On the dualization of scalars into (d-2)-forms in supergravity. Momentum maps, R-symmetry and gauged supergravity
aa r X i v : . [ h e p - t h ] A ug IFT-UAM/CSIC- - arXiv:1605.05559 [hep-th] May th , , On the dualization of scalars into ( d − ) -forms insupergravity Momentum maps, R-symmetry and gauged supergravity
Igor A. Bandos ,a and Tomás Ortín ,b1 Department of Theoretical Physics, University of the Basque Country UPV/EHU,P.O. Box , Bilbao, Spain IKERBASQUE, Basque Foundation for Science,
Bilbao, Spain Instituto de Física Teórica UAM/CSICC/ Nicolás Cabrera, – , C.U. Cantoblanco, E- Madrid, Spain
Abstract
We review and investigate different aspects of scalar fields in supergravity the-ories both when they parametrize symmetric spaces and when they parametrizespaces of special holonomy which are not necessarily symmetric (Kähler and Qua-ternionic-Kähler spaces): their rôle in the definition of derivatives of the fermionscovariant under the R-symmetry group and (in gauged supergravities) under somegauge group, their dualization into ( d − ) -forms, their role in the supersymme-try transformation rules (via fermion shifts, for instance) etc. We find a generaldefinition of momentum map that applies to any manifold admitting a Killing vec-tor and coincides with those of the holomorphic and tri-holomorphic momentummaps in Kähler and quaternionic-Kähler spaces and with an independent defini-tion that can be given in symmetric spaces. We show how the momentum mapoccurs ubiquitously: in gauge-covariant derivatives of fermions, in fermion shifts,in the supersymmetry transformation rules of ( d − ) -forms etc. We also give thegeneral structure of the Noether-Gaillard-Zumino conserved currents in theorieswith fields of different ranks in any dimension. a E-mail:
Igor.Bandos [at] ehu.eus b E-mail:
Tomas.Ortin [at] csic.es ontents Review of symmetric σ -models . Coset representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H-covariant derivatives and the momentum map . . . . . . . . . . . . A more basic definition of the momentum map . . . . . . . . . . . . . H-covariant Lie derivatives . . . . . . . . . . . . . . . . . . . . . . . . . Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gauging of an H subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SL ( R ) /SO ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SU (
1, 1 ) /U ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E (+ ) /SU ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noether -forms and dualization . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SL ( R ) /SO ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E (+ ) /SU ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NGZ -forms and dualization in d = . Supersymmetry and the momentum map . . . . . . . . . . . . . . . . . . . . . Supersymmetry transformations of ( d − ) -forms . . . . . . . . . . . . Tensions of supersymmetric ( d − ) -branes . . . . . . . . . . . . . . . . Fermion shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N = d = . . N = d = The higher-dimensional, higher-rank NGZ -forms and dualization . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N = d = . . N = d = . . N = B , d =
10 supergravity . . . . . . . . . . . . . . . . . . . . . . Conclusions A Kähler–Hodge manifolds in N =
1, 2 , d = supergravity A. -form potentials from the Kähler–Hodge manifolds of N =
1, 2, d = B Quaternionic–Kähler manifolds in N = , d = supergravity B. -form potentials from the Quaternionic–Kähler manifolds in N = d = ntroduction One of the main features of supergravity theories is the presence of scalar fields. Inmany cases this presence can be traced to a compactification of some higher-dimensionalsupergravity theory and, then, the scalars encode a great deal of information aboutthe moduli space of the compactification. In gauged supergravity theories, the scalarpotential gives rise to symmetry and supersymmetry breaking and identify possiblevacua, and can be used to construct inflationary models. Furthermore, the gravitatingsolutions of supergravity theories can (or must, depending on the case) have activescalars. For instance, the supergravity generalizations of the Reissner-Nordström blackhole have them, giving rise to very interesting phenomena such as the attractor mecha-nism [ , , , ]. Actually, the fact that their values at infinity (which can be interpretedas their vacuum expectation values) do not occur in the charged black-hole entropyformula [ ] is, certainly, a major indication of the existence of a microscopic inter-pretation for the black-hole entropy. On top of this, the relation of the scalars withcompactification moduli plays a fundamental rôle in the microscopic interpretation ofthe black-hole entropy in the context of string theory [ ]. But there are solutions muchmore directly related to scalar fields: these are the domain walls and the ( d − ) -branesolutions. We will discuss the latter later.Scalar fields are, therefore, not just a nuisance one has to live with in supergravity,but a blessing, a fantastic tool whose use one has to master, in spite of the fact that, sofar, we have only found one scalar field in Nature.Scalar fields can be coupled non-linearly among themselves (non-linear σ -modelsand scalar potentials) or to other fields (scalar-dependent kinetic matrices) in verysimple ways, without having to include terms of higher-orders in derivatives, becausetheir transformations do not contain spacetime derivatives (even if they couple to agauge field). Using this property one can rewrite theories of higher-order in derivativesof other fields (for instance, theories of gravity with corrections of higher-order incurvature) as standard quadratic theories with couplings to scalar fields. A well-knownexample is the equivalence between f ( R ) theories of gravity and Jordan-Brans-Dickescalar-tensor theories of gravity.This versatility of scalar fields comes at a price, though, and the non-linearitiescreate their own problems. In this paper we want to address specially one of them:that of the dualization of scalars into ( d − ) -form potentials.A it is well known, supergravity theories, as the low-energy, effective field-theorylimits of superstring theories, contain a great deal of information about the p -dimensionalextended objects ( branes ) that occur in the latter. This information is encoded in the ( p + ) -form potentials they electrically couple to. For p ≤ d /2 −
2, these fields appearin the supergravity action as fundamental fields. For higher values, though, one has toconsider their electric-magnetic duals.In most cases, the supergravity theory cannot be completely reformulated in termsof the dual supergravity fields: even their field strengths can only be defined using thefundamental ones. We have learned to deal with all of them at the same time using the o-called “democratic formulations” [ ] or PST-type duality-symmetric actions [ , ]constructed with the use of Pasti-Sorokin-Tonin approach [ , ]. The technical reason is that, typically, the supergravity action contains potentialswithout derivatives and the standard procedure for dualization requires, as a firststep, the replacement of the potentials by their field strengths as independent variablesin the action. This problem is more acute for scalar fields, because they genericallyappear without derivatives in the σ -model metric and in the kinetic matrices.When the σ -model metric admits an isometry, it is possible to make it independentof the associated scalar coordinate by a change of variables. If the isometry is a globalsymmetry of the theory, the kinetic matrices may also be independent of that scalartoo and, then, one could dualize it into a ( d − ) -form potential. One could repeatthe procedure for additional commuting isometries but most σ -models do not have asmany commuting isometries as scalar fields, even if they have more isometries thanscalar fields, as it happens in N > d = i.e. all the symmetries of the equations ofmotion, including those that do not leave the action invariant. This implies that it isnot enough to dualize the scalars (even if possible), since they do not transform inlinear representations of the duality group and the dual ( d − ) -form potentials canonly transform linearly.The basic idea to solve this problem was proposed in Ref. [ ] for the case of theSL( , R )/SO( ) σ -model that occurs in N = B , d =
10 supergravity (as well as in manyother theories): the objects to be dualized are not the scalars but the Noether -forms j A = j A µ dx µ associated to the symmetry . In a background metric g µν , these are relatedto the Noether current densities j µ A satisfying on-shell the continuity equation ∂ µ j µ A = . )by j A µ ≡ j A ν g νµ p | g | . ( . )In terms of the Noether -forms, the continuity equations take the form d ⋆ j A = . )and can be locally solved by introducing ( d − ) -form potentials B A so that Observe that the democratic formulation of Ref. [ ] does not have manifest SL ( R ) invariance inthe IIB sector because only the RR - and -forms are considered and they are part of a doublet and atriplet. In this sense it is incomplete. The more complicated PST-type type IIB supergravity action of [ ]contains the complete set of higher forms. Here and in what follows the indices A , B , C , . . . run over the adjoint representation of the wholeglobal symmetry group. j A = dB A . ( . )The ( d − ) -form fields B A are the duals of the scalars. In the kind of theories weare interested in, the numbers of scalars and ( d − ) -forms do not coincide in generalbecause there are more global symmetries than scalars. However, there are constraintsto be taken into account that reduce the number of independent dynamical degrees offreedom associated to the latter, as we will see.When the scalars couple to other fields, these must transform under the global sym-metries of the σ -model as well. Some of the transformations may be electric-magneticdualities and only the equations of motion will be left invariant by them. Accordingly,there are no Noether currents for those transformations. As shown in Ref. [ ], in the -dimensional case it is always possible to use the Noether-Gaillard-Zumino (NGZ) -forms [ ] which are conserved on-shell, to define the -forms B A . We will study thehigher-dimensional, higher-rank analog of the NGZ -forms in Section .In order to describe systematically the procedure, it is convenient to start by review-ing the construction of the metrics, Killing vectors, Vielbeins and connection -formsetc. in symmetric spaces, since this is the kind of target spaces that occurs in mostextended supergravities. We will do this in Section . In the process we will (re-)discover structures which appear in the gauging of the theories (specially in the super-symmetric case) (covariant derivatives, fermion shifts etc.) In particular, we are goingto see that in all symmetric spaces there exists a generalization of the holomorphic andtriholomorphic momentum maps associated to the Kähler-Hodge and quaternionic-Kähler manifolds of N =
1, 2 supergravities in d = ( d − ) -forms dual to the NGZ currents. These generalizations share the sameproperties and deserve to be called momentum maps as well.Furthermore, we are also going to give an even more general definition of momen-tum map (Section . . ), valid for any manifold admitting one isometry, showing thatin symmetric spaces, Kähler-Hodge or quaternionic-Kähler spaces our general defini-tion is equivalent to the standard one. This is one of the main results of this paper.To end Section we will review some well-known examples which will be usefulin what follows.In Section we address the dualization of the scalars of a symmetric σ -model into ( d − ) -form potentials along the lines explained before. Then, in Section we willconsider the case in which the scalars of the symmetric σ -model are coupled to thevector fields of a generic -dimensional field theory of the kind considered by Gaillard Actually, using the embedding-tensor formalism, it can be argued that the ( d − ) -form potentials ofany field theory transform in the adjoint representation of the global symmetry group [ , ]. Some ofthe symmetries may not act on the scalars at all but, to simplify matters, we focus here on the symmetriesof the scalar σ -model. nd Zumino in Ref. [ ], introducing the NGZ current -form and studying its dual-ization into -forms. We will also consider there (Section . . ) the general form ofthe supersymmetry transformations of the -forms and the rôle played in them by themomentum map. It is because of this rôle that we expect the tensions of the stringsthat couple to the -forms to be determined by the momentum map (Section . . ). Wewill also show how the momentum map occurs in the fermion shifts of the supersym-metry transformation rules of the fermions of -dimensional extended supergravities(Section . . ).The higher-dimensional case in which the scalars are also coupled to potentials ofdifferent and higher ranks will be considered in Section and we will show throughexamples that the equation of conservation of the generalized NGZ current -form hasa universal form.Our conclusions are contained in Section . Review of symmetric σ -models Let us consider a homogeneous space M on which the Lie group G acts transitively,and where H ⊂ G, topologically closed, is the isotropy subgroup. Then M is homeo-morphic to the coset space G/H of equivalence classes under right multiplication byelements of H { g H } (G acts from the left on these equivalence classes) and can be giventhe structure of a manifold of dimension dim G − dim H. Furthermore, G can be seenas a principal bundle with base space M=G/H, structure group H, and projection G → G/H.In any homogeneous space G/H, the Lie algebra of G, as a vector space, can bedecomposed as the direct sum g = h ⊕ k , where h is the Lie subalgebra of H and k isits orthogonal complement. By definition of subalgebra [ h , h ] ⊂ h . ( . )G/H is said to be a reductive homogenous space if [ k , h ] ⊂ k , ( . )which means that k is a representation space of H. Finally, G/H is said to be symmetricand ( k , h ) is called a symmetric pair if it is reductive and [ k , k ] ⊂ h . ( . )The two components of a symmetric pair are mutually orthogonal with respect tothe Killing metric which is block-diagonal. In this review we follow Refs. [ , ] although a big part of material can be also found in theclassical papers Refs. [ , , , ]. ow, if G/H is a symmetric space (G connected, and H compact) and there is aG-invariant metric defined on it, then it is Riemannian symmetric space.The metrics of the scalar σ -models that appear in all supergravities in d ≥ supercharges are the metrics of some Riemannian symmetricspace. The metrics and the kinetic terms can be constructed using a G/H coset repre-sentative or by using a generic element of G and gauging an H subgroup. Let us startby reviewing the first method. . Coset representative
Let us introduce some notation: we denote by { T A } , { M i } and { P a } (where A , B , . . . =
1, . . ., dim G, i , j , . . . =
1, . . ., dim H and a , b , . . . =
1, . . ., d ≡ dim G − dim H), three basesof, respectively, g , h and k with { T A } = { M i } ∪ { P a } (see Ref. [ , ] and also Ref. [ ]).The structure constants are defined by [ T A , T B ] = f ABC T C , ( . )and, by definition of symmetric space, the only non-vanishing components are [ M i , M j ] = f ijk M k , [ P a , M i ] = f aib P b , [ P a , P b ] = f abi M i . ( . )The adjoint representation of g is defined by the matrices Γ Adj ( T A ) BC ≡ f ACB , ( . )and, obviously, their restriction to the indices i , j is the adjoint representation of h .Furthermore, the matrices Γ ( M i ) ab = f iba , ( . )provide another representation of h with representation space k .Only the diagonal blocks of the the Killing metric K AB ≡ Tr (cid:2) Γ Adj ( T A ) Γ Adj ( T A ) (cid:3) = f ACD f BDC , ( . ) K ab and K ij are non-vanishing ( K ai = K ij is the restriction of the Killing metric ofG to H and, for the kind of groups we are considering, it is proportional to the Killingmetric of H. Under the adjoint action of G, defined by g − T A g ≡ T B Γ Adj ( g − ) B A , ( . )The Killing metric metric is invariant due to the cyclic property of the trace That is: K ij = f iCD f jDC = f ikl f jlk + f iab f jba . Because f iab f jba = Tr [ Γ ( M i ) Γ ( M j )] ∝ f ikl f jlk . AB = Tr h Γ Adj ( g − T A gg − T B g ) i = K CD Γ Adj ( g − ) C A Γ Adj ( g − ) D B . ( . )Then, since the Killing metric is invertible ( K AB ) for the kind of groups we are consid-ering, we find that K BC Γ Adj ( g ) C D K DA = Γ Adj ( g − ) AB . ( . )Let us denote by u ( φ ) = u ( φ , . . ., φ d ) a coset representative of G/H in some localcoordinate patch. In practice it will be a matrix transforming in some representation r of G. The scalar fields of the σ -model will be mappings from spacetime to G/Hexpressed in these coordinates as the functions φ m ( x ) . Under a left transformation g ∈ G, u ( φ ) transforms into another element of G, which becomes a coset representative u ( φ ′ ) only after a right transformation with the inverse of h ∈ H, that is (see Refs. [ , , , ]) gu ( φ ) = u ( φ ′ ) h . ( . )For a given choice of coset representative u , h will depend on g and φ , but we will notindicate explicitly that dependence.The left-invariant Maurer–Cartan -form V ∈ g and can be expanded as follows: V ≡ − u − du = e a P a + ϑ i M i . ( . )The e a components can be used as Vielbeins in G/H and the ϑ i components play therole of connection . The Maurer–Cartan equations satisfied by V ( dV − V ∧ V = -form components: de a − ϑ i ∧ e b f iba = . ) d ϑ i − ϑ j ∧ ϑ k f jki − e b ∧ e c f bci = . )Comparing the first of these equations with Cartan’s structure equation with vanishingtorsion D e a = de a + ω ba ∧ e b = -form ω ba = − ϑ i f iba , ( . )which also justifies the identification of ϑ i with a connection. The curvature -form ofthis connection is, from the definition The elements of the basis of k and h will be in the representation r in which u transforms. However,for the sake of simplicity, we will write P a instead of Γ r ( P a ) etc. whenever this does not lead to confusion. See the transformation rules Eqs. ( . ). ba ( ω ) = − R ( ϑ ) i f iba , where R ( ϑ ) i ≡ d ϑ i − ϑ j ∧ ϑ k f jki . ( . )Then, Eqs. ( . ) tell us that R ( ϑ ) i = e b ∧ e c f bci , R ba ( ω ) = − e d ∧ e e f dei f iba . ( . )We have defined a Vielbein basis and an affine connection on G/H, but we have notdefined a Riemannian metric yet. We can do so if we are provided with a metric g ab on k : ds = g ab e a ⊗ e b = g ab e am e bn d φ m d φ n ≡ G mn ( φ ) d φ m d φ n . ( . )Its pullback over spacetime, conveniently normalized, can be used in the action for the σ -model: S = Z G mn ( φ ) d φ m ∧ ⋆ d φ n . ( . )In order to construct a Riemannian symmetric σ -model the metric G mn ( φ ) must beinvariant under the left action of G. Under the left multiplication by g ∈ G, u ( φ ′ ) = gu ( φ ) h − , and the components of the left-invariant Maurer–Cartan -form transformin the adjoint representation of H (the ϑ i as a connection of H): e a ( φ ′ ) = ( he ( φ ) h − ) a = Γ Adj ( h ) a b e b ( φ ) , ϑ i ( φ ′ ) = ( h ϑ ( φ ) h − ) i + ( dhh − ) i , ( . )where e ( φ ) = e a ( φ ) P a and ϑ ( φ ) = ϑ i ( φ ) M i . Infinitesimally, h ∼ + σ i ( φ ) M i , ⇒ e a ( φ ′ ) ∼ e a ( φ ) + σ i ( φ ) f ib a e b ( φ ) , ( . )and the Riemannian metric G mn ( φ ) will be invariant under the left action of G if themetric g ab on k is H-invariant: f i ( ac g b ) c = . )In all the relevant cases we can set g ab ∝ K ab , the projection on k of the Killing metricand we will do so from now on. More precisely, we will use this normalization: g ab = Tr [ P a P b ] . ( . )Observe that the H-invariance of g ab Eq. ( . ) automatically guarantees that thetorsionless connection ω ba in Eq. ( . ) is metric-compatible and, therefore, it is theLevi–Civita connection of the above metric. We will sometimes use g AB = Tr [ T A T B ] . n addition to the isometries corresponding to the left action of G, with Killingvectors k A , the resulting Riemannian metric is also invariant under the right actionof N(H)/H, N(H) being the normalizer of H in G. The Killing vectors associated tothe latter are just the vectors e a dual to the horizontal Maurer–Cartan -forms in thedirections of N(H)/H [ ].Our next task is to find the general expression of the Killing vectors k A whichdefines the transformation rule of the Goldstone fields. From the infinitesimal versionof gu ( φ ) = u ( φ ′ ) h with g = + σ A T A , h = − σ A W Ai M i , φ m ′ = φ m + σ A k Am , ( . )where W Ai is known as the H-compensator , we get after some straightforward manipu-lations k Aa = − Γ Adj ( u − ( φ )) a A , ( . ) W Ai = − k Am ϑ im − P Ai , ( . )where we have defined the momentum map P Ai P Ai ≡ Γ Adj ( u − ( φ )) i A . ( . )It transforms as P ′ Ai = Γ Adj ( h ( φ )) i j P B j Γ Adj ( g − ) B A , ( . )and it plays a crucial role in the gauging of the global symmetry group G, as we aregoing to see below. . . H-covariant derivatives and the momentum map
With the objects that we have found we can construct
H-covariant derivatives and
H-covariant Lie derivatives , which transform covariantly under the compensating H trans-formations associated to global G transformations. Let us start by discussing the for-mer.In supergravity theories, H coincides with the R-symmetry group, under whichall the fermions transform, and, therefore, all the derivatives of fermions must be H-covariant derivatives. Under a global G transformation of the scalars, these spacetimefields undergo an H scalar-dependent, compensating transformation that can be con-travariant ξ ′ = Γ s ( h ) ξ , or covariant, ψ ′ = ψ Γ s ( h − ) , in some representation s of H. or those fields, with the help of the pullback over the spacetime of the H connection ϑ im d φ m , (see the second of Eqs. ( . )), we define the H-covariant derivative by D ξ ≡ d ξ − ϑ i Γ s ( M i ) ξ , D ψ ≡ d ψ + ψϑ i Γ s ( M i ) . ( . )The H-covariant derivative satisfies the Ricci identities D ξ = − R ( ϑ ) i Γ s ( M i ) ξ , D ψ = ψ R ( ϑ ) i Γ s ( M i ) , ( . )where R ( ϑ ) i stands here for the curvature -form defined in Eq. ( . ).Using k Aa k B b f abi + P A j P Bk f jki = Γ Adj ( u − ) A ′ A Γ Adj ( u − ) B ′ B f A ′ B ′ i = f ABC P Ci , ( . )we find that the momentum map that we have defined above satisfies the equivariancecondition D A P Bi − D B P Ai − k Aa k Bb f abi + P A j P Bk f jki = f ABC P Ci , ( . )where D A = k Am D m . Using the explicit form of the curvature Eq. ( . ) it is easy toderive the following equation, which is sometimes used as definition of the momentummap D m P Ai = − R mni ( ϑ ) k A n . ( . )One is often interested in gauging the global symmetry group G (or a subgroupof G), making the supergravity theory invariant under transformations of the formEq. ( . ) with parameters σ A which are promoted to arbitrary spacetime functions σ A ( x ) . Under these transformations, the pullback of the second equation of ( . )acquires an additional term and the (spacetime pullback of) above H-covariant deriva-tives do not transform covariantly anymore. As usual, it is necessary to introducespacetime -forms A A and modify the above covariant derivatives as follows: D ξ ≡ d ξ − (cid:16) A A P Ai + ϑ i (cid:17) Γ s ( M i ) ξ , D ψ ≡ d ψ + ψ (cid:16) A A P Ai + ϑ i (cid:17) Γ s ( M i ) . ( . ) We will denote the pullback with the same symbol, ϑ i when this does not lead to confusion. In spite of the name, which is, admittedly, misleading, there is no true gauge symmetry in thisconstruction. The H-transformations are not arbitrary functions of the spacetime coordinates. Neitherthey are arbitrary functions of the scalar fields (the coordinates on G/H). The only arbitrary parametersin these transformations are the global parameters of the G transformation that needs to be compensatedto go back to the coset representative. Here we will not concern ourselves with the problem of matching the rank of the subgroup of G tobe gauged with the number of -forms available in the supergravity theory. In general this requires theexplicit or implicit introduction of the so-called embedding tensor . We will use it in Section . . . he structure of these gauge covariant derivatives is identical to the covariantderivatives that occur in gauged N =
1, 2 supergravities , even if the scalar manifolds(Kähler–Hodge and quaternionic–Kähler manifolds) are no coset spaces: the spinorsof these theories transform under U( ) Kähler transformations and the Kähler -formconnection plays the role of ϑ in Eq. ( . ). In N = ) compensating transformations and the pullback ofthe SU( ) connection of the quaternionic-Kähler manifold plays the role of ϑ . Associ-ated to these symmetries there are holomorphic and tri-holomorphic momentum mapswhich play the same role as P Ai . A more detailed comparison between these structuresand the ones that arise in symmetric spaces can be found in the appendices.Observe that these covariant derivatives cannot be obtained by the often-used (butgenerally wrong) replacement of the pullback by the “covariant pullback” of the Hconnection ϑ i in Eqs. ( . ) ϑ im d φ m → ϑ im D φ m , ( . )where D φ m ≡ d φ m − A A k Am , ( . )is the covariant derivative of the scalars, because, according to Eq. ( . ), ϑ im D φ m = (cid:16) A A P Ai + ϑ im d φ m (cid:17) + A A W Ai , ( . )and the H-compensator does not vanish in general. Something similar happens in theKähler–Hodge and quaternionic–Kähler manifolds of N =
1, 2, d = , ].Using the identity Eq. ( . ) and other results derived in this section one can com-pute the Ricci identities for the D covariant derivative D ξ = − (cid:2) F A P Ai + R ( ϑ ) imn D φ m D φ n (cid:3) Γ s ( M i ) ξ , D ψ = ψ (cid:2) F A P Ai + R ( ϑ ) imn D φ m D φ n (cid:3) Γ s ( M i ) , ( . )where F A = dA A − f BC A A B ∧ A C . ( . ) . . A more basic definition of the momentum map
Let us consider a d Riemannian manifold M , not necessarily symmetric, but admit-ting a set of Killing vectors k Aa , that is See, for instance, Refs. [ , , ]. The signature is irrelevant in this discussion, which also applies to pseudo-Riemannian spaces. ( a | k A | b ) = . )where ∇ a is the Levi-Civita covariant derivative. To each Killing vector one can asso-ciate an infinitesimal rotation in tangent space generated by P Aba ≡ ∇ a k Ab . ( . )The antisymmetry of P A ab = g ac P Aca follows from the Killing equation. Let { M i } be a basis of the Lie algebra of the holonomy group of M. Generically it will be SO ( d ) but for spaces of special holonomy it will be some subgroup H ⊂ SO ( d ) . In particular,for Riemannian homogeneous spaces G/H, the holonomy group is precisely H. Wecan, then, decompose P Aba in that basis, defining at the same time the coefficients asthe components of the momentum map P Aba ≡ P Ai Γ ( M i ) ba . ( . )It is not hard to show using the explicit expressions for the Killing vectors andconnection Eqs. ( . ) and ( . ) that the momentum map we have just defined reducesto the one defined in Eq. ( . ) for symmetric spaces. Furthermore, using the generalidentity for Killing vectors ∇ a ∇ b k Ac = k Ad R dabc , ⇒ ∇ a P Acb = k Ad R dabc , ( . )and decomposing both sides of this equation in the basis of the holonomy algebra ,we get a general version of Eq. ( . ) ∇ a P Ai = k Ad R dai ( ϑ ) . ( . )Finally, we can also show that the momentum map satisfies the equivariance prop-erty [ P A , P B ] ab = f ABC P Cab − k Ac k Bd R cdab . ( . )Under an infinitesimal isometry δ α x m = α A k Am , ( . )objects living in tangent space (vectors ψ a , say ) transform as δ α ψ a = α A ( P Aab + k Am ω mab ) ψ b , ( . ) That is R adbc = − R adi Γ ( M i ) cb , ( . )where the minus sign is chosen to match the sign of the H-curvature R ( ϑ ) in symmetric spaces. These variables arise naturally in N = d scalarmultiplets x m + θ e ma ψ a ) . See, e.g. [ , ] and references therein. nd, when gauging the isometry group, these compensating transformations must betaken into account in the construction of the covariant derivative, which is given by D m ψ a = ∇ m ψ a − A Am P Aab ψ b = ∂ m ψ a − (cid:16) A Am P Aab + ω mab (cid:17) ψ b . ( . )This covariant derivative reduces to the H-covariant derivative in Eq. ( . ) for sym-metric spaces. The above definition can be generalized to objects transforming in otherrepresentations of SO ( n ) in the obvious way. . . H-covariant Lie derivatives
The H-compensator can be understood as the “local” parameter of the H compensatingtransformation associated to the infinitesimal G transformation generated by T A or theKilling vector k A . To study the behavior under G global transformations of fieldstransforming under these H compensating transformations (something usually donethrough the Lie derivative) it is necessary to take into account the latter. This requiresan H-covariant generalization of the standard Lie derivative with respect to the Killingvector k A denoted by L k A . The equivariance property of the H-compensator L k A W Bi − L k B W Ai + W A j W Bk f jki = f ABC W Ci , ( . )plays an essential role.On fields transforming contravariantly ξ ′ = Γ s ( h ) ξ , or covariantly ψ ′ = ψ Γ s ( h − ) ,the H-covariant derivative is defined by L k A ξ ≡ L k A ξ + W Ai Γ s ( M i ) ξ , L k A ψ ≡ L k A ψ − ψ W Ai Γ s ( M i ) . ( . )The main properties satisfied by this derivative are [ L k A , L k B ] = L [ k A , k B ] , ( . ) L k A e a = . ) L k A u = L k A u − uW Ai M i = T A u . ( . )Infinitesimally, the H connection ϑ i transforms with the H-covariant derivative ofthe transformation parameters. Thus, an appropriate definition for its H-covariant Liederivative would be L k A ϑ i ≡ L k A ϑ i + D W Ai . ( . ) See Ref. [ ], which we follow here, and references therein. One could define the H-covariant Liederivative with respect to any vector but the crucial Lie-algebra property Eq. ( . ) only holds for Killingvectors. sing the definitions and Eq. ( . ), one can show that L k A ϑ i = . ) . . Final remarks
The H-covariant derivative of u ( φ ) , which transforms covariantly in some representa-tion r , is, according to the definition D u ≡ du + u ϑ i M i = u h u − du + u ϑ i M i i = − ue a P a , ( . )where we have used the expansion of the Maurer–Cartan -form Eq. ( . ).We can use this result to obtain a very convenient expression of the action of the σ -model directly in terms of the coset representative u ( φ ) , which transforms linearlyunder G (with the metric defined in Eq. ( . )): S = Z Tr [ u − D u ∧ ⋆ u − D u ] . ( . )The invariance under the left action of G on the coset representative is manifest inthis form. This expression connects this approach with the approach that we are goingto review in the next section.To end this section, when the coset space is of the kind SL( n )/SO( n ), there is an al-ternative but completely equivalent construction which is often used in supergravity .One defines the symmetric and H-invariant matrix M ≡ uu T , M ′ = M ( φ ′ ) = g M ( φ ) g T , ( . )and choosing a basis in which the P a are symmetric matrices and the M i are antisym-metric, it is not difficult to show thatTr [ M − d M ∧ ⋆ M − d M ] = [ u − D u ∧ ⋆ u − D u ] . ( . )The equations of motion from this action are obtained much more easily from theformulation that we are going to discuss in the next section because one does not haveto deal with the scalar dependence of the connection ϑ i . Using the cyclic property of the trace, it can also be written in the form S = Z Tr [ D uu − ∧ ⋆ D uu − ] . ( . ) This coset arises naturally in toroidal compactifications. . Gauging of an H subgroup σ -models on coset spaces G/H are often constructed by gauging a subgroup H of a σ -model constructed on the group manifold G. The latter has the action S G [ u ] = Z Tr [ u − du ∧ ⋆ u − du ] , ( . )where u ( ϕ ) = u ( ϕ , · · · , ϕ dim G ) is now a generic element of the group G in somerepresentation r and in some local coordinate patch that contains the identity. Thisaction is invariant under the left and right (global) action of the group G and, therefore,its global symmetry group is G × G.Now we want to gauge a subgroup H of the right symmetry group, under which u ′ = uh − ( x ) . Here h ( x ) stands for an arbitrary function of the spacetime coordinatesthat gives an element of H at each point. Such a function can be constructed by expo-nentiation of a linear combination the generators of h with coefficients σ i ( x ) which arearbitrary functions of the spacetime coordinates. After gauging, the global symmetrygroup will be broken to G × H.We introduce an h -valued spacetime gauge field A = A i µ dx µ M i transforming ex-actly as the ϑ i components of the left-invariant Maurer–Cartan -form of G/H Eq. ( . )and the H-covariant derivative D u ≡ du + uA , ( . )and we simply replace the exterior derivative d by D in the action Eq. ( . ) withoutadding a kinetic term for the gauge field: S Gauged [ u , A ] = Z Tr [ u − D u ∧ ⋆ u − D u ] . ( . )The gauged action describes only d = dim G − dim H degrees of freedom: on gen-eral grounds we expect that the gauge symmetry can be used to eliminate dim H ofthe scalars ϕ x , x =
1, . . . ,dim G, from the action, leaving only those that parametrizethe coset space G/H, that we have denoted by φ m . This is the so-called unitary gauge .As we are going to see, only d of the scalar equations of motion are independent, incomplete agreement with the general expectation.Let us first consider the equations of motion of the gauge field. These are purelyalgebraic: δ S Gauged δ A i = ⋆ Tr [ M i u − D u ] = . )and the solution is The representation of the coset element by a group element defined modulo H-transformations isalso characteristic of the harmonic superspace approach [ , ] as well as of the spinor moving frameformalism [ , ]. See Ref. [ ] for a recent application and more references. i = V i , ( . )where V = − u − du is the left-invariant Maurer–Cartan -form in G. In the unitarygauge, V depends only on the physical scalars φ m and becomes, automatically, theleft-invariant Maurer–Cartan -form in G/H, so V i = ϑ i .This solution can be substituted in the above action: this substitution A i = V i ( ϕ ) and the derivation of the equations of motion for the scalars from the action are twooperations that commute and the final result is the same. As a matter of fact, after thesubstitution, the scalar equations of motion are δ S Gauged [ ϕ , V ( ϕ )] δϕ x = δ S Gauged [ ϕ , A ] δϕ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A i = V i + δ S Gauged [ ϕ , A ] δ A i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A i = V i δ V i δϕ x , ( . )but the second term vanishes identically.In the unitary gauge, after the substitution A i = V i = ϑ i ( φ ) we recover the actionEq. ( . ). Classically, these two formulations are completely equivalent. However, inthis formulation the scalar equations of motion are easier to derive because we havejust shown that we can ignore the variations of the connection with respect to thescalars.The equations of motion of the scalars ϕ x , x =
1, . . . ,dimG are δ S Gauged δϕ x = V x A Tr [ T A D ⋆ ( u − D u )] = . )The invariance under local, right, H-transformations implies, according to Noether’ssecond theorem, the following dimH Noether identities relating the scalar equationsof motion: Tr [ M i D ⋆ ( u − D u )] = . )These are off-shell identities and are also valid in the unitary gauge after the substitu-tion A i = ϑ i . Therefore, they are valid in the case discussed in the previous section.Taking into account the gauge identities the only non-trivial equations of motionare those of the physical scalars φ m , which take the form δ S Gauged δφ m = e am Tr [ P a D ⋆ ( u − D u )] = e am g ab D ⋆ e b = . ) . Examples
In this subsection we are going to review a few examples which we will use repeatedlyin what follows: . The SL ( R ) /SO ( ) coset space, which occurs in many supergravities: N = B , d =
10 supergravity (the effective field theory of the type IIB superstring) N = d =
9, 8 supergravity (obtained from the former by toroidal dimensionalreduction), N = d = N = B , d =
10 supergravity the scalarfields that parametrize this coset are the dilaton ϕ and RR -form χ , combinedinto the axidilaton field τ = χ + ie − ϕ (see Eq. ( . ) below). In N = d = -dimensional dilaton φ and the dual of the -dimensional Kalb-Ramond -form a and τ = a + ie − φ . . The SU (
1, 1 ) /U ( ) coset, which is an often used completely equivalent alternativeform of the SL ( R ) /SO ( ) coset space: constructed by Schwartz in the SU (
1, 1 ) formulation [ , , ] and rewritten in the SL ( R ) formulation in which thedilaton and RR -form appear more naturally in Ref. [ ]. The supersymmetrytransformations of all the fields of N = B , d =
10 supergravity, including the -forms dual to the dilaton and RR -form, were given in the SU (
1, 1 ) formulationin Ref. [ , ] and [ ]. In Section . . we are going to study the dualizationof the scalars and the supersymmetry transformations of the dual -form fieldsproposed in that reference from the geometrical point of view taken here. . The E (+ ) /SU ( ) coset space of N = d = . . SL ( R ) /SO ( ) The group SL ( R ) is isomorphic to SO (
2, 1 ) . The Lie brackets of the three generators { T A } can be conveniently written in the form [ T A , T B ] = − ε ABD Q DC T C , ⇒ f ABC = − ε ABD Q DC , A , B , . . . =
1, 2, 3 , ( . )where Q = diag (+ + − ) . A -dimensional representation (the one we are going towork with) is provided by T = σ , T = σ , T = i σ , ( . )where the σ i s are the standard Hermitian, traceless, Pauli matrices satisfying σ i σ j = δ ij + i ε ijk σ k . The Killing metric is K AB = − Q AB , and g AB = Tr ( T A T B ) = Q AB . ( . )A convenient SL ( R ) /SO ( ) coset representative is provided by u = e − ϕ T e e ϕ /2 χ ( T + T ) = e − ϕ /2 e ϕ /2 χ e ϕ /2 , ( . ) nd the symmetric SL ( R ) matrix M constructed according to the recipe Eq. ( . ) isthe usual one M = e ϕ | τ | χχ , where τ ≡ χ + ie − ϕ , ( . )is sometimes called the axidilaton field.The coset representative u transforms according to the general rule u ′ ( x ) = u ( x ′ ) = gu ( x ) h − where g = a bc d ∈ SL ( R ) , so ad − bc = h = cos θ sin θ − sin θ cos θ ∈ SO ( ) .( . )In order to preserve the upper-triangular form of the coset representative u , the com-pensating h transformation must be such thattan θ = − ce − ϕ c χ + d , ( . )and this completely determines the transformation rules for the coordinates ϕ , χ : interms of τ they take the usual fractional-linear form τ ′ = a τ + bc τ + d . ( . )The generators P a , M are P ≡ T , M ≡ T . ( . )The components of the left-invariant MC -form in the above basis are given by e = d ϕ , e = − e ϕ d χ , ϑ = e , ( . )and the SL ( R ) -invariant metric of the coset space is ds = g ab e a e b = d τ d τ ∗ ( ℑ m τ ) . ( . ) (cid:16) Γ Adj ( u ) A B (cid:17) = e ϕ χ − e ϕ χ − χ e ϕ ( − | τ | ) + e − ϕ − e ϕ ( − | τ | ) − χ − e ϕ ( + | τ | ) + e − ϕ − e ϕ ( + | τ | ) . ( . ) he first two components of each of the three columns of the matrix (cid:16) Γ Adj ( u − ) AB (cid:17) = − χ χ e ϕ χ e ϕ ( − | τ | ) + e − ϕ e ϕ ( + | τ | ) − e − ϕ e ϕ χ e ϕ ( − | τ | ) e ϕ ( + | τ | ) . ( . )are the two components of each of the three Killing vectors, while the third row givesthe three components of the momentum map (one for each isometry). Using the Vier-beins, and in terms of ∂ τ = ( ∂ χ + ie ϕ ∂ ϕ ) we get the following explicit expressions forthe Killing vectors, momentum map and H-compensator k = τ∂ τ + c.c. , k = ( − τ ) ∂ τ + c.c. , k = ( + τ ) ∂ τ + c.c. , ( . ) ( P A ) = (cid:16) e ϕ χ , e ϕ ( − | τ | ) , e ϕ ( + | τ | ) (cid:17) , ( . ) ( W A ) = i ( , τ , − τ ) + c.c. , ( . ) . . SU (
1, 1 ) /U ( ) An SU (
1, 1 ) matrix can be parametrized by two complex numbers a , bu = (cid:18) a b ∗ b a ∗ (cid:19) , | a | − | b | = . )U ( ) acts on these two complex numbers by multiplication ( a , b ) → e − i ϕ ( a , b ) .Therefore, its right action on this matrix is u −→ u (cid:18) e − i ϕ e i ϕ (cid:19) . ( . )It is customary to introduce the vectors V α ± , α =
1, 2 where the subindices + , − refer to the U ( ) weight u ≡ ( V α − V α + ) , ⇒ ( V α + ) ∗ = σ αβ V β − . ( . )In terms of these vectors the constraint | a | − | b | = V α − V β + − V α + V β − = ε αβ , ε αβ V α − V β + = . )This constraint can be solved by a complex number z and a real one ξ a = cosh ρ + i ξ sinh ρρ , b = z cosh ρ , where ξ = | z | ρ tanh ρ − ρ . ( . )In these coordinates the general SU (
1, 1 ) matrix u behaves near the origin as u ∼ × + ξ i σ + ℜ e ( z ) σ + ℑ m ( z ) σ , ( . )from which we can read the generators and find the Lie algebra T = σ , T = σ , T = i σ , [ T A , T B ] = − ε ABD Q DC T C , ( . )where Q = diag (+ + − ) . The Lie algebra is the same as that of SL ( R ) , with adifferent normalization and the metrics K AB , g AB are proportional. The subgroup U ( ) is generated by T .As we are going to see, in many instances, the adjoint index A =
1, 2, 3 can bereplaced by a symmetric pair of indices α , β =
1, 2 which can be obtained from thecorresponding Pauli matrix by left multiplication with ε αβ .Using U ( ) we can always bring u to the gauge ξ = z : u = p − | z | (cid:18) z ∗ z (cid:19) , z = V − / V − . ( . )Then, this coset representative u ( z ) transforms according to the general rule u ′ ( z ) = u ( z ′ ) = gu ( z ) h − with g = (cid:18) u v ∗ v u ∗ (cid:19) ∈ SU (
1, 1 ) , so | u | − | v | = h = (cid:18) e i ϕ e − i ϕ (cid:19) ∈ U ( ) ,( . )where e i ϕ = u + vz ∗ | u + vz ∗ | , and z ′ = v + u ∗ zu + v ∗ z . ( . )The generators P a , M are P ≡ T , T ≡ M . ( . )Since M is diagonal and the P a are anti-diagonal, the left-invariant -forms can bedecomposed in terms of a complex Vielbein e = e + ie and a real connection ϑ as − u − du = e a P a + ϑ M = (cid:18) i ϑ e ∗ e − i ϑ (cid:19) , ( . ) nd using u − = (cid:18) − V β + ε βα + V β − ε βα (cid:19) we find that they are given by e = − ε αβ V α − dV β − = dz − | z | , ϑ = i ε αβ V α + dV β − = i zdz ∗ − z ∗ dz − | z | . ( . )The invariant metric is ds = g ab e a e b = dzdz ∗ ( − | z | ) . ( . )The Killing vectors and momentum maps can be computed by using the generalformulae Eqs. ( . ) and ( . ). Using the definition of the adjoint action of a group onits Lie algebra Eq. ( . ): T B Γ Adj ( u − ) B A = u − T A u , ⇒ Γ Adj ( u − ) B A = g BC Tr h T C u − T A u i , ( . )so k Aa = − g ab Tr h P b u − T A u i , ( . ) P A = Tr h Mu − T A u i . ( . )It is convenient to use the symmetric pairs αβ to label the Killing vectors and mo-mentum maps. We get, introducing a global factor of − i to make them real k ( αβ ) + ik ( αβ ) = iV α − V β − , ( . ) P αβ = V ( α + V β ) − , ( . )and k Aa = − ik ( αβ ) a ε αγ T A γ β , P A = − iP ( αβ ) ε αγ T A γ β . ( . ) The change of variables − i τ = − z + z , ( . )brings the metric of the SL ( R ) /SO ( ) coset Eq. ( . ) to this one. . . E (+ ) /SU ( ) The E (+ ) /SU ( ) coset representative can be written in the convenient form of aUsp (
28, 28 ) matrix U = u AB I J v ∗ AB I J v AB I J u ∗ AB I J , U † (cid:0) − (cid:1) U = (cid:0) − (cid:1) , U T (cid:0) − (cid:1) U = (cid:0) − (cid:1) , ( . )where all the indices are complex SU ( ) indices raised and lowered by complex conju-gation that occur in antisymmetric pairs AB , I J so that U − = ( u † ) I J AB − ( v † ) I J AB − ( v T ) I J AB ( u T ) I J AB . ( . )The Usp (
28, 28 ) condition implies for the 28 ×
28 matrices u and v the two conditions u † u − v † v = , v T u + u T v = . )E (+ ) acts on the AB indices and the compensating (scalar-dependent) SU ( ) trans-formations act on the I J indices. A parametrization in terms of independent scalarfields can be found, for instance in Ref. [ ].Each column of the above matrix provides a set of complex vectors labeled bythe pair I J transforming in the fundamental (i.e ) representation under E (+ ) . Theaction of this group on the fundamental representation in this complex basis can bedescribed as follows: consider, for instance the complex combinations of the electricand magnetic -form field strengths F AB given by F AB ≡ √ (cid:16) F ij − iG ij (cid:17) Γ ij AB , ( . )where ij are antisymmetric pairs of real SL ( ) indices, the F ij are the 28 electric fieldstrengths of the theory, the G ij are the 28 magnetic field strengths defined from theLagrangian of the theory L by G ij µν ≡ ⋆ ∂ L ∂ F ij µν , ( . )and the Γ ij s are the SO ( ) gamma matrices. Then, the infinitesimal action of E (+ ) onthe fundamental representation is given by δ F AB F ∗ AB = Λ [ A [ C δ B ] D ] Σ ABCD Σ ∗ ABCD − Λ [ C [ A δ D ] B ] F CD F ∗ CD , ( . ) here the Λ AB are the anti-Hermitian parameters of infinitesimal SU ( ) transforma-tions, (i.e. Λ ∗ AB = − Λ B A and Λ A A =
0) and where the off-diagonal infinitesimalparameters Σ ABCD are complex self-dual, that is ( Σ ABCD ) ∗ ≡ Σ ∗ ABCD = ε ABCDEFGH Σ EFGH . ( . )The generators of E (+ ) in this representation are, therefore, Λ [ A [ C δ B ] D ] Σ ABCD Σ ∗ ABCD − Λ [ C [ A δ D ] B ] = Λ E F T EF + Σ EFGH T EFGH ( . )where T EF = T EF ABCD − T EF CD AB , T EFGH = T EFGH ABCD T EFGH ABCD ,( . )where, in its turn, T EF ABCD = (cid:16) δ [ A | E δ F | B ] CD − δ FE δ ABCD (cid:17) , T EFGH ABCD = δ EFGH ABCD , T EFGH ABCD = ε EFGHABCD . ( . )The generators T EF of h = su ( ) will also be denoted M FE and the generators ofthe complement T EFGH will be denoted by P EFGH . In order to avoid confusion, inthis section the indices in the adjoint representation of E (+ ) will be A , B , . . . andcorrespond to the pairs EF plus the quartets EFGH . The metric in e (+ ) g AB will be g AB ≡ Tr ( T A T B ) . ( . )Using the explicit form of the generators T A written above one can also computeexplicitly the structure constants f ABC , the Killing metric K AB and the metric g AB . Thelatter’s and its inverse’s only non-vanishing components are g EF GH = ( δ E H δ G F − δ E F δ G H ) , g ABCD EFGH = ε ABDCEFGH , g EF GH = ( δ EG δ FH − δ E F δ G H ) , g ABCD EFGH = · ε ABDCEFGH . ( . )The Vielbein and H-connection are defined by U − dU = e I JKL P I JKL + ϑ IJ M IJ = ϑ I J KL e I JKL e ∗ I JKL − ϑ KL I J , ( . )and, using Eq. ( . ) one finds that they are given by ϑ I J KL = − ( u † ) I J AB du ABKL + ( v † ) I J AB dv AB KL , e I JKL = − ( u † ) I J AB dv ∗ AB KL + ( v † ) I J AB du ∗ ABKL . ( . )From the Maurer-Cartan equations it follows that D e I JKL ≡ de I JKL − ϑ I J MN ∧ e MNKL + e I JMN ∧ ϑ ∗ MN KL = . ) R I J KL ≡ d ϑ I J KL − ϑ I J mn ∧ ϑ MN KL = e I JMN ∧ e ∗ MNKL . ( . )Again, in order to compute the Killing vectors and momentum maps we use thesame reasoning as in the previous example, arriving to k A EFGH = − g EFGH ABCD Tr (cid:2) T ABCD U − T A U (cid:3) , P A EF = g EF GH Tr (cid:20) T GH U − T A U (cid:21) . ( . ) Noether -forms and dualization For each isometry of the metric G mn ( φ ) with Killing vector k Am ( φ ) , the σ -model actionEq. ( . ) has a global symmetry with δ A φ m = k Am ( φ ) . According to Noether’s firsttheorem, there is a current density j A µ associated to each of them which is conservedon-shell: j A µ = q | g | g µν G mn k Am ∂ ν φ n , ∂ µ j A µ = − k Am δ S δφ m . ( . )The Noether -form is defined by j A = G mn k Am d φ n , ( . )and, for the choice of metric Eq. ( . ), and using the expression Eq. ( . ), it can bewritten in the form j A = Tr h T A D uu − i = − g AB Γ Adj ( u ) B a e a , ( . ) hich can also be obtained from the explicitly right-invariant expression Eq. ( . ) forthe transformation δ A u = T A u . The above expression makes it easier to show that not all of these -forms areindependent: they satisfy dim H scalar-dependent relations: j A Γ Adj ( u ) Ai = Tr h M i u − D u i = . )where we have used Eq. ( . ) and the orthogonality of the basis of h and k in symmet-ric spaces. Observe that the above expression is not simply j i = j A Γ Adj ( u ) A a = Tr h P a u − D u i = − g ab e b . ( . )This expression and the previous one appear in the equations of motion of the scalarsEq. ( . ) and in the gauge identities Eq. ( . ). Thus, these can be rewritten, respec-tively in the form e ma D ⋆ h j A Γ Adj ( u ) A a i = . ) D ⋆ h j A Γ Adj ( u ) Ai i = . )Using the explicit form of the H-covariant derivatives, we find that these equationscan be rewritten in the following form, much more directly related to the conservationof the Noether -forms: e ma Γ Adj ( u ) A a d ⋆ j A = . ) Γ Adj ( u ) Ai d ⋆ j A = . )Combining these equations with the explicit form of the Killing vectors Eq. ( . )we can easily prove the off-shell relation k Am δ S Gauged δφ m = − d ⋆ j A . ( . )The moral of these results is that the equations of motion of the scalars can be seenas combinations (projections) of some more fundamental equations: the conservation The complete infinitesimal transformation of u must include the compensating H-transformationsthat act from the right: δ A u = T A u + uW Ai M i , ( . )but they can be safely ignored in the right-invariant expression. aws of the Noether currents. The latter can completely replace the former. But onlythe Noether -forms can be dualized.The Noether -forms are closed on-shell and, on-shell, they can be dualized byintroducing as many ( d − ) -forms B A related to them by ⋆ j A ≡ dB A ≡ H A , ( . )solving locally the conservation laws. As usual, the Bianchi identities of the originalfields become the equations of motion of the dual ones. The obvious candidate toBianchi identity is D (cid:16) Γ Adj ( u − ) a A g AB j B (cid:17) = . )by virtue of Eq. ( . ) and Cartan structure equation D e a = . ). Then, the equations of motion satisfied by the -forms are D (cid:16) Γ Adj ( u ) A a ⋆ H A (cid:17) = . )There are only dimG − dimH equations, which means that we cannot solve for allthe H A . We must also use the constraint Γ Adj ( u ) Ai H A = . )which follows from Eq. ( . ).It would be desirable to have a kinetic term for the ( d − ) -forms B A from whichthe equations of motion ( . ) could be derived. The simplest candidate would be S = Z M AB H A ∧ ⋆ H B , ( . )where the scalar-dependent matrix M AB is defined by M AB ≡ Γ Adj ( u ) A a Γ Adj ( u − ) aC g CB , ( . )and is obviously singular, with rank ( M AB ) = dimG − dimH because M AB Γ Adj ( u − ) i A = ∀ i . This means that the combinations Γ Adj ( u ) Ai H A which are constrained to vanish,do not enter the action. Observe that M AB transforms under global G transformationsaccording to M ′ AB = Γ Adj ( g ) A A ′ Γ Adj ( g ) B B ′ M A ′ B ′ , ( . )so the above action is invariant.Observe also that the σ -model action can be written in terms of the same singularmatrix as We use Γ Adj ( u ) Aa = g ab Γ Adj ( u − ) bB g BA . = Z M AB j A ∧ ⋆ j B . ( . )Furthermore, observe that, if we define Γ Adj ( u ) A a B A ≡ B a , Γ Adj ( u ) Ai B A ≡ B i , ( . )and we impose the restriction B i =
0, the above action can be rewritten in the form S = Z g ab H a ∧ ⋆ H b , where H a = DB a . ( . )Notice, however, that the restriction B i = . ).The equations of motion that follow from the above action are simply d (cid:16) M AB ⋆ H B (cid:17) = . )Projecting them with Γ Adj ( u − ) a A , we get the equations of motion ( . ) Γ Adj ( u − ) a A d (cid:16) M AB ⋆ H B (cid:17) = g ab D (cid:16) Γ Adj ( u ) Ab ⋆ H A (cid:17) = . )whose solution is g ab Γ Adj ( u ) Ab ⋆ H A ∝ e a . ( . )However, if we project the equations of motion with Γ Adj ( u − ) i A we get a non-trivialconstraint: Γ Adj ( u − ) i A d (cid:16) M AB ⋆ H B (cid:17) = − e a f abi g bc Γ Adj ( u ) Ac ⋆ H A = . )which, upon use of the previous solutions gives a non-trivial constraint that we do notwant: R ( ϑ ) i = . )We have not found any completely satisfactory way of solving this problem ingeneral.Notice that a similar problem appeared with dualization of -form potential A ofeleven dimensional supergravity. Its action [ ] contains, besides the kinetic term of A and interaction of A with fermions, the Chern-Simons type term dA ∧ A ∧ A , andthis makes impossible to construct a dual action including -form potential A insteadof A [ , ].However, there exists the duality invariant action of -d supergravity includingboth A and A potentials [ ]. It was constructed using the PST (Pasti-Sorokin-Tonin)approach [ , ] and reproduce a (nonlinear) duality relation between the (general-ized) field strengths of A and A as a gauge fixed version of the equations of motion. otice that this action can be presented formally as a sigma model action [ ] for a su-pergroup with fermionic generator associated to A and bosonic generators associatedto A [ ].Then it is natural to expect that the similar situation occurs in our case of dualiza-tion of “non-Abelian” scalars. Even if the non-existence of a consistent way to writedual action in terms of only ( d − ) forms dual to a scalars parametrizing a non-Abeliancoset were proved, this would not prohibit the existence of a PST-type action involvingboth the scalars and the ( d − ) forms and producing the duality equations ( . ) asa gauge fixed version of the equations of motion. Moreover, for the particular case ofSU (
1, 1 ) /U ( ) coset such action was constructed in [ ], where it was also incorporatedin the complete action of type IIB supergravity.The generalization of the action from Ref. [ ] for the generic case of scalars parametriz-ing a symmetric space G/H reads S PST = Z L PST , ( . ) L PST = g AB j A ∧ ⋆ j B + g ij F i ∧ ⋆ F j + ( − ) d g AB H A ∧ v i v ⋆ H B , ( . )where H A = H A + ⋆ j A + Γ Adj ( u − ) i A ⋆ F i ≡ H A + ⋆ j A + P i A ⋆ F i , ( . )the one-form v is constructed from the PST scalar a ( x ) , the (would be) auxiliary fieldof the PST formalism, v = da ( x ) √ ∂ a ∂ a , v µ = ∂ µ a ( x ) √ ∂ a ∂ a , ( . )and F i = dx µ F µ i ( x ) is an auxiliary one-form carrying the index of H -generators. Thecontraction symbol is defined, as usual, by i v j A = v µ j µ A , i v H A = ( d − ) ! dx ν ( D − ) ∧ . . . ∧ dx ν H ν ... ν ( d − ) µ v µ . ( . )We assume the derivative of the PST scalar to be a time-like vector so that the squareroot in denominator is well defined (in our mostly minus signature), v µ v µ =
1, and H A = v ∧ i v H A + ⋆ ( v ∧ i v ⋆ H A ) , ( . )is valid for any ( d − ) -form and, in particular, for our H A = dB A .The study of this action and derivation of the duality conditions from its equationsof motion is out of the scope of this paper. Here we would like to stress the rôle the omentum map ( . ) plays in it: the auxiliary one-form F i always enter the action incontraction F i P i A . Indeed, this is the case for H A ( . ), and, due to j A Γ Adj ( u ) Ai = . ), the first two terms of the Lagrangian ( . ) can also be collected in g AB ( j A + P i A F i ) ∧ ⋆ ( j B + P j A F j ) , L PST = g AB ( j A + P i A F i ) ∧ ⋆ ( j B + P j A F j ) + ( − ) d g AB H A ∧ v i v ⋆ H B . ( . )We hope to return to the study the properties of this action and its applications insupergravity context in future publications. . Examples . . SL ( R ) /SO ( ) A short calculation gives D uu − = d MM − = j A B AB T B , ( . )where the Noether -forms are given by j = e ϕ d | τ | , j = − e ϕ χ d | τ | + e ϕ ( + | τ | ) d χ , j = + e ϕ χ d | τ | + e ϕ ( − | τ | ) d χ . ( . )As expected, they are not independent: they are related by one (dim H) relation of theform Eq. ( . ) where Γ Adj ( u ) A is the third column of the SO (
2, 1 ) matrix in Eq. ( . ).The singular matrix M AB in the action Eq. ( . ) is (cid:16) M AB (cid:17) = e ϕ | τ | ( − | τ | ) χ − ( + | τ | ) χ ( − | τ | ) χ e − ϕ + ( − | τ | ) − ( − | τ | ) − ( + | τ | ) χ − ( − | τ | ) − e − ϕ + ( + | τ | ) .( . )The combinations of dual ( d − ) -form field strengths H A that occur in that actionare Γ Adj ( u ) A H A = H − χ ( H + H ) , Γ Adj ( u ) A H A = e ϕ χ [ H − χ ( H + H )] + e − ϕ ( H + H ) + e − ϕ ( H − H ) ,( . ) nd the constraint Eq. ( . ) is Γ Adj ( u ) A H A = e ϕ χ H + e ϕ ( H + H ) − e ϕ | τ | ( H − H ) = . )The relation between these three ( d − ) -form field strengths and the scalars (thepullbacks of the Vierbeins) is Γ Adj ( u ) A a H A = ⋆ e a , ( . )and, with the help of the above constraint we can invert it, expressing entirely the threefield strengths in terms of the scalars. The relations are equivalent to H A = ⋆ j A . . . E (+ ) /SU ( ) Using the same properties we used to find the Killing vectors and momentum mapsEqs. ( . ) we find the Noether -forms are given by j A ABCD e ABCD where the e ABCD are the Vielbein and the components are given by j A ABCD = Tr h T ABCD U − T A U i . ( . )The explicit expressions can be easily computed using the generators and (inverse)coset representatives given above. For instance, j EF ABCD = ε ABCDGHMN h ( v † ) GH FJ u ∗ EJ MN + ( u † ) GH EJ v ∗ FJ MN i − δ ABCDGHMN (cid:2) ( v T ) GH EJ u FJ MN + ( u T ) GH FJ v EJ MN (cid:3) . ( . )Observe that the -forms j EF ABCD e ABCD are purely imaginary. NGZ -forms and dualization in d = The bosonic action of all -dimensional ungauged supergravities (and many other in-teresting theories as well) is of the generic form S [ g µν , A Λ µ , φ m ] = Z d x p | g | (cid:8) R + G mn ( φ ) ∂ µ φ m ∂ µ φ n + ℑ m N ΛΣ F Λ µν F Σ µν − ℜ e N ΛΣ F Λ µν ⋆ F Σ µν (cid:9) , ( . )where the indices Λ , Σ = · · · , n (the total number of fundamental vector fields) andwhere N ΛΣ ( φ ) is known as the period matrix and it is symmetric and, by convention,has a negative definite imaginary part. The σ -model metric G mn ( φ ) is that of a Rie-mannian symmetric space in all N > his is the case that we want to consider here in order to apply the results derived inthe previous sections.First of all, we want to rewrite this action in differential-form language and usingthe coset representative u : S = Z n − ⋆ R + Tr [ u − D u ∧ ⋆ u − D u ] − ℑ m N ΛΣ F Λ ∧ ⋆ F Σ − ℜ e N ΛΣ F Λ ∧ F Σ o .( . )Then, we define the dual vector field strengths G Λ ≡ ℑ m N ΛΣ ⋆ F Σ + ℜ e N ΛΣ F Σ , or G Λ + ≡ N ∗ ΛΣ F Σ + . ( . )The last relation is known as a (linear) twisted self-duality constraint . Defining the sym-plectic vector of vector field strengths ( F M ) ≡ (cid:18) F Λ G Λ (cid:19) , ( . )the equations of motion and the Bianchi identities can be written together as d F M = . )which can be solved locally by assuming the existence of -form potentials A M F M = d A M . ( . )This set of equations is invariant under linear transformations F ′ M = S M N F N , S ≡ (cid:18) A BC D (cid:19) . ( . )As it is well known, the preservation of the twisted self-duality constraint requires thesimultaneous transformation of the period matrix according to the rule N ′ = ( C + D N )( A + B N ) − . ( . )The preservation of the symmetry of N and of the negative definiteness of ℑ m N (together with the preservation of the energy-momentum tensor) require S to be anSp(2 n , R ) transformation, that is S T Ω S = Ω , Ω ≡ (cid:18) − (cid:19) . ( . )Finally, if these transformations are going to be symmetries of the equations of motion,the form of the period matrix as a function of the scalars must be preserved, and thisrequires the above transformation rule for N to be equivalent to a transformation ofthe scalars: ′ ( φ ) = N ( φ ′ ) . ( . )This transformation of the scalars must be an isometry of the σ -model metric. Thus, thesymmetries of the equations of motion of the theory are the group G of the isometries ofthe σ -model metric which act on the vector fields embedded in the symplectic group .These isometries always leave invariant the scalars’ kinetic term but only some of themmay leave invariant the whole action because many involve electric-magnetic dualityrotations. Thus, there is a Noether -form for each isometry of the scalar sector, andwe are going to denote it by a σ index, j ( σ ) A = h T A D uu − i , ( . )but, in general, they do not have a standard completion to Noether -forms of the fulltheory. A completion does, nevertheless, exist in all cases and it was found by Gaillardand Zumino in Ref. [ ]. To construct the Noether–Gaillard–Zumino (NGZ) -formwe first need to define the infinitesimal generators of G in the representation in whichthey act on the vector fields, {T A } . By assumption T A ∈ sp ( n , R ) and δ A F M = T AM N F N . ( . )Then, the NGZ -forms are given by j A = j ( σ ) A − T A M N ⋆ ( F N ∧ A M ) , A M = Ω MN A N . ( . )Observe that the NGZ -forms are not invariant under the gauge transformationsof the -forms δ σ A M = d σ M precisely because the Noether -forms j ( σ ) A are. Let uscheck that they are conserved on-shell. First, the conservation equation takes the form d ⋆ j A = d ⋆ j ( σ ) A − T A M N F N ∧ F M , ( . )where we have used Eqs. ( . ). Now we are going to see that this equation is propor-tional to the projection of the scalar equations of motion with the Killing vectors. Thescalar equations of motion are δ S δφ m = δ S Gauged δφ m − F Λ ∂ G Λ ∂φ m = . )where S Gauged is the σ -model action normalized as in Eq. ( . ). Contracting with theKilling vectors we get k Am δ S δφ m = − (cid:20) d ⋆ j ( σ ) A + F Λ k Am ∂ G Λ ∂φ m (cid:21) = . ) We are going to ignore the possibility of scalars which do not couple to the vector fields becausethis simply does not happen in N > here we have used Eq. ( . ). The infinitesimal transformation rule for the periodmatrix follows from Eq. ( . ) and Eq. ( . ) k Am ∂ N ΛΣ ∂φ m = T A ΛΣ − N ΛΩ T A ΩΣ + T A ΛΩ N ΩΣ − N Λ T A Ω∆ N ∆Σ , ( . )and, replacing it in Eq. ( . ), we find the conservation equations as combinations ofthe equations of motion: k Am δ S δφ m = − h d ⋆ j ( σ ) A − T A M N F M ∧ F N i = . )The conservation of the NGZ -forms can be solved locally by the introduction of -forms B A such that ⋆ j A = dB A , ⇒ ⋆ j ( σ ) A = dB A + T AM N F N ∧ A M ≡ H A , ( . )where H A are the -form field strengths, gauge invariant under δ σ A M = d σ M , δ σ B A = d σ A − T AM N F N ∧ d σ M , ( . )and satisfying the Bianchi identities dH A − T A MN F M ∧ F N = . )The equations of motion have the same form as in the general case studied inSection .Observe that the NGZ currents are subject to the same constraint as the Noethercurrents, Eq. ( . ), because k Am Γ Adj ( u ) Ai = . ). Together,they lead to the constraints T i MN Γ ( u − ) M P Γ ( u − ) N Q F P ∧ F Q = H A Γ Adj ( u ) Ai . ( . ) We use the following infinitesimal form for the symplectic matrix S : S MN ∼ δ MN + σ A T A M N , where T AP [ M Ω N ] P = . )The different block components of T A M N are defined by (cid:16) T A M N (cid:17) = T A ΛΣ T A ΛΣ T A ΛΣ T A ΛΣ , with T A ΛΣ = −T A ΣΛ , T A ΛΣ = T A ΣΛ , T A ΛΣ = T A ΣΛ .( . ) Otherwise, the NGZ currents would represent too many degrees of freedom. . Supersymmetry and the momentum map . . Supersymmetry transformations of ( d − ) -forms In supersymmetric theories the ( d − ) -form fields dual to the scalars, B A , must trans-form under supersymmetry and the algebra of the supersymmetry transformationsacting on these fields, δ ǫ B A , must close on shell.In the N =
1, 2, d = , ] these transformations were found to haveleading terms with a common structure that can be generalized to all N and, actually,to all d : δ ǫ B A µ ··· µ ( d − ) ∼ P Ai ( M i ) I J ¯ ǫ J γ [ µ ··· µ ( d − ) ψ µ ( d − ) ] I + D m P Ai ( M i ) I J ¯ ǫ J γ µ ··· µ ( d − ) λ m I + · · · ( . )In this expression I , J are R-symmetry indices (that is, a representation of H), ψ µ I arethe gravitini, λ m I are dilatini or, more generally, the supersymmetric partners of thescalars, labeled here by m , n , p , ( M i ) I J are the generators of the Lie algebra of H inthe same representation, and the P Ai are the momentum maps of the isometries of thecoset space G/H or the holomorphic and tri-holomorphic momentum maps of Kähler-Hodge and quaternionic-Kähler spaces in in N =
1, 2, d = Henceforth,the index A is a “global” adjoint G index. D is the H -covariant derivative actingon the momentum map (or the Kähler- or S U ( ) -covariant derivatives in the Kähler-Hodge and quaternionic-Kähler cases and, according to the previous observation, onlythe i index has to be covariantized for. The additional terms in this supersymmetrytransformation rule are proportional to other p -forms of the theory and are associatedto the Chern-Simons terms in the ( d − ) -form field strengths H A .The second term in this proposal adopts slightly different forms depending on thetheory under consideration. First of all, one can always apply the main property ofthe momentum map Eq. ( . ) which also appears in different guises: Eqs. ( . ) incoset spaces G/H, (A. ) and (A. ) in Kähler-Hodge spaces and, finally, (B. ) inquaternionic-Kähler spaces. Then, one can use different properties of the H-curvatureso that it does not appear explicitly: Eq. ( . ) in coset spaces G/H, the Kähler-Hodgecondition that identifies the Kähler -form J with the curvature of the complex bundlein Kähler-Hodge spaces, and the condition that relates the curvature of the SU ( ) connection to the hyperKähler structure Eq. (B. ) in quaternionic-Kähler manifolds.Furthermore, observe that in the second term of Eq. (B. ), the hyperini ζ α carrya single Sp ( n h ) index α but their product with the Quadbein U α I u carries an R-symmetry index I plus a hyperscalar index u , according to the general expectation.The example in Section . . provides additional confirmation of the universality ofthe above supersymmetry transformation rule. The N = and B. , respectively. These cases are reviewed in Appendices A and B. he above proposal looks different from the exact result obtained in superspacefor the N = d = ], but one has to take into account that the -forms and their -form field strengths in that reference carry “local” (H=SU ( ) ) in-dices instead of “global” adjoint E (+ ) indices. The relation between these two sets ofvariables is B A ≡ B B Γ Adj ( u ) B A , ( . )and it is not difficult to see that the supersymmetry transformation rules Eq. ( . ) splitinto δ ǫ B i µ ··· µ ( d − ) ∼ ( M i ) I J ¯ ǫ J γ [ µ ··· µ ( d − ) ψ µ ( d − ) ] I + · · · ( . ) δ ǫ B a µ ··· µ ( d − ) ∼ f abi ( M i ) I J ¯ ǫ J γ µ ··· µ ( d − ) e bm λ m I + · · · ( . )In the particular case of N = d = ]. Using Weyl spinor notation and taking into account that . The superpartners of scalar fields λ ABCD α I are split on two Weyl spinors of theform λ ABCD I α = ǫ ABCDI JKL χ α JKL , and ¯ λ ABCD ˙ α I = − δ [ AI ¯ χ ˙ α BCD ] . ( . ) . The structure constants corresponding to the commutators [ h , h ] ∈ k , genericallydenoted by f abi in our review, take in this case the form f ABCD EFGH I J = ( δ I [ A ǫ BCD ] EFGHJ − δ I [ E ǫ FGH ] ABCDJ ) , ( . )and the h generators and M i I J T KL I J = δ K J δ I L − δ K L δ I J ,we find that the above generic equations, that can be extracted from the superfieldresults of Ref. [ ], take the explicit form δ ǫ B IJ µν ∝ T IJ K L i ( ǫ K σ [ µ ¯ ψ L ν ] + ψ K [ µ σ ν ] ¯ ǫ L ) , ( . ) δ ǫ B I JKL µν ∝ ǫ [ I σ µν χ JKL ] − ǫ I JKLABCD ¯ ǫ A ˜ σ µν ¯ χ BCD . ( . )On the other hand, the corresponding relation for the -form field strengths H A ≡ H B Γ Adj ( u ) B A , ( . )together with Eq. ( . ) explain, from a technical point of view, why the H i were foundin Ref. [ ] to be dual to fermion bilinears. . . Tensions of supersymmetric ( d − ) -branes The supersymmetry transformations of the ( d − ) -forms into the gravitini determinethe tension of the 1/2-supersymmetric ( d − ) -branes that couple to them in a κ -symmetric action (see, for instance, Ref. [ ]), if any. The explicit construction of theU-duality- invariant and κ -symmetric actions of the 1/2-supersymmetric ( d − ) -branes,in the same spirit as the construction of the SL ( R ) - invariant actions for all branes intype IIB d =
10 supergravity in Ref. [ ] or for 0-branes in N =
2, 8, d = ] has only been carried out for the N = d = ]. Nevertheless,some general lessons can be learned from those results and from the general form ofthe supersymmetry transformations of ( d − ) -form potentials Eq. ( . ).On general grounds, ( d − ) -branes will be characterized by charges in the adjointrepresentation of G, q A and the Wess-Zumino term in their effective world-volumeaction will contain the leading term q A Z B A . ( . )Then, the supersymmetry transformation rule Eq. ( . ) requires the presence ofa scalar-dependent factor in the kinetic term that can be identified with the tension T ( d − ) : Z d ( d − ) ξ T ( d − ) q | g ( d − ) | , ( . )which we conjecture to be of the form T ( d − ) = q | q A q B P Ai P B j g ij | . ( . )Observe that the rank dim H matrix P Ai P B j g ij is related to the matrix M AB definedin Eq. ( . ) by P Ai P B j g ij = g AB − g AC g BD M CD , ( . )and for ( d − ) -brane charges in the conjugacy class q A q B g AB = -and S -branes of N = B , d =
10 supergravity [ ]) T ( d − ) = q | q A q B M CD | , ( . )which is the expression one would have guessed from Eq. ( . ).Clearly, more work is needed in order to find the complete κ -invariant worldvolumeactions, find the U-duality-invariant ( d − ) -brane tensions and, eventually, prove theabove conjecture, but we think that our arguments concerning the general structure ofthe supersymmetry transformation Eq. ( . ) give some support to it. T-duality-invariant Wess-Zumino terms for the κ -symmetric world-volume effective actions of allbranes in maximal supergravity in any dimension have been proposed in Ref. [ ]. . . Fermion shifts
The holomorphic and triholomorphic momentum maps (resp. P A and P Ax ) also ap-pear naturally in the so-called fermion shifts of the supersymmetry transformations ofthe fermions of gauged N =
1, 2, d = A Λ as gauge fields for perturbative symmetries of theaction), the supersymmetry transformations of the gravitini, gaugini and hyperini of N = d = δ ǫ ψ I µ = D µ ǫ I + h T + µν ε I J − S x η µν ε IK ( σ x ) K J i γ ν ǫ J , δ ǫ λ Ii = i D Z i ǫ I + h(cid:0) G i + + W i (cid:1) ε I J + i W i x ( σ x ) I K ε KJ i ǫ J , δ ǫ ζ α = i U α I u D q u ǫ I + N α I ǫ I , ( . )where the fermion shifts are given by S x = g L Λ P Λ x , W i = g L ∗ Λ k Λ i = − i g G ij ∗ f ∗ Λ j ∗ P Λ , W i x = g G ij ∗ f ∗ Λ j ∗ P Λ x , N α I = g U α Iu L ∗ Λ k Λ u . ( . )As usual in N >
1, the scalar potential is given by an expression quadratic in thefermion shifts: V ( Z , Z ∗ , q ) = − S ∗ x S x + G ij ∗ W i W ∗ j ∗ + G ij ∗ W i x W ∗ j ∗ x + N α I N α I = g h − ℑ m N ΛΣ P Λ P Σ + L ∗ Λ L Σ ( H uv k Λ u k Σ v − P Λ x P Σ x )+ G ij ∗ f Λ i f ∗ Σ j ∗ P Λ x P Σ x i . ( . )The presence of the momentum map on all those terms is due to their transforma-tions properties under global duality transformations. To make this fact manifest and See, for instance, Refs. [ , , ]. The momentum maps carry an index Λ here which associatesthem to the vector field that gauges the corresponding global symmetry. It is understood in this notationthat only the momentum maps associated to the gauge symmetries occur in these expressions. Thisnotation is considerably improved by the introduction of the embedding tensor, as awe are going to see. ain more insight in the structure of these terms, it is convenient to use the embed-ding tensor ϑ M A . This object relates each symmetry generator ( A index) to the vectorfield of the theory that gauges it ( M index). Introducing the embedding tensor in thefermion shifts restores the (formal) symplectic invariance of the theory : S x = V M ϑ M A P Ax , W i = − i D i V ∗ M ϑ M A P A , W i x = D i V ∗ M ϑ M A P Ax , N α I = g U α I u V ∗ M ϑ M A k Au , ( . )while the scalar potential must take the form V ( Z , Z ∗ , q ) = − M MN ϑ M A ϑ N B P A P B + V ∗ M V N ϑ M A ϑ N B ( H uv k Au k Bv − P Ax P Bx )+ G ij ∗ D i V M D j ∗ V ∗ N ϑ M A ϑ N B P Ax P Bx . ( . )Our general definition of momentum map shares the same transformation prop-erties and, therefore, the momentum maps should occur in all the fermion shifts ofall theories. The expressions given in the literature, though, are written in a differentlanguage which obscures this point. Here we are going to show in several exampleshow the momentum map allows one to rewrite the fermion shifts in a universal way ifone makes use of the embedding tensor.Let us consider first the N > d = ] that can describe all thesetheories simultaneously and in a language very close to that of the N = d = We just need to know some details of this formulation: the N = V M that describes the scalars in the vector multiplets and its Kähler-covariantderivative D i V M are now generalized to V MI J = −V MJI and V Mi where the indices I , J = · · · , N and i , j = · · · , n V (the number of vector multiplets). The fermions in thesupergravity multiplet are ψ µ I , χ I JK , χ I JKLM (antisymmetric in all the SU ( N ) indices, The embedding tensor and its associated formalism were introduced in Refs. [ , , ]. They weredeveloped in the context of the maximal -dimensional supergravity in Refs. [ , ], but its use is byno means restricted to that context (see Chapter in Ref. [ ] and references therein. The details of such a general gauged theory have not yet been worked out in the literature. The gauge coupling constant g is also replaced by the embedding tensor, since it can describe severalgauge groups with different coupling constants. So far, this formalism has been used only in ungauged supergravities. Our proposals for the fermionshifts should help to extend this formulation to the most general gauged theories. nd χ I JKLM = ε I JKLMNOPQ χ OPQ for N = The fermions in the generic vectorsupermultiplet are λ iI and λ i I JK (again, antisymmetric in all the SU ( N ) indices, and λ i I JK = ε I JKL λ iL for N = N >
4. However, inthis formalism, several fields of the N = λ I = ε I J ··· J χ J ··· J and λ I JK = ε I JKLMN χ LMN . In practice, in N =
6, it is easier to work with λ I and χ I JK , which fit in the general pattern.Combining this knowledge with the fermion shifts of the N = it is not difficult to guess the form of the generic fermion shifts: δ ǫ ψ µ I ∼ · · · + V M IK ϑ M A P A i ( M i ) K J γ µ ǫ J , ( . ) δ ǫ χ I JK ∼ · · · + V M [ I J | ϑ M A P A i ( M i ) L | K ] ǫ L ( . ) δ ǫ λ iI ∼ · · · + V Mi ϑ M A P A i ( M i ) J I ǫ J , ( . )where we have boldfaced the H indices to distinguish them from those labeling thevector supermultiplets. For the N =
3, 5 cases there are additional fermion fieldswhich are independent of ψ µ I , χ I JK , λ i I and whose fermion shifts are more difficult toguess. We have found the following possibilities: . For the SU ( ) singlets λ i = ε I JK λ i I JK of N = δ ǫ λ i ∼ · · · + ε I JK V Mi ϑ M A P A i ( M i ) I L δ LJ ǫ K . ( . ) . For the SU ( ) singlet χ = ε I ··· I χ I ··· I of N = δ ǫ χ ∼ · · · + ε I I I I I V M I I ϑ M A P A i δ I J ( M i ) J I ǫ I . ( . )In many gauged supergravities (see, for instance the N = d = , ]or the N = d = , ]), the fermion shifts are given in terms of the“dressed structure constants” of the gauge group. In the SO ( ) -gauged N = d = ], and in the conventions used there, these are defined by f ijk ≡ L im L jn L pk f mn p , where f mn p = ǫ mnp , ( . ) χ IJKLM is only relevant as an independent field for N =
5, because it is also related to another fieldfor N =
6, as awe are going to see. Evidently, the fermion shifts in the hyperini will not be generalized, as there are no hypermultipletsin
N 6 = d = We thank Mario Trigiante for enlightening conversations on this point. nd where L im is the SL ( R ) /SO ( ) coset representative ( m , n , p =
1, 2, 3 are indicesin the fundamental (vector) representation of SL ( R ) and i , j , k =
1, 2, 3 are indices inthe fundamental representation of SO ( ) ) and L mi = ( L − ) mi is its inverse. The dressed structure constants can be rewritten in terms of the three momentummaps P mn and the three Killing vectors k m a (where a labels the generators of the coset)using the definition of the adjoint action of the group on the algebra, adapted to theseconventions: f ijk = L im L jn ( T m ) n p ( L − ) pk = L im Γ Adj ( L − ) m α ( T α ) jk = L im h P ml ( T l ) jk − k ma ( T a ) jk i ,( . )where we have used the (transposed of the) definition of the adjoint action of the groupon the algebra Eq. ( . ).Similar identities can be used in other supergravities and we hope the use of themomentum map can be of help in writing all gauged supergravity theories in a ho-mogenous language. . Examples . . N = d = supergravity The bosonic fields of pure N = d = τ = χ + ie − ϕ , and six vector fields A Λ µ Λ = · · · , 6. The bosonic action is S = Z d x q | g | (cid:26) R + ∂ µ τ∂ µ τ ∗ ( ℑ m τ ) − e − ϕ F Λ µν F Λ µν + χ F Λ µν ⋆ F Λ µν (cid:27) , ( . )and, comparing with the generic action Eq. ( . ) we find that the period matrix is givenby N ΛΣ = − τδ ΛΣ . ( . )The action of this theory is invariant under SO ( ) rotations of the vector fields,whose symplectic infinitesimal generators, labeled by a are ( T a M N ) = (cid:18) T a ΛΣ T a ΛΣ (cid:19) , where T a ΛΣ = − T a ΣΛ . ( . )The equations of motion are also invariant under the SL ( R ) group of simultaneouselectric-magnetic rotation of all the electric field strengths F Λ into the dual magnetic In our notation L im = u mi , the transposed. Observe that here ℑ m τ = e − ϕ instead of e − ϕ as in Section . . . nes G Λ defined in Eq. ( . ). The symplectic generators associated to these transfor-mations are the tensor products of those in Eq. ( . ) by the identity in dimensions.More explicitly ( T M N ) = (cid:18) δ ΛΣ − δ ΛΣ (cid:19) , ( T M N ) = (cid:18) δ ΛΣ δ ΛΣ (cid:19) , ( T M N ) = (cid:18) δ ΛΣ − δ ΛΣ (cid:19) . ( . )We will denote the generators of SL ( R ) with the label α to distinguish them fromthose of the SO ( ) group. Observe that there are no scalars associated to SO ( ) . Theonly scalar, the axidilaton, is invariant under SO ( ) . Since this group is a symme-try of the action, the NGZ current coincides with the Noether current, has no scalarcontribution and is given by j a = − T a M N ⋆ ( F N ∧ A M ) . ( . )Using Maxwell equations and Bianchi identities of the vector fields we find that d ⋆ j a = − T a M N F N ∧ F M , ( . )which vanishes identically due to the antisymmetry of the SO ( ) generators. In thiscase, evidently, there is nothing to be dualized and there are no -forms B a .The NGZ currents of the SL ( R ) electric-magnetic duality group are non-trivial,though: j α = j ( σ ) α − T α M N ⋆ ( F N ∧ A M ) , α =
1, 2, 3 . ( . )It is necessary to use the equations of motion of the scalars (and not just the Maxwellequations and Bianchi identities) to show that they are conserved on-shell. They aredualized into -forms B α according to the general prescription. We will not give thedetails here. . . N = d = supergravity The above general scheme can be applied to N = d = H EF = dB EF − h F EA A FA + A EA F FA − δ E F (cid:16) F AB A AB + A AB F AB (cid:17)i ,( . ) H EFGH = dB EFGH − (cid:16) F [ EF A GH ] + ε EFGHABCD F AB A CD (cid:17) . ( . ) The higher-dimensional, higher-rank NGZ -forms anddualization In d > p >
1. The global symmetries of the theory can act on thesefield as rotations or, when the rank and dimension allow it ( d = ( p + ) ), as electric-magnetic transformations. The latter are not symmetries of the action but, nevertheless,as in the -dimensional case, a generalized Noether-Gaillard-Zumino (NGZ) current -form j A which is conserved on shell can be defined for each and all the generators ofthe full duality group.The equation that expresses this conservation can be written in a universal form: let F I , H m , G a , . . . be, respectively, the -, -, -, . . . form field strengths of the n , n , n , . . .fundamental -, -, -, . . . fields of the theory and let ˜ F I , ˜ H m , ˜ G a , . . . their dual ( d − ) -, ( d − ) -, ( d − ) -, . . . form field strengths. As the indices chosen show, if the fun-damental field strengths F I , H m , G a , . . . transform linearly under the duality group as δ A F I = T A I J F J , δ A H m = − T Anm H n , δ A G a = T Aab G b , . . . , the dual field strengths musttransform in the conjugate representations, that is δ A ˜ F I = − T A JI ˜ F J , δ A ˜ H m = T Amn ˜ H n , δ A ˜ G a = − T Aba ˜ G b , . . . The only exception to these transformation rules are the electric-magnetic transformations. In d =
4, for instance, they relate F I to ˜ F I and the pair ( F M ) ≡ (cid:16) F I ˜ F I (cid:17) transforms as a Sp ( n , R ) vector according to δ A F M = T A M N F N with T A M N ∈ sp ( n , R ) . In d =
6, electric-magnetic duality transformations relate H m to ˜ H m and the pair ( H M ) ≡ (cid:16) ˜ H m H m (cid:17) transforms as a SO ( n , n ) vector according to δ A H M = T A M N H N with T A M N ∈ so ( n , n ) etc.It is not difficult to see through the - and -dimensional examples we are going topresent next that the equation satisfied by the Noether current -forms is always, upto conventional coefficients, of the form − k Ax δ S δφ x = d ⋆ j A + T A I J F J ∧ ˜ F I + T Amn ˜ H n ∧ H m · · · = . )and in the exceptional cases mentioned above, one should replace T A I J F J ∧ ˜ F I by T A M N F M ∧ F N , T Amn ˜ H n ∧ H m by T AM N H M ∧ H N etc.On-shell, the above equation would take the form d ⋆ j NGZA = . )but it is not possible to give a general form of this current because, in each theory, thefield strengths contain different Chern-Simons terms, all of them duality-invariant. Inthe -dimensional example that follows, we have found the explicit form, but in the -dimensional one, we have not.The dualization of the NGZ current -forms into ( d − ) -form potentials proceedsas in the -dimensional case. . Examples . . N = d = supergravities The bosonic action of any -dimensional ungauged supergravity-like theory with scalars φ x and Abelian vector fields A I (in particular, N = d = ] S = Z n ⋆ R + G xy d φ x ∧ ⋆ d φ y − a I J F I ∧ ⋆ F J + C I JK F I ∧ F J ∧ A K o , ( . )where G xy ( φ ) is the σ -model metric, a I J ( φ ) is the kinetic matrix of the vector fields and C I JK is a constant, symmetric tensor. In supergravity theories these three couplingsare related in a very precise way, but we will not need to use this structure for ourpurposes.The equations of motion of the vector fields are d ( a I J ⋆ F J − C I JK F J ∧ A K ) = . )and can be solved locally by a I J ⋆ F J − C I JK F J ∧ A K ≡ d ˜ A I , ( . )where the ˜ A I are the magnetic -forms dual to the vector fields. Their gauge-invariantfield strengths are ˜ F I = d ˜ A I + C I JK F J ∧ A K , ⇒ d ˜ F I = C I JK F J ∧ F K , ( . )and are related to the vector field strengths by˜ F I = a I J ⋆ F J . ( . )The equations of motion of the scalars are − δ S δφ z = G zw (cid:2) d ⋆ d φ w + Γ xyw d φ x ∧ ⋆ d φ y (cid:3) + ∂ z a I J F I ∧ ⋆ F J . ( . )If the action is invariant under the global transformations generated by δ A φ x = k Ax ( φ ) , δ A A I = T A I J A J , ( . )which implies that the functions k Ax ( φ ) are Killing vectors of the σ -model metric G xy ,the kinetic matrix satisfies k Ax ∂ x a I J = − T AK ( I a J ) K , ( . )and the symmetric tensor satisfies AL ( I C JK ) L = . )we find that − k Az δ S δφ z = d ⋆ j ( σ ) A − T AK I a JK F I ∧ ⋆ F J . ( . )In order to dualize the Noether currents, we first have to replace the Hodge dual ofthe vector field strengths by the ˜ F I : − k Az δ S δφ z = d ⋆ j ( σ ) A − T AK I F I ∧ ˜ F K , ( . )and, then, using the invariance of the C I JK tensor Eq. ( . ) we get − k Az δ S δφ z = d h ⋆ j ( σ ) A − T AK I ( A I ∧ ˜ F K + F I ∧ ˜ A K ) i = . )As usual, we solve locally this equation by introducing -form potentials D A ⋆ j ( σ ) A − T AK I ( A I ∧ ˜ F K + F I ∧ ˜ A K ) ≡ dD A , ( . )with gauge-invariant field strengths and duality relation K A = dD A − T AK I ( A I ∧ ˜ F K + F I ∧ ˜ A K ) , ⋆ j ( σ ) A = K A . ( . ) . . N = d = supergravity This example is based on the results found in Ref. [ ]. The possible electric fields inan -dimensional theory are scalars φ x , -forms A I , -forms B m , and -forms C a . Themost general Abelian, massless, ungauged supergravity-like theory in dimensionswith this field content can be written in the form S = Z n − ⋆ R + G xy d φ x ∧ ⋆ d φ y + M I J F I ∧ ⋆ F J + M mn H m ∧ ⋆ H n − ℑ m N ab G a ∧ ⋆ G b − ℜ e N ab G a ∧ G b − dC a ∧ ∆ G a − ∆ G a ∧ ∆ G a − d mnp B m ∧ dB n ∧ dB p + d mnp B m ∧ H n ∧ H p + d i I m d iJ n A I ∧ A J ∧ ∆ H m ∧ dB n o . ( . )where G xy , M I J , M mn , N ab are scalar-dependent kinetic matrices ( N ab complex and therest real), the field strengths are defined by I = dA I . ( . ) H m = dB m − d mI J F I ∧ A J , ( . ) G a = dC a + d a I m F I ∧ B m − d a I m d mJK A I ∧ F J ∧ A K , ( . ) d mI J , d a I m being constant deformation parameters and ∆ G a etc. denote all the terms inthe corresponding field strength but dC a etc.The -forms can be dualized in -forms C a with field strengths and duality relations G a ≡ dC a + d aI m F I ∧ B m − d aI m d mJK A I ∧ F J ∧ A K , G a = −ℑ m N ab G a ∧ ⋆ G b − ℜ e N ab G a ∧ G b ≡ R a , ( . )where the d aI m are constant independent parameters. The electric and magnetic -forms and the deformation parameters can be collected in symplectic vectors: ( C i ) ≡ (cid:18) C a C a (cid:19) , ( d i I m ) ≡ (cid:18) d a I m d aI m (cid:19) , ( . )with the symplectic indices i , j to be raised and lowered with the symplectic metric ( Ω ij ) = ( Ω ij ) = (cid:0) − (cid:1) .The -forms B n can be dualized into -forms ˜ B m with field strength and dualityrelation˜ H m ≡ d ˜ B m + d i Im C i ∧ F I + d mnp B n ∧ ( H p + ∆ H p ) + d i I m d iJ n A I ∧ A J ∧ ∆ H n ,˜ H m = M mn ⋆ H n , ( . )where the new deformation d mnp = d [ mnp ] must be related to the other deformationsby d i ( I | m d i | J ) n = − d mnp d pI J . ( . )Finally, the -forms A I can be dualized into -forms ˜ A I with field strength andduality relation ˜ F I ≡ d ˜ A I + · · · ,˜ F I = M I J ⋆ F J , ( . )where the dots stand for a very long expression that can be found in Ref. [ ]. s in the previous example, let us assume that the equations of motion are invariantunder the global transformations generated by δ A φ x = k Ax ( φ ) , δ A A I = T A I J A J , δ A B m = − T Anm B n , δ A C i = T Ai j C j , ( . )where the matrices T A I J , T Amn and the matrices (cid:16) T Ai j (cid:17) ≡ (cid:18) T Aab T Aab T A ab T A ab (cid:19) , ( . )which must be generators of the symplectic group, T Ai [ j Ω k ] i = . )are different representations of the same Lie algebra as the one generated by the vectors k Ax ( φ ) : [ T A , T B ] = f ABC T C , [ k A , k B ] = − f ABC k C . ( . )As in the previous case, this implies that the functions k Ax ( φ ) are Killing vectors ofthe σ -model metric G xy , the kinetic matrices satisfy k Ax ∂ x M I J = − T AK ( I M J ) K , k Ax ∂ x M mn = T A ( m p M n ) p , k Ax ∂ x N ab = − T A ab − N ac T Acb + T A ac N cb + N ac T Acd N db , ( . )and the deformation tensors d mI J , d i Im , d mnp are invariant under the δ A transformations: δ A d mI J = − T Anm d nI J − T AK ( I d n | J ) K = δ A d i I m = T Ai j d j I m − T A J I d i J m + T Amn d i I n = δ A d mnp = T A [ mq d np ] q = . )Since, in general, these symmetries are not symmetries of the action, we proceed asin the -dimensional case, contracting the equations of motion of the scalars, given by Observe that the transformations involving the -forms include electric-magnetic rotations. -formsin dimensions transform as the -forms in -dimensions with groups which must be embedded in thesymplectic group. The transformation rule of the period matrix is unusual because our definition of the dual -formfield strength differs by a sign from the usual one. δ S δφ x = G xy [ d ⋆ d φ y + Γ zwy d φ z ∧ ⋆ d φ w ] − ∂ x M I J F I ∧ ⋆ F J − ∂ x M mn H m ∧ ⋆ H n − G a ∂ x R a , ( . )with the Killing vectors of the σ -model metric G xy ( φ ) , k Ax ( φ ) . Using the Killing equa-tion, we get − k Ax δ S δφ x = d ⋆ j ( σ ) A − k Ax ∂ x M I J F I ∧ ⋆ F J − k Ax ∂ x M mn H m ∧ ⋆ H n − G a k Ax ∂ x R a .( . )Using now Eqs. ( . ) and the duality relations for the field strengths, we arrive to − k Ax δ S δφ x = d ⋆ j ( σ ) A + T A J I F I ∧ ˜ F J + T Amn ˜ H n ∧ H m + T Ai j G j ∧ G j = . )on shell. It is not difficult to see that the exterior derivative of the expression inthe l.h.s. of the equation vanishes due to the Bianchi identities satisfied by the fieldstrengths and due to the invariance of the deformation tensors expressed in the rela-tions Eqs. ( . ). This means that it should be possible to rewrite this equation as theconservation of a higher-dimensional NGZ current, that is d ⋆ j NGZA = j NGZA ≡ j ( σ ) A + ∆ j A , ( . )where ∆ j A has a very long a complicated form.A local solution of this conservation equation is provided by ⋆ [ j ( σ ) A + ∆ j A ] = dD A where D A is a -form potential D A . Then, reasoning as in the -dimensional case, thegauge-invariant -form field strength K A and its duality relation will be given by K A ≡ dD A + ⋆ ∆ j A , K A = ⋆ j ( σ ) A , ( . )and its Bianchi identity will adopt the universal form dK A = T A I J F J ∧ ˜ F I + T Amn ˜ H n ∧ H m + T Ai j G j ∧ G i . ( . ) . . N = B , d = supergravity Our last example concerns the dualization of the scalars of N = B , d =
10 super-gravity [ , , ], the effective field theory of the type IIB superstring. They are thedilaton ϕ and the RR 0-form χ and, combined in the axidilaton τ = χ + ie − ϕ theyparametrize the SL ( R ) /SO ( ) described in Section . . . They are dualized into -form potentials satisfying a constraint [ , ] according to the general rules and the eld strengths, whose form depends very strongly on conventions, satisfy a Bianchiidentity of the universal form proposed above.Here we want to focus on the supersymmetry transformation rules of the -forms,constructed in Refs. [ , ] in the SU (
1, 1 ) /U ( ) formulation used in [ ] and studied inSection . . . We want to compare them with the general form proposed in Section . .They are given by δ ǫ A αβµ ··· µ = V ( α + V β ) − ¯ ǫγ [ µ ··· µ ψ µ ] + c.c. − iV α + V β + ¯ ǫγ µ ··· µ λ C + c.c. + · · · ( . )where we have omitted terms proportional to other p -form fields, which are related tothe Chern-Simons terms in the -form field strengths. Comparing now with Eqs. ( . )and ( . ) we see that the terms constraining the gravitini are multiplied by the mo-mentum map while the terms containing the dilatini are proportional to the Killingvectors, as expected according to our general arguments. Conclusions
In this paper we have reviewed the general problem of dualizing the scalars of a d -dimensional theory into ( d − ) -form potentials preserving the dualities of the theoryin a manifest form and taking into account their possible couplings to the potentialsof the theory. We have not considered the dualization in presence of a scalar poten-tial, since doing this properly, requires the full tensor hierarchy machinery, which liesoutside of the scope of this paper. In general, the dualization procedure has to be necessarily incomplete: the non-linearly interacting scalars cannot be replaced completely by the ( d − ) -form poten-tials, as often happens in supergravity theories with most potentials. Nevertheless, onemay hope to find a PST-like formulation for them. For the particular case of scalarsparametrizing the coset SU( , )/U( ) the PST-type action was constructed in Ref. [ ]as a part of type IIB supergravity action. Here we have presented the generalization ofthe action of Ref. [ ] for the generic symmetric space G/H; the properties of this actionand its applications will be considered elsewhere.Since we need to dualize conserved charges and some of the symmetries one hasto consider in supergravity theories leave invariant the equations of motion but notthe action, it is necessary to consider the Noether-Gaillard-Zumino current, whosegeneralization to theories in higher dimensions and with higher-rank potentials wehave studied. The general d = ] provides a quite complete example of how to proceed inthat case. uring this study we have found it necessary to extend the concept of momen-tum map to all symmetric spaces. The holomorphic and triholomorphic momentummaps defined in Kähler and quaternionic-Kähler spaces play a very important rôlein N =
1, 2, d = -forms dual tothe scalars (and, therefore, in the tensions of the strings that couple to them) and inthe covariant derivatives of the fermions in gauged supergravities. We have shownthrough examples that the generalized momentum map satisfies similar equations andplays exactly the same rôle in N > d > ( d − ) -forms dualto the scalars and the fermion shifts.In N = N , d , mostlybecause of historical reasons: some of them have been constructed by dimensionalreduction, some others in superspace or using other approaches. This complicatesunnecessarily working with them and establishing relations between them via com-pactifications, truncations, gaugings etc. As a theories of dynamical supergeometry,it should be possible to describe them in a more N - and d -independent form. A bigstep in this direction was taken in Ref. [ ], specially for -dimensional theories, whichwere described in an almost N -independent fashion, but neither the gaugings nor thehigher-rank form fields were considered there. We hope the extension of the concept ofmomentum map proposed here and its systematic use (specially in the construction offermion shifts and scalars potentials) will be useful to rewrite all gauged supergravitiesin a more homogeneous form. Acknowledgments
TO would like to thank M. Trigiante for interesting and stimulating conversations dur-ing the Workshop
Theoretical Frontiers in Black Holes and Cosmology held at the Inter-national Institute of Physics in Natal (Brazil) in June . BI is thankful to MIAPPand to the organizers of the MIAPP program ”Higher spin theory and duality” for thehospitality in Munich at the final stage of this project. This work has been supported inpart by the Spanish Ministry of Science and Education grant FPA - -C - (IBand TO), the Centro de Excelencia Severo Ochoa Program grant SEV- - (TO), he Spanish Consolider-Ingenio program CPAN CSD - (TO) and by theBasque Country University program UFI / (IB). TO wishes to thank M.M. Fernán-dez for her permanent support. A Kähler–Hodge manifolds in N =
1, 2 , d = supergrav-ity In this Appendix we want to review briefly the definition of the holomorphic momen-tum map and other structures which have their parallel in the main text in the contextof Kähler–Hodge (KH) manifolds, which are not necessarily symmetric or even ho-mogenous spaces. We adopt the notation and conventions of Refs. [ , ].A Kähler manifold is a complex, Hermitian manifold whose fundamental -form J J ≡ J ij ∗ dZ i ∧ dZ ∗ j ∗ = i G ij ∗ dZ i ∧ dZ ∗ j ∗ , (A. )is closed d J = )This equation implies the vanishing of the torsion, the identification of the Hermitianconnection with the Levi-Civita connection and the local existence of a real function,the Kähler potential K ( Z , Z ∗ ) , such that i J ij ∗ = G ij ∗ = ∂ i ∂ j ∗ K . (A. ) K is defined up to Kähler transformations, which have the form K ′ = K + λ ( Z ) + λ ∗ ( Z ∗ ) , (A. )were λ ( Z ) is an arbitrary holomorphic function of the complex coordinates Z i .In N =
1, 2, d = G ij ∗ plays the role of the σ -model metric.The Kähler (connection) -form is defined by Q ≡ i ( ∂ i K dZ i − c.c. ) , (A. )transforms under Kähler transformations as a U ( ) connection Q ′ = Q + i ( ∂λ − ∂ ∗ λ ∗ ) , (A. )and the Kähler -form can be seen as its Kähler-invariant curvature J ≡ d Q . (A. ) Kähler–Hodge manifold is a Kähler manifold M on which a complex line bundle L → M has been defined such that its first Chern class (given by the Ricci -form R of the fiber’s Hermitian metric) is equal to the Kähler -form J .As we are going to show, in the KH manifolds of N =
1, 2, d = -form connection Q and its curvature J play the same as the H = U ( ) connection ϑ and its curvature R ( ϑ ) defined in Eqs. ( . ) and ( . ): Q → − ϑ , J → − R ( ϑ ) , (A. )even though there is no coset structure. The requirement that the Kähler manifold isactually Kähler–Hodge is crucial.The fermionic fields of N =
1, 2 supergravity are sections of the associated U ( ) bundle, which means that, under Kähler transformations Eq. (A. ), they transform as ψ ′ = e − q ( λ − λ ∗ ) ψ , (A. )if their weight is the real number q . The Kähler-covariant derivative on fields of Kählerweight q is given by D ψ = d ψ + iq Q ψ , (A. )where here Q is the spacetime pullback of the Kähler -form. This definition shouldbe compared with that of the H-covariant derivative Eq. ( . ).Let us now assume that the theory we are considering has some global symmetrytransformation group acting on the scalars. These transformations must necessarilybe holomorphic isometries of the Kähler metric generated by Killing vectors K A ≡ k Ai ( Z ) ∂ i + k ∗ Ai ∗ ( Z ∗ ) ∂ i ∗ but they must also preserve the entire KH structure.First of all, this implies that the transformations generated by the Killing vectorswill leave the Kähler potential invariant up to Kähler transformations: L k A K ≡ k Ai ∂ i K + k ∗ Ai ∗ ∂ i ∗ K = λ A ( Z ) + λ ∗ A ( Z ∗ ) , (A. )for certain holomorphic functions λ A ( Z ) . This, in its turn, implies that all the fieldswhich transform as in Eq. (A. ) will transform as ψ ′ = e − q ( λ A − λ ∗ A ) ψ , (A. )under the transformation generated by K A . This is similar to the H compensatingtransformations described in Section . . and it is clear that the imaginary part ofthe holomorphic functions λ A plays the same role as the H-compensator defined inEq. ( . ) i ( λ A − λ ∗ A ) → − W A . (A. )Taking another Lie derivative in Eq. (A. ) we find the following equivariance prop-erty k A λ B − L k B λ A = − f ABC λ C , (A. )which is identical to that of the H-compensator Eq. ( . ).Secondly, the Kähler -form J must also be preserved L k A J = i k A d J + d ( i k A J ) = )Eq. (A. ) and the above equation imply the local existence of real functions P A (theholomorphic momentum maps) such that i k A J = d P A . (A. )Comparing this equation with Eq. ( . ) and taking into account the correspondencesEq. (A. ) we find that the holomorphic momentum map plays the same role as themomentum map defined in Eq. ( . ): P A → P A . (A. )A local solution of Eq. (A. ) is i P A = k Ai ∂ i K − λ A = − ( k ∗ Ai ∗ ∂ i ∗ K − λ ∗ A ) = i k A Q − i ( λ A − λ ∗ A ) , (A. )where we have taken into account Eq. (A. ). This equation should be compared withEq. ( . ) that relates the H-connection, the H-compensator and the momentum map.Furthermore, the holomorphic Killing vectors can be obtained form the momentummap ( Killing prepotential ) ∂ i P A = ik ∗ A i . (A. )In N = d = V M and the symplectic generators T A M N : P A = h V ∗ | T A V i = T MA N V ∗ M V N . (A. )If we now gauge the group of holomorphic isometries generated by the Killingvectors k Ai we can follow the same rules as in symmetric spaces to construct the gauge-covariant derivatives, adding to the pullback of the H-connection (Kähler connection)the product A A P A where A A is the spacetime gauge field: Q → Q − gA A P A = Q i D Z i + Q i ∗ D Z ∗ i ∗ − gA A ℑ m λ A . (A. )The momentum map also occurs in the fermion shifts of the fermions’ supersym-metry transformation rules. The details depend on the theory and its R-symmetrygroup and can be found, for instance, in Ref. [ ]. . -form potentials from the Kähler–Hodge manifolds of N =
1, 2 , d = supergravity The dualization of the complex scalars of N = N = d = . To finish the job, though, a supersymmetry trans-formation rule must be provided for the dual -form fields, at least to lowest (zeroth)order in fermions. The supersymmetry algebra must close on shell and up to dualityrelations between the magnetic and electric vector fields and between the -forms andthe NGZ currents.This was first done in the N = d = ]. After the use ofthe expression for the momentum maps Eq. (A. ), the supersymmetry transformationrules found there can be written in the form δ ǫ B A µν = − i P A ¯ ǫ I γ [ µ ψ I ν ] − i k ∗ A i ¯ ǫ I γ µν λ iI + c.c. + T A M N A M [ µ | δ ǫ A N | ν ] . (A. )The commutator of two of these supersymmetry transformations gives [ δ η , δ ǫ ] = δ g.c.t. ( ξ ) + δ gauge ( Λ ) + δ gauge ( Λ ) . (A. )where ξ µ are the parameters of general coordinate transformations, Λ M are the -formparameters of the gauge transformations of the gauge fields A M µ and Λ A are the -form parameters of the gauge transformations of the -form fields B A µν . Their explicitexpressions can be found in Ref. [ ].In the actual computation of the commutator, the derivative of the momentum map,which gives the corresponding Killing vector and the scalar part of the NGZ currentappears naturally. Upon dualization, that term gives the contraction of the -form fieldstrength H A with ξ µ , which is a general coordinate transformation of B A up to a gaugetransformation.The supersymmetry transformation rule for the -forms of N = d = ] and fits into the same pattern (the differences are basicallydue to the different conventions) δ ǫ B A µν = i P A ¯ ǫ ∗ γ [ µ ψ ν ] + ∂ i P A ¯ ǫγ µν χ i + c.c. − T AM N A M [ µ δ ǫ A N ν ] . (A. ) B Quaternionic–Kähler manifolds in N = , d = su-pergravity The structures constructed for symmetric spaces can also be generalized to quaternionic-Kähler (QK) spaces, 4 m -dimensional Riemannian spaces whose holonomy group isSU ( ) × Sp ( m ) . QK manifold is a 4 m -dimensional Riemannian manifold that satisfies the follow-ing properties: . It admits a triplet of complex structures J xmn , x =
1, 2, 3 satisfying the algebra ofthe unit imaginary quaternions J xm p J y pn = − δ xy δ mn + ε xyz J zmn . (B. )(Observe that this property implies the property that characterizes complex struc-tures ( J x ) = − ∀ x .) . The Riemannian metric g mn is Hermitian with respect to the three complex struc-tures: g mn = J ( x ) m p J ( x ) nq g pq . (B. )We can define a triplet of Kähler -forms (hyperKähler -form) J xmn ≡ J xm p g np . (B. ) . There is a SU ( ) bundle over the QK space with connection -form A xm d φ m andit is required that the hyperKähler -form is covariantly constant with respect toit: D m J xnp ≡ ∇ m ( ω ) J x np + ε xyz A ym J znp = )where ∇ m ( ω ) stands for the covariant derivative with the Levi-Civita connection ω . . The SU ( ) curvature, defined by F x ≡ d A x + ε xyz A y ∧ A z , (B. )is proportional to the hyperKähler structure F x = κ J x . (B. )In N = d = κ = −
1. (If κ = ( ) component of the curvature -form of the Levi-Civita connection (obtained throughthe projection with the hyperKähler structure) and the hyperKähler -form mn pq ( ω ) J x pq = − m κ J xmn , (B. )plays the same role as the relation between the Kähler -form and the Ricci -form inKähler-Hodge manifolds. It establishes a bridge between symmetric spaces and QKspaces: here, A will play the role of the H = SU ( ) connection ϑ and the hyperKählerstructure J x will play the role of the curvature R ( ϑ ) thanks to the above property. A x → ϑ x , J x → κ R x ( ϑ ) . (B. )The fermionic fields of N = d = ( ) bundleand, under an transformation with infinitesimal parameter λ x they transform as δ λ ψ I = i λ x σ I J ψ J , (B. )and the SU ( ) -covariant derivative acting on them is given by D ψ I ≡ d ψ I − i A x σ I J ψ J . (B. )Compare this definition with that of the H-covariant derivative Eq. ( . ).Now let us assume the existence of an isometry group of the QK manifold preserv-ing the hyperKähler structure J x . This means that the transformations generated bythe corresponding Killing vectors k A (known as triholomorphic Killing vectors) leaveinvariant J x up to an SU ( ) transformation L k A J x = − δ λ A J x , or L k A J x ≡ L k A J x − ε xyz λ yA J z = )for some SU ( ) infinitesimal parameters λ xA which play the role of the H-compensatorsdefined in Eq. ( . ) λ xA → W Ax . (B. )In order to determine λ xA we observe that the H-compensator has to be universal :all the objects that define the QK geometry must be invariant under the action ofthe isometry and the same compensating SU ( ) transformation. In particular, for theSU ( ) connection L k A A xm = L k A A xm + D m λ xA = k An F xnm + D m ( k An A xn + λ xA ) = )This equation implies that k An F xnm is the SU ( ) -covariant derivative of an objectthat we can identify with the (triholomorphic) momentum map: Here we ignore the U ( ) component of the R-symmetry group, which we have discussed in theprevious Appendix. An F xnm = κ D m P Ax , (B. ) κ P Ax = k An A xn + λ xA . (B. )The last equation should be compared with Eq. ( . ) while the first should be com-pared with Eq. ( . ). Using the relation between the curvature F x and the hyperKählerstructure J x Eq. (B. ) one can multiply both sides of the first equation by J x and obtain k Am = − κ J x mn D n P Ax . (B. )In this equation the triholomorphic momentum map plays the role of triholomorphicKilling prepotential.The construction of gauge-covariant derivatives using the momentum map followsthe same pattern as in symmetric spaces (see, for instance, Ref. [ ]). The momentummap also appears in all the fermion shift terms of the supersymmetry transformationrules of the fermions of the N = d = ]. B. -form potentials from the Quaternionic–Kähler manifolds in N = , d = supergravity Since the hyperscalars do not couple to the vector fields, their dualization is speciallysimple: the NGZ currents are equal to the Noether current of the σ -model. The super-symmetry transformation rules for the dual -forms are given by [ ] δ ǫ B A µν = − P A J I ¯ ǫ I γ [ µ ψ J | ν ] + i U α J u D u P A J I ¯ ǫ I γ µν ζ α + c.c. , (B. )where u , v label the real coordinates of the QK manifold (the hyperscalars q u , U α J u arethe inverse Vielbein of the QK manifold (the tangent space index being splint into aSU ( ) index J and an Sp ( m ) index α and where P A J I ≡ i P Ax ( σ x ) I J . 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