KCL-TH-06-06 UUITP-10/06 HIP-2006-28/TH USITP-06-03 hep-th/0607156 February 1, 2008 7 0 On the covariance of the Dirac-Born-Infeld-Myers action 0 2 n a J P.S. Howe1, U. Lindstr¨om2,3 and L. Wulff4 0 3 2 1 Department of Mathematics, King’s College, London, UK v 6 2 Department of Theoretical Physics, Uppsala, Sweden 5 1 3 HIP-Helsinki Institute of Physics, P.O. Box 64 FIN-00014 University of Helsinki, Suomi-Finland 7 0 6 4 Department of Theoretical Physics, Stockholm University, Sweden 0 / h t - p e h : v Abstract i X r a A covariant version of the non-abelian Dirac-Born-Infeld-Myers action is presented. The non- abelian degrees of freedom are incorporated by adjoining to the (bosonic) worldvolume of the brane a number of anticommuting fermionic directions corresponding to boundary fermions in the string picture. The proposed action treats these variables as classical but can be given a matrix interpretation if a suitable quantisation prescription is adopted. After gauge-fixing and quantisation of the fermions, the action is shown to be in agreement with the Myers action derived from T-duality. It is also shown that the requirement of covariance in the above sense leads to a modified WZ term which also agrees with the one proposed by Myers. Contents 1 Introduction 1 2 The geometry of M and the non-abelian gauge field 2 c 3 The DBIM action 5 4 The Wess-Zumino term 7 5 Comparison of WZ terms 11 6 Discussion 15 A Appendix 16 1 Introduction An intriguing feature of string theory is the fact that one can have coincident D-branes. The lowest-order action for such stacks of branes includes the Yang-Mills action for the non-abelian gauge fields which arise when one takes the coincidence limit. Since the gauge part of the action for a single D-brane is Born-Infeld one would expect that a non-abelian generalisation of this action shouldberequiredinthecase of coincident branes. Althoughtherehasbeenalot ofwork on this topic it is still not completely clear what this action is and how it should incorporate invariance principles. It is the purpose of this note to propose such an action for both the Dirac-Born-Infeld and the WZ terms which should be present. The derivation we give is strictly speaking only valid in a certain approximation, which we explain below, but we shall argue that it is not unreasonable to expect that it can be extended beyond this. Many features of the bosonic terms in the non-abelian action are known. Some years ago, Tseytlin [1] put forward the proposal that the ordinary Born-Infeld action could be generalised to the non-abelian case by using the same formula with an overall symmetrised trace. Although this does not incorporate all the terms in the effective string action [2] it is nevertheless a well- defined object to work with. Subsequently, starting from the Tseytlin action, Myers derived a non-abelian Dirac-Born-Infeld action by demanding that lower-dimensional brane actions be consistent with T-duality [3]. He also used T-duality to derive a non-abelian Wess-Zumino term and showed that this has the property that higher degree RR forms can couple to a Dp-brane giving rise to a dielectric effect. Similar results were obtained from matrix theory [4, 5]. We shall show that Myers’s results can be understood from the point of view of invariance under diffeomorphisms of the brane and gauge symmetries. TheMyers version of the DBI action was derived in thephysical gauge where(p+1) of the coor- dinates of the target space are identified with those of the brane and the transverse coordinates are taken to bethe scalar fields. These are then promoted to matrices in the non-abelian theory. This procedure clearly breaks diffeomorphism symmetry for the brane. Although non-abelian gauge invariance is maintained through the use of covariantised pull-backs it is not at all obvi- 1 ous that the Myers action is invariant with respect to gauge transformations of the background gauge fields. This is also true for the Wess-Zumino term which involves the RR potentials as well as the B field. In fact, it is not clear what the non-abelian generalisation of the modified field strength which appears in the action for a single D-brane should be. A proposal for this was made in [6] in the supersymmetric context and this is the one which we shall use here. In this paper we shall derive non-abelian DBI and WZ actions in a formalism which is inspired by the use of boundary fermions to describe non-abelian degrees of freedom in open string theory [7, 8, 9, 10]. We shall work in the approximation in which these fermions are taken to be classical variables. Mathematically this amounts to extending the worldvolume of the brane by a number of fermionic directions and replacing the brane embedding in the target spacetime by a generalised embedding defined as a map from the extended worldvolume to the target space [6]. Although the formalism is not a fully-fledged matrix formalism, the results we derive can be compared tothoseintheliterature ifwereplace thePoisson bracket (in thefermionicpartofthe space)bythematrixcommutatorandimposethesymmetrisedtraceprescription. Thisapproach can be justified to some extent in the world-sheet picture. In the papers cited above it is shown how quantisation of the fermions leads to the fermions being replaced by gamma matrices, so that functions of them become matrices, and how correlation functions of products of operators involving the fermions become the path-ordered trace of products of matrices. Since we are concerned with operators at the same boundary point it is natural to adopt the prescription that the path-ordered trace goes over to the symmetrised trace in this case. Moreover, it is canonical practice to replace Poisson brackets by commutators in the quantisation procedure. The DBI action we propose has a very simple structure. Since the extended space is actually a superspace it is natural to replace the Born-Infeld determinant with a superdeterminant. The matrix in this superdeterminant is the sum of the pull-back of the target space metric and an abelian two-form field strength modified by the pull-back of the B-field. The non-abelian gauge field emerges from the expansion of the abelian gauge field in the fermionic coordinates. This action is manifestly invariant under diffeomorphisms and gauge transformations and we show explicitly that it reproduces the Myers DBI action in the physical gauge. Our WZ action looks very similar to the Myers WZ action. We show that the couplings of the branes to scalar commutators can be motivated by diffeomorphism invariance while the couplings to the higher rank RR forms are then required by RR gauge symmetry. However, our formalism is not manifestly invariant and one has to work to prove these results. Although the structure of our WZ action is very similar to Myers’s there is a difference in that his involves contractions of forms with the scalar commutator whereas ours involves a similar contraction but in the fermionic directions. We show explicitly how the terms collect together to gives the Myers result. Our conventions for differential forms are given in the appendix. 2 The geometry of M and the non-abelian gauge field d We shall be interested in a p-brane specified by an embedding f : M M where M is the → worldvolume of the brane and M the target space, both spaces being bosonic. In order to incor- poratenon-abelian degrees of freedomweextend theformerto asuperspace,M, thecoordinates of which we denote by yM = (xm,ηµ). The coordinates of the target space are denoted by xm. The embedding is replaced by a generalised embedding f : M M. The spcace M is equipped → with an abelian gauge field such that the modified field strength A b c c 2 K := d f∗B , (2.1) A− where B is the NS two-form potential on the targetbspace, is invariant under gauge transforma- tions of both objects. It is furthermore assumed that K is non-singular. µν Following the ideas developed in [6] we use the field K to specify horizontal subspaces in the tangent spaces of M. If ω is a one-form on M then we define its horizontal component to be c c ω = ω K νω , (2.2) m m m ν − where b K µ := K Nνµ (2.3) m mν and Nµν := (K )−1 . (2.4) µν We can view this as a change of basis if we also identify ω = ω . For a vector v we have µ µ b vm = vm vµ = vµ +vmK µ (2.5) m b We define the horizontal component obf K itself to be , F := K K NµνK . (2.6) mn mn mµ νn F − We note that K has no mixed components in the hatted basis. It is straightforward to compute the transformation properties of various objects under diffeomorphisms. In particular, we have δK µ = vn( K µ K µ)+vν∂ K µ+ vµ Nµν∂ vn m n m m n ν m m ν nm D −D D − F δNµν = vm Nµν +vρ∂ Nµν 2Nρ(µ∂ vν) +2vmNρ(µ∂ K ν) (2.7) m ρ ρ ρ m b D − b b b which implies that b b b b δω = vn ω +vν∂ ω + vnω +Nµν∂ vn ω m n m ν m m n ν nm µ D D F δω = vn ω +vν∂ ω +∂ vnω +∂ vνω vn∂ K νω . (2.8) µ n µ ν µ µ n µ ν µ n ν b b D b b b b b −b b The derivative b is definbed byb b b b b b b b b m D := ∂ K µ∂ . (2.9) m m m µ D − 3 The transformation rule of ω shows that this object is not preserved under general diffeomor- m phisms, only those for which ∂ vn = 0. Transformations which do not satisfy this constraint are µ needed in order to reach thebphysical gauge where the generalised embedding has the form b xm = (xm,xm′(x,η)) . (2.10) However, we do not want to make this gauge choice at this stage as the power of covariance would be lost. Note that any object which has no mixed components in the hatted basis will transform homogeneously under diffeomorphisms. This holds for . mn F We now turn to the emergence of a non-abelian gauge field from the abelian one we have introduced. The requirement that K be non-singular will be satisfied for any background B if µν (d ) is non-singular. We can then use a vertical diffeomorphism to bring to the standard µν µ A A form 1 = η . (2.11) µ µ A 2 The transformations of are A δ = ∂ a+vN(∂ ∂ )+b (2.12) M M N M N N M A A − A whereaistheabeliangauge parameter andb isthepull-back of thegauge parameterforgauge M transformations of the B field. As stated above we can use vµ to go to the standard gauge for . The residual vertical diffeomorphisms are then given by µ A vµ = δµν(∂ a+b vn∂ ) (2.13) ν ν ν n − − A We shall denote in the standard gauge by A ; it transforms as m m A δA = ∂ a+(A ,a)+b +vnF . (2.14) m m m m nm where the Poisson bracket (,) is defined by e (f,g) := δµν∂ f∂ g . (2.15) µ ν F := ∂ A ∂ A +(A ,A ) is the non-abelian field strength tensor, and b denotes the mn m n n m m n m − covariant pull-back of b with respect to the Yang-Mills derivative. This is given by e b := D xmb = (∂ xm+A µ∂ xm)b , (2.16) m m m m m µ m where e A µ := δµν∂ . (2.17) m ν m A The relation between the non-abelian field strength tensor and , in the standard gauge, is F given by 4 = F B B NµνB . (2.18) mn mn mn mµ νn F − − This formula may be taken as the definitioen of thee approperiately modified non-abelian field strength tensor in the presence of a B field. For later use we note the relation between the hatted and tilded bases, valid in the standard gauge. The fermionic components of a one-form ω are the same while ω = ω +B Nµνω . (2.19) m m mµ ν b e e 3 The DBIM action In this section we present the Lagrangian for the DBIM action in the presence of the additional fermionic variables. We set L := g +K (3.1) MN MN MN where g denotes the pull-back of the target-space metric to M. The Lagrangian is then MN simply c = sdetL . (3.2) MN L − p ThisLagrangianobviouslytransformsasadensityunderdiffeomorphismsofM andismanifestly invariant under gauge transformations of both and B. We shall now show that it coincides A with the Myers action [3] in the physical gauge provided that we interprect functions of η as matrices, replace Poisson brackets by commutators and replace integration over the fermionic variables with the symmetrised trace over all matrix factors. The superdeterminant is sdetL = det(L L LµνL )(detL )−1 , (3.3) MN mn mµ νn µν − where Lµν := (L )−1. If we introduce E := g B , then µν MN MN MN − L = E +(dA) (3.4) mn mn mn while L = δ +E (3.5) µν µν µν and L = E A (3.6) mµ mµ mµ − in the standard gauge, A = 1η . We remind the reader that A = ∂ A .We therefore have µ 2 µ mµ µ m 5 L L LµνL = E +(dA) (E A )(δ +E )−1(E +A ) . (3.7) mn mµ νn mn mn mµ mµ µν µν νn nν − − − We want to express this in terms of F and replace the ordinary pull-back on m indices by mn the covariant one defined by D = ∂ +A µ∂ . After a straightforward piece of algebra one m m m µ indeed finds that L L LµνL = E +F E (δ +E )−1E , (3.8) mn mµ νn mn mn mµ µν µν νn − − where e e e E = D xmD xnE mn m n mn E = D xm∂ xnE . (3.9) emν m ν mn e We shall now compute the second term on the RHS of (3.8) in the physical gauge, xm = (xm,xm′(x,η)). In this gauge E = D xm∂ xnE mν m ν mn e = Dmxm∂νxn′Emn′ = ∂νxn′(Emn′ +Amµ∂µxm′Em′n′) = Emn′∂νxn′ , (3.10) e while Eµν = ∂µxm′∂νxn′Em′n′ . (3.11) We therefore have Emµ(δµν +Eµν)−1Eνn = Emp′∂µxp′(δµν +Eµν)−1∂νxq′Eq′n . (3.12) Expanding out theeinverse we find e e e ∂µxp′(δµν +Eµν)−1∂νxq′ = Mp′q′ Mp′r′Er′s′Ms′q′ +... (3.13) − where Mm′n′ := δµν∂ xm′∂ xn′ = (xm′,xn′) . (3.14) µ ν The series of terms is easily summed; the final result is p′q′ Lmn LmµLµνLνn = Fmn +Emn+Emp′ (Q−1 1)E−1 Eq′n (3.15) − − (cid:16) (cid:17) e e e 6 where Qm′n′ =δm′n′ +Mm′p′Ep′n′ , (3.16) and where E−1 denotes the inverse of Em′n′. In order to complete the picture we need to compute detL and show that it equals det−1Q µν in the physical gauge. This is a straightforward exercise using the exptrln formula for the determinant. For example, one has δµνEµν = δµν∂µxm′∂νxn′Em′n′ = Mm′n′Em′n′ = tr(ME) (3.17) − in the physical gauge. One therefore obtains detL = exp( trln(1+ME)) = (detQ)−1 (3.18) µν − as required. So we indeed find that sdetLMN = det (Emn +Fmn +Emp′[(Q−1 1)E−1]p′q′Eq′n)det Q . (3.19) − − − q p It is quite remarkable that this exepression agreees precisely with Myers’es result [3] provided that one interprets it in the way we have suggested. 4 The Wess-Zumino term In [3] Myers gives an expression for the WZ term for a Dp-brane modified to the non-abelian case. The most remarkable feature of this term is that p-branes can couple to higher-degree RR potential forms. The mechanism for this is that one can lower the degree of a pulled-back form by two by contracting a pair of transverse indices with Mm′n′, which is the commutator in Myers. In this way, for example, a five-form RR field can give rise to a three-form and thus couple to a D2-brane. However, it is far from obvious that Myers’s WZ term is gauge-invariant or invariant under diffeomorphisms of the brane. In this section we construct a WZ term in our model which has these properties, although not manifestly. Its structure is very similar to the Myers WZ term, although the match up of the terms is not quite straightforward. The Wess-Zumino term for a Dp-brane is 1 = √N exp( i )eF C , (4.1) WZ N L −2 (cid:20) (cid:21)p+1,0 X b where N := det Nµν. The subscript (p+1,0) indicates that the (p+1,0)-form component is to be projected out, a (p,q)-form being a (p+q)-form on M with even degree p and odd degree q in the hatted basis. The RR potentials C are pulled back to M with the hatted pull-back, e.g. c b c 7 C = xm∂ xnC , (4.2) mν m ν mn D where b = ∂ K µ∂ . (4.3) m m m µ D − The point of using this pull-back rather than the one defined with the straightforward gauge covariant derivative is that it is invariant with respect to the abelian gauge transformations of both and B. Theoperation i appearingin (4.1) denotes thecontraction of a form with Nµν. N A Again this is invariant under gauge transformations of and B. However, the full expression A is neither manifestly covariant under diffeomorphisms of M nor under gauge transformations of the RR fields. c First supposethat theWZ form for a Dp-brane, dividedby√N, transforms in aregular fashion, i.e. without a term involving ∂ vm, and let L be the dual of this form, then we can take the µ Lagrangian, regarded as a function, to be L =√NL, and we have WZ b δL = (vm +vµ∂ )L+ vmL (4.4) m µ m D D while b b b δ√N = (vm +vµ∂ )√N +√NK (vmNµρ∂ K ν Nµρ∂ vν) . (4.5) m µ µν ρ m ρ D − Combining these two rebsults it isbeasy to see that b b δ(√NL) =( 1)M∂ (vM(√NL)) . (4.6) M − We therefore see that if L transforms regularly then L will transform in the desired fashion WZ under diffeomorphisms of M. The transformations of the pulled-back RR forms do include irregular terms and so the problem is to show that these all cancel between the different terms in the action involving thecsame RR field but different powers of i . In fact, we can use this N to argue that these terms must be present. As a simple example, consider the C terms in the 3 action for a D2-brane. If we focus on only the irregular parts of the transformation of C we 3 have δC 3Nµν∂ vq C (4.7) mnp ν q[m |µ|np] ∼ F Clearly we need to add somethingb to the Lagranbgian prboportional to N to cancel this varia- tion. The obvious expression to try is i C , which involves the part of C with two fermionic N F indices. Since has no fermionic indices, this term involves C . The irregular terms in the µνp F transformation of this field are b b δC 2∂ vqC +Nρσ∂ vq C . (4.8) µνp (µ |q|ν)p ρ qp µνσ ∼ F The terms of interest in thebtransformatbiobn of i C arebtherefbore N F b 8 δ(NµνC ) NµνNρσ∂ vqC µν[m np] ρ µνσ q[m np] F ∼ F F +2Nµν∂ vqC . (4.9) b µ b bqν[mFnp] The first line on the RHS vanishes as it involves binb three dimensions, while the second F ∧F is equal to +2 multiplied by the RHS of (4.7). Expanding out the exponential in (4.1) we see that precisely the right coefficient is generated in order for these two terms to cancel. The remaining terms can easily be seen to be the regular terms in the transformation of the part of the D2-brane action which involves C . 3 This line of reasoning can easily be extended to the general case. For a given brane, a given RR field will appear in a sequence of terms with increasing powers of and i . It is not difficult N F to verify that the irregular variations in a given term cancel against those coming from the two adjacent terms. We can therefore conclude that the WZ term is invariant up to a total derivative under abelian gauge transformations of and B and diffeomorphisms of M. Since A the non-abelian gauge transformations arise from the vertical diffeomorphisms combined with the abelian gauge transformations of in the standard gauge we are therefore assuredcthat the A WZ term will be invariant under these. Theaboveconsiderationsindicatetheneedforthei termsbutdonotmixRRformsofdifferent N rank. ThefullstructureisrequiredbydemandinggaugeinvarianceforthebackgroundRRfields. Since the WZ Lagrangian is a sum of standard WZ terms of different rank one might think that this is obvious but closer inspection shows that it is not, because the i operation does not N commute with exterior differentiation. The proof of gauge invariance is not difficult however. We shall focus on the IIA case for simplicity. The gauge transformations of a (p+1)-form potential is δC = dΛ Λ H (4.10) p+1 p p−2 − where H is the NS three-form, and Λ is used to denote the gauge parameters. Pulled back to M, H becomes dK. So the gauge transformation of the WZ form is − c δ WZ = √N e−21iN (d(ΛeF)+dN−1ΛeF) , (4.11) L p+1,0 h X i b b where N−1 := K (4.12) −F Now is horizontal and is not acted on by i . It therefore plays no essential roˆle in the proof N F that this expression gives rise to a total derivative. From this point of view it may as well be absorbed into the parameters Λ; that is, we regard ΛeF as the redefined Λ. Each Λ appears in pairs of terms which have the same degree. A general pair of terms in (4.11) has the form 1 1 ( )n√N in(dΛ) in+1(dN−1Λ) . (4.13) −2 N p+1,2n− 2n+2 N p+1,2n+2 (cid:18) (cid:19) b b 9