Emergent D5-brane Background from D-strings
aa r X i v : . [ h e p - t h ] M a r Emergent D5-Brane Background from D-Strings
Frank F errari and Antonin R ovai
Service de Physique Th´eorique et Math´ematiqueUniversit´e Libre de Bruxelles and International Solvay InstitutesCampus de la Plaine, CP 231, B-1050 Bruxelles, Belgique [email protected], [email protected]
We solve the worldsheet theory describing the near-horizon dynamics of a D-stringin the presence of a very large number N of D5-branes. The model is pre-geometric inthe sense that the near-horizon worldsheet Lagrangian does not have dynamical fieldsassociated with the dimensions transverse to the D5-branes. The solution at large N is shown to be given by a classical action for the D-string moving in a curved ten-dimensional spacetime. The four dimensions transverse to the D5-branes emerge fromthe quantum loops of the original strongly coupled quantum worldsheet field theory.By comparing with the Dirac-Born-Infeld plus Chern-Simons action for a D-stringin a general type IIB background, we identify the string-frame metric, dilaton andRamond-Ramond three-form field-strength and find a match with the near-horizongeometry of a stack of D5-branes.July 17, 2018 Introduction
The AdS/CFT correspondence [1–3] and its generalizations (see e.g. [4, 5] for re-views and references) offer a framework in which, at least in principle, geometry andquantum gravity can be studied from ordinary quantum field gauge theories. In thisframework, a d -dimensional field theory is postulated to be dual, in the large N limit,to a classical gravitational theory defined on a higher D > d dimensional curvedspacetime. The D − d additional dimensions are emerging from strongly coupledquantum physics in the gauge theory. The study of the emerging geometry from thepure field theory point of view is unfortunately rather difficult in general. It requiresto compute observables in the strong coupling regime of the field theory and to extractfrom them informations about the dual geometry.Recently, it was proposed in [6] to focus on the field theory models describingthe near-horizon worldvolume dynamics of a fixed number of D-branes, called theprobe branes, in the presence of a very large number N of other D-branes, called thebackground branes. These models are pre-geometric, because they do not containdynamical fields associated with the dimensions transverse to the background branes.According to the AdS/CFT lore, in the large N limit, they should be equivalent toclassical worldvolume theories describing the motion of the probe D-branes in theten dimensional supergravity solution sourced by the background branes. This non-trivial supergravity solution should thus emerge, alongside the transverse dimensionsof space on which it lives, from the large N solution of the probe branes worldvolumemodel. Much more details on these ideas can be found in [6–8].The power of this approach is that it is easier to extract some strong couplinginformation from the large N probe brane models than from the large N backgroundbrane theories [6]: the latter are complicated matrix models, whereas the former arevector-like models. It is then possible to explicitly derive the emergence of space froma microscopic calculation in the strongly coupled vector-like worldvolume theory. Inall the examples that have been studied so far [6, 7, 9], the solution at large N canbe described by a classical action matching the non-abelian D-brane action for theprobe branes in the correct ten-dimensional curved background. The scalar fieldsassociated with the emerging classical dimensions of space are composite variablesin the original microscopic description whose quantum fluctuations are suppressed inthe large N limit by the quantum loops of the vector variables.The aim of the present work is to apply the above ideas to derive the emergentnear-horizon geometry of a large number of D5-branes in type IIB string theory bystudying the corresponding probe D-string worldsheet model. We start in Section 2by presenting in details the relevant D-string microscopic pre-geometric worldsheettheory. We solve the model at large N in Section 3 and find that the solution isexpressed in terms of a classical action which contains the right dynamical fields todescribe the motion of the D-strings in a ten-dimensional background. In Section 4, wecompare the expansion of this action around a flat worldsheet with the corresponding2erms derived from the well-known Dirac-Born-Infeld plus Chern-Simons D-stringaction in arbitrary supergravity background. This allows us to identify the string-frame metric, dilaton and Ramond-Ramond three-form field-strength from a purelyfield theoretic calculation. The result matches perfectly the near-horizon supergravitysolution sourced by the background D5-branes. In principle, we need the action describing the dynamics of the open string modesof a system composed of a fixed number K of D-strings and a large number N ofD5-branes, in an appropriate decoupling, near-horizon limit [10]. All the branes arechosen to be parallel to each other, and we work in the Euclidean. In particular, themodel preserves eight supercharges and there is an SO(2) × SO(4) × SO(4) ′ globalsymmetry group corresponding to rotations in spacetime preserving the brane config-uration: SO(2) is associated with rotations on the D-string worldsheet, SO(4) ′ withrotations on the D5-brane worldvolume transverse to the D-strings and SO(4) withrotations transverse to both the D-strings and the D5-branes.This action could be studied by evaluating appropriate low-energy limits of openstring disk diagrams with various boundary conditions. This is a very tedious andcomplicated procedure, which was performed in the case of the D(-1)/D3 systemin [11, 12]. In our case, the result is a sum of a worldsheet action for the D-stringsand a worldvolume action for the D5-branes, with couplings between the D-stringand D5-brane degrees of freedom. However, the na¨ıve action obtained in this way forthe degrees of freedom living on the D5-branes would not be renormalizable and thuscould be used only in the infrared. In other words, the full description of the D5-braneis not field theoretic. Fortunately, an explicit description of the D5-brane degrees offreedom and their couplings to the D-strings will not be required for our purposes.Indeed, due to supersymmetry, these couplings are not expected to contribute to theterms in the effective action on which we shall focus. A similar non-renormalizationtheorem was discussed in [13] in the case of D-particles. We refer to [6] for a detaileddiscussion of these issues in the case of the D(-1)/D3 system and to [14] for a generaldiscussion in non-supersymmetric contexts.We thus focus on the D-string worldsheet Lagrangian, without referring any longerto possible couplings to the D5-brane fields. The form of the Lagrangian is stronglycontrained by (4 ,
4) supersymmetry and global symmetries. The simplest way to de-rive it is to proceed in two steps. First, we perform the dimensional reduction ofthe U( K ) N = 1 gauge theory in six dimensions down to two dimensions. The the-ory must contain one hypermultiplet in the adjoint, corresponding to D1/D1 stringdegrees of freedom, and N hypermultiplets in the fundamental corresponding to theD1/D5 strings. Second, a suitable scaling limit must be implemented on this action,which corresponds to the correct low energy limit associated with the standard Mal-3acena near-horizon limit [1, 10, 15]. We present these two steps successively in thenext two subsections. The six dimensional U( K ) gauge theory has a spacetime symmetry group SO(6),R-symmetry group SU(2) + , internal symmetry group SU(2) − and flavor symmetrygroup SU( N ). The field content is as follows. The vector multiplet is composedof the gauge potential A r , 1 ≤ r ≤ α , which is a doublet ofSU(2) + . The adjoint hypermultiplet contains the adjoint scalars a µ in the vector ofSO(4) ∼ SU(2) + × SU(2) − and a fermionic doublet ¯Λ ˙ α of SU(2) − . The fundamentalhypermultiplets contain SU(2) + scalar doublets ( q αf , ˜ q αf ) and Weyl fermions ( χ f , ˜ χ f )in the fundamental and antifundamental of SU( N ). The chirality of the Weyl spinorsin the vector multiplet on the one hand and in the hypermultiplets on the other handmust be opposite.After the dimensional reduction down to two dimensions, the six-dimensionalspacetime symmetry group SO(6) yields the two-dimensional spacetime symmetrygroup SO(2) and a new global SO(4) ′ ∼ SU(2) ′ + × SU(2) ′− . The fields of the six-dimensional theory are then reorganized into representations of SO(2) and SO(4) ′ ,while their transformation laws under U( K ), U( N ), SU(2) + and SU(2) − are leftunchanged.More precisely, the six dimensional vector multiplet yields the two dimensionalgauge field A I , four scalars φ m transforming in the vector representation of SO(4) ′ and spinors Λ αζ , Λ α ˙ ζ transforming in the representations (1 / , / and (0 , / − / of SO(4) ′ × SO(2). The adjoint hypermultiplet yields four scalars a µ and spinors ¯Λ ˙ αζ ,¯Λ ˙ α ˙ ζ transforming in the representations (1 / , − / and (0 , / / of SO(4) ′ × SO(2).Finally, the fundamental hypermultiplets yield the scalars q αf and ˜ q αf together withfermions ( χ ζf , ˜ χ fζ ) in the (1 / , − / and ( χ ˙ ζf , ˜ χ f ˙ ζ ) in the (0 , / / representationsof SO(4) ′ × SO(2) respectively. Let us note that the model is also invariant underworldsheet parity transformations which act by exchanging the SU(2) ′ + and SU(2) ′− factors of SO(4) ′ . All these symmetry properties are summarized in Table 1 in theAppendix.The (4 ,
4) supersymmetric Lagrangian resulting from the dimensional reduction4an then be written as˜ L = ℓ g s tr U( K ) (cid:18) ℓ I K + 12 F IJ F IJ + ∇ I φ m ∇ I φ m − (cid:2) φ m , φ n (cid:3)(cid:2) φ m , φ n (cid:3) + ∇ I a µ ∇ I a ν − (cid:2) φ m , a µ (cid:3)(cid:2) φ m , a µ (cid:3) + 2 i (cid:2) a µ , a ν (cid:3) D µν − D µν D µν − αζ σ mζ ˙ ζ (cid:2) φ m , Λ ˙ ζα (cid:3) − i Λ αζ ∇ w Λ αζ + 2 i Λ α ˙ ζ ∇ ¯ w Λ ˙ ζα + 2 ¯Λ ζ ˙ α σ mζ ˙ ζ (cid:2) φ m , ¯Λ ˙ α ˙ ζ (cid:3) − i ¯Λ ζ ˙ α ∇ ¯ w ¯Λ ˙ αζ + 2 i ¯Λ ˙ α ˙ ζ ∇ w ¯Λ ˙ α ˙ ζ − iσ µα ˙ α Λ αζ (cid:2) a µ , ¯Λ ˙ αζ (cid:3) + 2 iσ µα ˙ α Λ α ˙ ζ (cid:2) a µ , ¯Λ ˙ α ˙ ζ (cid:3)(cid:19) + 12 ∇ I ˜ q αf ∇ I q αf − φ m ˜ q αf φ m q αf −
12 ˜ χ fζ σ mζ ˙ ζ φ m χ ˙ ζf + 12 ˜ χ f ˙ ζ ¯ σ ˙ ζζm φ m χ fζ + i ˜ χ fζ ∇ ¯ w χ fζ − i ˜ χ f ˙ ζ ∇ w χ ˙ ζf − √ q αf Λ αζ χ ζf + 1 √ q αf Λ ˙ ζα χ f ˙ ζ − √ χ ζf Λ αζ q αf + 1 √ χ f ˙ ζ Λ α ˙ ζ q αf + i q αf D µν σ βµνα q βf . (2.1)We have introduced the usual complex worldsheet coordinates w and ¯ w , with asso-ciated covariant derivatives ∇ w = ( ∇ − i ∇ ) / ∇ ¯ w = ( ∇ + i ∇ ) /
2. We havealso introduced a self-dual auxiliary field D µν in the adjoint of U( K ). This fieldallows us to write the quartic interactions between the scalars ( a, q, ˜ q ) in a simpleway which will be useful for performing the scaling limit in the next subsection. Theoverall normalization as well as the constant term in the Lagrangian are fixed by theD-string tension, the string length being related to the fundamental string tension by ℓ = 2 πα ′ . The scalar fields a µ and φ m in the worldsheet Lagrangian (2.1) are associated withthe motion of the D-strings parallel and transverse to the D5-branes respectively. Thecorresponding coordinates are X µ = ℓ a µ , Y m = ℓ φ m . (2.2)To implement the standard decoupling, near-horizon limit [1, 10, 15], we take ℓ s → X µ , Y m /ℓ = φ m and the six-dimensional ’t Hooft coupling N ℓ g s fixed, as usual. Supersymmetry then dictates that the fermionic superpartners Λ α and ¯ ψ ˙ α = ℓ ¯Λ ˙ α of φ m and X µ respectively must also be kept fixed. Introducing5 α = ℓ Λ α , the Lagrangian (2.1) then simplifies in the scaling to L p = 12 ℓ g s tr U( K ) (cid:18) I K + ∇ I X µ ∇ I X ν + 2 i (cid:2) X µ , X ν (cid:3) D µν − ℓ − (cid:2) Y m , X µ (cid:3)(cid:2) Y m , X µ (cid:3) − i ¯ ψ ζ ˙ α ∇ ¯ w ¯ ψ ˙ αζ + 2 i ¯ ψ ˙ α ˙ ζ ∇ w ¯ ψ ˙ α ˙ ζ + 2 ℓ − ¯ ψ ζ ˙ α σ mζ ˙ ζ [ Y m , ¯ ψ ˙ α ˙ ζ ] − iℓ − ψ αζ σ µα ˙ α (cid:2) X µ , ¯ ψ ˙ αζ (cid:3) + 2 iℓ − ψ α ˙ ζ σ µα ˙ α (cid:2) X µ , ¯ ψ ˙ α ˙ ζ (cid:3)(cid:19) + 12 ∇ I ˜ q αf ∇ I q αf − ℓ Y m ˜ q αf Y m q αf − ℓ ˜ χ fζ σ mζ ˙ ζ Y m χ ˙ ζf + 12 ℓ ˜ χ f ˙ ζ ¯ σ ˙ ζζm Y m χ fζ + i ˜ χ fζ ∇ ¯ w χ fζ − i ˜ χ f ˙ ζ ∇ w χ ˙ ζf − √ ℓ ˜ q αf ψ αζ χ ζf + 1 √ ℓ ˜ q αf ψ ˙ ζα χ f ˙ ζ − √ ℓ ˜ χ ζf ψ αζ q αf + 1 √ ℓ ˜ χ f ˙ ζ ψ α ˙ ζ q αf + i q αf D µν σ βµνα q βf . (2.3)This Lagrangian will be our starting point for the probe D-string worldsheet theory.It is important to emphasize that the scalar fields Y m are non-dynamical auxiliaryvariables in (2.3). Indeed, they do not have a kinetic term and could be trivially inte-grated out. However, as explained in the next Section, these fields become dynamicaldue to the quantum corrections and play a central rˆole both in the mathematics andthe physical interpretation of the solution of the model at large N [6]. N We now solve the model defined by the Lagrangian (2.3) along the lines of [6,7,9]. Thecrucial property is that the fields ( q, ˜ q, χ, ˜ χ ) carry only one U( N ) index and are thusvector-like variables. The large N path integral over these fields can then always beperformed exactly, using standard techniques for large N vector models [16–20]. Theidea is to rewrite the action by introducing suitable auxiliary fields, in order to makethe vector variables appear only quadratically. In our case, the relevant auxiliaryfields are precisely the variables ( Y m , ψ αζ , ψ α ˙ ζ , D µν ) which we have already includedwhen writing (2.3). The path integral over the vector variables is then Gaussian. Theresult is an effective action for the auxiliary fields, which become dynamical throughthe quantum loops of the vector variables. Moreover, and most importantly, thiseffective action is automatically proportional to N because the vector fields have N components. At large N , it can thus be treated classically.The resulting structure is thus perfectly consistent with the D-string seen as mov-ing in a higher dimensional classical non-trivial background. Indeed, the fields Y m can be interpreted as the emerging coordinates which behave classically at large N and the metric on the emerging space will be related to the kinetic term for the Y m .6et us now carry out this procedure explicitly for our model, mainly focusing onthe case of a single D-string probe. The effective action
N S eff is given by e − NS eff = Z d q d˜ q d χ d ˜ χ e − S p , (3.1)where S p is the action for the Lagrangian (2.3). In order to derive the emergentgeometry, we can focus on the bosonic part of the effective action and thus set thefermionic fields ψ and ¯ ψ to zero. Note, however, that computing the fermionic termsin the effective action could also be done straightforwardly.The integral (3.1) is Gaussian and yields S eff ( A, X, Y, D ) = 12
N ℓ g s Z d w tr U( K ) (cid:16) I K + ∇ I X µ ∇ I X µ + 2 i (cid:2) X µ , X ν (cid:3) D µν − ℓ − (cid:2) Y m , X µ (cid:3)(cid:2) Y m , X µ (cid:3)(cid:17) + ln ∆ q, ˜ q − ln ∆ χ, ˜ χ , (3.2)where the determinants ∆ q, ˜ q and ∆ χ, ˜ χ are given by∆ q, ˜ q = det (cid:0) − I K ⊗ I ∇ + ℓ − Y m Y m ⊗ I + iD µν ⊗ σ µν (cid:1) , (3.3)∆ χ, ˜ χ = det (cid:18) − i I K ⊗ I ∇ ¯ w ℓ − Y m ⊗ σ m − ℓ − Y m ⊗ ¯ σ m i I K ⊗ I ∇ w (cid:19) . (3.4)At large N , the field D µν is fixed in terms of the other variables by the saddle pointequation δS eff δD µν = 0 . (3.5)If we specialize to the case K = 1 of a single D-string probe, then the solution issimply D µν = 0. This follows from the vanishing of the linear term in D in theexpansion of (3.3) around D = 0 or, equivalently, from the commuting nature of the X µ and SO(4) invariance. We thus get S eff ( A, X, Y ) = 1
N ℓ g s Z d z (cid:16) ∂ I X µ ∂ I X µ (cid:17) + 2 ln det (cid:0) −∇ + ℓ − Y m Y m (cid:1) − ln det (cid:18) − i I ∇ ¯ w ℓ − Y m σ m − ℓ − Y m ¯ σ m i I ∇ w (cid:19) . (3.6)7 .2 The effective action up to cubic order We are going to use (3.6) up to order three in an expansion in the field strength F IJ and around constant values of the coordinate worldsheet fields, X µ = x µ + ℓ ǫ µ , Y m = y m + ℓ ǫ m . (3.7)Eventually, we shall match N S eff up to this order in the next Section with the D-stringaction in a general type IIB background.The explicit computation of the expansion is straightforward. Let us introducethe radial coordinate r defined by r = y m y m . (3.8)We write ln ∆ q, ˜ q = 2 ln det K B + 2 tr ln( I + K − B ϕ ) , (3.9)ln ∆ χ, ˜ χ = ln det K F + tr ln( I + K − F ξ ) , (3.10)in terms of the bosonic and fermionic propagators K − B ( w, w ′ ) = Z d p (2 π ) e ip · ( w − w ′ ) p + ℓ − r , (3.11) K − F ( w, w ′ ) = Z d p (2 π ) e ip · ( w − w ′ ) p + ℓ − r (cid:18) I p w ℓ − y m σ m − ℓ − y m ¯ σ m − I p ¯ w (cid:19) (3.12)and with ϕ = 2 ℓ − y m ǫ m + ǫ m ǫ m − i∂ I A I − iA I ∂ I + A I A I , (3.13) ξ = (cid:18) I A ¯ w ǫ m σ m − ǫ m ¯ σ m − I A w (cid:19) . (3.14)We then expand the traces in (3.9) and (3.10) usingln( I + δ ) = − ∞ X k =1 ( − k k δ k (3.15)and compute the resulting one-loop Feynman integrals.At zeroth and first order, the contributions from the bosonic and fermionic de-terminants cancel each other and only the constant D-string tension term in (3.6)remains. At second and third order, we obtain a non-local effective action, which wewrite as a power series in derivatives by expanding the associated Feynman integrals8or small external momenta. Overall, we get N S eff ( A, X, Y ) = Z d w (cid:16) ℓ g s + ℓ g s ∂ I ǫ µ ∂ I ǫ µ + N ℓ πr ∂ I ǫ m ∂ I ǫ m + N ℓ πr F IJ F IJ − N ℓ πr y m ǫ m ∂ I ǫ n ∂ I ǫ n − N ℓ πr ǫ m y m F IJ F IJ − iN ℓ πr ǫ IJ y m ǫ mnlp ǫ n ∂ I ǫ l ∂ J ǫ p (cid:17) + · · · (3.16)where the · · · stand for terms of quartic or higher order and terms with more than twoderivatives. Of course, the result is consistent with the symmetries of the microscopictheory discussed in Section 2.1, including the worldsheet parity which must comeaccompanied by a parity transformation in the directions y m . The action for a D-string moving in a general type IIB supergravity background isthe sum of a Dirac-Born-Infeld term and a Chern-Simons term [21, 22], S = 1 ℓ g s Z d ξ e − Φ p det [P( G + B ) + ℓ F ] + iℓ g s Z h P( C B + C ) + ℓ C F i , (4.1)where the fields Φ, G , B , C and C are the dilaton, string-frame metric, Kalb-Ramond two-form and Ramond-Ramond potentials respectively, F is the worldsheetfield strength and P denotes the pull-back of the spacetime fields to the worldsheet.Working in the static gauge and writing the fields Z i , 1 ≤ i ≤
8, corresponding tothe coordinates transverse to the D-string worldsheet as Z i = z i + ℓ ǫ i , (4.2)we can expand (4.1) in powers of ǫ i and F . Following the basic idea that underliesthe present work as well as our previous studies [6, 7, 9], this expansion should matchwith the similar expansion (3.16) of the effective action describing the solution ofthe large N microscopic model of the D-strings in the presence of the N D5-branes.Morevover, we should be able to derive the precise supergravity background sourcedby the D5-branes from the coefficients in the expansion (3.16). Let us check that thisis indeed the case.The zeroth order Lagrangian derived in this way from (4.1) reads L (0) = 1 ℓ g s h e − Φ p det( G IJ + B IJ ) + i ǫ IJ (cid:0) C B IJ + ( C ) IJ (cid:1)i , (4.3)where the capital Latin indices 1 ≤ I, J, . . . ≤ e − Φ p det( G IJ + B IJ ) = 1 , C B IJ + ( C ) IJ = 0 . (4.4)9aking these constraints into account, the Lagrangian derived from (4.1) at first orderin ǫ is, up to an irrelevant total derivative, L (1) = 12 g s F IJ (cid:0) − E IJ + iC ǫ IJ (cid:1) , (4.5)where the matrix E IJ is the inverse of G IJ + B IJ . Using the fact that L (1) = 0 in(3.16) and worldsheet parity, we get B IJ = 0 , C = 0 . (4.6)The conditions (4.4) thus reduce to e − Φ p det G IJ = 1 , ( C ) IJ = 0 . (4.7)Taking these results into account as well as the fact that G iI = 0 from ISO(2) in-variance along the worldsheet, the second order Lagrangian derived from (4.1) thenreads L (2) = ℓ g s (cid:20)(cid:0)
12 ˜ G IJ G ij + ˜ G I [ J ˜ G K ] L B iL B jK (cid:1) ∂ I ǫ i ∂ J ǫ j + 14 ˜ G IJ ˜ G KL F IK F JL − iǫ IJ ∂ [ i ( C ) j ] I ǫ i ∂ J ǫ j (cid:21) , (4.8)where the matrix ˜ G IJ is the inverse of the two-by-two matrix G IJ . Comparing with(3.16) and using again parity invariance yields G IJ = s πr ℓ g s N δ IJ , ∂ [ i ( C ) j ] I = 0 (4.9)and then e Φ = s πr ℓ g s N (4.10)by using (4.7). Comparing the terms ∂ I ǫ i ∂ J ǫ j in (3.16) and (4.8), using (4.9) and theISO(2) symmetry to fix B Ii = 0, we find G µν = s πr ℓ g s N δ µν , G mn = r ℓ g s N πr δ mn . (4.11)We thus find that the components G IJ and G µν match, which shows that the metrichas the expected SO(6) isometry of the background sourced by D5-branes. We cancontinue the same analysis at third order. Using the constraints on the background10hat we have already derived, (4.1) yields the third order Lagrangian up to twoderivative terms, L (3) = ℓ g s (cid:20) ∂ i (cid:0) ˜ G IJ G jk (cid:1) ǫ i ∂ I ǫ j ∂ J ǫ k + iǫ IJ ∂ [ i ( C ) jk ] ǫ i ∂ I ǫ j ∂ J ǫ k + 12 ∂ i ( ˜ G IJ ˜ G KL ) ǫ i F IL F JK + ˜ G IK ˜ G JL B ij F KL ∂ I ǫ i ∂ J ǫ j (cid:21) . (4.12)Matching with (3.16) and using the ISO(2) invariance and SO(6) isometry of thebackground, we then obtain B ij = 0 , F = d C = 16 ǫ mnlp ∂ p e − d y m ∧ d y n ∧ d y l . (4.13)Overall, we have derived the following type IIB supergravity background e Φ = s πr ℓ g s N , d s = e Φ (d w I d w I + d x µ d x µ ) + e − Φ d y m d y m ,F = d C = 16 ǫ mnlp ∂ p e − d y m ∧ d y n ∧ d y l , B = C = 0 , (4.14)which perfectly matches with the well-known near-horizon geometry of N D5-branes[23].
Acknowledgements
We would like to thank Micha Moskovic for useful discussions. This work is sup-ported in part by the Belgian Fonds de la Recherche Fondamentale Collective (grant2.4655.07) and the Belgian Institut Interuniversitaire des Sciences Nucl´eaires (grant4.4511.06 and 4.4514.08). A.R. is a Research Fellow of the Belgian Fonds de laRecherche Scientifique-FNRS.
A Notations and conventions
We work in Euclidean signature throughout this paper. All our conventions are chosenconsistently with those of [6, 7, 9].
A.1 Symmetries and indices
For K D-strings parallel to N D5-branes, the symmetry group is SO(2) × SO(4) × SO(4) ′ . The SO(2) factor corresponds to rotations in the two-dimensional space paral-lel to the D-strings, the SO(4) factor corresponds to rotations in the four-dimensional11pin(4) Spin(4) ′ U(1) U( N ) U( K ) I, J, . . . (0 ,
0) (0 ,
0) 1 µ, ν, . . . (1 / , /
2) (0 ,
0) 0 m, n, . . . (0 ,
0) (1 / , /
2) 0 α, β, . . . (upper or lower) (1 / ,
0) (0 ,
0) 0 ˙ α, ˙ β, . . . (upper or lower) (0 , /
2) (0 ,
0) 0 ζ , ξ, . . . (upper or lower) (0 ,
0) (1 / ,
0) 0 ˙ ζ, ˙ ξ, . . . (upper or lower) (0 ,
0) (0 , /
2) 0 f, f ′ , . . . (lower) (0 ,
0) (0 ,
0) 0
N 1 f, f ′ , . . . (upper) (0 ,
0) (0 ,
0) 0 ¯N 1 i, j, . . . (lower) (0 ,
0) (0 ,
0) 0 i, j, . . . (upper) (0 ,
0) (0 ,
0) 0 A I (0 ,
0) (0 ,
0) 1 X jµi = ℓ A jµi (1 / , /
2) (0 ,
0) 0 Y jmi = ℓ φ jmi (0 ,
0) (1 / , /
2) 0 ψ jαζi = ℓ Λ jαζi (1 / ,
0) (1 / ,
0) 1 / ψ jα ˙ ζi = ℓ Λ jα ˙ ζi (1 / ,
0) (0 , / − / ¯ ψ ˙ α jζi = ℓ ¯Λ ˙ α jζi (0 , /
2) (1 / , − / ¯ ψ ˙ α j ˙ ζi = ℓ ¯Λ ˙ α j ˙ ζi (0 , /
2) (0 , /
2) 1 / D jµνi (1 ,
0) (0 ,
0) 0 q αfi (1 / ,
0) (0 ,
0) 0
N K ˜ q αfi (1 / ,
0) (0 ,
0) 0 ¯N ¯K χ ζfi (0 ,
0) (1 / , − / N K χ ˙ ζfi (0 ,
0) (0 , /
2) 1 / N K ˜ χ fiζ (0 ,
0) (1 / , − / ¯N ¯K ˜ χ fi ˙ ζ (0 ,
0) (0 , /
2) 1 / ¯N ¯K Table 1: Conventions for the transformation laws of indices and fields. For maximumclarity, we have indicated all the indices associated to each field, whereas in the maintext the gauge U( N ) and U( K ) indices are usually suppressed. The representations ofSpin(4) = SU(2) + × SU(2) − and Spin(4) ′ = SU(2) ′ + × SU(2) ′− are indicated accordingto the spin in each SU(2) factor. The (1 / , /
2) of SU(2) + × SU(2) − correspondsto the fundamental representations of SO(4). The U(1) group corresponds to theworldsheet rotations under which positive and negative chirality spinors have charge1 / − / ′ fac-tor corresponds to rotations in the four-dimensional space transverse to both theD-strings and the D5-branes.We use indices 1 ≤ I, J, . . . ≤ ≤ µ, ν ≤ ≤ α, β ≤ ≤ m, n ≤ ≤ ζ , ξ ≤ ′ respectively. For right-handed spinors we use dotted indices ˙ α , ˙ ζ etc. The SO(4)spinor indices are raised and lowered according to the standard conventions λ α = ǫ αβ λ β , (A.1) ψ ˙ α = ǫ ˙ α ˙ β ψ ˙ β , (A.2)with ǫ = − ǫ = − ǫ = ǫ = 1, and similarly for the SO(4) ′ spinor indices ζ , ξ, etc.We also have gauge group indices f, f ′ and i, j for U( N ) and U( K ) respectively.We often suppress these indices if there is no ambiguity.The branes are located in R according to Table 2. We denote w I the twocoordinates parallel to the D-strings and by ( z i ) = ( x µ , y m ) the coordinates transverseto the D-strings. We also define the radial coordinate r by r = y m y m . (A.3)1 2 3 4 5 6 7 8 9 10D5 × × × × × × D1 × × w w x x x x y y y y (A.4)Table 2: Location of the D1- and D5-branes in R . The third row indicates thenotation we use for the various types of coordinates. A.2 Four-dimensional algebra
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