy of the symplectic form on R but also the vanishing of a certain surfaceterm on �R �in our case the Gribov horizon�.

The practicality of equivariant localization principle stems from the freedom of choosingcertain auxiliary structures used in the localization process. One such structure is an invariantalmost complex structure compatible with the symplectic form. We will show that by choosing anappropriate almost complex structure, the surface term in question may be set equal to zero on thefirst Gribov horizon �in fact, on all Gribov horizons�. This makes possible the use of equivariantlocalization principle to localize the path integrals for the gauge theory on the gauge slice. We willalso see that the exclusion of Gribov horizons is related to the question of independence of thepath integral from the gauge fixing parameter � �gauge independence�.

There are certain topological field theories whose actions can be obtained by gauge fixing thezero action which possesses a very large transitive Abelian gauge symmetry29 �also Refs. 30 and31�. Since the starting action is zero for these theories, the total action after gauge fixing is thegauge fixing action itself. Therefore, it should be possible to apply our methods to calculate, e.g.,the partition functions of these theories via localization. We will apply this strategy to the partitionfunction of supersymmetric quantum mechanics.32–34 In this case, we do not have any reason torestrict the path integral inside the first Gribov horizon; we must integrate over the whole space offield histories. We will see that this can possibly lead to gauge dependence of path integrals, ifthere are critical points which lie on Gribov horizons. However, we will also see that for fieldconfigurations with periodic boundary conditions, there are no such critical points, and this willlead to a gauge independent result.

Most of our calculations will be done in a finite dimensional setting, and then the results willbe formally generalized to infinite dimensions. It would be interesting to specify infinite dimen-sional functional spaces where this correspondence becomes rigorous. This will be done else-where. In the present work, we will concentrate on the formal algebraic and geometric aspects ofthe problem.

The outline of the paper is as follows. In Sec. II, we give an algebraic introduction toequivariant cohomology and work out an example which will be used in giving a geometricmeaning to antighost and auxiliary Lautrup–Nakanishi field, constructing the symplectic manifoldin question and establishing the connection between the BRST complex and the Cartan model ofequivariant cohomology. In Sec. III, we will consider geometric Cartan model based on a sym-plectic manifold with Hamiltonian group actions. We will introduce the equivariant symplecticform and review the basics of equivariant integration and non-Abelian localization principle. Wewill end the section with a discussion of complications that afflict the localization principle whenthe equivariant symplectic form has singularities. In Sec. IV, we will establish the relation betweenthe BRST cohomology and the Cartan model of equivariant cohomology, from both algebraic andgeometric perspectives. In Sec. V, we will generalize the discussion of the previous section toYang–Mills theory. We will identify the ghost operator as a symplectic form, construct the momentmap, and interpret the gauge fixing action as a closed equivariant form. In Sec. VI, we willinterpret the Faddeev–Popov method as equivariant localization. We will introduce the complexstructure which will be used in localization and solve the problems caused by the singularities ofthe symplectic form. In Sec. VII, we will illustrate the applicability of our method on a simpleexample from topological field theory. Namely, we will calculate the partition function of SUSYquantum mechanics by equivariant localization.

II. CARTAN MODEL

In this section, we will review the basics of algebraic version of equivariant cohomology. Werefer the reader to Ref. 20 for a more detailed account of the subject.

Let �A ,d� be a graded commutative differential algebra. Let G be a group which acts on �A ,d�as a group of linear maps that preserve the grading. Thus, if

033512-3 Equivariant symplectic geometry of gauge fixing J. Math. Phys. 49, 033512 �2008�


130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

A = �i

A�i� �2�

and �g denotes the action of g�G, then �g :A�i�→A�i�. We will denote the Lie algebra of G by g.�A ,d� is called a G* algebra if there is a graded Lie algebra g̃, which is generated by three typesof derivations of A: for each ��g, one has derivations £� of degree 0, derivations �� of degree −1,and the differential d of degree 1 as the generators. The graded commutation relations of thesegenerators are

�£�,£�� = £��,��, ��,��+ = 0,

�£�,�� = ��,��, ��,d�+ = £�,

�£�,d� = 0, �d,d�+ = 0. �3�

If the graded algebra is assumed to have a topological structure which allows limits to be taken,then one also requires the compatibility conditions of the actions

�g£��g−1 = £Adg�, �4�

�g��g−1 = �Adg�, �5�

�gd�g−1 = d . �6�

A very important example of a G* algebra arises when a connected Lie group G acts on amanifold M. Then, ��M� is a G* algebra. In this case, ��M� is the graded commutative differ-ential algebra whose grading is simply given by the exterior degree. The action of G is given bythe pullback action, while g̃ is generated by the Lie derivatives £a along the fundamental vectorfields Va corresponding to the generators ea of g, contractions �a by Va, and the exterior derivatived.

In the following, we will work at the infinitesimal level, so for us a G* algebra will be adifferential graded commutative algebra with a g̃ action as given above. We will denote thestructure constants of g in a basis �ea by Cab

c , i.e., �ea ,eb�=Cabc ec.

Let S�g*� be the symmetric tensor algebra over g*. Then, for any G* algebra A, the spaceC�G ,A�=S�g*� � A is a graded algebra where the degree of an element of the form

�ea1¯ ear� � ap �7�

is given by p+2r. Here, ap is a homogenous element of A with degree p and ea’s are the basiselements of g*.

C�G ,A� admits a G action given by

�g = Adg−1*

� �g. �8�

Here, �g is the action of G on A and Ad* is the coadjoint action of G on S�g*�. Infinitesimally, theaction of g is given by

1 � £� + C� � 1, � � g �9�

Here, C� is the coadjoint action of g on S�g*� given by

C��ea1¯ ear� =

n=1

r

ea1¯ ean−1�− �aCab

aneb�ean+1¯ ear. �10�

Note that



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

�eaCa��ea1¯ ear� = 0. �11�

The Cartan derivative is defined as

D = 1 � d − iea� �a. �12�

Note also that the combination ea � �a is independent of basis. D is a graded derivation of degree1; it increases the degree of a homogenous element by 1, and for a homogenous element a�C�G ,A� of degree p, one has

D�ab� = �Da�b + �− 1�pa�Db� . �13�

A simple calculation shows that

D2 = − iea� £a. �14�

Therefore, on the space CG�A�= �C�G ,A��G of G-invariant elements of C�G ,A�, one has

D2 = 0. �15�

So, all the ingredients of a cohomology theory are present. The resulting cohomology is called theequivariant cohomology. The space CG�A� is called the Cartan model of the equivariant cohomol-ogy of A. In what follows, we will concentrate on the case A=��M�. In our local considerations,we will take M to be a vector space V. In this case, it will be convenient to identify ��V� withC�V� � ∧V*. Thus,

C�G,��V�� = S�g*� � C�V� � ∧ V*. �16�

The appropriate graded algebra to use in the field theory context is given by the direct generali-zation of this finite dimensional model,

C�b,, c̄� = C�ba� � C�i� � ∧ �c̄i� ,

a = 1, . . . ,dim G, i = 1, . . . ,dim V . �17�

Here, C�ba�, C�i�, and ∧�c̄i� are the sets of formal power series in the commuting fields ba, i,and the anticommuting field c̄i, respectively. The field ba carries a Lie algebra index and trans-forms under the coadjoint action of the group G. Notice that the fields i and c̄i have the sameindex structure. The gradings of b, , and c̄ are 2, 0, and 1, respectively.

Let us consider an example of a Cartan model which will turn out to be crucial for ourconstructions in the following. Let G=V, considered as translation group, and M =V. Then, theaction of V on C�V ,��V��=S�V*� � ��V� is given by

�v = 1 � Tv, �18�

where Tv is a translation in V by a vector v�V. So, we have an action of V on C�V ,��V��. Theinfinitesimal action is given by

1 � £v. �19�

The Cartan differential is given by

D = 1 � d − iea� �a. �20�

The Cartan model is the V �translation� invariant part of C�V ,��V��=S�V*� � C�V� � ∧V*,

CV��V�� = S�V*� � ∧ V*. �21�

In the field theoretical model C�b , , c̄�, the group action on the generators is given by



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

�vba = 0, �22�

�va = a + va, �23�

�vc̄a = 0. �24�

Infinitesimally,

£vba = 0, �25�

£va = − va, �26�

£vc̄a = 0. �27�

The action of the Cartan derivative on the generators is given by

Dba = 0, �28�

Da = c̄a, �29�

Dc̄a = − iba. �30�

Finally, the Cartan model is the translationally invariant part �i.e., the part annihilated by all £v’s�of C�ba� � C�a� � ∧ �c̄a�,

CG�b, c̄� = C�ba� � ∧ �c̄a� . �31�

III. INTEGRATION AND LOCALIZATION

In this section, we consider the geometric model CG��M�� for equivariant cohomology. Wewill assume M to be a compact symplectic manifold without boundary. The symplectic form on Mwill be denoted by �. We will assume that the action of the compact connected group G on M issymplectic �canonical� and Hamiltonian. We will also assume that the Lie algebra g of G issemisimple. One may relax the compactness condition on M by considering equivariant forms ofcompact support or of rapid decrease.

A. Cartan model on a symplectic manifold

If �g denotes the action of g�G on M, then

�g*� = � . �32�

Infinitesimally, the action of u�g is given by the fundamental vector field Vu. Thus, we have

£Vu� = 0. �33�

Let �Ta be a basis for g, then we will denote VTasimply by Va, and the corresponding Lie

derivative and contraction by £a and �a, respectively.The action of G is assumed to be Hamiltonian. This means that for each u�g, there will be

a map �u�C�M�, linear in u, and satisfying

d�a = − �a� , �34�



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

��a,�b = Cabc �c. �35�

Here, Cabc are the structure constants of g, �a=�Ta

, and �, is the Poisson bracket induced by thesymplectic form �.

Cartan model of equivariant cohomology HG�M� is the cohomology based on the gradeddifferential algebra

CG��M�� = �S�g*� � ��M��G, �36�

D = d − iba�a. �37�

A generic equivariant form in CG��M�� can be written as

k

�kPk�b� , �38�

where �k is a k form in ��M� and Pk�b� is a polynomial in b �i.e. an element of S�g*��.A result of fundamental importance in equivariant cohomology is as follows: assuming that

the action of G on M is free, the equivariant cohomology is equivalent to the de Rham cohomol-ogy of the quotient manifold M /G,20

HG�M� = HdR�M/G� . �39�

In this sense, the equivariant cohomology may be regarded as the generalization of the de Rhamcohomology of quotient manifolds to the case of non-free group actions, for which M /G is not asmooth manifold.

Let us examine some examples of equivariant forms. An important example is the equivariantsymplectic form20

�̄ = � − iba�a. �40�

It is easy to check that this is a closed equivariant 2-form. For example, G-invariance can be seenas follows:

£a�̄ = £a� − ibb£a�b + ifabc bb�c = 0, �41�

since £a�=0 by the hypothesis that the action is symplectic, and

− ibb£a�b + ifabc bb�c = − ibb��a,�b − fab

c �c� = 0, �42�

by the definition of the Poisson bracket and the hypothesis that the action is Hamiltonian. Clos-edness follows from a similar calculation,

D�̄ = D�� − iba�a� = d� − iba�a� − ibad�a = 0. �43�

Conversely suppose that M is a manifold on which a group G acts. Let �̄ be an equivariant2-form, which is necessarily of the form

�̄ = � − iba�a. �44�

If �̄ is closed, then we have

D�̄ = d� − ibaia� − ibad�a = 0. �45�

This implies

d� = 0, �a� = − d�a. �46�

Moreover, the G-invariance of �̄ implies



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

£a� = 0, £a�b = Cabc �c. �47�

If � is assumed to be nondegenerate, then these observations imply that � is a symplectic formand G has a symplectic and Hamiltonian action on M.

Another example of an equivariant form is the 4-form

b · b � �b,b� , �48�

where �,� is the Cartan–Killing metric on g*. The G-invariance follows from the Ad-invariance ofthe Cartan–Killing metric. It is also easy to see that �b ,b� is a closed equivariant form. This 4-formand the equivariant symplectic form will play important roles in the interpretation of the gaugefixing action as a closed equivariant form.

B. Equivariant localization

The integral of an equivariant form is defined as1,2

� k

�kPk�b� =� db1¯ dbNe−��/2�b·bPk�b��

M�k. �49�

Note that the exponential of the 4-form b ·b is a convergence factor, which regulates the integra-tion over b’s.

A very important property of this definition is the vanishing of the integral of an exactequivariant form

� D =� db1¯ dbNe−��/2�b·bPk�b��

Md�k, �50�

which vanishes, by Stoke’s theorem, if M is compact and without boundary or if �k is of compactsupport or of rapid decrease. In particular, for any closed equivariant r-form and any equivariant1-form �, the form

�etD� − 1� �51�

is exact,

�etD� − 1� = �tD� +t2

2!D�D� + ¯ , �52�

= D�t� +t2

2!�D� + ¯ , �53�

=D��− 1�r �t� +t2

2!�D� + ¯ � . �54�

Consequently,

� =� etD�. �55�

This formula is the basis of Witten’s non-Abelian localization principle.1 The evaluation of theintegral on the right hand side in the large t limit localizes the integral on the critical points ofba�a�,



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

d�ba�a�� = 0. �56�

More explicitly, we have two sets of equations,

bad��a�� = 0, �57�

�a� = 0. �58�

These equations can be simplified if � is chosen in a certain way.1 Let J be a G-invariant almostcomplex structure compatible with the symplectic form � �i.e., ��JX ,JY�=��X ,Y�� and choose

� = JdI . �59�

Here,

I = � · � , �60�

where �=�aTa. The compatibility of J with � implies that the metric g defined by

g�X,Y� = ��X,JY� �61�

is positive definite. With this choice of �, and under the assumption made about J, one can arriveat the following two conclusions whose proofs rely heavily on the positive definiteness of g:1

�a� = 0 ⇔ dI = a

�ad�a = 0, �62�

and that under the assumption of free action of G on �−1�0�,

�bad��a��q = 0 ⇔ ba = 0 for q � �−1�0� . �63�

The first conclusion implies two types of critical points: ordinary critical points of I whichsatisfy �a=0 for all a and higher critical points for which dI=0 but I�0. We will denote the setof ordinary critical points by �−1�0�. The second conclusion implies the compactness of theordinary critical point set in g��−1�0� along the g direction. In particular, the convergence factore−��/2�b·b is not needed to make sense of the integral. In fact, it can be argued1 that on �−1�0�, theequivariant integral is independent of the convergence factor �i.e., independent of ��. However, forhigher critical points, dependence on � cannot be avoided. Moreover, singularities of g may verywell lead to � dependence of the equivariant integral, even for ordinary critical points. Since g isderived from the symplectic form �, the singularities of the latter invalidate conditions �62� and�63�. Depending on the integrand, this may give rise to � dependence in the equivariant integral.As we will see below, � dependence is something we do not want.

C. Gauge theory

In our discussion of gauge fixing as equivariant localization,

−� ddx tr�c̄� · �c� �64�

will be interpreted as a �pre�symplectic form and the part of the gauge fixing action given by

−� ddx tr�− ib� · A − c̄� · �c� �65�

will become an equivariant symplectic form. The Boltzmann factor corresponding to the remain-ing part of Sgf,



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

exp�−�

2� dnxb · b� , �66�

which is the source of gauge dependence, will be interpreted as the equivariant convergence factor.A detailed discussion of algebraic and geometric constructions which are responsible for theseidentifications will be given in the next section. For now, we will be contented with some remarksconcerning the gauge dependence and degeneracies of the �pre�symplectic form.

The possibility of omitting the convergence factor is closely related to the stability of theBRST symmetry of the gauge fixed action.2 We will investigate in Sec. VI the conditions underwhich the convergence factor may be omitted in the gauge fixed path integral. However, noticeonce more that the validity of �62� and �63� is guaranteed only when the metric g is positivedefinite. In our discussion of gauge fixing, we will take the symplectic form � to be the ghostLagrangian and both J and g will be given in terms of the ghost operator. Thus, in order to ensurethe validity of �62� and �63�, we will have to restrict the region of integration to a subset where �is nondegenerate. However, with the restriction of the equivariant integral to a subregion R�Mwith �R��, we must question the validity of the equivariant localization principle since thedifference

� �R� = �R

�1 − etD�� , �67�

which, in general, reduces to a surface integral over �R, may not vanish. One can safely use theequivariant localization principle in the region R only when � �R�=0.

Summarizing, in all our applications of equivariant localization, we will require the validity ofthe conditions �62� and �63�. If they are not satisfied at certain singular points �e.g., points whereg is singular or points where � is degenerate�, then we will avoid them by restricting the equiva-riant integral to a region which contains no singularity. However, in such a region, equivariantlocalization principle may no longer be applicable due to a nonvanishing surface term. In suchcases, the validity of localization principle should be checked by showing that � �R�=0 on therestricted region.

By a Gribov horizon, we will mean a connected region in the space of all gauge fields wherethe ghost operator is degenerate. As was argued by Gribov in Ref. 13 these horizons may bethought of as a collection of bounding surfaces, with the property that the kth horizon lies entirelyin the region bounded by �k+1�th horizon. Moreover, on the first horizon, there is only one zeromode of the ghost operator, on the second there are two, on the third there are three, and so on. Ina region lying between two consecutive horizons, the ghost operator is nondegenerate. As wasobserved by Gribov,13 these horizons should be avoided in path integrals. Gribov horizons containgauge fixed field configurations in whose neighborhoods there are other gauge fields which obeythe same gauge fixing condition but differ from each other by infinitesimal gauge transformations�i.e., they lie on the same gauge orbit�; clearly, such configurations lead to overcounting in the pathintegral and therefore should be eliminated altogether. The standard procedure is to restrict thepath integral to the interior of the first Gribov horizon. However, on this region, � is nondegen-erate and the conditions �62� and �63� are satisfied. We will see that when the integral is restrictedinside the first Gribov horizon, � �R�=0 for =e�̄ and for an appropriate choice of the almostcomplex structure J.

IV. BRST COMPLEX AS A CARTAN MODEL

A. Algebraic model

In this section, we will construct an algebraic model for equivariant cohomology and relate itto the BRST cohomology. Geometrically oriented readers may safely skip this section and proceed

2Differentiation with respect to � inserts a BRST exact term to the path integral. Consequently, nonvanishing of the �derivative implies nonvanishing of the expectation value of a BRST exact term and the BRST symmetry is broken.



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

to the next one where a geometric version of the same construction is discussed. Let B�K ,O�= ∧k* � C�O� be the minimal sector �no antighosts and no auxiliary fields� of the BRST com-plex. Here, we are assuming that K is a connected group which acts freely on a manifold N andthat O is a K-orbit in N. k denotes the Lie algebra of K and C�O� is the set of smooth functionson O. In field theory context, we will use the generalization of the algebraic model

B�K,O� = C�qa� � ∧ �cb� ,

a,b = 1, . . . ,dim O = dim K , �68�

where N will be identified with the space of gauge connections, O with a gauge orbit, and K withthe gauge group. Here, qa are the coordinates on O.

The action of the BRST differential s on the generators is given by

sf�q� = ca�eaf�q�, f � C�O� , �69�

sca = − 12cbccfbc

a . �70�

Here, £eais the Lie derivative in the direction of the fundamental vector field ea corresponding to

the generator Ta of k, and �Ta ,Tb�= fabc Tc.

Now, let us consider the tensor product of graded algebras,

C�G,A� � B�K,O� = S�g*� � A � B�K,O� . �71�

Then, we write

A = �i

Ai, �72�

B�K,O� = �j

Bj , �73�

where Ai is the space of homogenous elements of A with degree i and, likewise, Bj are theelements of B with ghost number j. Then, A=A � B�K ,O� is a graded algebra,

A = �kAk, �74�

Ak = span�ai� bj:ai � Ai, bj � Bj, i + j = k . �75�

We define the product on A as

�ai� bj��ar

� bs� = �− 1� jr�aicr� bjbs� . �76�

From this definition, it follows that �i� AkAl�Ak+l and �ii� A is supercommutative. Thus, A is agraded commutative algebra.

Proposition 4.1: A is a G* algebra.Proof: We already remarked that A is a graded commutative algebra. We will define the action

of G on A=A � B�K ,O� as

�g = �g � 1. �77�

This is clearly a well defined action. The associated Lie superalgebra is generated by

£� � 1, �78�

�� 1, �79�



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

d � 1 + �− 1�F� s . �80�

Here, £� denotes the infinitesimal action of ��g on A and F is the number operator for the gradingof A. The commutation relations of �−1�F with the generators are

��− 1�F,£�� = ��− 1�F,��+ = ��− 1�F,d�+ = 0. �81�

Consequently, one gets

�£� � 1,£c � 1� = £��,c� � 1, �82�

�£� � 1,�c � 1� = ��,c� � 1, �83�

�£� � 1,d � 1 + �− 1�F� s� = 0, �84�

�� 1,�c � 1�+ = 0, �85�

�� 1,d � 1 + �− 1�F� s�+ = £� � 1, �86�

�d � 1 + �− 1�F� s,d � 1 + �− 1�F

� s�+ = 0. �87�

Thus, we conclude that A=A � B�K ,O� is a G* algebra.As a corollary to this proposition, we have the following.Proposition 4.2: C�G ,A� � B�K ,O�=C�G ,A�.The Cartan derivative on C�G ,A� is defined according to the standard construction as

D = 1 � d � 1 + 1 � �− 1�F� s − iba

� �a � 1. �88�

The Cartan model is given by

CG�A� = �S�g*� � A � B�K,O��G = �S�g*� � A�G� B�K,O� , �89�

since the action of G on B�K ,O� is trivial. Moreover, if g is Abelian, then

CG�A� = S�g*� � AG� B�K,O� . �90�

Now, let us consider the special case C�G ,A�=C�k ,��k��, where, as we did in the example atthe end of Sec. II, we take k as an Abelian group acting on itself by translations. In this case,

C�k,A� = S�k*� � �C�k� � ∧ k*� � �C�O� � ∧ k*� . �91�

The Cartan derivative on the generators of C�k ,A� is

Dba = 0, �92�

Da = c̄a, �93�

Dc̄a = − iba, �94�

Dca = − 12cbccCbc

a , �95�

Df�q� = cb£ebf�q�, f � C�O� . �96�

In the corresponding Cartan model, we have AG= ∧k*. So, the Cartan differential on the genera-tors of CG�A�=S�k*� � ∧k* � �∧k* � C�O�� is



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

Dba = 0, �97�

Dc̄a = − iba, �98�

Dca = − 12cbccCbc

a , �99�

Df�q� = cb�ebf�q� . �100�

However, this is nothing but the nonminimal sector of the BRST complex, except the fact that thegrading is different. The comparison between the two gradings is given in the table below, whereF is the number operator in the Cartan model and � is the ghost number operator in the BRSTcomplex.

b c̄a ca x

F 2 1 1 0

� 0 − 1 1 0

. �101�

The fact that D is a differential in both complexes follows from the simple observation that F iseven if and only if � is even.

B. Geometric model

Now, we will construct a geometric version of our algebraic model. We know that in ageometric setting, the Cartan derivative acting on the coordinate functions gives the exteriordifferentials of the latter. So, from

D = c̄ , �102�

we deduce that c̄ should be interpreted as the exterior differential of . Similarly,

Df�q� = df�q� = ca£eaf�q� �103�

suggests the interpretation of c’s as the 1-forms dual to the fundamental vector fields correspond-ing to the action of the gauge group K. This is in accordance to the standard geometric interpre-tation of ghost fields10 �also Refs. 27 and 28�. So, the underlying manifold can be taken to beO��k, where O� is an orbit of the gauge group K. If we think of this manifold as a trivial vectorbundle over O� with a typical fiber given by the gauge Lie algebra k, then group action has thenice interpretation of fiber translations. Notice that the dimensions of the base manifold and thefiber are the same. Consequently, our vector bundle is even dimensional. Now, our aim is to turna subset of this vector bundle into a symplectic manifold on which the action of fiber translationsis symplectic and Hamiltonian. In the next section, we will show that Faddeev–Popov gauge fixingin Yang–Mills theory involves a field theoretic version of this construction.

Explicitly, our geometric construction goes as follows. We have a manifold Q and a group Kwith a free action on Q. Let k be the Lie algebra of K and assume dim k=n. Let fab

c be thestructure constants of k. In the field theory context, Q and K will be the space of connections andthe gauge group, respectively. Let us label the orbits O� of the K-action by the index �. Since theaction is free, the fundamental vector fields ea form a global frame for the tangent bundle TO�.Similarly, the dual 1-forms ca form a global frame for the cotangent bundle.

Now, let us consider M =Q�k. The coordinates in Q and k will be denoted by x and ,respectively. The direct product K�k has a natural action on M. Here, we think of k as anadditive group acting on itself by translations. It is this action, and not the action of K �gaugegroup�, that will be used in the construction of the equivariant cohomology. The fundamentalvector field corresponding to the action of u�k on M is



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

Vu = ua �

�a = uaVa, �104�

where Va=� /�a and we will use the notation c̄a�da for the dual basis. Moreover, the Liederivative along Va will be denoted by £a and the contraction �Va

by �a.Under the action of K, M is fibered into orbits of the form U�=O��k. Let �ab be an n�n

matrix valued function on Q, which satisfies

£ec�ab − £eb

�ac − fcbd �ad = 0. �105�

The meaning of this condition will become clear below. Denote the restriction of �ab to O� by�ab

��. We will define Gribov horizons as the connected components of the solution set of theequation det �ab=0. More precisely, the ith Gribov horizon will be defined as the locus where �ab

has precisely k vanishing eigenvalues. The ith Gribov horizon will be denoted by �i−1. Further-more, we will assume that Gribov horizons are boundaries and that �i−1 lies entirely in the regionbounded by �i. The region bounded by �i−1 and �i will be denoted by Ci−1. In particular, the regionbounded by the first Gribov horizon �1 will be denoted by C0. Each �i−1 is naturally embedded inM as �i−1�k. We will denote the intersections Ci�O� and �i�O� by Ci

�� and �i��, respectively.

Thus, in particular, �1�� is the surface where the first Gribov horizon intersects the gauge orbit �

and C0�� is the portion of the gauge orbit bounded by �1

��. The 2-form

� = �ab�x�c̄a ∧ cb � �2�M� �106�

is clearly degenerate at each point of M. Consider the restriction of � to U�,

��U�= �ab

��q�c̄a ∧ cb, �107�

where q’s are the coordinates in O�. This restricted 2-form is degenerate only on �i��k�U�. We

will also refer to �i��k’s as Gribov horizons. In the rest of this section, we will assume �� to

be nondegenerate. This assumption is justified if we restrict � to C0��k�U�. We will have more

to say about this restriction at the end of this section and also in Sec. VI.Now, thanks to the condition £ec

�ab−£ea�cb− fca

d �db=0 and the fact that �� /�pc��ab=0, wehave

d��U�= £ec

�ab��cc ∧ c̄a ∧ cb − �ad

��c̄a ∧ dcd = − £ec�ab

��c̄a ∧ cc ∧ cb + 12 fcb

d �ad��c̄a ∧ cc ∧ cb

= − 12 �£ec

�ab�� − £eb

�ac�� − fcb

d �ad��c̄a ∧ cc ∧ cb = 0. �108�

Here, we also made use of the Mauer–Cartan equation dcd=− 12 fcb

d cc∧cb. Thus, each C0��k

becomes a symplectic manifold with symplectic form ��C0��k. On each C0

��k, the action of kis free and symplectic. The first assertion is true since the action of k on itself and hence onC0

��k is free. The second assertion follows from a simple computation,

£a�� = �bc�q�£ac̄b ∧ cc = 0, �109�

since £ac̄b=£adpb=d�ab=0. We can also show that this action is Hamiltonian by noticing that the

equation defining the moment map,

d�c = − �c� , �110�

is equivalent to

£eb�a = − �ab,

��a

�b = 0. �111�

Here, the second equation implies that the moment map is independent of fiber coordinates. Then,the first equation can be integrated to give the moment map as a function of q’s. Moreover,



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

��a,�b = ��

�a ,�

�b = 0. �112�

Hence, we conclude that the action is Hamiltonian. Thus, we have proved the following.Proposition 4.3: For each �, the action of the additive group k on C0

��k is free, symplectic,and Hamiltonian.

One can form the Cartan model of equivariant cohomology based on the action of the additivegroup k on C0

��k �in fact, also on U��,

Ck��C0�� k�� = �S�k*� � ��C0

�� k��k, �113�

=S�k*� � ��C0�� k��k. �114�

Thus, the generators of the Cartan model consist of q, c, c̄, and b. In particular, the tensorcomponents of the differential form part of an element of Cartan model should be independent of’s. This model for the equivariant cohomology is a geometric version of the algebraic modelconstructed at the beginning of this section by adding a trivial pair to the minimal sector of theBRST complex. In fact, the action of the Cartan derivative,

D = d − iba�a, �115�

on the generators of the Cartan model is given by

Dqa = dqa = cb£ebqa, �116�

Dba = 0, �117�

Dc̄a = − ibc�cdpa = − iba, �118�

Dca = dca = − 12 fbc

a cb ∧ cc, �119�

where we used the Mauer–Cartan equations in the last line.Now, using this relation between BRST cohomology and equivariant cohomology, and the fact

that for free group actions the latter is equivalent to the de Rham cohomology of the quotient, weget �for the standard treatment, see Ref. 27� the following.

Proposition 4.4:

HBRSTi � HdR

i �K�, i � 0. �120�

Proof: Notice that both the BRST cohomology and the equivariant cohomology contain thetrivial pair �b , c̄�, which does not affect the cohomology of the minimal BRST sector. Thus,

Hmin BRSTi � Hnonmin BRST

i � Hki �U�� . �121�

However, recall that the action of k on U� is free. Therefore, using the fundamental characteriza-tion of the equivariant cohomology, we have

Hki �U�� HdR

i �U�/k� � HdRi �O�� HdR

i �K� . �122�

Thus, we conclude

HBRSTi � HdR

i �K� . �123�

Before we end this section, we want to take a closer look at the singularities of �. We willdenote the image of �−1�0��O��k under the natural projection on O� by �̃−1�0�. Consider �r

��

and let �Xkbk=1

r be the zero modes of �ab, i.e., �abXkb=0. Let Xk=Xk

beb, then



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

�Xk� = − �abc̄aXk

b = 0 �124�

and

£Xk� = d�Xk

� = 0. �125�

Consequently, £Xk�n=0. On the other hand,

£Xk�n = � £Xk

��det ��c̄1 ∧ ¯ ∧ c̄1 ∧ c1 ∧ ¯ ∧ cn�

= � �£Xk�det ��c̄1 ∧ ¯ ∧ c̄1 ∧ c1 ∧ ¯ ∧ cn

� a=1

n

�det ��c̄1 ∧ ¯ ∧ c̄n ∧ c1 ∧ ¯ £Xkca

¯ ∧ cn

= � �£Xk�det �� + eaXk

a�det ��c̄1 ∧ ¯ ∧ c̄n ∧ c1 ∧ ¯ ∧ cn

= � �£Xk�det ��c̄1 ∧ ¯ ∧ c̄n ∧ c1 ∧ ¯ ∧ cn, �126�

where we used

£Xkca = ��Xk

d + d�Xk�ca = − 1

2 �Xk�fbc

a cb ∧ cc� + dXa = − fbca Xbcc + cbeb�Xa� , �127�

which, together with the complete antisymmetry of fbca , implies

c̄1¯ c̄n ∧ c1 ∧ ¯ £Xk

ca¯ ∧ cn = ea�Xa�c̄1 ∧ ¯ ∧ c̄1 ∧ c1 ∧ ¯ ∧ cn. �128�

So, we have £Xk�det ��=0 and, consequently, Xk is tangent to �r

��. Moreover,

£�Xk,Xl�� = d��Xk,Xl�

� = d�£Xk�Xl

− �Xl£Xk

�� = 0, �129�

which implies �ab�Xk ,Xl�b=0. So, �Xk ,Xl�b=FklmXm

b and �Xk ,Xl�=FklmXm. Thus, the vector fields

�Xkk=1r form an integrable distribution on �r

��. On each leaf of the corresponding foliation, themoment map is constant,

£Xk�a = Xk

b£eb�a = − �abXk

b = 0. �130�

So, we conclude that if �̃−1�0� intersects �r��, then the points of intersection form a submanifold

�union of the leaves with �=0� of the horizon.Conversely, if a subset N of �̃−1�0� is a smooth connected manifold in O� then N should lie

on a horizon. If X is a vector field tangent to N, then £X�a=0 implies, by a calculation similar to�132�,13 �abXb=0. Thus, X must be tangent to a horizon. We conclude that N lies on a horizon.

Summarizing, we have proved the following.Proposition 4.5: If �̃−1�0� intersects a horizon, then it does so at certain leaves of the foliation

of the horizon generated by the zero modes of �ab. Conversely, if a subset of �̃−1�0� forms asubmanifold of O�, then that submanifold lies on a horizon.

V. GENERALIZATION TO YANG–MILLS THEORY

As we remarked earlier in Yang–Mills theory, Q will be the space of all connections on aprincipal G-bundle over the Euclidean space Rn. As is well known, this is an affine space. We willassume that G is compact and its Lie algebra g is semisimple. The gauge group K will beidentified with the space of all smooth maps from Rn into G. We will interpret the ghost operatoras a 2-form on Q�k. The ghost fields ca�x� will be interpreted as 1-forms dual to the fundamentalvector fields that generate infinitesimal gauge transformations,



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

eax =� dny� �

�y��x − y��ab + fda

b A�d �y��x − y��

�A�b �y�

, �131�

�eax,eby� =� dnzfax,bycz ecz, �132�

where fax,bycz = fab

c ��x−z��x−y� are the structure constants of the gauge algebra.In order to generalize the construction of the last section to the case of Yang–Mills theory, we

have to specify what is the field that generalizes fiber coordinates. Then, the differential of thisfield can be identified with the antighost. In particular, the field corresponding to fiber coordinatesmust be bosonic, so that its differential will be anticommuting, and must have the same indexstructure as the antighost. In the standard gauge fixed Yang–Mills action, there is one such field,namely, the auxiliary field ba�x�. However, b will be used as the generator of S�k*� and, therefore,is not the correct choice. Instead, we introduce a noninteracting free field a�x� into the Yang–Mills action. Such a modification of the action will clearly not affect the value of the partitionfunction since is Gaussian and do not interact with other fields. The introduction of as a freefield into the Yang–Mills action allows us to interpret the functional integral of the theory as anequivariant integral, i.e., as an integral of a differential form followed by an integral over S�k*�.More precisely, consider the partition function of the gauge fixed theory,

Z =� DADbDcDc̄De−SYM−SgfR�� . �133�

Here, R�� is the Gaussian regulator for integration over ; since neither SYM nor Sgf depend on, choosing the regulator such that �DR��=1 gives us the usual partition function for the gaugefixed Yang–Mills action. Assume that the gauge symmetry is not anomalous and write the measurefor the gauge field as

DA = J��d��DU , �134�

where J�� is the gauge invariant Jacobian factor, J��d�� is the measure on the space of gaugeorbits, and DU is the gauge invariant measure on the gauge orbit. Then, the expectation value ofa product W of Wilson loops can be written as

�M/K

J��d��We−SYM��

DUDbDcDc̄De−SgfR�� . �135�

Now, the integral

��

DUDbDcDc̄De−SgfR�� 136�

is the integral of a differential form on U�=O��k, followed by an integration over S�k*�. In fact,the measures DUDc and DDc̄ are the ones appropriate for integration of exterior forms on O�

and k, respectively,

��

DUDcDc̄D → �U�=O��k

. �137�

Now, we want show that Sgf is an equivariant form on U�. We start by showing that the ghostoperator is a closed 2-form on U�. We just need to check the validity of the condition �105� for theghost operator �� whose integral kernel is



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

�ax,by = �ab�y��y − x� + ��y − x�� · Aab��x� − Aab�x� · �y��y − x� . �138�

Thus, we have to show

ecz��ax,by� − eby��ax,cz� = fcz,bydu �ax,du. �139�

The calculations, which are straightforward but a bit tedious, are given in the Appendix. Noticethat the above condition means that for each �ax�, �ax,by is a 1-cocycle in the cohomology of thegauge Lie algebra.

Next, we want show that −� ·Aa�x� is the moment map for fiber translations. We need to check�111� for �ax=−� ·Aa�x�,

eyb�− � · Aa�x�� = − �x��ab�x��y − x� + fadbA�

d �x��y − x��

= − �y��y − x� − �� · Aab�x��y − x� − Aab · �y��y − x�� = − �ax,by �140�

and

�

�b�y��− � · Aa�x�� = 0. �141�

Thus, the negative of the action for the ghost fields is an equivariant symplectic form

− Sghost =� dnx�c̄a��ca + iba��A�a� = � − ib · � , �142�

where b ·� is short for �dnxbxa�ax.

More generally, the gauge fixing action is an equivariant form

− Sgf = −�

2b · b + � − ib · � = −

�

2b · b + �̄ . �143�

So, the integral

� Db�U�

R��e−Sgf =� Db�U�

R��e−��/2�b·b+�̄ �144�

is almost the integral of an equivariant form. The problem is the Gaussian regulator R, which isnot an equivariant form. Moreover, we will have to address the fact that � is degenerate on theGribov horizons. In the next section, we will see how one can handle these issues.

VI. GAUGE FIXING AND EQUIVARIANT LOCALIZATION

Now, we are ready to interpret Faddeev–Popov method as the equivariant localization.

A. Treatment of the regulator

The complication due to the regulator can be handled by incorporating the latter in thedefinition of equivariant integration. However, it is important for our purposes to make sure thatthis modification does not spoil the equivariant localization principle.

Proposition 6.1: For any equivariant form � in our Cartan model,

� R��D� = 0. �145�

Proof: Let us write � as



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

� = I

�I�q�bI, �146�

where

�I�q� = ��I�KL�q�dqKdL, �147�

and I, K, and L are multi-indices with a fixed value of �K�+ �L�. Since ��I�KL does not depend onp, we have

d��I�KLdqKdL� = dq��I�KLdqK�dL. �148�

So,

� R��D� =� R��dq��I�KLdqKdL�bI − R��I�KL�a�dqKdL�babI

=� R��dq��I�KLdqKdL�bI = �k

e−��b,b�bI�M

dq��I�KLdqK��k

R��dL = 0.

�149�

As a simple corollary to this proposition, we have the modified localization formula as follows.Proposition 6.2:

� R�� =� R��etD� �150�

for D�=0 and any equivariant 1-form � in our Cartan model.Proof:

� R��etD� − 1� =� R��tD� +t2

2D�D� + ¯ =� R��D�t� +

t2

2�D� + ¯

=� R��D��t� +t2

2�D� + ¯ � = 0. �151�

B. Choice of J and �

The 1-form � will be chosen in the form J�dI�, where J is a k-invariant almost complexstructure compatible with �� and

I = a

�a2. �152�

Let us assume for a while that �� is nondegenerate. We will consider the degenerate case later inthis section. As an invariant almost complex structure on U�, we will take

J�ca� = − 12 ��−1�abc̄b, �153�

J�c̄c� = 2�cdcd, �154�

where all the indices are raised and lowered by �. Thus,

J =1

2��−1�abc̄b

� ea − 2�abcb�

�

�a . �155�

Proposition 6.3: J is k-invariant and compatible with �



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

Proof: k-invariance of J follows from the fact that £a annihilates all the terms appearing in J.J is an almost complex structure, as can be seen from

J2�ca� = J�− 12 ��−1�abc̄b� = − ��−1�ab�bccc = − ca, �156�

J2�c̄a� = J�2�abcb� = − �ab��−1�bcc̄c = − c̄a. �157�

So, J2=−1. Compatibility of J with � follows from

�ab�J�c̄a� ∧ J�cb�� = − �ab�2�accc� ∧ 12 ��−1�bdc̄d = − �ab��−1�bd�accc ∧ c̄d = �dcc̄

d ∧ cc = �

�158�

and the fact that

�ab�Jc̄a� � cb − �ab�Jcb� � c̄a = �ab�2�accc� cb + 1

2 ��−1�bcc̄c� c̄a� = 2��T��cbcc

� cb + 12 c̄a

� c̄a

�159�

is a positive definite metric.The invariant 1-form � is given by

� = J�dI� = 2J��bd�b� = 2J��b�£ea�b�ca� = − 2�b�£ea

�b� 12 ��−1�adc̄d = �b�ba��−1�adc̄d = �bc̄b

�160�

and

�b� = c

�cdpc� �

�pb = �b. �161�

Then we have −�c��−1�bceb and the following.Proposition 6.4:

� = �U�, d� = �, D� = �̄ . �162�

Proof:

�U� = �U��adc̄a ∧ cd� = �adc̄a�c��−1�bc�bd = �ac̄a = � , �163�

d� = d�b ∧ c̄b = �− �b�� ∧ c̄b = − �b��acc̄a ∧ c̄c� ∧ c̄b = − �bcc

c ∧ c̄b = � , �164�

and,

D� = �d − iba�a�� = � − iba�a = �̄ . �165�

The following expression in terms of a coordinate basis will be useful later:

d� = d�b ∧ c̄b =��b

�qi dqi ∧ c̄b. �166�

The top form in etd� is given by

det�t��b

�qi . �167�

The following is another useful identity:



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

� ∧ �n−1 =1

n�U�n =

1

n�U��a1b1

¯ �anbn�c̄a1 ∧ cb1 ∧ ¯ ∧ c̄an ∧ cbn�

=�n

n�U��a1b1

¯ �anbn��c̄a1 ∧ ¯ ∧ c̄an� ∧ �cb1 ∧ ¯ ∧ cbn��

=�n

n�U��a1b1

¯ �anbn�a1¯an�b1¯bnc̄1 ∧ ¯ ∧ c̄n ∧ c1 ∧ ¯ ∧ cn�

= �n�n − 1�!�det ��c̄1 ∧ ¯ ∧ c̄n� ∧ �U�c1 ∧ ¯ ∧ cn� . �168�

Here and in the following, �n= �−1�n�n−1�/2 and �n= �−1�n�n+1�/2. Notice that the almost complexstructure J and the metric g are singular at the points where � is singular. Now, restricting all thesestructures to C0

��k, where � is nondegenerate, we get a positive definite metric g, which isdevoid of singularities.

C. Faddeev–Popov as equivariant localization

Since g has no singularities on C0��k, the equations,

bad��a�� = 0, �169�

�a� = �a = 0, ∀ a , �170�

that define the localizing manifold enjoy the nice properties discussed in Sec. III. In particular, theunique solution of the first equation on �0

−1�0�=�−1�0�� C0��k� is b=0. Moreover, from the

second equation, we see that higher critical points do not contribute to localization. We will showbelow that the contribution of �−1�0� is gauge independent �i.e., independent of ��. Now, we mustshow that equivariant localization principle holds on C0

��k, that is, we must show that thedifference

� �� dnb�C0

��kR��e−��/2�b·be�̄�etD� − 1� �171�

vanishes. As we will see shortly, this is proportional to a surface term on �1��. However, before we

do that, let us try a simpler localization, still based on equivariant cohomology, to illustrate theproblems encountered in restricting the path integral to C0

��k.Let us note that

− 12�b · b = D� , �172�

where � is the equivariant 3-form

� = −i

2�bac̄a, d� = 0. �173�

So, one may write the partition function as

�C0

��ke�̄R�� =� dnb�

C0��k

R��eD�e−ib·�+�. �174�

The term eD� can be omitted in this integral if one can show that



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

�� dnb�C0

��kR��eD� − 1�e−ib·�+� = 0. �175�

If this was the case, then one could integrate over b to get a delta function and thus localize theintegral. However, it is not clear that the above difference is zero. The problem is a nonvanishingsurface term,

�� =� dnb�C0

��kD�� + 1

2�D� ¯ �e�̄�R�� =� dnb�C0

��kd�� + 1

2�D� ¯ �e�̄�R��

= −� dnb�C0

��k��de�̄�F�b2�R�� , �176�

where

F�b2� = 1 + 12D� + ¯ �177�

is a power series in b2=b ·b, and we used D�̄=0=d�. Thus,

�� = −� dnb�C0

��k�d�e−ib·�e−��F�b2�R�� = −� dnb�

C0��k

1

�n − 1�!�d�e−ib·��n−1�F�b2�R�� ,

�178�

noting

d�e−ib·��n−1� = � c̄a1 ∧ ¯ ∧ c̄an−1d�e−ib·��a1b1¯ �an−1bn−1

cb1 ∧ ¯ ∧ cbn−1� , �179�

integrating over the fiber k, and using Stokes theorem for the integral over C0��,

�� = �� dnb�F�b2�

2�n − 1�!ba�

�1��

�aa1¯an−1e−ib·��a1b1¯ �an−1bn−1

cb1¯ cbn−1, �180�

which is, in general, nonzero. As expected, ��=0 is guaranteed only for the special case �=0�Landau gauge�.

Now, let us go back and try to localize the integral by using the invariant 1-form �. Aconsistent use of equivariant localization principle on C0

��k requires the vanishing of the dif-ference

� �� dnb��C0

��kR��e�̄�etD� − 1� , �181�

=� �dnb��C0

��kR��e�̄�et�̄ − 1� . �182�

Here, �dnb�=dnbe−�1/2��b·b. So,

� = 0 ⇔� �dnb��C0

��kR��e�t+1��̄ =� �dnb��

C0��k

R��e�̄ �183�

or



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

� = 0 ⇔�

�t� �dnb��

C0��k

R��et�̄ = 0. �184�

This is very similar to the situation encountered in topological field theories, where ��0 signalsthe breakdown of BRST symmetry,35

�


C0��k

R��et�̄ =� �dnb��C0

��kR��̄et�̄ =� �dnb��

C0��k

R��D�et�̄

=� �dnb��C0

��kR��D��et�̄� =� �dnb��

C0��k

R��d��et�̄� .

�185�

The top exterior form in d��et�̄� is

tn−1

�n − 1�!d�e−itb·�� ∧ �n−1� = �ntn−1d�e−itb·��det ��c̄1 ∧ ¯ ∧ c̄n� ∧ �U�c1 ∧ ¯ ∧ cn�� .

�186�

After integrating over k, we get

�


C0��k

R��et�̄ = �ntn−1� �dnb��1

��e−itb·� det��U�c1 ∧ ¯ ∧ cn�� = 0,

�187�

since det��=0 on �1��. Thus we conclude �=0 and the equivariant localization principle is valid

on C0��k.

Now, we can localize the equivariant integral on the critical point set �0−1�0�. Assuming that

the gauge slice intersects C0�� only once, we have �0

−1�0�= �q��k, where q�� is the element ofC0

�� with ��q��=0, i.e., the point of intersection. We do not expect contributions from theboundary since the integrand is proportional to det �. In the equivariant integral, one can nowlocalize the integration over C0

��k on a neighborhood of �0−1�0� of the form B�k, where B is

a ball with center at q��. The integral over b can also be localized on a neighborhood S0 of b=0,

� dnb�C0

��ke−�1/2��b·b+�̄etD�R�� = �

S0

dnb�B�k

e−�1/2��b·b+�̄etD�R�� . �188�

Since k acts freely on B�k, we get

−�

2b · b � Hk

4�B � k� � H4��B � k�/k� = H4�B� = 0. �189�

In fact, as we saw earlier,

−1

2�b · b = D�, � = −

i

2�b · c . �190�

Now, we have the following �see also Sec. 2 of Ref. 1�.Proposition 6.5:

�S0

dnb�B�k

eD�e�t+1�D�R�p� = �S0

dnb�B�k

e�t+1�D�R�� . �191�



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

Proof: Define a k-invariant �i.e., independent of � bump function u, which is equal to 1 in aneighborhood B��k, B��B, of q�� and 0 outside B�k, then the partition function can be writtenas

�S0

dnb�B�k

eD�e�t+1�D�R��u�q� . �192�

Now, consider the difference

�S0

dnb�B�k

�eD� − 1�e�t+1�D�R��u�q�

= �S0

dnb�B�k

D�� + 12�D� + ¯ �e�t+1�D�R��u�q�

= �S0

dnb�B�k

d�� + 12�D� + ¯ �e�t+1�D�u�q��R��

+ �S0

dnb�B�k

�� + 12�D� + ¯ �e�t+1�D��du�q��R�� . �193�

Performing the integration over in the first integral gives a surface term proportional to u;therefore, the first integral vanishes. On the other hand, in the large t limit, the second integral canbe restricted onto B��k, where du=0; consequently, the second integral vanishes as well.1

After letting t→ t−1, we get

� e�̄e�t−1�D�R�� =� dnb�C0

��ketD�R�� =� dnb�

C0��k

etd�−itba�a�R��

= �C0

��k��n��t�a��etd�R�� = �

C0��k

��n��t�a�etd�R�� , �194�

but, as we noted earlier, the top form in etd� is

det�t��a

�qi � . �195�

Therefore, the integrand is

��n��t�a�det�t��a

�qi � = ��q − q��sgn det�t��a

�qi � . �196�

Using

��c

�qi = eib£eb

�c = eib�bc �197�

and choosing det�eib��0, we get the integrand equal to

��q − q��sgn det�� . �198�

However, det��0 inside the first Gribov horizon. Therefore, the integrand is simply a deltafunction around the point of intersection qr and, consequently, we get



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

� R�p�e−�̄ = 1. �199�

If there are additional Gribov copies, we obtain a sum over delta functions and the integralgives the intersection number n� of the gauge slice with C0

��. In this case, a further truncation ofthe ordinary critical points may be necessary.34,36–40 We will not address this issue in this paper.

VII. APPLICATION TO TOPOLOGICAL FIELD THEORIES

Our methods can readily be used to explain the localization properties of certain topologicalfield theories. Here, for illustrative purposes, we will consider the simplest such theory, namely,the supersymmetric quantum mechanics.32,33

A. Supersymmetric quantum mechanics

For simplicity we will consider the one dimensional case. According to the Baulieu–Singermethod29 �see also Refs. 30 and 41�, the action of SUSY quantum mechanics can be interpreted asthe gauge fixing action corresponding to the zero action on the space of histories of a particleliving on R. The zero action, being invariant under an arbitrary shift

q�t� → q�t� + �q�t� , �200�

needs gauge fixing. Notice that the action of this Abelian gauge group is transitive and that thereis only one gauge orbit. If the gauge condition/moment map is chosen as

dq

dt+

�V

�q, �201�

where V is a polynomial, then the standard Faddeev–Popov procedure gives the gauge fixingaction �which is equal to the total action� as

� dt� �

2b2 − ib�dq

dt+

�V

�q − �̄� d

dt+

�2V

�q2 �� . �202�

We will assume that V� and V� do not have simultaneous zeros. We will employ periodic boundarycondition for q�t� with period T,

q�0� = q�T� . �203�

The action of BRST/Cartan differential on the generators is given by

Dq = � , �204�

D� = 0, �205�

D�̄ = − ib , �206�

Db = 0. �207�

First, let us characterize the Gribov horizons of this action. The Gribov horizon is defined asthe locus where the presymplectic form is degenerate. So, we look for the zero modes f of theghost operator

df

dt+

�2V

�q2 f = 0. �208�

The solution as a functional of q is given by



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

f�t� = f�0�exp�− �0

t

dt�V��q�t�� . �209�

Since we employ periodic boundary conditions, these zero modes will be present only when

�0

T

dt�V��q�t�� = 0. �210�

This condition specifies the Gribov horizons of our model. Thus, in a region lying between twoconsecutive Gribov horizons, histories obey

�0

T

dt�V��q�t�� 0. �211�

Notice that static configurations qs with V��qs��0 do not lie on Gribov horizons. Moreover, forcertain potentials, Gribov horizons are absent. For example, there are no Gribov horizons for apotential V�q� with V��q��0, ∀q.

The ordinary critical points are given by the zeros of the moment map

dq

dt+

�V

�q= 0. �212�

These can be recognized as the instantons of the theory,

I = �0

T

dt�2 = �0

T

dt�dq

dt 2

+ � �V

�q 2

+ 2�dq

dt

�V

�q , �213�

=�0

T

dt�dq

dt 2

+ � �V

�q 2

+ 2�V�q�T�� − V�q�0�� , �214�

=�0

T

dt�dq

dt 2

+ � �V

�q 2

, �215�

but for an ordinary critical point, we have

I = �0

T

dt�dq

dt 2

+ � �V

�q 2

= 0. �216�

Hence,

dq

dt= 0, �217�

V��q� = 0. �218�

So, the ordinary critical points are time independent and coincide with the critical points of thepotential V. Since V� and V� do not have common zeros, we see that none of the critical points lieon a Gribov horizon. Then, the partition function is given by



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

Z = qc

sgn det� d

dt+ V��qc�� . �219�

This is the result derived in Ref. 31 by the Nicolai map construction.42 If we relax the assumptionthat V� and V� do not vanish simultaneously, we get critical points that lie on Gribov horizons. Inthis case, one can avoid the Gribov horizons by restricting the integral to regions lying betweenthe horizons. Since the integrand is proportional to det � �after the anticommuting fields areintegrated out�, the contribution of a horizon to Z is zero. The surface terms will also vanish sincethey are all proportional to det � and we still get the same answer for the partition function.

If one relaxes the periodicity condition, then one gets nontrivial solutions of �=0 �instantons�.These nontrivial instantons lie on Gribov horizons �i.e., they have zero modes�. Again, theircontributions to Z vanish since the integrand is proportional to det �. However, for path integralswith BRST exact �i.e., equivariantly exact� insertions, the surface terms are no longer proportionalto det �. So, one cannot ensure the gauge independence of the equivariant integral and this signalsthe breakdown of the BRST symmetry/SUSY. In fact, it is well known that in this case, theinstanton effects break the supersymmetry.31,34,43,44

ACKNOWLEDGMENTS

The author would like to thank O. T. Turgut for useful conversations.

APPENDIX: CLOSEDNESS OF THE GHOST OPERATOR

In this appendix, we will show that the ghost operator �� obeys the condition �105�, whichimplies that ��U�

is a closed form. We will assume that the space-time is the Euclidean space Rn.Then, the gauge group K can be identified as the space of smooth maps from Rn into a compactLie group G. We will assume that the Lie algebra g of G is semisimple. The structure constants fab

c

are antisymmetric in the lower indices and obey the Jacobi identity

fabd fdc

e + fcad fdb

e + fbcd fda

e = 0. �A1�

We will take �ab as the Cartan–Killing form and use it to raise and lower the Lie algebra indices.The fact that G is compact and g is semisimple implies that fcab=�cdfab

d is totally antisymmetric.In the adjoint representation, the gauge connection is given by

A�ab = A�c facb, �A2�

where A�c is the connection in the fundamental representation. As a consequence of the Jacobi

identity, we get

fdabAdc� − fdacAdb

� = fdbcAad� . �A3�

The infinitesimal action of K on Q is given by the fundamental vector fields

eax =� dny� �

�y��x − y��ab + fda

b A�d �y��x − y��

�A�b �y�

, �A4�

�eax,eby� = fabc ��x − y�ecx. �A5�

The structure constants of the gauge algebra are

fax,bycz = fab

c ��x − y��x − z� . �A6�

The ghost operator �� can be written explicitly as



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

��ab = ��ab + ��A�ab� + A�ab��. �A7�

The integral kernel of this operator is

�ax,by = �ab�y��y − x� + ��y − x�� · Aab��x� − Aab�x� · �y��y − x� . �A8�

Now, we want to verify that

ecz��ax,by� − eby��ax,cz� = fcz,bydu �ax,du. �A9�

Let us start with the right hand side,

fcz,bydu �ax,du =� dnufdcb��z − y��z − u��ax,du = fdcb��z − y��ax,dz

= fdcb��ad�z��z − x� + ��z − x�� · Aad��x� − Aad�x� · �z��z − x��

= facb��z − y��z��z − x� + ��z − y��z − x�� · �fdcbAad��x�

− fdcbAad�x� · ��z − y��z��z − x� . �A10�

On the other hand,

ecz�ax,by = ��y − x��x�faeb��x��z − x��c

e + A�ec�z��z − x�� − faeb��x��z − x��ce

+ A�ec�z��z − x��y��y − x�

= facb��y − x��x��z − x� − �x��z − x��y��y − x��

+ faebA�ec�z��y − x��x��z − x� − ��z − x��y��y − x�� . �A11�

Using the distributional identities

��y − x��x��z − x� − ��z − x��y��y − x��f��z�

= ��z − x��x��y − x� − ��y − x��z��z − x��f��z�

= ��z − y��z − x�� · f��x� − f��x��z − y��z��z − x� �A12�

and

��y − x��x��z − x� − �x��z − x��y��y − x� + �y ↔ z� = ��z − y��z��z − x� , �A13�

we can write

ecz��ax,by� − eby��ax,cz� = facb��z − y��z��z − x� + ��z − y��z − x��fadb� · Adc − fadc� · Adb�

− �fadbA�dc − fadcA�eb��z − y��z��z − x� . �A14�

Finally, using �A3�, we get

ecz��ax,by� − eby��ax,cz� = facb��z − y��z��z − x� + ��z − y��z − x�� · �fdcbAad�

− �fdcbA�ad��z − y��z��z − x� , �A15�

which equals fcz,bydu �ax,du.

1 E. Witten, J. Geom. Phys. 9, 303 �1992�.2 C. Beasley and E. Witten, J. Diff. Geom. 70, 183 �2005�, e-print arXiv:hep-th/0503126.3 M. Blau and G. Thompson, Nucl. Phys. B 439, 367 �1995�.4 L. Jeffrey and F. Kirwan, Topology 34, 291 �1995�.5 L. D. Faddeev and V. N. Popov, Phys. Lett. 25B, 29 �1967�.6 B. de Witt, Phys. Rev. 162, 1195 �1967�.7 E. S. Fradkin and I. V. Tyutin, Phys. Rev. D 2, 2841 �1970�.8 G. ’t Hooft, Nucl. Phys. B 33, 173 �1971�.



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

http://dx.doi.org/10.1016/0393-0440(92)90034-X

http://dx.doi.org/10.1016/0550-3213(95)00058-Z

http://dx.doi.org/10.1016/0040-9383(94)00028-J

http://dx.doi.org/10.1016/0370-2693(67)90067-6

http://dx.doi.org/10.1103/PhysRev.162.1195

http://dx.doi.org/10.1103/PhysRevD.2.2841

http://dx.doi.org/10.1016/0550-3213(71)90395-6

9 C. Becchi, A. Rouet, and R. Stora, Commun. Math. Phys. 42, 127 �1975�; I. Tyutin, Lebedev Institute Preprint No. N39,1975.

10 L. Bonora and P. Cotta-Ramusino, Commun. Math. Phys. 87, 589 �1983�.11 J. Thierry-Mieg, J. Math. Phys. 21, 2384 �1980�.12 B. Lautrup, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 35 �1966�; N. Nakanishi, Prog. Theor. Phys. 35, 1111 �1966�; 37,

618 �1967�.13 V. N. Gribov, Nucl. Phys. B 139, 1 �1978�; I. M. Singer, Commun. Math. Phys. 60, 7 �1978�.14 I. A. Batalin and G. A. Vilkovisky, Phys. Lett. 102B, 27 �1981�; 120B, 166 �1983�; Phys. Rev. D 28, 2567 �1983�.15 A. Nersessian, Mod. Phys. Lett. A 10, 3043 �1995�.16 A. S. Schwarz and O. Zaboronsky, Commun. Math. Phys. 183, 463 �1997�.17 J. Kalkman, Commun. Math. Phys. 153, 447 �1993�.18 H. Kanno, Z. Phys. C 43, 477 �1989�.19 R. Bott, in Quantum Field Theory: Perspective and Prospective, edited by C. DeWitt-Morette and J. B. Zuber �Kluwer,

Dordrecht, 1999�.20 V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory �Springer-Verlag, Berlin, 1999�.21 M. F. Atiyah and R. Bott, Topology 23, 1 �1984�.22 J. J. Duistermaat and G. J. Heckman, Invent. Math. 69, 259 �1982�.23 E. Witten, Dynamics of Quantum Field Theory, Lecture 11 in Quantum Fields and Strings: A Course for Mathematicians,

edited by P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, and E. Witten�AMS, Providence, RI, 1999�, Vol. 2.

24 S. Cordes, G. W. Moore, and S. Ramgoolam, Nucl. Phys. B, Proc. Suppl. 41, 184 �1995�.25 R. J. Szabo, e-print arXiv:hep-th/9608068.26 M. Libine, e-print arXiv:math.SG/0709.3615.27 M. Henneaux and C. Teitelboim, Quantization of Gauge Systems �Princeton University Press, Princeton, NJ, 1992�.28 J. A. De Azcarraga and J. M. Izquiedro, Lie groups, Lie algebras, Cohomology and Some Applications in Physics

�Cambridge University Press, Cambridge, 1995�.29 L. Baulieu and I. M. Singer, Commun. Math. Phys. 125, 227 �1989�.30 L. Baulieu and I. M. Singer, Nucl. Phys. B, Proc. Suppl. B5, 12 �1988�; R. Brooks, D. Montano, and J. Sonnenschein,

Phys. Lett. B 214, 91 �1988�.31 D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, Phys. Rep. 209, 129 �1991�.32 E. Witten, Nucl. Phys. B 202, 253 �1982�.33 E. Witten, J. Diff. Geom. 17, 661 �1982�.34 P. Salomonson and J. W. Van Holten, Nucl. Phys. B 196, 509 �1982�.35 E. Witten, Commun. Math. Phys. 117, 353 �1988�.36 D. Zwanziger, Nucl. Phys. B 192, 259 �1981�.37 M. A. Semenov-Tyan-Shanskii and V. A. Franke, Zap. Nauchn. Semin. LOMI 120, 159 �1982�.38 C. Parrinello and G. Jona-Lasinio, Phys. Lett. B 251, 175 �1990�.39 G. Dell’Antonio and D. Zwanziger, Commun. Math. Phys. 138, 291 �1991�.40 P. van Baal, Nucl. Phys. B 369, 259 �1992�; e-print arXiv:hep-th/9511119; e-print arXiv:hep-th/9711070.41 D. Birmingham, M. Rakowski, and G. Thompson, Phys. Lett. B 214, 381 �1988�.42 H. Nicolai, Phys. Lett. 89B, 341 �1980�; Nucl. Phys. B 176, 419 �1980�; Phys. Lett. 117B, 408 �1982�.43 V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 223, 445 �1983�.44 S. Ouvry and G. Thompson, Nucl. Phys. B 344, 371 �1990�.



130.18.123.11 On: Thu, 18 Dec 2014 05:54:07

http://dx.doi.org/10.1007/BF01614158

http://dx.doi.org/10.1007/BF01208267

http://dx.doi.org/10.1063/1.524385

http://dx.doi.org/10.1143/PTP.35.1111

http://dx.doi.org/10.1016/0550-3213(78)90175-X

http://dx.doi.org/10.1007/BF01609471

http://dx.doi.org/10.1016/0370-2693(81)90205-7

http://dx.doi.org/10.1103/PhysRevD.28.2567

http://dx.doi.org/10.1142/S0217732395003173

http://dx.doi.org/10.1007/BF02506415

http://dx.doi.org/10.1007/BF02096949

http://dx.doi.org/10.1007/BF01506544

http://dx.doi.org/10.1016/0040-9383(84)90021-1

http://dx.doi.org/10.1007/BF01399506

http://dx.doi.org/10.1016/0920-5632(95)00434-B

http://dx.doi.org/10.1007/BF01217907

http://dx.doi.org/10.1016/0920-5632(88)90366-0

http://dx.doi.org/10.1016/0370-2693(88)90458-3

http://dx.doi.org/10.1016/0370-1573(91)90117-5

http://dx.doi.org/10.1016/0550-3213(82)90071-2

http://dx.doi.org/10.1016/0550-3213(82)90505-3

http://dx.doi.org/10.1007/BF01223371

http://dx.doi.org/10.1016/0550-3213(81)90202-9

http://dx.doi.org/10.1016/0370-2693(90)90249-6

http://dx.doi.org/10.1007/BF02099494

http://dx.doi.org/10.1016/0550-3213(92)90386-P

http://dx.doi.org/10.1016/0370-2693(88)91381-0

http://dx.doi.org/10.1016/0370-2693(80)90138-0

http://dx.doi.org/10.1016/0550-3213(80)90460-5

http://dx.doi.org/10.1016/0370-2693(82)90570-6

http://dx.doi.org/10.1016/0550-3213(83)90065-2

http://dx.doi.org/10.1016/0550-3213(90)90366-L

Documents

Equivariant symplectic geometry of gauge fixing in Yang–Mills theory