The group structure of non-Abelian NS-NS transformations
aa r X i v : . [ h e p - t h ] M a y UG-FT-267/10CAFPE-137/10May 2010
The group structure of non-AbelianNS-NS transformations
Bert Janssen and Airam Marcos-Caballero Departamento de F´ısica Te´orica y del Cosmos andCentro Andaluz de F´ısica de Part´ıculas ElementalesUniversidad de Granada, 18071 Granada, Spain
ABSTRACT
We study the transformations of the worldvolume fields of a system of multiple coincidingD-branes under gauge transformations of the supergravity Kalb-Ramond field. We find thatthe pure gauge part of these NS-NS transformations can be written as a U ( N ) symmetry ofthe underlying Yang-Mills group, but that in general the full NS-NS variations get mixed upnon-trivially with the U ( N ). We compute the commutation relations and the Jacobi identitiesof the bigger group formed by the NS-NS and U ( N ) transformations. E-mail address: [email protected] E-mail address: [email protected] Introduction
Although the new physics associated with systems of multiple coinciding D-branes has beenknown for more than a decade and a half [1], there are still a few open issues concerningthe effective action that describes the low-energy dynamics of the system. One of the famousunsolved problems is the construction of a non-Abelian Born-Infeld action (see for example [2]and references thereof). The form of the Chern-Simons action is much better known, throughthe work of [3]-[11]. Still, also here are a few open issues left, such as the gauge invariance ofthe action. The invariance under U ( N ) gauge symmetry is straightforward to show, as theentire action is written in terms of objects that transform in the adjoint representation and istraced over all Yang-Mills indices. However, the Chern-Simons term is also the one responsiblefor the couplings of the multiple D-brane system to the background fields of supergravity andthe invariance of this part of the action under the gauge transformations of the supergravitybackground field has drawn little or no attention.A few early attempts to prove the gauge invariance were made in [4, 12], but the first syste-matic approach was done is a series of papers by J. Adam et al., proving the explicit invarianceunder Ramond-Ramond (R-R) and massive gauge transformations [13, 14], under Neveu-Schwarz-Neveu-Schwarz (NS-NS) transformations [15] and unifying it in a global picture in[16].One of the remarkable results of these papers is that the NS-NS gauge transformations aremuch more complicated than the R-R or the massive gauge transformations, as they not onlyaffect the supersymmetry background fields, but also the worldvolume fields living on the D-branes. The reason why this is so, is a mixture of Abelian and non-Abelian effects: already inthe case of a single (Abelian) D-brane, the Born-Infeld vector transforms with the pullback ofthe gauge parameter Σ µ of the NS-NS transformation of the Kalb-Ramond field B µν . Howeverin the non-Abelian case this pullback is performed with U ( N ) covariant derivatives, whichunder T-duality generates non-trivial transformation rules for the embedding scalars, in verymuch the same way as the dielectric couplings to higher-form R-R potentials appear in theaction of multiple coinciding D-branes [11].This leads immediately to a second issue: where the R-R and massive transformations aremerely a straightforward generalization of their Abelian counterparts in the single D-branesystem, the non-Abelian NS-NS transformation rules involve non-trivial commutator termsthat are not present in the Abelian case [14, 15].Even though the explicit invariance of the non-Abelian Chern-Simons term of the effec-tive worldvolume action under the NS-NS transformations is proven in [15], the complicatedtransformation rules remain surprising and the aim of this letter is to study their structurein more detail. The main result of this paper is that the non-Abelian NS-NS transformationsno longer have a U (1) group structure, but that they intertwine non-trivially with the U ( N )Yang-Mills group of the worldvolume theory. We will show that part of the NS-NS transfor-mations can be written as a U ( N ) gauge transformation, but that there is also a non-trivialpart that can not. We will construct explicitly the algebra spanned by the NS-NS and the U ( N ) transformations by computing the commutation relations and the Jacobi identities.The organization of this paper is as follows: in section 2 we will derive the NS-NS trans-formation rules for the worldvolume fields, following the argument of [15]. In section 3 wewill construct and comment the NS-NS rules for matrix functions and composite objects suchas commutators and covariant derivative, needed in the next sections. In section 4 we willconstruct the algebra formed by the NS-NS and U ( N ) transformations and check the Jacobi2dentities in section 5. We also review some useful issues of non-commutative algebra in theappendix A. Finally, we summarize our results in the conclusions. In this section we will review quickly the derivation of the NS-NS transformation rules for theworldvolume fields, as done in [15], based on the T-duality between D p - and D( p − N multiple coinciding D q -branes consists of a ( q + 1)-dimensional Yang-Mills (Born-Infeld) vector V a , that acts as the gauge field of the U ( N )symmetry group of the the system, and a set of 9 − q matrix-valued transverse scalars X i ,that transform in the adjoint representation of U ( N ). From the target space point of view, thelatter have the interpretation of the non-Abelian embedding scalars of the D p -branes, whilethe former arises as the potential caused by the charged endpoints of open strings ending onthe branes.The T-duality between the D p -brane and the D( p − − p embedding scalars and the ( p + 1)-dimensional Born-Infeldvector of the D p -brane by Y i and ˆ V ˆ a , and the 10 − p embedding scalars and the p -dimensionalBorn-Infeld vector of the D( p − X ˆ ı and V a respectively. Then the T-duality rulesthat relate the field contents of both theories after dualising in a worldvolume direction x ofthe D p -brane are given by [17]ˆ V a −→ V a , Y i −→ X i , ˆ V x −→ X x , Y x = σ x , (2.1)where we have decomposed ˆ V ˆ a and X ˆ ı as ˆ V ˆ a = ( ˆ V a , ˆ V x ) and X ˆ ı = ( X i , X x ) and σ x is theworldvolume coordinate of the D p -brane in the x -direction. The last equation is then merelyan expression of the fact that we write the actions in the static gauge, at least the directionin which the T-duality is performed, while the first two state that the BI vector componentsand the transverse scalars in directions different from the T-dualised one are the same inboth actions. The non-trivial part of the T-duality rules is contained in the third equation,that matches the degrees of freedom of the two theories by mapping the x -component of theD p -brane BI vector with the extra embedding scalar of the D( p − V ˆ a transforms asa vector under general coordinate transformations ˆ ζ ˆ a in the worldvolume (i.e. reparametrisa-tions of the worldvolume directions), as a gauge field under U (1) transformations ˆ χ and withthe pull-back of a shift under NS-NS gauge transformations of the supergravity Kalb-Ramondfield δB ˆ µ ˆ ν = 2 ∂ [ˆ µ Σ ˆ ν ] : δ ˆ V ˆ a = ˆ ζ ˆ b ∂ ˆ b ˆ V ˆ a + ∂ ˆ a ˆ ζ ˆ b ˆ V ˆ b − ∂ ˆ a ˆ χ − Σ ˆ µ ∂ ˆ a Y ˆ µ . (2.2)On the other hand, the Abelian embedding scalars transform as scalars under worldvolume co-ordinate transformations and as target space coordinates under target space diffeomorphisms δx ˆ µ = − ξ ˆ µ : δX ˆ µ = ζ b ∂ b X ˆ µ − ξ ˆ µ . (2.3)3t can easily be checked that indeed not only the field content transforms as in (2.1), but alsotheir variations dualise as δ ˆ V a −→ δV a , δ ˆ V x −→ δX x , δY i −→ δX i , (2.4)provided that the transformation parameters map to each other as (ˆ µ = ( µ, x ))ˆ ζ a −→ ζ a , Σ µ −→ Σ µ , ˆ χ −→ χ + Σ x X x , ˆ ζ x −→ Σ x , Σ x −→ ξ x . (2.5)The non-Abelian case is a bit more involved: here the non-Abelian Born-Infeld vectorstill transforms as a vector under worldvolume reparametrisations, but is now promoted toa U ( N ) Yang-Mills gauge field and, more importantly, the pullback of the NS-NS parameterΣ ˆ µ needs now to be done through U ( N )-covariant derivatives: δ ˆ V ˆ a = ˆ ζ ˆ b ∂ ˆ b ˆ V ˆ a + ∂ ˆ a ˆ ζ ˆ b ˆ V ˆ b − ˆ D ˆ a ˆ χ − Σ ˆ µ ˆ D ˆ a Y ˆ µ . (2.6)The transformation rules for the non-Abelian scalars X i get even more corrections: besidestransforming as scalars and as coordinates under worldvolume and target space general co-ordinate transformations respectively, they also transform as adjoint scalars of U ( N ) and,surprisingly, acquire also a transformation under the NS-NS transformation Σ ˆ µ [14, 15]: δX ˆ µ = ζ b ∂ b X ˆ µ − ξ ˆ µ + i [ χ, X ˆ µ ] + i Σ ˆ ρ [ X ˆ ρ , X ˆ µ ] . (2.7)The last term is quite unexpected from the Abelian point of view, as it consists purely ofa commutator. Yet it arises very naturally from the T-dualisation of the last term in thevariation of δV x in (2.6), due to the covariant derivative in the pullback,ˆ D x Y i = i [ ˆ V x , Y i ] −→ i [ X x , X i ] , (2.8)as T-duality assumes x to be an isometry direction and hence ∂ x Y i = 0. Note that thismechanism that generates these transformations is exactly the same as the one that generatesthe dielectric coupling terms in the Chern-Simons term of multiple coinciding D-branes [11].In a certain sense the NS-NS transformation rules can be though of as “dielectric gaugetransformations”. It should then be clear that the presence of this term is crucial for thetransformation rules for the variations (2.4) to hold also in the non-Abelian case and hencefor the consistency of the set-up. It is precisely this term that will change the group structureof the non-Abelian NS-NS transformations. In the previous section we saw that form invariance under T-duality of the non-Abelian D-brane actions implies that a gauge transformation of the background Kalb-Ramond field δB µν = 2 ∂ [ µ Σ ν ] in the target space induces a simultaneous transformation of the embeddingsscalars and the Born-Infeld vector in the worldvolume, according to δB µν = 2 ∂ [ µ Σ ν ] , δX µ = i Σ ρ [ X ρ , X µ ] , δV a = − Σ µ D a X µ , (3.1) We omit hats on the indices henceforth, as we will not apply T-duality transformations in the rest of thepaper. The notation should be self-explicatory X ) of the embedding coordinates X µ transformsunder NS-NS gauge transformations as δ Φ( X ) = ∂ µ Φ( X ) δX µ = ∂ µ Φ( X ) i Σ ρ [ X ρ , X µ ] = i Σ µ [ X µ , Φ( X )] , (3.2)where in the last step we used the matrix identity (A.6).The variation of the U ( N ) covariant derivative D a X µ is given by δD a X µ = ∂ a ( δX µ ) + i [ δV a , X µ ] + i [ V a , δX µ ]= ∂ a ( i Σ ρ [ X ρ , X µ ]) + i [ − Σ ρ D a X ρ , X µ ] + i [ V a , i Σ ρ [ X ρ , X µ ]]= ∂ a ( i Σ ρ [ X ρ , X µ ]) + i Σ ρ [ ∂ a X ρ , X µ ] + i Σ ρ [ X ρ , ∂ a X µ ] + i Σ ρ [ D a X ρ , X µ ]+ i [Σ ρ , X µ ] D a X ρ − Σ ρ [ V a , [ X ρ , X µ ]] − [ V a , Σ ρ ][ X ρ , X µ ]= i Σ ρ [ X ρ , D a X µ ] + i [ X µ , Σ λ ] D a X λ − i [ X µ , X λ ] D a Σ λ , (3.3)where the last two terms can be unified as δD a X µ = i Σ ρ [ X ρ , D a X µ ] + 2 i [ X µ , X λ ] ∂ [ λ Σ ν ] D a X ν . (3.4)The variation of the commutator [ X µ , X ν ] can be calculated as δ [ X µ , X ν ] = [ δX µ , X ν ] + [ X µ , δX ν ] (3.5)= [ i Σ ρ [ X ρ , X µ ] , X ν ] + [ X µ , i Σ ρ [ X ρ , X ν ]]= i Σ ρ [[ X ρ , X µ ] , X ν ] + i [Σ ρ , X ν ][ X ρ , X µ ]+ [ X µ , i Σ ρ ][ X ρ , X ν ] + i Σ ρ [ X µ , [ X ρ , X ν ]]= i Σ ρ [ X ρ , [ X µ , X ν ]] + i [ X µ , Σ ρ ][ X ρ , X ν ] − i [ X µ , X ρ ][Σ ρ , X ν ] . Again the last two terms can be taken together as δ [ X µ , X ν ] = i Σ ρ [ X ρ , [ X µ , X ν ]] + 2 i [ X µ , X ρ ] ∂ [ ρ Σ λ ] [ X λ , X ν ] . (3.6)Finally, it will be useful to compute also the variation of a double commutator [[ X µ , X ν ] , X λ ].Using the above result we have that δ [[ X µ , X ν ] , X λ ] = i Σ ρ h X ρ , [[ X µ , X ν ] , X λ ] i − i h [ X µ , X ν ] , X ρ i [Σ ρ , X λ ] (3.7)+ i h [ X µ , X ν ] , Σ ρ i [ X ρ , X λ ] − i h [[ X µ , X ρ ][Σ ρ , X ν ] , X λ i + i h [[ X µ , Σ ρ ][ X ρ , X ν ] , X λ i . We therefore see that the NS-NS transformations treat embedding scalars, their covariantderivatives and their commutators on different footing, even if they all sit in the adjoint of U ( N ). A priori there is of course no reason to expect that the NS-NS gauge symmetry wouldrespect the same structures as the U ( N ) group. In the next section however we will showthat there is a certain relation between the two groups.5 The algebra of NS-NS transformations
A first hint of the group structure of the NS-NS transformations (3.1) comes from realisingthat there is a class of transformations that leave B µν invariant, but act non-trivially onthe worldvolume fields. Indeed, taking the NS-NS parameter to be exact, Σ µ = ∂ µ Λ, thetransformation rules (3.1) take the form δB µν = 0 , δX µ = i∂ ρ Λ[ X ρ , X µ ] , δV a = − ∂ µ Λ D a X µ . (4.1)The only non-trivial symmetry in the worldvolume theory is the U ( N ) gauge group, suchthat one would expect the above transformations to be U ( N ) gauge symmetries. Indeed, withthe aid of (A.6) and (A.7), these rules can be rewritten as δB µν = 0 , δX µ = i [Λ , X µ ] , δV a = − D a Λ , (4.2)i.e. as U ( N ) gauge transformation with parameter Λ. Also the more complicated NS-NStransformation rules for the commutator (3.6) and the covariant derivative (3.4) reduce tothe standard transformation rules for objects in the adjoint representation of U ( N ): δ [ X µ , X ν ] = i [Λ , [ X µ , X ν ]] , δD a X µ = i [Λ , D a X µ ] . (4.3)We therefore see that the pure gauge part of the NS-NS transformations is in fact a U ( N ) symmetry. However a general NS-NS transformation can not be written as a partof the U ( N )-algebra and it would be interesting to analyse the complete structure of theintertwining NS-NS and U ( N ) transformations. We will therefore calculate the commutatorsof the NS-NS variations amongst each other, NS-NS with U ( N ) and U ( N ) with itself.The last case is in fact trivial, as we are considering the commutation rules of the U ( N )(sub-)algebra, [ δ χ , δ χ ] = δ χ with χ = i [ χ , χ ] . (4.4)Less trivial are the commutators involving NS-NS transformations. The mixed U ( N ) andNS-NS commutator, acting on the scalars, is given by[ δ Σ , δ χ ] X µ = δ Σ (cid:16) i [ χ, X µ ] (cid:17) − δ χ (cid:16) i Σ ρ [ X ρ , X µ ] (cid:17) (4.5)= − Σ ρ [ X ρ , [ χ, X µ ]] + [ χ, X ρ ][Σ ρ , X µ ] − [ χ, Σ ρ ][ X ρ , X µ ] + [ χ, Σ ρ [ X ρ , X µ ]] . Using the ( U ( N )) Jacobi identities and the decomposition rules for nested commutators, wesee that the above commutator can be written as a U ( N ) transformation[ δ Σ , δ χ ] X µ = i [ ˜ χ, X µ ] with ˜ χ = i Σ ρ [ X ρ , χ ] . (4.6)The same commutator can be checked acting on the Born-Infeld vector V a to yield[ δ Σ , δ χ ] V a = − D a ˜ χ, (4.7)with ˜ χ given by the same expression.Finally, the commutator of two NS-NS transformations is non-trivial as well. Acting onthe scalars we find that[ δ Σ (1) , δ Σ (2) ] X µ = − Σ (1) λ [ X λ , Σ (2) ρ ][ X ρ , X µ ] + Σ (2) λ [ X λ , Σ (1) ρ ][ X ρ , X µ ] (4.8)+ i Σ (2) ρ (cid:16) i Σ (1) λ [ X λ , [ X ρ , X µ ]] + i [ X ρ , Σ (1) λ ][ X λ , X µ ] − i [ X ρ , X λ ][Σ (1) λ , X µ ] (cid:17) + i Σ (1) ρ (cid:16) i Σ (2) λ [ X λ , [ X ρ , X µ ]] + i [ X ρ , Σ (2) λ ][ X λ , X µ ] − i [ X ρ , X λ ][Σ (2) λ , X µ ] (cid:17) . δ Σ (1) , δ Σ (2) ] X µ = Σ (1) λ Σ (2) ρ [ X µ , [ X λ , X ρ ]] + [ X µ , Σ (1) λ Σ (2) ρ ][ X λ , X ρ ]= i h i Σ (1) λ Σ (2) ρ [ X λ , X ρ ] , X µ i . (4.9)In other words, the commutator of two NS-NS variations is again a U ( N ) transformation withparameter ¯ χ = i Σ (1) λ Σ (2) ρ [ X λ , X ρ ] . (4.10)Again the same result is obtained for the commutator acting on V a :[ δ Σ (1) , δ Σ (2) ] V a = − D a (cid:16) i Σ (1) λ Σ (2) ρ [ X λ , X ρ ] (cid:17) . (4.11)We find therefore that the U ( N ) and the NS-NS transformations form a larger algebragiven by [ δ χ , δ χ ] = δ χ with χ = i [ χ , χ ] , [ δ Σ , δ χ ] = δ ˜ χ with ˜ χ = i Σ ρ [ X ρ , χ ] , [ δ Σ , δ Σ ] = δ ¯ χ with ¯ χ = i Σ (1) λ Σ (2) ρ [ X λ , X ρ ] . (4.12)It is worthwhile to check the above algebra on the more complicated transformation rulesfor the commutator and the covariant derivative. As we mentioned, it will not yield manynew insights, but it will give us more confidence in the derived results.The calculation simplifies greatly if we write in general the NS-NS transformation rules as δZ = δ Z + δ ∗ Z, (4.13)where δ Z is the “standard part” δ Z = i Σ ρ [ X ρ , Z ] , (4.14)for any Z , and δ ∗ Z is the correction terms that appear for Z being [ X µ , X ν ] or D a X µ , δ ∗ [ X µ , X ν ] = i [ X µ , Σ ρ ][ X ρ , X ν ] − i [ X µ , X ρ ][Σ ρ , X ν ] ,δ ∗ D a X µ = i [ X µ , Σ λ ] D a X λ − i [ X µ , X λ ] D a Σ λ . (4.15)The trick now is to realise that the δ ∗ part commutes with both δ and U ( N ) variations.Indeed, we have that δ ∗ ( δ χ Z ) = δ ∗ (cid:16) i [ χ, Z ] (cid:17) = i [ δ ∗ χ, Z ] + i [ χ, δ ∗ Z ] = δ χ ( δ ∗ Z ) , (4.16)as the correction terms δ ∗ vanish for χ . Hence the mixed commutator [ δ Σ , δ χ ] reduces to thestandard part [ δ Σ , δ χ ] Z = [ δ , δ χ ] Z + [ δ ∗ , δ χ ] Z = δ ˜ χ Z, (4.17)as the second term vanishes identically. Similarly the commutator [ δ Σ , δ Σ ] simplified to[ δ Σ , δ Σ ] Z = [ δ , δ ] Z + [ δ ∗ , δ ∗ ] Z. (4.18)The first term in the right-hand side is again the standard commutator δ ¯ χ Z , and althoughthe second term has to be calculated for each case separately, it can be shown to vanishidentically for both [ X µ , X ν ] and D a X µ . Hence we see indeed that the algebra (4.12) is alsosatisfied for these objects, in spite of their involved transformation rules.7 Jacobi identities
Finally, in order to ensure the consistency of the algebra (4.12), we will check the Jacobiidentities. There are four identities to be checked, and from the structure of the algebra it isclear that all will result in a U ( N ) transformation. We will prove that the Jacobi identitiesare satisfied by showing that the resulting U ( N ) transformations have zero parameter.Indeed, the different identities, acting on a generic object Z , yield U ( N ) transformations[ δ χ , [ δ χ , δ χ ]] Z + [ δ χ , [ δ χ , δ χ ]] Z + [ δ χ , [ δ χ , δ χ ]] Z = i [ χ , Z ] , [ δ Σ , [ δ χ , δ χ ]] Z + [ δ χ , [ δ Σ , δ χ ]] Z + [ δ χ , [ δ χ , δ Σ ]] Z = i [ ˜ χ, Z ] , [ δ Σ , [ δ Σ , δ χ ]] Z + [ δ χ , [ δ Σ , δ Σ ]] Z + [ δ Σ , [ δ χ , δ Σ ]] Z = i [ ¯ χ, Z ] , [ δ Σ , [ δ Σ , δ Σ ]] Z + [ δ Σ , [ δ Σ , δ Σ ]] Z + [ δ Σ , [ δ Σ , δ Σ ]] Z = i [ ˆ χ, Z ] , (5.1)with the parameters χ , ˜ χ , ¯ χ and ˆ χ given by χ = [ χ , [ χ , χ ]] + [ χ , [ χ , χ ]] + [ χ , [ χ , χ ]] = 0 , ˜ χ = Σ ρ [[ χ , χ ] , X ρ ] + [ χ , X ρ ][Σ ρ , χ ] − [ χ , Σ ρ ][ X ρ , χ ] − [ χ , Σ ρ [ X ρ , χ ]] + [ χ , Σ ρ [ X ρ , χ ]] , ¯ χ = − Σ (1) λ [ X λ , Σ (2) ρ [ X ρ , χ ]] + Σ (2) λ [ X λ , Σ (1) ρ [ X ρ , χ ]] − [ χ, Σ (1) λ Σ (2) ρ [ X λ , X ρ ]]+ Σ (2) ρ [ X ρ , X λ ][Σ (1) λ , χ ] − Σ (2) ρ [ X ρ , Σ (1) λ ][ X λ , χ ] − Σ (1) ρ [ X ρ , X λ ][Σ (2) λ , χ ] + Σ (1) ρ [ X ρ , Σ (2) λ ][ X λ , χ ] , ˆ χ = − Σ (1) ν [ X ν , Σ (2) λ Σ (3) ρ [ X λ , X ρ ]] + Σ (2) λ Σ (3) ρ [ X λ , X ν ][Σ (1) ν , X ρ ] − Σ (2) λ Σ (3) ρ [ X λ , Σ (1) ν ][ X ν , X ρ ] − Σ (3) ρ [ X ρ , Σ (1) ν Σ (2) λ [ X ν , X λ ]] + Σ (1) ν Σ (2) λ [ X ν , X ρ ][Σ (3) ρ , X λ ] − Σ (1) ν Σ (2) λ [ X ν , Σ (3) ρ ][ X ρ , X λ ] − Σ (2) λ [ X λ , Σ (3) ρ Σ (1) ν [ X ρ , X ν ]] + Σ (3) ρ Σ (1) ν [ X ρ , X λ ][Σ (2) λ , X ν ] − Σ (3) ρ Σ (1) ν [ X ρ , Σ (2) λ ][ X λ , X ν ] . (5.2)The parameter χ vanishes identically due to the Jacobi identity of U ( N ), but with a littlematrix algebra also the other expressions can quite easily be shown to be proportional to the U ( N ) Jacobi identities and therefore to vanish identically:˜ χ = Σ ρ (cid:16) [[ χ , χ ] , X ρ ] + [[ X ρ , χ ] , χ ] + [[ χ , X ρ ] , χ ] (cid:17) = 0 , ¯ χ = − Σ (1) λ Σ (2) ρ (cid:16) [ X λ , [ X ρ , χ ]] + [ χ, [ X λ , X ρ ]] + [ X ρ , [ χ, X λ ]] (cid:17) = 0 , ˆ χ = − Σ (1) ν Σ (2) λ Σ (3) ρ (cid:16) [ X ν , [ X λ , X ρ ]] + [ X ρ , [ X ν , X λ ]] + [ X λ , [ X ρ , X ν ]] (cid:17) = 0 . (5.3)8 Conclusions
From the derivation performed in [14, 15] it is known that the background gauge transfor-mation of the Kalb-Ramond field δB µν = 2 ∂ [ µ Σ ν ] induces non-trivial, “dielectric”, transfor-mations of the worldvolume field content of the non-Abelian action of a system of multiplecoinciding D-branes. These NS-NS transformations acting on the embedding scalars take theform of a pure commutator, δX µ = i Σ ρ [ X ρ , X µ ], and arises from T-dualising the transfor-mation rules for the Born-Infeld vector δV a = − Σ µ D a X µ , in the same way as the dielectriccouplings arise in the Myers action.It has been shown already in [15] that the Chern-Simons action of the system of multipleD-branes is invariant under the NS-NS transformations of both the background and theworldvolume fields. In this letter we have shown that the group structure of the NS-NStransformations is more involved than in the Abelian case, due to the fact that the non-Abelian NS-NS transformations rules imply a non-trivial mixture with the U ( N ) Yang-Millssymmetry.A first hint of this can be seen if we take the NS-NS parameter to be exact, Σ µ = ∂ µ Λ.In that case, the supergravity part is untouched, δB µν = 0, but the worldvolume fields dotransform non-trivially under a U ( N ) gauge transformation with parameter Λ. In otherwords, the exact part of the NS-NS transformations can be written as a part of the U ( N )Yang-Mills symmetry of the worldvolume theory.However, this reduction to U ( N ) variations can not be done for a general NS-NS transfor-mation with arbitrary parameter Σ µ and the full group structure of the intertwining U ( N )and NS-NS transformations is revealed by the full algebra[ δ χ , δ χ ] = δ χ with χ = i [ χ , χ ] , [ δ Σ , δ χ ] = δ ˜ χ with ˜ χ = i Σ ρ [ X ρ , χ ] , [ δ Σ , δ Σ ] = δ ¯ χ with ¯ χ = i Σ (1) λ Σ (2) ρ [ X λ , X ρ ] . (6.1)We see that the U ( N ) algebra of the Yang-Mills is a non-trivial sub-algebra of the full alge-bra, which also involves non-trivial commutators between NS-NS and U ( N ) transformationsand between NS-NS amongst each other. The surprising issue however is that all the re-sulting commutators turn out to be U ( N ) transformations, the inherent gauge symmetryof the worldvolume theory. In a certain sense, the U ( N ) symmetry “non-Abelianises” theother gauge symmetries present (in this case the NS-NS transformations), but only in a verymild way: modulo a U ( N ) transformation, they still all behave as if they were U (1) gaugesymmetries in the Abelian theory.A few comments are in order: first, as already mentioned in [15], there seems to be aremarkable difference between the NS-NS gauge transformations of B µν and the R-R trans-formations of C µν , at least at the level of the worldvolume theory, in spite of the fact that B µν and C µν form a doublet under S-duality. The reason for this is of course that S-dualitytakes us out of the perturbative regime in which we can trust the description we have beenworking with. However it should be clear that if we believe in S-duality in Type IIB stringtheory, there should be a “non-Abelianisation” (in the sense of (6.1)) of the R-R gauge trans-formations, not only of the R-R two-form C µν , but via T-duality of all other R-R potentialsas well.Secondly, it is well known that (parts of the) NS-NS transformations mix with (part ofthe) general coordinate transformations under T-duality. Knowing on the one hand that a9atisfactory description of general coordinate transformations in non-commutative (or matrix-valued) geometry is still an open issue, and keeping in mind the non-trivial structure of thenon-Abelian NS-NS transformations on the other hand, it will be clear that one might findinteresting and surprising results on non-commutative geometry and gauge transformationsif one tried to T-dualise the algebra (6.1). We leave these ideas for future investigations. A The non-Abelian formalism
There are many ways to generalise functions Φ( x ) of the Abelian coordinates x µ to matrixfunctions Φ( X ) of the matrix-valued coordinates X µ . One of the most common prescriptions(and the one traditionally used in the context of multiple coinciding D-brane systems) is thesymmetrised prescription, through the non-Abelian Taylor expansion of Φ( X ),Φ( X ) = ∞ X n =0 n ! ∂ µ . . . ∂ µ n Φ | x =0 X µ . . . X µ n , (A.1)where the matrix multiplication is taken to be the symmetrised product of n Lie algebraelements A , . . . , A n , A . . . A n = 1 n ! X σ ∈ S n A σ (1) . . . A σ ( n ) . (A.2)As the product between X ’s is not defined within the structure of the algebra, strictly speak-ing Φ( X ) is not an element of the u ( N ) algebra. However with the above definitions, Φ( X )becomes an element of the tensor algebra T ( u ( N )) of u ( N ). In addition to this, the commu-tator structure can be transferred to T ( u ( N )), imposing the following equivalence relation AB − BA ∼ [ A, B ] , (A.3)yielding the so-called the universal enveloping algebra U ( u ( N )) of u ( N )). Note that thesymmetrised prescription has to be imposed on U ( u ( N )), rather then on T ( u ( N )), due to theequivalence relation (A.3), as otherwise all commutators of the Lie algebra would vanish.The non-Abelian functions defined in this way have a series of useful properties. Forinstance, the variation of the non-Abelian functions is given by δ Φ( X ) = ∞ X n =0 n X i =1 ∂ µ . . . ∂ µ n Φ | x =0 X µ . . . δX µ i . . . X µ n = ∂ µ Φ( X ) δX µ , (A.4)where we used ∂ µ Φ( X ) as a shorthand for its non-Abelian Taylor expansion. Given that thescalars transform under u ( N ) gauge transformations as δX µ = i [ χ, X µ ], the variation of thenon-Abelian function Φ( X ) can then be written as δ χ Φ( X ) = ∂ µ Φ( X ) i [ χ, X µ ] = i [ χ, Φ( X )] , (A.5)where in the last step we used the properties of the symmetrised prescription and the com-mutator to prove that [Φ( X ) , X µ ] = ∂ ρ Φ( X )[ X ρ , X µ ] . (A.6)In the same way we can also define a covariant derivative of Φ( X ) via D a Φ( X ) = ∂ a Φ( X ) + i [ V a , Φ( X )] = ∂ µ Φ( X ) D a X µ . (A.7)10imilarly the commutator of two non-Abelian functions is given by[Φ ( X ) , Φ ( X )] = i [ X µ , X ν ] ∂ µ Φ ( X ) ∂ ν Φ ( X ) . (A.8)It can easily be checked that this definition satisfied the Jacobi identity[Φ , [Φ , Φ ]] + [Φ , [Φ , Φ ]] + [Φ , [Φ , Φ ]] = 0 . (A.9) Acknowledgements
The authors wish to thank J. Adam and J. Mas for useful discussions. The work of B.J. ispartially supported by the Spanish Ministerio de Ciencia e Innovaci´on under contract FIS2007-63364, by the Junta de Andaluc´ıa group FQM 101 and the Proyecto de ExcelenciaP07-FQM-03048.
References [1] E. Witten, Nucl. Phys. B460 (1996) 335, hep-th/9510135.[2] A.A. Tseytlin,
Born-Infeld action, supersymmetry and string theory , hep-th/9908105.[3] M. Douglas,
Branes within Branes , hep-th/9512077.[4] M. Green, C. Hull, P. Townsend, Phys. Lett. B382 (1996) 65, hep-th/9604119.[5] H. Dorn, Nucl. Phys. B494 (1997) 105, hep-th/9612120.[6] A. Tseytlin, Nucl. Phys. B501 (1997) 41, hep-th/9701125.[7] M. Douglas, Adv. Theor. Math. Phys. 1 (1998) 198, hep-th/9703056.[8] C. Hull, JHEP 9810 (1998) 011, hep-th/9711179.[9] M. Garousi, R. Myers, Nucl. Phys. B542 (1999) 73, hep-th/9809100.[10] W. Taylor, M. Van Raamsdonk, Nucl. Phys. B573 (2000) 703, hep-th/9910052.[11] R. Myers, JHEP 9912 (1999) 022, hep-th/9910053.[12] C. Ciocarlie, JHEP 0107 (2001) 028, hep-th/0105253.[13] J. Adam, J. Gheerardyn, B. Janssen, Y. Lozano, Phys. Lett. B589 (2004) 59,hep-th/0312264.[14] J. Adam, J. Gheerardyn, B. Janssen, Y. Lozano,
On the gauge invariance of the non-Abelian Chern-Simons action for D-branes , in N. Alonso et al (Ed.),