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Seiberg-Witten maps from the point of view of consistent deformations of gauge theories

a r X i v :h e p -t h /0106188v 3 7 S e p 2001ULB-TH/01-17

hep-th/0106188

Glenn Barnich a,?,Maxim Grigoriev b and Marc Henneaux a,c a Physique Th′e orique et Math′e matique,Universit′e Libre de Bruxelles,Campus Plaine C.P.231,B–1050Bruxelles,Belgium b Lebedev Physics Institute,53Leninisky Prospect,Moscow 117924,Russia c Centro de Estudios Cientif′?cos,Casilla 1469,Valdivia,Chile Abstract Noncommutative versions of theories with a gauge freedom de?ne (when they exist)consistent deformations of their commutative counterparts.General aspects of Seiberg-Witten maps are discussed from this point of view.In particular,the

existence of the Seiberg-Witten maps for various noncommutative theories is related

to known cohomological theorems on the rigidity of the gauge symmetries of the

commutative versions.In technical terms,the Seiberg-Witten maps de?ne canonical

transformations in the antibracket that make the solutions of the master equation for

the commutative and noncommutative versions coincide in their anti?eld-dependent

terms.As an illustration,the on-shell reducible noncommutative Freedman-Town-

send theory is considered.

?Research Associate of the Belgium National Fund for Scienti?c Research

1Introduction

A remarkable feature of Yang-Mills theory is the(formal)rigidity of its gauge structure. Namely,there is no consistent deformation of the Yang-Mills action

I Y M[Aμ]=?

1Actually,rigidity holds only if the gauge group has no Abelian factor.When there is an Abelian factor,there can be deformations of(1.1)that truly deform the gauge structure;these involve conserved currents[3].One can easily show that such a possibility does not arise in the noncommutative deformation of U(N)Yang-Mills theory because of the derivative structure of the coupling,see[5].

where commutative and noncommutative versions of Wilson lines were compared and in [21],where the SW map was computed in the framework of Kontsevich’s approach to deformation quantization.A reference that also uses cohomological arguments(though not in the BV formalism)is[22].An explicit inverse SW map was given in[23].

2Seiberg-Witten maps:general considerations

2.1Noncommutative gauge theories as consistent deformations

of commutative gauge theories

Consider a gauge theory with action I0[??]= d n x L0([??]),where the notation f=f([y]) means that the function f depends on the variable y and a?nite number of its derivatives. We put a hat on the?elds in anticipation of changes of?eld variables performed below to unhatted?elds.We denote the in?nitesimal gauge transformations byδ0,???i=R i0([??],[??]) (the dependence on the gauge parameters??αand their derivatives is of course linear).One form of the Noether identities expressing gauge invariance is

δL0

2g2I2+...,

(2.3)

?R i=R i

0+gR i1+

g2Z2α+...,

(2.5)

such that the Noether identities and the reducibility equations are preserved order by order in the deformation parameter g,

δ?L

but we require that each term L i in the power expansion?L=L0+gL1+···be a local function,i.e.,contains a?nite number of derivatives.

With these de?nitions,noncommutative extensions of commutative gauge theories are clearly(when they exist)consistent deformations of their commutative counterparts,the deformation parameter g multiplying the matrixθμνde?ning the non-commutativity of the coordinates(see below).

2.2Gauge structure

As explained e.g.in[16],the most general gauge transformation is obtained by adding to the transformationsδ0,???i=R i0(or R i),in which the gauge parameters are chosen arbitrarily(in particular,can be functions of the?elds and their derivatives),an arbitrary antisymmetric combination of the equations of motion.Because the most general gauge transformation explicitly refers to the dynamics,the algebra of all the gauge transfor-mations of pure Maxwell theory,say,is di?erent from the algebra of Born-Infeld theory. However,in both cases,the relevant information about the gauge transformations is con-tained in the transformationδAμ=?μ?,which is identical for the two theories.Only the “on-shell trivial”gauge transformations,involving the equations of motion,are di?erent. For this reason,one says that pure Maxwell theory and Born-Infeld theory have identical gauge structures.

More generally,one says that two gauge theories have identical gauge structures if it is possible to choose their generating sets such that they coincide.The generating sets are precisely the subsets of the gauge algebra that contain the relevant information about all the gauge symmetries,in the sense that any gauge transformation can be obtained from the generating set by choosing appropriately the gauge parameters(possibly,as functions of the?elds)and adding if necessary an on-shell trivial gauge symmetry[16].There is a huge freedom in the choice of generating sets.Furthermore,one may also rede?ne the reducibility relations2.Moreover,the equivalence may become manifest only after one has rede?ned the?eld variables.So in order to show that two theories have identical gauge structures,one must establish the existence of a?eld transformation such that the two generating sets can be made to coincide(through rede?nitions of the gauge parameters and addition of on-shell trivial gauge transformations).

One virtue of the anti?eld formalism is that all this freedom can be neatly taken into account trough canonical transformations(see e.g.[24,12,16]).For this reason,we review how consistent deformations are formulated in the anti?eld(BV)formalism.

2.3Cohomological reformulation of consistent deformations in

the BV formalism

In the framework of the BV formalism,all the information on the invariance of the action and the algebra of gauge symmetries is encoded in an extended action satisfying the so-called master equation.The problem of consistent deformations of gauge theories can then be reformulated[25](see[26]for a review)as the problem of deforming the solution of the master equation in the space of local functionals(while maintaining the master equation

itself).In the present context,local functions are formal power series in the deformation parameter,each term depending on the original?elds,the ghosts,ghosts for ghosts,their anti?elds and a?nite number of derivatives of all these?elds.Local functionals are identi?ed with equivalence classes of local functions up to total divergences(see[27]for more details).We shall denote the solution of the(classical)master equation for the original and deformed theories by S0and?S,respectively.One has

S0=I0+ d n x???i R i0([??],[?C])+?C?αZ0α([??],[?ρ])+...,

(2.8)

2(?S,?S)=0.

(2.11)

Here,the?Cαare the ghosts replacing the gauge parameters,the?ρA are the ghosts for ghosts,while???i,?C?αand?ρ?A are the associated anti?elds,of respective anti?eld number 1,2,3(see e.g.[9,16,17]for details).

Note that the anti?eld-independent part of the solution of the master equation is just the classical action.The information about the gauge symmetries,their algebra,the reducibility equations etc is contained in the anti?eld-dependent part.Thus,if the original and deformed theories have the same gauge structure,S0and?S have the same anti?eld-dependent part and vice-versa.The advantage of the cohomological reformulation is that standard techniques of deformation theory can now be applied.

In particular,(2.11)implies that

(?S,

??S

?g |g=0=S1is a cocycle of the

BRST di?erential s0=(S0,·)of the undeformed theory,

(S0,S1)=0.

(2.13)

A deformation is trivial if it can be undone through a canonical,i.e.,antibracket preserving,transformation3:

?S[?φ([φ],[φ?];g),?φ?([φ],[φ?];g);g]=S

0[φ,φ?],

(2.14)

Here–and throughout below–,the original?elds and ghosts are collectively denoted by ?φ,while?φ?denotes collectively all the anti?elds.Thus,a generic canonical transformation mixes the original?elds,the ghosts and the anti?elds.Equivalently,this means that?S can be obtained from the original S0through the inverse canonical transformation

?S[?φ,?φ?;g]=S

0[φ([?φ],[?φ?];g),φ?([?φ],[?φ?];g)].

(2.15)

This is the case i?the cocycle??S

?g =(?S,?Ξ).

(2.16)

Indeed,if(2.16)holds,then a canonical transformation with the required properties is given by

φA(x)=P exp( g0dg′ ·,?Ξ(g′) )?φA(x),φ?A(x)=P exp( g0dg′ ·,?Ξ(g′) )?φA(x).

(2.17)

Conversely,if the deformed action can be obtained from the undeformed one by an canon-ical transformation for any g,then the passage from?S(g)to?S(g+δg)is an in?nitesimal canonical transformation(by the group property of canonical transformations)and(2.16) holds.It will be useful in the sequel to introduce the generating functional F[φ,?φ?;g]of “second type”in ghost number?1such that

?φA(x)=δL F

δ?φA(x),

(2.18)

(see[12]and appendix A of[28]for material on antibracket preserving transformations).

It follows from(2.12)and(2.16)that a necessary condition for the existence of a non trivial deformation is the existence of a non trivial cohomology class for the deformed theory.Because every cocycle of the BRST di?erential?s=(?S,·)of the deformed theory gives to lowest order in g a cocycle of the BRST di?erential s0=(S0,·)of the undeformed one,and furthermore,every coboundary of the undeformed theory can be extended to a coboundary of the deformed theory,it follows that non trivial deformations are controlled by the local BRST cohomology of the undeformed theory.In particular,if this cohomology is empty in the relevant subspace of the space of local functionals,it follows that the deformation is trivial.By relevant subspace,we mean the subspace of local functionals of ghost number0possibly restricted through additional requirements like global symmetries or power counting restrictions,depending on the problem at hand.

Elements of H0(s0)are called non trivial in?nitesimal deformations.One can further-more show that the obstruction to extending in?nitesimal deformations are controlled by the antibracket map

(·,·)M:H0(s(0))?H0(s(0))?→H1(s(0))

(2.19)

but this will not be needed here.

2.4SW maps in the context of deformation theory

Given a non trivial consistent deformation of a gauge theory,one may ask whether the gauge symmetries,their algebra and their reducibilities are equivalent to the one of the undeformed theory through allowed rede?nitions of the most general type.The allowed rede?nitions of the gauge symmetries can involve rede?nitions of the gauge parameters that contain the?elds themselves,as well as the possible addition of on-shell trivial symmetries [16].Note that if the classical action?I is equivalent through?eld rede?nitions to the action I0,then the gauge symmetries and their structure are of course equivalent,whereas the

converse does not hold(e.g.,as explained above,Maxwell theory and Born-Infeld theory are based on inequivalent actions but their gauge structures are identical).

Because the gauge symmetries and their structure are described in the master equation through terms with strictly positive anti?eld number,this question amounts to the question of the existence of a canonical transformation?φ[φ,φ?;g],?φ?[φ,φ?;g]such that

?S[?φ[φ,φ?;g],?φ?[φ,φ?;g];g]=S

0[φ,φ?]+V[?,g]

(2.20)

That is,if one can absorb all the anti?eld-dependence through a canonical transformation, the only e?ect of the deformation will be indeed(after rede?nitions)just to change the original action I0[?]into I e?[?;g]≡I0[?]+V[?,g],

I0[?]→I e?[?;g]=I0[?]+V[?;g],

(2.21)

V[?;g]=gV1[?]+

?g =

δ?φ?,?φ?;g]=S0[φ,

δL F

for some C′[?]that depends only on the original?elds and not on the anti?elds(and which is of course annihilated by s0,s0C′=0).Then(2.20)can be achieved through a succession of canonical transformations.

Indeed,let?z=(?φ,?φ?)and consider the canonical transformation z1=exp(g)(·,Ξ(1))?z, so that?z=z1?g(z1,Ξ(1))+O(g2).It follows that

?S[?z([z1];g);g]=S

0[z1]+gV1[?1]+g2?S(2)[z1]+O(g3),

(2.27)

for some?S(2)[z1]and s0V1=0.More generally,assume that the Seiberg-Witten map has been constructed to order k,i.e.,that we have constructed a canonical transformation z k=z k[?z;g]such that

?S[?z[z k;g];g]=S

0[z k]+

i=1

g i V i[?k]+g k+1?S(k+1)[z k]+O(g k+2),

(2.28)

with s0V i=0.The equation1

(though somewhat cumbersome)procedure for explicitly removing the anti?eld-dependent terms and?nding the generators of the successive canonical transformations that bring the gauge structure back to its original form through the use of homotopy formulas.A similar situation prevails for the noncommutative Freedman-Townsend model,as we shall analyse in section4.

Finally,it is also possible to analyze along the above lines a situation where for instance the deformation of the action and the gauge symmetries are non trivial,while the algebra and higher order structure constants of the gauge symmetries are equivalent.

2.6Weak Seiberg-Witten gauge equivalence

By expanding the condition(2.20)for the existence of the Seiberg-Witten maps according to the anti?eld number,one recovers formulas familiar from the Yang-Mills context(but modi?ed to weak relations).

For instance,if we denote by f i([?],g)and gα([?],[C],g)the expression of the hatted ?elds and ghosts in terms of the original?elds and ghosts when the anti?elds are set equal to zero in(2.18),

f i([?],g)=δL F

δ?C?α φ?=0

(2.32)

the condition(2.20)becomes at anti?eld number zero

?I[f;g]=I e?[?;g].

(2.33)

In anti?eld number1,one gets

?δ?λ??i≈δλ??i,

(2.34)

where the even gauge parameter?λcorresponds to the odd function gα([?],[C],g)whileλcorresponds to Cα.The relation(2.34)generalizes,in the open,reducible algebra case, the key relation of[1]that de?nes the Seiberg-Witten maps.

Finally,in anti?eld number2,one gets an integrability condition for(2.34),as well as possible(admissible)rede?nition of the reducibility functions.The integrability condition is the BRST version of the Wess-Zumino type consistency condition[35]that one can deduce directly from the weak Seiberg-Witten equivalence condition(2.34).In higher anti?eld number,one gets higher-order integrability conditions related to the existence of higher-order structure functions.

3Noncommutative Freedman-Townsend model

3.1Preliminaries

We assume from now on the space-time manifold to be R n with coordinates xμ,μ= 1,...,n.The Weyl-Moyal star-product is de?ned through

f?g(x)=exp(i∧12)f(x1)g(x2)|x

1=x2=x ,∧12=

for a real,constant,antisymmetric matrixθμν.The parameter g is the deformation pa-rameter.Standard formulas are recovered by taking g=1,while the commutative case corresponds to g=0.

Let M=m A(x)T A with T A generators of the Lie algebra u(N),i.e.,antihermitian matrices.The coe?cients m A(x)are real,commuting or anticommuting?elds.If her-mitian conjugation for the multiplication of Z2graded functions is de?ned by(mn)?= (?)|m||n|nm,then hermitian conjugation of matrix valued function also satis?es(MN)?= (?)|m||n|NM.We denote the invariant metric Tr T A T B by g AB,Tr T A T B=g AB.It is non-degenerate.

The graded Moyal bracket de?ned by

[M?,N]=M?N?(?)|M||N|N?M

(3.2)

is again a u(N)valued function,because(M?N)?=(?)|M||N|N?M.This is a straight-forward extension of the reasoning of[36]in the case where one allows the functions to belong to a Z2graded algebra.Furthermore,the covariant derivative and associated?eld strength are de?ned as follows:

M=?μM+[?Aμ?,M],

[?Dμ,?Dν]M=[?Fμν?,M],

?F μν=?μ?Aν??ν?Aμ+[?Aμ?,?Aν].

(3.3)

A key property of the Moyal star-product is

M?N=MN+?μΛμ.

(3.4)

As a consequence,if boundary terms are neglected,

d n x Tr M?N= d n x Tr MN= d n x Tr N?M(?)|M||N|,

(3.5)

d n x Tr M?N?O= d n x Tr O?M?N(?)|O|(|M|+|N|),

(3.6)

d n x Tr M?[N?,O]= d n x Tr[M?,N]?O,

(3.7)

d n x Tr?DμM?N=? d n x Tr M??DμN.

(3.8)

3.2Action and gauge algebra of noncommutative FT model The noncommutative U(N)Freedman-Townsend model exists in four dimensions.It is most conveniently formulated in?rst order form.The action is

?I= d4x Tr ?12?Aμ??Aμ

(3.9)

in Minkowski space-time,with signature(?+++),?0123=1and?0123=?1.The action is invariant under the gauge transformations

?δ?λ?B

μν

=?D[μ?λν],?δ?

=0,

(3.10)

with gauge parameters?λμ.The gauge algebra is abelian,[?δ?

λ1,?δ?

λ2

]=0and reducible

on-shell.Indeed,if?λμ=?Dμ?η,then the gauge transformation reduces to on-shell trivial transformations,

?δ?D?η?B

μν

?μνρσ[

δ?I

?μνρσBμνFρσ+

only deformations of the Freedman-Townsend model are precisely exhausted by the ad-dition of functions of the Aμand their derivatives.This implies,in particular,that the noncommutative Freedman-Townsend model must be amenable to the form

?I=I

0+V[A;g].

(3.16)

Note that the?eld Aμis auxiliary in(3.12)–i.e.,can be eliminated through its own equation of motion.It remains auxiliary in the deformed theory in the sense that its equations of motion can still be solved as formal power series in g.Note also that the equations of motion for Bμνare left unchanged in the deformation and imply Fμν=0.By solving this constraint,A=g?1dg,the action becomes that of the non-linear sigma model modi?ed by higher dimensionality operators.

4Seiberg-Witten map for Freedman-Townsend model

4.1Existence of the Seiberg-Witten map

The solution of the master equation for the noncommutative Freedman-Townsend model is given by

(4.1)?S= d4x Tr ?12?Aμ??Aμ+

+?B?μν??D[μ?Cν]+?C?μ??Dμ?ρ+

2?μνρσ?Bμν?Fρσ+

[?B?μν,?B?ρσ]?μνρσ?ρ .

It follows from the cohomological analysis of the previous section that the Seiberg-Witten maps exists:by a canonical transformation,one can transform the functional(4.1) into the solution of the master equation for the commutative theory,plus V i([Aμ])-terms that are strictly gauge-invariant4.

One can explicitly construct the canonical transformation order by order in the?elds, using standard cohomological weapons(homotopy formula for the free BRST di?erential etc).We have not been able to sum the formal series obtained in this recursive manner in a concise and useful way,however,except in the u(1)-case,to which we shall therefore now exclusively turn.

4.2SW map in the U(1)case

In the u(1)-case,the solution of the master equation for the commutative model is given by

S0[φ,φ?]= d4x ?12AμAμ+B?μν?[μCν]+C?μ?μρ , (4.3)

Our goal is to?nd a canonical transformation(2.18)such that(2.24)is satis?ed for?S[?φ,?φ?] given by(4.1)and S0[φ,φ?]by(4.3).The searched-for generating functional F[φ,?φ?]takes the form

(4.4)F[φ,?φ?]= d n x ?A?μfμ([A],[H])+?B?[μν]f[μν]([A],[B])+?C?μ(Cμ+2[Cα?,fμ]α)

2θμνf?1?νh,

(4.5)

with

f?1g(x)=

sin∧12

2[f?,h].

(4.7)

It is easy to check that in anti?eld nubmber higher than1,the generating functional(4.4) satis?es(2.24)for arbitrary fμ([A],[H]).

Identifying terms of anti?eld number1in(2.24)one gets,

fμν=Bμν+2[Bμα?,fν]α?2[Bνα?,fμ]α?

igθαβ

for l=0,1,...and Hμρσ=?[μBρσ],a particular expression for the contracting homotopy σ0is given by

(4.13)

σ0f= 10dt?zα](ty,tz).

We leave it to the reader to check that the generating functional F does the job of transforming(4.1)into(4.3)no matter how the gauge-invariant functions fμ([A],[H])is chosen(it must just be invertible).That there is an ambiguity in the SW map,character-ized by addition of the gauge-invariant functions to fμ,fμνand also to the higher anti?eld number terms in F(so that there are in fact maps)is not surprising:any rede?nition that involves only gauge invariant quantities preserves the gauge structure.

To summarize,a particular class of solutions for the SW map for the noncommutative abelian Freedman-Townsend model has been obtained.The new feature of this map, compared with the SW map for Yang-Mills models,is that the generating functional F[φ,?φ?]is quadratic in some of the anti?elds,so that the transformations of some of the ?elds contain the anti?elds.This is related to the fact that the equations of motion appear in the reducibility identities(while the Yang-Mills gauge structure is de?ned not just on-shell,but also o?-shell).This feature is easily incorporated at no cost in the anti?eld formalism.

Finally,we note that the Lagrangian of the non-commutative Freedman-Townsend model can be mapped to the Lagrangian of the commutative one up to second order inθ. It is an interesting question to investigate whether this holds to all orders.In this context, we recall that the2-dimensional commutative and noncommutative WZW models are known to be equivalent not just in their gauge structure but also in their action[43,44].

5Conclusion

The conclusion in[45]is“that there should be an underlying geometric reason for the Seiberg-Witten map.”The analysis of this paper shows that there is at least a deformation theoretic reason for the existence of this map in the following sense.

Consistent deformations of gauge theories with non trivial deformations of the gauge structure are in general severely constrained.The appropriate framework to study these constraints in the general case(reducibility,closure only on-shell)is the anti?eld-anti-bracket formalism.The cohomology that controls non trivial deformations of the gauge structure is the local BRST cohomology,and how non trivial cocycles depend on the anti?elds.

By analyzing explicitly the noncommutative Freedman-Townsend model,these consid-erations have been shown to apply beyond the original Yang-Mills framework.Because they do not depend on the precise deformation considered,they also apply for defor-mations that involve for instance more complicated star-products than the Weyl-Moyal star-product.The analysis can also be straightforwardly extended to models with higher rank p-forms as discussed in[37].

The cohomological theorems of[3,4,27],which guarantee the existence of the SW maps were studied initially with quantum motivations in mind(they control perturbative renormalizability–i.e.,gauge invariance of the needed counterterms–as well as candidates anomalies for general(e?ective)theories with the same gauge structure as Yang-Mills

models).The present paper clearly indicates their relevance in the classical context as well.

Acknowledgments

M.H.is grateful to Kostas Skenderis for informative discussions at an early stage of this work.This research has been partially supported by the“Actions de Recherche Concert′e es”of the“Direction de la Recherche Scienti?que-Communaut′e Fran?c aise de Belgique”,by IISN-Belgium(convention4.4505.86),by Proyectos FONDECYT1970151 and7960001(Chile),by the European Commission RTN programme HPRN-CT-00131, in which the authors are associated to K.U.Leuven and by INTAS grant00-00262.The work of M.G.is partially supported by the RFBR grant99-01-00980.He is grateful to the Free University of Brussels for kind hospitality.

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