Electric/magnetic duality for chiral gauge theories with anomaly cancellation
Jan De Rydt, Torsten T. Schmidt, Mario Trigiante, Antoine Van Proeyen, Marco Zagermann
CCERN-PH-TH/2008-172KUL-TF-08/15MPP-2008-97arXiv:0808.2130
Electric/Magnetic Duality for Chiral GaugeTheories with Anomaly Cancellation
Jan De Rydt , , Torsten T. Schmidt , Mario Trigiante ,Antoine Van Proeyen and Marco Zagermann Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,Celestijnenlaan 200D B-3001 Leuven, Belgium Physics Department,Theory Unit, CERN,CH 1211, Geneva 23, Switzerland Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6,80805 M¨unchen, Germany Dipartimento di Fisica & INFN, Sezione di Torino, Politecnico di TorinoC. so Duca degli Abruzzi, 24, I-10129 Torino, Italy
Abstract
We show that 4D gauge theories with Green-Schwarz anomaly cancellation and possible generalizedChern-Simons terms admit a formulation that is manifestly covariant with respect to electric/magneticduality transformations. This generalizes previous work on the symplectically covariant formulationof anomaly-free gauge theories as they typically occur in extended supergravity, and now also includesgeneral theories with (pseudo-)anomalous gauge interactions as they may occur in global or local N = 1supersymmetry. This generalization is achieved by relaxing the linear constraint on the embeddingtensor so as to allow for a symmetric 3-tensor related to electric and/or magnetic quantum anomaliesin these theories. Apart from electric and magnetic gauge fields, the resulting Lagrangians also featuretwo-form fields and can accommodate various unusual duality frames as they often appear, e.g., instring compactifications with background fluxes. e-mails: { Jan.DeRydt, Antoine.VanProeyen } @fys.kuleuven.be, { schto, zagerman } @mppmu.mpg.de,[email protected] a r X i v : . [ h e p - t h ] F e b ontents N = 1 supersymmetry 73 The embedding tensor and the symplectically covariant formalism 10 B -field and a new constraint . . . . . . . . . 183.2.5 Generalized Chern-Simons terms . . . . . . . . . . . . . . . . . . . . . 193.2.6 Variation of the total action . . . . . . . . . . . . . . . . . . . . . . . . 19 G µν M . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Introduction
In field theories with chiral gauge interactions, the requirement of anomaly-freedom imposesa number of nontrivial constraints on the possible gauge quantum numbers of the chiralfermions. The strongest requirements are obtained if one demands that all anomalous one-loop diagrams due to chiral fermions simply add up to zero.These constraints on the fermionic spectrum can be somewhat relaxed if some of theanomalous one-loop contributions are instead cancelled by classical gauge-variances of cer-tain terms in the tree-level action. The prime example for this is the Green-Schwarz mech-anism [1]. In its four-dimensional incarnation, it uses the gauge variance of Peccei-Quinnterms of the form a F ∧ F , with a ( x ) being an axionic scalar field and F some vector fieldstrengths, under gauged shift symmetries of the form a ( x ) → a ( x ) + c Λ( x ), where Λ( x ) isthe local gauge parameter and c a constant. Gauge variances of this form may cancel mixedAbelian/non-Abelian as well as cubic Abelian gauge anomalies in the quantum effective ac-tion. The Abelian gauge bosons that implement the gauged shift symmetries of the axionsvia St¨uckelberg-type gauge couplings correspond to the anomalous Abelian gauge groups andgain a mass due to their St¨uckelberg couplings. If their masses are low enough, these pseudo-anomalous gauge bosons might be observable and could possibly play the rˆole of a particulartype of Z (cid:48) -boson. The phenomenology of such St¨uckelberg Z (cid:48) -extensions of the StandardModel was studied in various works [2, 3, 4, 5, 6, 7, 8, 9, 10], which were in part inspired byintersecting brane models in type II orientifolds, where the operation of a 4D Green-Schwarzmechanism is quite generic [11]. In [18, 19, 20], however, it has recently been pointed out that in these orientifold com-pactifications, the Green-Schwarz mechanism is often not sufficient to cancel all quantumanomalies. In particular, the cancellation of mixed Abelian anomalies between anomalousand non-anomalous Abelian factors in general needs an additional ingredient, so-called gen-eralized Chern-Simons terms (GCS terms), in the classical action. GCS terms are of theschematic form A ∧ A ∧ dA and A ∧ A ∧ A ∧ A , where the vector fields A are not all thesame. It is quite obvious that GCS terms are not gauge invariant, and it is precisely thisgauge variance that can be used in some cases to cancel possible left-over gauge variancesfrom quantum anomalies and Peccei-Quinn terms. Interestingly, these GCS terms indeed dooccur quite generically in the above-mentioned orientifold compactifications [18, 20]. Phe-nomenologically, they provide extra trilinear (and quartic) couplings between anomalous andnon-anomalous gauge bosons, which, given a low St¨uckelberg mass scale, may lead to Z (cid:48) -bosons with possibly observable new characteristic signals [18, 19, 20].In [26], it is shown how models with all three ingredients (each of which individually For more details on intersecting brane models, see, e.g. the reviews [12, 13, 14, 15, 16, 17] and referencestherein. See also [21, 22, 23, 24, 25]. N = 1 supersymmetry. This compatibility is non-trivial, because a violation of gauge symmetries usually also triggers a violation of the on-shellsupersymmetry, as is best seen by recalling that in the Wess-Zumino gauge the preserved su-persymmetry is a combination of the original superspace supersymmetry and a gauge trans-formation. Due to the presence of the quantum gauge anomalies, one therefore also has totake into account the corresponding supersymmetry anomalies of the quantum effective ac-tion, as they have been determined by Brandt for N = 1 supergravity in [27, 28]. A recentapplication of the theories studied in [26] to globally supersymmetric models with interestingphenomenology appeared in [29].While in [18, 26] the general interplay of all the above three ingredients is discussed, itshould be emphasized that not all three ingredients necessarily need to be present in a gaugeinvariant theory. This is obvious from the original St¨uckelberg Z (cid:48) -models [3, 4, 5, 6, 7, 8, 9, 10],which do not have GCS terms. However, one can also construct purely classical theories, inwhich only the last two ingredients (ii) and (iii), i.e. the gauged shift symmetries and theGCS terms, are present and the fermionic spectrum is either absent or non-anomalous. Infact, it was in such a context that GCS terms were first discussed in the literature. Moreconcretely, their possibility was first discovered in extended gauged supergravity theories [30],which are automatically free of quantum anomalies due to the incompatibility of chiral gaugeinteractions with extended 4D supersymmetry. The ensuing papers [31, 32, 33, 34, 35, 36,37, 38, 39, 40] likewise remained focused on – or were inspired by – the structures found inextended supergravity. Recently, axionic gaugings and GCS terms were also considered in thecontext of global N = 1 supersymmetry in [41]. In all these cases, the absence of quantumanomalies restricts the form of the possible gauged axionic shift symmetries.Another very important example in this context is the work [42], which combines classi-cally gauge invariant local Lagrangians that may also include Peccei-Quinn and GCS termswith the concept of electric/magnetic duality transformations. In four spacetime dimen-sions, a field theory with n Abelian vector potentials and no charged matter fields admitsreparametrizations in the form of electic/magnetic duality transformations. Those transfor-mations that leave the set of field equations and Bianchi identities invariant are the rigid (orglobal) symmetries of the theory and form the global symmetry group G rigid . In section 3.2,we will discuss how, in general, G rigid is contained in the direct product of the symplectic4uality transformations that act on the vector fields and the isometry group of the scalarmanifold of the chiral multiplets: G rigid ⊆ Sp (2 n, R ) × Iso( M scalar ).Note, however, that the Lagrangians that encode the field equations are in general notinvariant under such rigid symmetry transformations, as the latter may involve nontrivialmixing of field equations and Bianchi identities. Moreover, the fields before and after asymmetry transformation are, in general, not related by a local field transformation.In order to gauge a rigid symmetry in the standard way (i.e., in order to introduce chargesfor some of the fields), one needs to be able to go to a symplectic duality frame in whichthe symmetry leaves the action invariant. This automatically implies that the symmetryis also implemented by local field transformations. This would then allow the introductionof minimal couplings and covariant field strengths for the electric vector potentials in theLagrangian in the usual way. This standard procedure obviously singles out certain dualityframes and breaks the original duality covariance.In [42], it was shown how one can nevertheless reformulate 4D gauge theories in such away as to maintain, formally, the full duality covariance of the original ungauged theory. Inorder to do so, the authors consider electric and magnetic gauge potentials ( A µ Λ , A µ Λ ) (Λ =1 , . . . , n ) at the same time and combine them into a 2 n -plet, A µM ( M = 1 , . . . , n ) of vectorpotentials. Introducing then also a set of antisymmetric tensor fields, an intricate system ofgauge invariances can be implemented, which ensures that the number of propagating degreesof freedom is the same as before. The coupling of the electric and magnetic vector potentialsto charged fields is then encoded in the so-called embedding tensor Θ M α = (Θ Λ α , Θ Λ α ),which enters the covariant derivatives of matter fields, φ , schematically,( ∂ µ − A µM Θ M α δ α ) φ . (1.1)Here, α = 1 , . . . , dim( G rigid ) labels the generators of the rigid symmetry group, G rigid , actingas δ α φ on the matter fields. In general, the gauge group also acts on the vector fields via(2 n × n )-matrices, ( X M ) N P ≡ X MN P ≡ Θ M α ( t α ) N P , (1.2)where the ( t α ) N P are in the fundamental representation of Sp (2 n, R ).The embedding tensor has to satisfy a quadratic constraint in order to ensure the closureof the gauge algebra inside the algebra of G rigid . In [42], this fundamental constraint issupplemented by one additional constraint linear in the embedding tensor, which can be5ritten in terms of the above-mentioned tensor X MN P , as X ( MN Q Ω P ) Q = 0 , (1.3)where Ω P Q is the symplectic metric of Sp (2 n, R ). This constraint is sometimes called the“representation constraint”, as it suppresses a representation of the rigid symmetry group inthe tensor X MN P . Together with the quadratic constraint, it ensures mutual locality of allphysical fields that are present in the action. The full physical meaning of this additionalconstraint, however, always remained a bit obscure, and was inferred in [42] from identitiesthat are known to be valid in N = 8 or N = 2 supergravity.In this paper, we propose a physical interpretation of this representation constraint andrecognize it as the condition for the absence of quantum anomalies . Quantum anomalies areautomatically absent in extended 4D supergravity theories, and so it is no surprise, that theinternal consistency of N = 8 or N = 2 supergravity always hinted at the validity of theconstraint (1.3).We then go one step further and show that if quantum anomalies proportional to aconstant, totally symmetric tensor, d MNP , are present, the representation constraint (1.3)has to be relaxed to X ( MN Q Ω P ) Q = d MNP , with d MNP = Θ
M α Θ N β Θ P γ d αβγ , (1.4)to allow for a gauge invariant quantum effective action. Here d αβγ is a symmetric tensor thatwill be defined by the anomalies. We show explicitly how the framework of [42] has to bemodified in such a situation and that the resulting gauge variance of the classical Lagrangianprecisely gives the negative of the consistent quantum anomaly encoded in d MNP .Our work can thus be viewed as a generalization of [42] to theories with quantum anoma-lies or, equivalently, as the covariantization of [18, 26] with respect to electric/magnetic du-ality transformations, and includes situations in which pseudo-anomalous gauge interactionsare mediated by magnetic vector potentials. While already interesting in itself, our resultspromise to be very useful for the description of flux compactifications with chiral fermionicspectra, as e.g. in intersecting brane models on orientifolds with fluxes, because flux com-pactifications often give 4D theories which appear naturally in unusual duality frames and This constraint was considered in [42] for general N and in particular for N = 1 gauged supergravity andgeneralizes an analogous condition originally found in [30]. In the context of rigid N = 1 supersymmetry, itselectric version already appeared in [41]. A subtlety arises for generators δ α that have a trivial action on the vector fields, i.e., ( t α ) M N = 0. In thatcase the mutual locality of the corresponding electric/magnetic components of the embedding tensor shouldbe imposed as an independent quadratic constraint. The tensor d MNP is the one that defines the consistent anomaly in the form given in equation (3.61). Asthe gauge symmetry in the matter sector is implemented by minimal couplings to the gauge potentials dressedwith an embedding tensor, as can be seen from (1.1), the tensor d MNP must be of the form (1.4). N = 1 supersymmetry In this section, we summarize the results of [26] which will later motivate our proposedgeneralization (1.4) of the original constraint (1.3).In a generic low energy effective field theory, the kinetic and the theta angle terms ofvector fields, A µ Λ , appear with scalar field dependent coefficients , L g.k. = 14 e I ΛΣ ( z, ¯ z ) F µν Λ F µν Σ − R ΛΣ ( z, ¯ z ) ε µνρσ F µν Λ F ρσ Σ . (2.1)Here, F µν Λ ≡ ∂ [ µ A ν ]Λ + X ΣΩΛ A µ Σ A ν Ω denotes the non-Abelian field strengths with X ΣΩΛ = X [ΣΩ]Λ being the structure constants of the gauge group. We use the metric signa-ture ( − + ++) and work with real ε = 1. As usual, e denotes the vierbein determinant.The second term in (2.1) is often referred to as the Peccei-Quinn term, and the functions I ΛΣ ( z, ¯ z ) and R ΛΣ ( z, ¯ z ) depend nontrivially on the scalar fields, z i , of the theory. One cancombine these functions to a complex function N ΛΣ ( z, ¯ z ) = R ΛΣ ( z, ¯ z ) + i I ΛΣ ( z, ¯ z ). In a su-persymmetric context, N ΛΣ ( z, ¯ z ) has to satisfy certain conditions, depending on the amountof supersymmetry. In N = 1 global and local supersymmetry, which will be the subject of theremainder of this section, N ΛΣ = N ΛΣ (¯ z ) simply has to be antiholomorphic in the complexscalars of the chiral multiplets.If, under a gauge transformation with gauge parameter Λ Ω ( x ), acting on the field strengthsas δ (Λ) F Λ µν = Λ Ξ F Ω µν X ΩΞΛ , some of the z i transform nontrivially, this may induce a corre-sponding gauge transformation of N ΛΣ (¯ z ). In case this transformation is of the form of a To compare notations between this paper, ref. [26] and ref. [42], note that the vector fields were denotedas W µA in [26], and are here and in [42] denoted as A µ Λ (upper greek letters are electric indices). In [26],the kinetic matrix for the vector multiplets is, as in most of the N = 1 literature, denoted as f AB , whichcorresponds to − i N ∗ ΛΣ in this paper. The structure constants f ABC of [26] correspond to the X ΛΣΩ = f ΛΣΩ here, and the axionic shift tensors C AB,C of [26] are now called X ΛΣΩ = X Λ(ΣΩ) = C ΣΩ , Λ . To compareformulae of [42] to those here and in [26], the Levi-Civita symbol ε µνρσ appears in covariant equations withopposite sign (but ε = 1 is valid in both cases due to another orientation of the spacetime directions). δ (Λ) N ΛΣ = Λ Ω δ Ω N ΛΣ , δ Ω N ΛΣ = X ΩΛΓ N ΣΓ + X ΩΣΓ N ΛΓ , (2.2)the kinetic term (2.1) is obviously gauge invariant. This is what was assumed in the actionof general matter-coupled supergravity in [43]. If, however, one takes into account also other terms in the (quantum) effective action, amore general transformation rule for N ΛΣ (¯ z ) may be allowed: δ Ω N ΛΣ = − X ΩΛΣ + X ΩΛΓ N ΣΓ + X ΩΣΓ N ΛΓ . (2.3)Here, X ΩΛΣ is a constant real tensor symmetric in the last two indices, which can be recog-nized as a natural generalization in the context of symplectic duality transformations [41, 26].Closure of the gauge algebra requires the constraint X ΩΛΣ X ΓΞΩ + 2 X Σ[ΞΩ X Γ]ΛΩ + 2 X Λ[ΞΩ X Γ]ΣΩ = 0 . (2.4)If X ΩΛΣ is non-zero, this leads to a non-gauge invariance of the Peccei-Quinn term in L g.k. : δ (Λ) L g.k. = 18 ε µνρσ X ΩΛΣ Λ Ω F µν Λ F ρσ Σ . (2.5)For rigid parameters, Λ Ω = const . , this is just a total derivative, but for local gauge parame-ters, Λ Ω ( x ), it is obviously not.In order to understand how this broken invariance can be restored, it is convenient tosplit the coefficients X ΩΛΣ into a sum, X ΩΛΣ = X (s)ΩΛΣ + X (m)ΩΛΣ , X (s)ΩΛΣ = X (ΩΛΣ) , X (m)(ΩΛΣ) = 0 , (2.6)where X (s)ΩΛΣ is completely symmetric, and X (m)ΩΛΣ denotes the part of mixed symmetry. Termsof the form (2.5) may then in principle be cancelled by the following two mechanisms, or acombination thereof:(i) As was first realized in a similar context in N = 2 supergravity in [30] (see also the sys-tematic analysis [31]), the gauge variation due to a non-vanishing mixed part, X (m)ΩΛΣ (cid:54) = 0,may be cancelled by adding a generalized Chern-Simons term (GCS term) that containsa cubic and a quartic part in the vector fields, L GCS = 13 X (CS)ΩΛΣ ε µνρσ (cid:18) A µ Ω A ν Λ ∂ ρ A Σ σ + 38 X ΓΞΣ A µ Ω A ν Λ A ρ Γ A σ Ξ (cid:19) . (2.7) This construction of general matter-couplings has been reviewed in [44]. There, the possibility (2.3) wasalready mentioned, but the extra terms necessary for its consistency were not considered. X (CS)ΩΛΣ , which has the same mixed symmetrystructure as X (m)ΩΛΣ . The cancellation occurs provided the tensors X (m)ΩΛΣ and X (CS)ΩΛΣ are,in fact, the same. It was first shown in [41] that such a term can exist in rigid N = 1supersymmetry without quantum anomalies.(ii) If the chiral fermion spectrum is anomalous under the gauge group, the anomaloustriangle diagrams lead to a non-gauge invariance of the quantum effective action Γ forthe gauge symmetry: δ (Λ)Γ = (cid:82) d x Λ Λ A Λ of the form A Λ = − ε µνρσ (cid:20) d ΩΣΛ ∂ µ A ν Σ + (cid:18) d ΩΣΓ X ΛΞΣ + 32 d ΩΣΛ X ΓΞΣ (cid:19) A µ Γ A ν Ξ (cid:21) ∂ ρ A σ Ω , (2.8)with a symmetric tensor d ΩΛΣ . If X (s)ΩΛΣ = d ΩΛΣ , (2.9)this quantum anomaly cancels the symmetric part of (2.5). This is the Green-Schwarzmechanism.In [26], it was studied to what extent a general gauge theory of the above type (i.e.,with gauged axionic shift symmetries, GCS terms and quantum gauge anomalies) can becompatible with N = 1 supersymmetry. The results can be summarized as follows: if onetakes as one’s starting point the matter-coupled supergravity Lagrangian in eq. (5.15) ofreference [44], an axionic shift symmetry with X ΛΣΩ (cid:54) = 0 satisfying the closure condition(2.4) can be gauged in a way consistent with N = 1 supersymmetry if(i) a GCS term (2.7) with X (CS)ΩΛΣ = X (m)ΩΛΣ is added,(ii) an additional term bilinear in the gaugini, λ Σ ( x ), and linear in the vector fields isadded : L extra = −
14 i A µ Ω X ΩΛΣ ¯ λ Λ γ γ µ λ Σ , (2.10)(iii) the fermions in the chiral multiplets give rise to quantum anomalies with d ΩΛΣ = X (s)ΩΛΣ . The consistent gauge anomaly, A Λ is of the form (2.8). The exact result forthe supersymmetry anomaly can be found in [28] or eq. (5.8) of [26]. These quantumanomalies precisely cancel the classical gauge and supersymmetry variation of the newLagrangian L old + L GCS + L extra , where L old denotes the original Lagrangian of [44]. More precisely, the anomalies have a scheme dependence. As reviewed in [18] one can choose a schemein which the anomaly is proportional to a symmetric d ΩΛΣ . Choosing a different scheme is equivalent to thechoice of another GCS term (see item (i)). We will always work with a renormalization scheme in which thequantum anomaly is indeed proportional to the symmetric tensor d ΩΛΣ according to (2.8). A superspace expression for the sum L GCS + L extra is known only for the case X (s)ΛΣΩ = 0, i.e., for the casewithout quantum anomalies [41]. The embedding tensor and the symplectically covariant for-malism
In this section, we recapitulate the results of [42], which describe a symplectically covariantformulation of (classically) gauge invariant field theories. Correspondingly, we will assumethe absence of quantum anomalies in this section.
In the absence of charged fields, a gauge invariant four-dimensional Lagrangian of n Abelianvector fields A µ Λ (Λ = 1 , . . . , n ) only depends on their curls F µν Λ ≡ ∂ [ µ A ν ]Λ . Defining thedual magnetic field strengths G µν Λ ≡ ε µνρσ ∂ L ∂F ρσ Λ , (3.1)the Bianchi identities and field equations read ∂ [ µ F νρ ]Λ = 0 , (3.2) ∂ [ µ G νρ ] Λ = 0 . (3.3)The equations of motion (3.3) imply the existence of magnetic gauge potentials, A µ Λ , via G µν Λ = 2 ∂ [ µ A ν ]Λ . These magnetic gauge potentials are related to the electric vector poten-tials, A µ Λ , by nonlocal field redefinitions. The electric Abelian field strengths, F µν Λ , and theirmagnetic duals, G µν Λ , can be combined into a 2 n -plet, F µν M , such that F M = ( F Λ , G Λ ).This allows us to write (3.2) and (3.3) in the following compact way: ∂ [ µ F νρ ] M = 0 . (3.4)Apparently, equation (3.4) is invariant under general linear transformations F M → F (cid:48) M = S M N F N , where S M N = (cid:32) U ΛΣ Z ΛΣ W ΛΣ V ΛΣ (cid:33) , (3.5)but only for symplectic matrices S M N ∈ Sp (2 n, R ) a relation of the type (3.1) is possible.The admissible rotations S M N thus form the group Sp (2 n, R ): S T Ω S = Ω , (3.6)10ith the symplectic metric, Ω MN , given byΩ MN = (cid:32) ΛΣ Ω ΛΣ (cid:33) = (cid:32) δ ΣΛ − δ ΛΣ (cid:33) . (3.7)We define Ω MN via Ω MN Ω NP = − δ M P . Note that the components of Ω MN should not bewritten as Ω ΛΣ etc., as these are different from (3.7).Starting with a kinetic Lagrangian of the form (2.1), an electric/magnetic duality trans-formation leads to a new Lagrangian, L (cid:48) ( F (cid:48) ), which is of a similar form, but with a new gaugekinetic function N ΛΣ → N (cid:48) ΛΣ = ( V N + W ) ΛΩ (cid:2) ( U + Z N ) − (cid:3) ΩΣ . (3.8)The subset of Sp (2 n, R ) symmetries (of field equations and Bianchi identities) for whichthe Lagrangian remains unchanged in the sense that L (cid:48) ( F (cid:48) ( F )) = L ( F ) and (3.8) is imple-mented by transformations of the fields on which N depends, are invariances of the action.In a different duality frame, the Lagrangian might have a different set of invariances.From the spacetime point of view, these are all rigid (“global”) symmetries. Sometimesthese global symmetries can be turned into local (“gauge”) symmetries. For the conventionalgaugings one has to restrict to the transformations that leave the Lagrangian invariant, whichimplies that Z ΛΣ in the matrices S M N of (3.5) has to vanish. In the context of symplecticallycovariant gaugings [42], however, this restriction can be lifted, and we will come back to thesein section 3.2. The standard way to perform a gauging of a symmetry of interest is thereforeto first switch to a symplectic duality frame in which the symmetries of interest act on F µν M = ( F µν Λ , G µν Λ ) by lower block triangular matrices (i.e. those with Z = 0) such thatthey become (as rigid symmetries) invariances of the action.The gauging requires the introduction of gauge covariant derivatives and field strengthsand can be implemented solely with the electric vector fields A µ Ω and the correspondingelectric gauge parameters Λ Ω . The gaugeable symplectic transformation, S , must be of theinfinitesimal form S M N = δ M N − Λ Ω S Ω M N . (3.9)According to our definition (3.5), these infinitesimal symplectic transformations act on thefield strengths by multiplication with the matrices S Λ M N from the left. Following the con-ventions of [42], however, we will use matrices X Ω M N to describe the infinitesimal symplecticaction via multiplication from the right: δF µν M = F (cid:48) µν M − F µν M = − Λ Ω F µν N X Ω N M , i.e. X Ω N M = S Ω M N . (3.10)11or standard electric gaugings, we then have δ (cid:32) F Λ µν G µν Λ (cid:33) = − Λ Ω (cid:32) X ΩΞΛ X ΩΛΞ X ΩΞΛ (cid:33) (cid:32) F Ξ µν G µν Ξ (cid:33) , (3.11)where X ΩΣΛ = − X ΩΛΣ = f ΩΣΛ are the structure constants of the gauge algebra, and X ΣΞΓ = X Σ(ΞΓ) give rise to the axionic shifts mentioned in section 2 (compare (3.8) with (2.3) for theparticular choice of S given in (3.9)).The gauging then proceeds in the usual way by introducing covariant derivatives ( ∂ µ − A µ Λ δ Λ ),where the δ Λ are the gauge generators in a suitable representation of the matter fields. Onealso introduces covariant field strengths and possibly GCS terms as described in section 2. Aswe assume the absence of quantum anomalies in this section, we have to require X (ΛΣΓ) = 0. We will now turn to the more general gauging of symmetries. The group that will be gaugedis a subgroup of the rigid symmetry group. What we mean by the rigid symmetry group isa bit more subtle in N = 1 supergravity (or theories without supergravity) than in extendedsupergravities. This is due to the fact that in extended supergravities the vectors are super-symmetrically related to scalar fields, and therefore their rigid symmetries are connected tothe symmetries of scalar manifolds.In N = 1 supersymmetry, the rigid symmetry group, G rigid , is a subset of the product ofthe symplectic duality transformations that act on the vector fields and the isometry group ofthe scalar manifold of the chiral multiplets: G rigid ⊆ Sp (2 n, R ) × Iso( M scalar ). The relevantisometries are those that respect the K¨ahler structure (i.e. generated by holomorphic Killingvectors) and that also leave the superpotential invariant (in supergravity, the superpotentialshould transform according to the K¨ahler transformations). Elements ( g , g ) of Sp (2 n, R ) × Iso( M scalar ) that are compatible with (3.8) in the sense that the symplectic action (3.8) of g on the matrix N is induced by the isometry g on the scalar manifold, are rigid (“global”)symmetries provided they also leave the rest of the theory (deriving from scalar potentials,etc.) invariant [45]. The rigid symmetry group, G rigid , is thus a subgroup of Sp (2 n, R ) × Iso( M scalar ). The generators of G rigid will be denoted by δ α , α = 1 , . . . , dim( G rigid ). These generatorsact on the different fields of the theory either via Killing vectors δ α = K α = K iα ∂∂φ i defininginfinitesimal isometries on the scalar manifold, or with certain matrix representations , e.g. Note that this may include cases where either the symplectic transformation g or the isometry g istrivial. Another special case is when the isometry g is non-trivial, but N does not transform under it, ashappens, e.g, when N = i is constant. G rigid is in general a genuine subgroup of Sp (2 n, R ) × Iso( M scalar ),even in the latter case of constant N . The structure constants defined by [ δ α , δ β ] = f αβγ δ γ lead for the matrices to [ t α , t β ] = − f αβγ t γ . α φ i = − φ j ( t α ) ji .On the field strengths F µν M = ( F µν Λ , G µν Λ ), these rigid symmetries must act by multi-plication with infinitesimal symplectic matrices ( t α ) M P , i.e., we have( t α ) [ M P Ω N ] P = 0 . (3.12)In order to gauge a subgroup, G local ⊂ G rigid , the 2 n -dimensional vector space spanned bythe vector fields A µM has to be projected onto the Lie algebra of G local , which is formallydone with the so-called embedding tensor Θ M α = (Θ Λ α , Θ Λ α ). Equivalently, Θ M α completelydetermines the gauge group G local via the decomposition of the gauge group generators, whichwe will denote by ˜ X M , into the generators of the rigid invariance group G rigid :˜ X M ≡ Θ M α δ α . (3.13)The gauge generators ˜ X M enter the gauge covariant derivatives of matter fields, D µ = ∂ µ − A µM ˜ X M = ∂ µ − A µ Λ Θ Λ α δ α − A µ Λ Θ Λ α δ α , (3.14)where the generators δ α are meant to either act as representation matrices on the fermionsor as Killing vectors on the scalar fields, as mentioned above. On the field strengths of thevector potentials, the generators δ α act by multiplication with the matrices ( t α ) N P , so that(3.13) is represented by matrices ( X M ) N P whose elements we denote as X MN P , see (1.2),and whose antisymmetric part in the lower indices appears in the field strengths F µν M = 2 ∂ [ µ A ν ] M + X [ NP ] M A µN A ν P , X NP M = Θ
N α ( t α ) P M . (3.15)The symplectic property (3.12) implies X M [ N Q Ω P ] Q = 0 , X MQ [ N Ω P ] Q = 0 . (3.16)In the remainder of this paper, the symmetrized contraction X ( MN Q Ω P ) Q will play an im-portant rˆole. We therefore give this tensor a special name and denote it by D MNP : D MNP ≡ X ( MN Q Ω P ) Q . (3.17)Note that this is really just a definition and no new constraint. Using the definition (3.17), These matrices might be trivial, e.g., for Abelian symmetry groups that only act on the scalars (and/orthe fermions) and that do not give rise to axionic shifts of the kinetic matrix N ΛΣ . X ( MN ) Q Ω RQ + X RM Q Ω NQ = 3 D MNR , i.e. X ( MN ) P = Ω P R X RM Q Ω NQ + D MNR Ω RP . (3.18) The embedding tensor Θ
M α has to satisfy a number of consistency conditions. Closure ofthe gauge algebra and locality require, respectively, the quadratic constraintsclosure: f αβγ Θ M α Θ N β = ( t α ) N P Θ M α Θ P γ , (3.19)locality: Ω MN Θ M α Θ N β = 0 ⇔ Θ Λ[ α Θ Λ β ] = 0 , (3.20)where f αβγ are the structure constants of the rigid invariance group G rigid , see footnote11. Another constraint, besides (3.19) and (3.20), was inferred in [42] from supersymmetryconstraints in N = 8 supergravity D MNR ≡ X ( MN Q Ω R ) Q = 0 . (3.21)This constraint eliminates some of the representations of the rigid symmetry group andis therefore sometimes called the “representation constraint”. As we pointed out in theintroduction, one can show that the locality constraint is not independent of (3.19) and(3.21), apart from specific cases where ( t α ) M N has a trivial action on the vector fields.However, we will neither use the locality constraint (3.20) nor the representation con-straint (3.21). We will, instead, need another constraint in section 3.2.4, whose meaning wewill discuss in section 4. Before coming to that new constraint, we thus only use the closureconstraint (3.19). This constraint reflects the invariance of the embedding tensor under G local and it implies for the matrices X M the relation[ X M , X N ] = − X MN P X P . (3.22)This clearly shows that the gauge group generators commute into each other with ‘structureconstants’ given by X [ MN ] P . However, note that X MN P in general also contains a non-trivialsymmetric part, X ( MN ) P . The antisymmetry of the left hand side of (3.22) only requires thatthe contraction X ( MN ) P Θ P α vanishes, as is also directly visible from (3.19). Therefore onehas X ( MN ) P Θ P α = 0 → X ( MN ) P X P QR = 0 . (3.23)14riting (3.22) explicitly gives X MQP X NP R − X NQP X MP R + X MN P X P QR = 0 . (3.24)Antisymmetrizing in [ M N Q ], we can split the second factor of each term into the antisym-metric and symmetric part, X MN P = X [ MN ] P + X ( MN ) P , and this gives a violation of theJacobi identity for X [ MN ] P as X [ MN ] P X [ QP ] R + X [ QM ] P X [ NP ] R + X [ NQ ] P X [ MP ] R = − (cid:0) X [ MN ] P X ( QP ) R + X [ QM ] P X ( NP ) R + X [ NQ ] P X ( MP ) R (cid:1) . (3.25)Other relevant consequences of (3.24) can be obtained by (anti)symmetrizing in M Q . Thisgives, using also (3.23), the two equations X ( MQ ) P X NP R − X NQP X ( MP ) R − X NM P X ( QP ) R = 0 ,X [ MQ ] P X NP R − X NQP X [ MP ] R + X NM P X [ QP ] R = 0 . (3.26) The violation of the Jacobi identity (3.25) is the prize one has to pay for the symplecticallycovariant treatment in which both electric and magnetic vector potentials appear at the sametime. In order to compensate for this violation and in order to make sure that the numberof propagating degrees of freedom is the same as before, one imposes an additional gaugeinvariance in addition to the usual non-Abelian transformation ∂ µ Λ M + X [ P Q ] M A µP Λ Q andextends the gauge transformation of the vector potentials to δA µM = D µ Λ M − X ( NP ) M Ξ µNP , D µ Λ M = ∂ µ Λ M + X P QM A µP Λ Q , (3.27)where we introduced the covariant derivative D µ Λ M , and new vector-like gauge parame-ters Ξ µNP , symmetric in the upper indices. The extra terms X ( P Q ) M A µP Λ Q and the Ξ-transformations contained in (3.27) allow one to gauge away the vector fields that correspondto the directions in which the Jacobi identity is violated, i.e., directions in the kernel of theembedding tensor (see (3.23)).It is important to notice that the modified gauge transformations (3.27) still close on thegauge fields and thus form a Lie algebra. Indeed, if we split (3.27) into two parts, δA µM = δ (Λ) A µM + δ (Ξ) A µM , (3.28)15he commutation relations are[ δ (Λ ) , δ (Λ )] A µM = δ (Λ ) A µM + δ (Ξ ) A µM , [ δ (Λ) , δ (Ξ)] A µM = [ δ (Ξ ) , δ (Ξ )] A µM = 0 , (3.29)with Λ M = X [ NP ] M Λ N Λ P , Ξ µP N = Λ ( P D µ Λ N )2 − Λ ( P D µ Λ N )1 . (3.30)To prove that the terms that are quadratic in the matrices X M in the left-hand side of (3.29)follow this rule, one uses (3.26).Due to (3.23) and (3.27), however, the usual properties of the field strength F µν M = 2 ∂ [ µ A ν ] M + X [ P Q ] M A µP A ν Q (3.31)are changed. In particular, it will no longer fulfill the Bianchi identity, which now must bereplaced by D [ µ F νρ ] M = X ( NP ) M A [ µN F νρ ] P − X ( P N ) M X [ QR ] P A [ µN A ν Q A ρ ] R . (3.32)Furthermore, F µν M does not transform covariantly under a gauge transformation (3.27).Instead, we have δ F µν M = 2 D [ µ δA ν ] M − X ( P Q ) M A [ µP δA ν ] Q = X NQM F µν N Λ Q − X ( NP ) M D [ µ Ξ ν ] NP − X ( P Q ) M A [ µP δA ν ] Q , (3.33)where the covariant derivative is (both expressions are useful and related by (3.26)) X ( NP ) M D µ Ξ ν NP = ∂ µ (cid:0) X ( NP ) M Ξ ν NP (cid:1) + A µR X RQM X ( NP ) Q Ξ ν NP , D µ Ξ ν NP = ∂ µ Ξ ν NP + X QRP A µQ Ξ ν NR + X QRN A µQ Ξ ν P R . (3.34)Therefore, if we want to deform the original Lagrangian (2.1) and accommodate electric andmagnetic gauge fields, F µν M cannot be used to construct gauge-covariant kinetic terms.For this reason, the authors of [42] introduced tensor fields B µν α , later in [46] to bedescribed by B µν MN , symmetric in ( M N ), and with them modified field strengths H µν M = F µν M + X ( NP ) M B µν NP . (3.35)16e will consider gauge transformations of the antisymmetric tensors of the form δB µν NP = 2 D [ µ Ξ ν ] NP + 2 A [ µ ( N δA ν ] P ) + ∆ B µν NP , (3.36)where ∆ B µν NP depends on the gauge parameter Λ Q , but we do not fix it further at thispoint. Together with (3.33), this then implies δ H µν M = X NQM Λ Q H µν N + X ( NP ) M ∆ B µν NP . (3.37) The first step towards a gauge invariant action is to replace F µν Λ in L g . k . , (2.1), by H µν Λ ,which then yields the new kinetic Lagrangian L g . k . = e I ΛΣ H µν Λ H µν Σ − R ΛΣ ε µνρσ H µν Λ H ρσ Σ , (3.38)where again I ΛΣ and R ΛΣ denote, respectively, Im N ΛΣ and Re N ΛΣ . Using G µν Λ ≡ ε µνρσ ∂ L ∂ H ρσ Λ = R ΛΓ H µν Γ + 12 eε µνρσ I ΛΓ H ρσ Γ , (3.39)the Lagrangian and its transformations can be written as L g . k . = − ε µνρσ H Λ µν G ρσ Λ ,δ L g . k . = − ε µνρσ G µν Λ δ H Λ ρσ + ε µνρσ Λ Q (cid:0) H Λ µν X Q ΛΣ H Σ ρσ − H Λ µν X Q ΛΣ G ρσ Σ − G µν Λ X Q ΛΣ G ρσ Σ (cid:1) , (3.40)where, in the third line, we used the infinitesimal form of (3.8): δ (Λ) N ΛΣ = Λ M (cid:104) − X M ΛΣ + 2 X M (ΛΓ N Σ)Γ + N ΛΓ X M ΓΞ N ΞΣ (cid:105) . (3.41)When we introduce G µν M = (cid:0) G µν Λ , G µν Λ (cid:1) with G µν Λ ≡ H µν Λ , (3.42)we can rewrite the second line of (3.40) in a covariant expression, and when we also use (3.37)we get δ L g . k . = ε µνρσ (cid:2) − G µν Λ (cid:0) Λ Q X P Q Λ H ρσP + X ( NP )Λ ∆ B ρσNP (cid:1) + G µν M G ρσN Λ Q X QM R Ω NR (cid:3) . (3.43) Note that F µνN in the second line of (3.33) can be replaced by H µνN due to (3.23). L g . k . in (3.38) is still not gauge invariant. This shouldnot come as a surprise because (3.41) contains a constant shift (i.e., the term proportionalto X M ΛΣ ), which requires the addition of extra terms to the Lagrangian as was reviewed insection 2 for purely electric gaugings. Also the last term on the right hand side of (3.41)gives extra contributions that are quadratic in the kinetic function. In the next steps we willsee that besides GCS terms, also terms linear and quadratic in the tensor field are requiredto restore gauge invariance. We start with the discussion of the latter terms. B -field and a new constraint The second step towards gauge invariance is made by adding topological terms linear andquadratic in the tensor field B µν NP to the gauge kinetic term (3.38), namely L top ,B = ε µνρσ X ( NP )Λ B µν NP (cid:16) F ρσ Λ + X ( RS )Λ B ρσRS (cid:17) . (3.44)Note that for pure electric gaugings X ( NP )Λ = 0, as we saw in (3.11). Therefore, in this casethis term vanishes, implying that the tensor fields decouple.We recall that, up to now, only the closure constraint (3.19) has been used. We are nowgoing to impose one new constraint : X ( NP ) M Ω MQ X ( RS ) Q = 0 . (3.45)We will later show that this constraint is implied by the locality constraint (3.20) and theoriginal representation constraint of [42], i.e. (1.3), but also by the locality constraint andthe modified constraint (1.4) that we discussed in the introduction. The constraint thus saysthat X ( NP )Λ X ( RS )Λ = X ( NP )Λ X ( RS )Λ . (3.46)A consequence of this constraint that we will use below follows from the first of (3.18) and(3.23): X ( P Q ) R D MNR = 0 . (3.47)The variation of L top ,B is δ L top ,B = ε µνρσ X ( NP )Λ (cid:2) H µν Λ δB ρσNP + B ρσNP δ F µν Λ (cid:3) (3.48)= ε µνρσ X ( NP )Λ (cid:2) H µν Λ δB ρσNP + 2 B ρσNP (cid:0) D µ δA ν Λ − X ( RS )Λ A Rµ δA Sν (cid:1)(cid:3) . .2.5 Generalized Chern-Simons terms As in [42], we introduce a generalized Chern-Simons term of the form (these are the last twolines in what they called L top in their equation (4.3)) L GCS = ε µνρσ A µM A ν N (cid:18) X MN Λ ∂ ρ A σ Λ + 16 X MN Λ ∂ ρ A σ Λ + 18 X MN Λ X P Q Λ A ρP A σQ (cid:19) . (3.49)Modulo total derivatives one can write its variation as (using (3.24) antisymmetrized in[ M N Q ] and the definition of D MNP in (3.17)) δ L GCS = ε µνρσ (cid:2) F µν Λ D ρ δA σ Λ − F µν Λ X ( NP )Λ A ρN δA σP − D MNP A µM δA ν N (cid:0) ∂ ρ A σP + X RSP A ρR A σS (cid:1)(cid:3) . (3.50)These variations can be combined with (3.48) to δ ( L top ,B + L GCS ) = ε µνρσ (cid:2) H µν Λ D ρ δA σ Λ + H µν Λ X ( NP )Λ (cid:0) δB ρσNP − A ρN δA σP (cid:1) − D MNP A µM δA ν N (cid:0) ∂ ρ A σP + X RSP A ρR A σS (cid:1)(cid:3) . (3.51) We are now ready to discuss the symmetry variation of the total Lagrangian L V T = L g . k . + L top ,B + L GCS , (3.52)built from (3.38), (3.44) and (3.49). We first check the invariance of (3.52) with respect to theΞ-transformations. We see directly from (3.43) that the gauge-kinetic terms are invariant.The second line of (3.51) also clearly vanishes inserting (3.27) and using (3.47). This leaves uswith the first line of (3.51), which, using (3.36) and (3.27), can be written in a symplecticallycovariant form: δ Ξ L V T = − ε µνρσ H µν M X ( NP ) Q Ω MQ D ρ Ξ σNP . (3.53)The B -terms in H , see (3.35), are proportional to X ( RS ) M and thus give a vanishing contri-bution due to our new constraint (3.45). For the F terms we can perform an integration byparts and then (3.32) gives again only terms proportional to X ( RS ) M leading to the sameconclusion. We therefore find that the Ξ-variation of the total action vanishes.We can thus further restrict to the Λ M gauge transformations. According to (3.33), the D ρ δA σ Λ -term in (3.51) can then be replaced by Λ Q X NQ Λ H ρσN (see again footnote 13),which can then be combined with the first term of (3.43) to form a symplectically covariant Integration by parts with the covariant derivatives is allowed as (3.24) can be read as the invariance ofthe tensor X and (3.16) as the invariance of Ω. δ L V T = ε µνρσ (cid:2) G µν M Λ Q X NQR Ω MR H ρσN + G µν M G ρσN Λ Q X QM R Ω NR + ( H − G ) µν Λ X ( NP )Λ ∆ B ρσNP − D MNP A µM D ν Λ N (cid:0) ∂ ρ A σP + X RSP A ρR A σS (cid:1)(cid:3) . (3.54)We observe that if the H in the first line was a G , eqs. (3.16) and (3.18) would allow oneto write the first line as an expression proportional to D MNP . This leads to the first line in(3.55) below. The second observation is that the identity (
H − G ) Λ = 0 allows one to rewritethe second line of (3.54) in a symplectically covariant way, so that, altogether, we have δ L V T = ε µνρσ (cid:2) G µν M Λ Q X NQR Ω MR ( H − G ) ρσN + G µν M G ρσN Λ Q D QMN − ( H − G ) µν M Ω MR X ( NP ) R ∆ B ρσNP − D MNP A µM D ν Λ N (cid:0) ∂ ρ A σP + X RSP A ρR A σS (cid:1)(cid:3) . (3.55)By choosing ∆ B ρσNP = − Λ N G ρσP − Λ P G ρσN , (3.56)the result (3.55) becomes δ L V T = ε µνρσ (cid:2) Λ Q D MNQ (cid:0) G µν M ( H − G ) ρσN + G µν M G ρσN (cid:1) − D MNP A µM D ν Λ N (cid:0) ∂ ρ A σP + X RSP A ρR A σS (cid:1)(cid:3) , (3.57)which is then proportional to D MNP , and hence zero when the original representation con-straint (3.21) of [42] is imposed.Our goal is to generalize this for theories with quantum anomalies. These anomaliesdepend only on the gauge vectors. The field strengths G , (3.39), however, also depend on thematrix N which itself generically depends on scalar fields. Therefore, we want to considermodified transformations of the antisymmetric tensors such that G does not appear in thefinal result.To achieve this, we would like to replace (3.56) by a transformation such that X ( NP ) R ∆ B ρσNP = − X ( NP ) R Λ N G ρσP + Ω RM D MNQ Λ Q ( H − G ) ρσN . (3.58)Indeed, inserting this in (3.55) would lead to δ L V T = ε µνρσ (cid:2) Λ Q D MNQ F µν M F ρσN − D MNP A µM D ν Λ N (cid:0) ∂ ρ A σP + X RSP A ρR A σS (cid:1)(cid:3) , (3.59)20here we have used (3.47) to delete contributions coming from the B µν NP term in H µν M (cf.(3.35)).The first term on the right hand side of (3.58) would follow from (3.56), but the secondterm cannot in general be obtained from assigning transformations to B ρσNP (compare with(3.18)). Indeed, self-consistency of (3.58) requires that the second term on the right handside be proportional to X ( NP ) R , which imposes a further constraint on D MNP . We will see insection 4.3 how we can nevertheless justify the transformation law (3.58) by introducing otherantisymmetric tensors. For the moment, we just accept (3.58) and explore its consequences.Expanding (3.59) using (3.15) and (3.27) and using a partial integration, (3.59) can berewritten as δ L V T = −A [Λ] , (3.60)where A [Λ] = − ε µνρσ Λ P D MNP ∂ µ A ν M ∂ ρ A σN − ε µνρσ Λ P (cid:0) D MNR X [ P S ] N + 32 D MNP X [ RS ] N (cid:1) ∂ µ A ν M A ρR A σS . (3.61)This expression formally looks like a symplectically covariant generalization of the electricconsistent anomaly (2.8). Notice, however, that at this point this is really only a formalanalogy, as the tensor D MNP has, a priori, no connection with quantum anomalies. We willstudy the meaning of this analogy in more detail in the next section. To prove (3.60), oneuses (3.47) and the preservation of D MNP under gauge transformations, which follows frompreservation of X , see (3.24), and of Ω, see (3.16), and reads X M ( N P D QR ) P = 0 . (3.62)For the terms quartic in the gauge fields, one needs the following consequence of (3.62):( X RSM X P QN D LMN ) [ RSP L ] = − ( X RSM X P M N D LQN + X RSM X P LN D QMN ) [ RSP L ] = − ( X RSM X P LN D QMN ) [ RSP L ] , (3.63)where the final line uses (3.25) and again (3.47).Let us summarize the result of our calculation up to the present point. We have usedthe action (3.52) and considered its transformations under (3.27) and (3.36), where ∆ B µν NP was undetermined. We used the closure constraint (3.19) and one new constraint (3.45). Weshowed that the choice (3.56) leads to invariance if D MNP vanishes, which is the represen-tation constraint (3.21) used in the anomaly-free case studied in [42]. However, when weuse instead the more general transformation (3.58) in the case D MNP (cid:54) = 0, we obtain thenon-vanishing classical variation (3.60). The corresponding expression (3.61) formally looks21ery similar to a symplectically covariant generalization of the electric consistent quantumanomaly.In order to fully justify and understand this result, we are then left with the followingthree open issues, which we will discuss in the following section:(i) The expression (3.61) for the non-vanishing classical variation of the action has to berelated to quantum anomalies so that gauge invariance can be restored at the level ofthe quantum effective action, in analogy to the electric case described in section 2. Thiswill be done in section 4.1.(ii) The meaning of the new constraint (3.45) that was used to obtain (3.60) has to beclarified. This is subject of section 4.2.(iii) We have to show how the transformation (3.58), which also underlies the result (3.60),can be realized. This will be done in section 4.3.
In section 3, we discussed the algebraic constraints that were imposed on the embeddingtensor in ref. [42] and that allowed the construction of a gauge invariant Lagrangian withelectric and magnetic gauge potentials as well as tensor fields. Two of these constraints,(3.19) and (3.20), had a very clear physical motivation and ensured the closure of the gaugealgebra and the mutual locality of all interacting fields. The physical origin of the thirdconstraint, the representation constraint, (3.21), on the other hand, remained a bit obscure.In order to understand its meaning, we specialize it to its purely electric components, X (ΛΣΩ) = 0 . (4.1)Given that the components X ΛΣΩ generate axionic shift symmetries (remember the first termon the right hand side of (3.41)), we can identify them with the corresponding symbols X ΛΣΩ in section 2, and recognize (4.1) as the condition for the absence of quantum anomalies forthe electric gauge bosons (see (2.9)). It is therefore suggestive to interpret (3.21) as thecondition for the absence of quantum anomalies for all gauge fields (i.e. for the electric andthe magnetic gauge fields), and one expects that in the presence of quantum anomalies, thisconstraint can be relaxed. We will show that the relaxation consists in assuming that the22ymmetric tensor D MNP defined by (3.17) is of the form D MNP = d MNP , (4.2)for a symmetric tensor d MNP which describes the quantum gauge anomalies due to anomalouschiral fermions. In fact, one expects quantum anomalies from the loops of these fermions, ψ ,which interact with the gauge fields via minimal couplings¯ ψγ µ ( ∂ µ − A µ Λ Θ Λ α δ α − A µ Λ Θ Λ α δ α ) ψ . (4.3)Therefore, the anomalies contain – for each external gauge field (or gauge parameter) – anembedding tensor, i.e. d MNP has the following particular form: d MNP = Θ
M α Θ N β Θ P γ d αβγ , (4.4)with d αβγ being a constant symmetric tensor. In the familiar context of a theory with aflat scalar manifold, constant fermionic transformation matrices, t α , and the correspondingminimal couplings, the tensor d MNP is simply proportional to d MNP ∝ Θ M α Θ N β Θ P γ
Tr( { t α , t β } t γ } , (4.5)where the trace is over the representation matrices of the fermions. We showed that the generalization of the consistent anomaly (2.8) in a symplecticallycovariant way leads to an expression of the form (3.61) with the D MNP -tensor replaced by d MNP . Indeed, the constraint (4.2) implies the cancellation of this quantum gauge anomalyby the classical gauge variation (3.60). Note that it is necessary for this cancellation that theanomaly tensor d MNP is really constant (i.e., independent of the scalar fields). We expectthis constancy to be generally true for the same topological reasons that imply the constancyof d ΛΓΩ in the conventional electric gaugings [27, 28]. In this way we have already addressedthe first issue of the end of the previous section. We are now going to show how the constraint(4.2) suffices also to address the other two issues, (ii) and (iii). The possibility to impose a relation such as (4.2) is by no means guaranteed for all types of gauge groups(see e.g. [47] for a short discussion in the purely electric case studied in [26]). One might wonder how the magnetic vector fields A µ Λ can give rise to anomalous triangle diagrams, asthey have no propagator due to the lack of a kinetic term. However, it is the amputated diagram with internalfermion lines that one has to consider. .2 The new constraint We now comment on the constraint (3.45): X ( NP ) M Ω MQ X ( RS ) Q = 0 . (4.6)We will show that this equation holds if the locality constraint is satisfied, and (4.2) is imposedon D MNP with d MNP of the particular form given in (4.4). To clarify this, we introduce asin [42] the ‘zero mode tensor’ Z Mα = 12 Ω MN Θ N α , i.e. (cid:40) Z Λ α = Θ Λ α ,Z Λ α = − Θ Λ α . (4.7)One then obtains, using (3.18), the definition of X in (3.15) and (4.4) that X ( NP ) M = Z Mα ∆ αNP , (4.8)for some tensor ∆ αNP = ∆ αP N . Due to the fact that we allow the symmetric tensor D MNP in(3.17) to be non-zero and impose the constraint (4.2), this tensor ∆ αNP is not the analogousquantity called d αMN in [42] , but can be written as∆ αNP = ( t α ) N Q Ω P Q − d αβγ Θ N β Θ P γ . (4.9)However, the explicit form of this expression will not be relevant. We will only need that X ( NP ) M is proportional to Z Mα .Now we will finally use the locality constraint (3.20), which implies Z Λ[ α Z Λ β ] = 0 , i.e. Z Mα Z Nβ Ω MN = 0 . (4.10)This then leads to the desired result (4.6).The tensor Z Mα can be called zero-mode tensor as e.g. the violation of the usual Jacobiidentity (second line of (3.25)) is proportional to it. We now show that it also defines zeromodes of D MNR . Indeed, another consequence of the locality constraint is X MN P Ω MQ Θ αQ = 0 → X MN P Z Mα = 0 , X QM P Ω QS X SN R = 0 . (4.11) Note that the components of Ω MN have signs opposite to those of Ω MN as given in (3.7). We use ∆ αMN in this paper to denote the analogue (or better: generalization) of what was called d αMN in [42], because d αMN is reserved in the present paper to denote the quantity Θ M β Θ N γ d αβγ (cf eq. (4.20))related to the quantum anomalies. D MNR Z Rα = 0 . (4.12)Note that we did not need (4.2) to achieve this last result, but that the equation is consistentwith it. Finally, in this section we will justify the transformation (3.58), without requiring furtherconstraints on the D -tensor. That transformation gives an expression for X ( NP ) R ∆ B ρσNP that is not obviously a contraction with the tensor X ( NP ) R (due to the second term on theright hand side of (3.58)). We can therefore in general not assign a transformation of B ρσNP such that its contraction with X ( NP ) R gives (3.58). To overcome this problem, we will haveto change the set of independent antisymmetric tensors. The B µν MN cannot be consideredas independent fields in order to realize (3.58). We will, as in [42], introduce a new set ofindependent antisymmetric tensors, denoted by B µν α for any α denoting a rigid symmetry.The fields B µν NP and their associated gauge parameters Ξ NP appeared in the relevantformulae in the form X ( NP ) M B µν NP or X ( NP ) M Ξ NP , see e.g. in (3.27), (3.33), (3.35) and(3.44). With the form (4.8) that we now have, this can be written as X ( NP ) M B µν NP = Z Mα ∆ αNP B µν NP . (4.13)We will therefore replace the tensors B µν MN by new tensors B µν α using∆ αMN B µν MN → B µν α . (4.14)and consider the B µν α as the independent antisymmetric tensors. There is thus one tensorfor every generator of the rigid symmetry group. The replacement thus implies that X ( NP ) M B µν NP → Z Mα B µν α . (4.15)We also introduce a corresponding set of independent gauge parameters Ξ µ α through thesubstitution: ∆ αMN Ξ µMN → Ξ µ α . (4.16)This allows us to reformulate all the equations in the previous sections in terms of B µν α and25 µ α . For instance we will write: δA µM = D µ Λ M − Z Mα Ξ µ α , (4.17) H µν M = F µν M + Z Mα B µν α , (4.18) L top ,B = ε µνρσ Z Λ α B µν α (cid:16) F ρσ Λ + Z Λ β B ρσ β (cid:17) . (4.19)We will show that considering B µν α as the independent variables, we are ready to solvethe remaining third issue mentioned at the end of section 3. To this end, we first note thatall the calculations in section 3 remain valid when we use (4.15) and (4.17)-(4.19) to expresseverything in terms of the new variables B µν α and Ξ µ α , because the equations (3.45) and(3.47) we used in section 3 are now simply replaced by (4.10) and (4.12), respectively.If we now set, following (4.4), d MNP = Θ
M α d αNP , d αNP = d αβγ Θ N β Θ P γ , (4.20)then we can define (bearing in mind (4.8)) δB µν α = 2 D [ µ Ξ ν ] α + 2∆ α NP A [ µN δA ν ] P + ∆ B µν α , ∆ B µν α = − αNP Λ N G µν P + 3 d αNP Λ N ( H − G ) µν P , (4.21)to reproduce (3.58), where the left-hand side of (3.58) is replaced according to (4.15). Herethe covariant derivative is defined as D [ µ Ξ ν ] α = ∂ [ µ Ξ ν ] α + f αβγ Θ P β A [ µP Ξ ν ] γ . (4.22)Of course, (4.21) is only fixed modulo terms that vanish upon contraction with the embeddingtensor. In this section we have seen, so far, that it is possible to relax the representation constraint(3.21) used in ref. [42] to the more general condition (4.2) if one allows for quantum anomalies.The physical interpretation of the original representation constraint (3.21) of [42] is thus theabsence of quantum anomalies.Due to these constraints we obtained the equation (4.8), which allowed us to introducethe B µν α as independent variables. All the calculations of section 3.2 are then valid with thesubstitutions given in (4.15) and (4.16). We did not impose (4.8) in section 3.2, and thereforewe could at that stage only work with B µν NP . However, now we conclude that we need the B µν α as independent fields and will further only consider these antisymmetric tensors.26he results of this section can alternatively be viewed as a covariantization of the resultsof [18, 26] with respect to electric/magnetic duality transformations. To further check theconsistency of our results, we will in the next section reduce our treatment to a purely electricgauging and show that the results of [26] can be reproduced.
Let us first explicitly write down D MNP in its electric and magnetic components: D ΛΣΓ = X (ΛΣΓ) , D ΛΣΓ = X ΛΣΓ − X (ΣΓ)Λ , D ΛΣΓ = − X ΓΛΣ + 2 X (ΛΣ)Γ ,D ΛΣΓ = − X (ΛΣΓ) . (4.23)In the case of a purely electric gauging, the only non-vanishing components of the em-bedding tensor are electric: Θ M α = (Θ Λ α , . (4.24)Therefore also X Λ N P = 0 and (4.4) implies that the only non-zero components of D MNP = d MNP are D ΛΣΩ . Therefore, (4.23) reduce to D ΛΣΩ = X (ΛΣΩ) , X (ΣΩ)Λ = 0 , X ΩΛΣ = 0 . (4.25)The non-vanishing entries of the gauge generators are X ΛΣΓ and X ΣΩΛ = − X ΣΛΩ = X [ΣΩ]Λ ,the latter satisfying the Jacobi identities since the right hand side of (3.25) for M N QR allelectric indices vanishes. The X [ΣΩ]Λ can be identified with the structure constants of thegauge group that were introduced e.g. in (2.2). The X ΛΣΩ correspond to the shifts in (2.2).The first relation in (4.25) then corresponds to (2.9).The locality constraint is trivially satisfied and the closure relation reduces to (2.4) asexpected.At the level of the action L VT , all tensor fields drop out since, when we express everythingin terms of the new tensors B µν α , these tensors always appear contracted with a factorΘ Λ α = 0. In particular, the topological terms L top ,B vanish and the modified field strengthsfor the electric vector fields H µν Λ reduce to ordinary field strengths: H µν Λ = 2 ∂ [ µ A ν ]Λ + X [ΩΣ]Λ A µ Ω A ν Σ . (4.26) We have not discussed the complete embedding into N = 1 supersymmetry here, which would include allfermionic terms as well as the supersymmetry transformations of all the fields. This is beyond the scope ofthe present paper. X (CS)ΩΛΣ = X (m)ΩΛΣ .Finally, the gauge variation of L VT reduces to minus the ordinary consistent gauge anomaly,as we presented it in (2.8).This concludes our reinvestigation of the electric gauging with axionic shift symmetries,GCS terms and quantum anomalies as it follows from our more general symplectically covari-ant treatment. We showed that the more general theory reduces consistently to the knowncase of a purely electric gauging. G µν M For completeness, we will show in this section that G µν M (as defined in (3.39) and (3.42))is the object that transforms covariantly on-shell, rather than H µν M . We consider the totalaction (3.52), where now L top ,B is given by (4.19), and in L g . k . , the expression (4.18) isused. We write the general variation of this action under generic variations δA µM , δB µν α of A µM , B µν α . The variation of L g . k . has a contribution only from H Λ , since the matrix N is inert under variations of A µM and B µν α , and thus will be given by the first term in theexpression of δ L g . k . in (3.40). Summing this variation with the variation of the topologicalterms (3.51) we find: δ L V T = ε µνρσ (cid:2) G µν M D ρ δA σN Ω MN + ( H µν Λ − G µν Λ ) (cid:0) Z Λ α δB ρσ α − X ( NP )Λ A ρN δA σP (cid:1) − D MNP A µM δA ν N (cid:0) ∂ ρ A σP + X RSP A ρR A σS (cid:1)(cid:3) . (4.27)This allows us to determine the equations of motion for the independent tensor fields B µν α : δ L V T δB µν α ≈ ⇔ ( H − G ) µν Λ Z Λ α = ( H − G ) µν Λ Θ Λ α ≈ , (4.28)which tells us that the equations of motion imply that just some H µν Λ are identified on-shellwith the corresponding G µν Λ . More precisely, these are the tensors H µν Λ that are singled outby the contraction with Θ Λ α ; they thus correspond to those magnetic vectors A µ Λ that enterthe action. From (4.28), together with the constraint (4.2) and the particular form (4.4) for d MNP , we also see that ( H µν P − G µν P ) D P MN ≈ . (4.29)The properties (4.28) and (4.29) will be used next to prove that the tensor which is actuallyon-shell covariant under gauge-induced duality transformations is G µν M and not H µν M .Given the complete gauge variation for the antisymmetric tensor fields (4.21), we can write Identifications on shell are indicated by ≈ . H µν M and G µν M , which generalize thosefound in [42, 36] for D MNP = 0: δ H µν M = − Λ Q X QP M H µν P + Λ Q (cid:0) X ( QP ) M + Ω MN D NP Q (cid:1) ( H µν P − G µν P ) ,δ G µν Λ = − Λ Q X QP Λ G µν P + Λ Q ˆ X P Q Λ ( H µν P − G µν P ) ,δ G µν Λ = − Λ Q X QP Λ G µν P + 12 ε µνρσ I ΛΣ Λ Q ˆ X P Q Σ ( H ρσP − G ρσP )+ R ΛΣ Λ Q ˆ X P Q Σ ( H µν P − G µν P ) , (4.30)where we have used the following short-hand notation:ˆ X P QM ≡ X P QM + Ω MN D NP Q . (4.31)The first line of (4.30) follows from (3.37) and (3.58). The second transformation is a com-ponent of the first one since G µν Λ = H µν Λ , and for the transformation of G µν Λ we use (3.41).From (4.28) and (4.29) we see that, on-shell, the terms containing ( H µν P − G µν P ) ˆ X P QM vanish. Therefore we conclude that, as opposed to H µν M , the tensor G µν M is on-shell gaugecovariant and the gauge algebra closes on it modulo field equations. Consistency of courserequires that field equations transform into field equations, and indeed it can be shown that: δ ( H µν Λ − G µν Λ ) = Λ Q (cid:16) ˆ X P Q Λ + R ΛΣ ˆ X P Q Σ (cid:17) ( H µν P − G µν P )+ ε µνρσ I ΛΣ Λ Q ˆ X P Q Σ ( H ρσP − G ρσP ) . (4.32) Let us now briefly illustrate the above results by means of a simple example. We considera theory with a rigid symmetry group embedded in the electric/magnetic duality group Sp (2 , R ). The embedding in the symplectic transformations is given by t M N = (cid:32) − (cid:33) , t M N = (cid:32) (cid:33) , t M N = (cid:32) (cid:33) , (5.1)i.e. t = 1. Let us consider the following subset of duality transformations: S M N = δ M N − Λ P X P N M , with generators X P M N = (cid:32) X P (cid:33) , (5.2)29here Λ P is the rigid transformation parameter. The tensor X is related to the embeddingof the symmetries in the symplectic algebra using the embedding tensor, X P M N = (cid:88) α =1 Θ P α t αM N . (5.3)We have thus chosen the embedding tensorΘ P = 0 , Θ P = X P , Θ P = 0 . (5.4)We now want to promote S M N to be a gauge transformation, i.e., we take the Λ N = Λ N ( x )spacetime dependent and the X P M N are the gauge generators. This obviously correspondsto a magnetic gauging, as (4.25) is violated, and therefore requires the formalism that wasdeveloped in [42] and reviewed in section 3.2. The locality constraint (3.20) is automaticallysatisfied, as only the index value α = 2 appears, and closure of the gauge algebra spanned bythe X P M N requires that we impose (3.19), where only the right-hand side is non-trivial. Itrequires Θ = 0, and thus the only gauge generators that are consistent with this constraintare X P M N = ( X M N , X M N ) , with X M N = 0 , X M N = (cid:32) X (cid:33) . (5.5)Note that this choice still violates the original linear representation constraint (3.21), as (4.23)gives D = − X (cid:54) = 0. However, as we saw in section 3, this does not prevent us fromperforming the gauging with generators X P M N given in (5.5). We introduce a vector A µM which contains an electric and a magnetic part, A µ and A µ . Note that only the magneticvector couples to matter via covariant derivatives since the embedding tensor projects out theelectric part. In what follows, we also assume the presence of anomalous couplings betweenthe magnetic vector and chiral fermions. As we will now review, this justifies the nonzero X (cid:54) = 0, since it will give rise to anomaly cancellation terms in the classical gauge variationof the action. More precisely, we will have to require thatΘ = X , − X = d = ( X ) ˜ d , (5.6)where we introduced ˜ d as the component of d αβγ .To show this, we first introduce a kinetic term for the electric vector fields: L g . k . = e I H µν H µν − R ε µνρσ H µν H ρσ , (5.7)30here we introduced the modified field strength (4.18) H µν = 2 ∂ [ µ A ν ]1 + 12 X B µν , (5.8)which depends on a tensor field B µν and therefore transforms covariantly under δA µ = ∂ µ Λ + X A µ Λ − X Ξ µ ,δB µν = 2 ∂ [ µ Ξ ν ]2 + 4 A [ µ ∂ ν ] Λ − ∂ [ µ A ν ] 1 − Λ G µν ,δA µ = ∂ µ Λ . (5.9)This follows from (4.21) since the only nonzero component of ∆ MN is ∆ = 2 and for d MN we have only d = −
1. One can check that δ H µν = − X Λ ( H + G ) µν , with H µν = F µν = 2 ∂ [ µ A ν ]1 , G µν ≡ RH µν + 12 e I ε µνρσ H ρσ . (5.10)Under gauge variations, the real and imaginary part of the kinetic function transform asfollows (cf. (3.41)): δ I = 2Λ X RI , δ R = Λ X (cid:0) R − I (cid:1) . (5.11)Then it’s a short calculation to show that δ L g . k . = ε µνρσ Λ X G µν ∂ ρ A σ . (5.12)This is consistent with (3.43).In a second step, we add the topological term (4.19) L top ,B = ε µνρσ X B µν ∂ [ ρ A σ ] 1 . (5.13)The gauge variation of this term is equal to (up to a total derivative) δ L top ,B = − Λ X ε µνρσ ( ∂ µ A ν ) (2 ∂ ρ A σ + G ρσ ) . (5.14)The generalized Chern-Simons term (3.49) vanishes in this case. Combining (5.12) and (5.14),one derives δ ( L g . k . + L top ,B ) = − Λ X ( ∂ µ A ν ) ( ∂ ρ A σ ) ε µνρσ . (5.15)31his cancels the magnetic gauge anomaly whose form can be derived from (3.61), A [Λ] = − ε µνρσ Λ d ( ∂ µ A ν ) ( ∂ ρ A σ ) , (5.16)if we remember that X = − D = − d . Note that the electric gauge fields do notappear which corresponds to the fact that the electric gauge fields do not couple to the chiralfermions.A simple fermionic spectrum that could yield such an anomaly (5.16) is given by, e.g., threechiral fermions with canonical kinetic terms and quantum numbers Q = ( − , ( − , (+2)under the U (1) gauged by A µ . Indeed, with this spectrum, we would have Tr( Q ) = 0, i.e.,vanishing gravitational anomaly, but a cubic Abelian gauge anomaly d ∝ Tr( Q ) = +6. In this paper we have shown how general gauge theories with axionic shift symmetries, gen-eralized Chern-Simons terms and quantum anomalies [26] can be formulated in a way that iscovariant with respect to electric/magnetic duality transformations. This generalizes previouswork of [42], in which only classically gauge invariant theories with anomaly-free fermionicspectra were considered. Whereas the work [42] was modelling extended (and hence au-tomatically anomaly-free) gauged supergravity theories, our results here can be applied togeneral N = 1 gauged supergravity theories with possibly anomalous fermionic spectra. Suchanomalous fermionic spectra are a natural feature of many string compactifications, notablyof intersecting brane models in type II orientifold compactifications, where also GCS termsfrequently occur [18]. Especially in combination with background fluxes, such compactifica-tions may naturally lead to 4D actions with tensor fields and gaugings in unusual dualityframes. Our formulation accommodates all these non-standard formulations, just as ref. [42]does in the anomaly-free case.At a technical level, our results were obtained by relaxing the so-called representationconstraint to allow for a symmetric three-tensor d MNP that parameterizes the quantumanomaly. In contrast to the other constraints for the embedding tensor, this modified rep-resentation constraint is not homogeneous in the embedding tensor, which is a novel featurein this formalism. Also our treatment gave an interpretation for the physical meaning of the“representation” constraint: In its original form used in [42], it simply states the absenceof quantum anomalies. It is interesting, but in retrospect not surprising, that the extendedsupergravity theories from which the original constraint has been derived in [42], need thisconstraint for their internal classical consistency.It would be interesting to embed our results in a manifestly supersymmetric framework.Likewise, it would be interesting to study explicit N = 1 string compactifications within the32ramework used in this paper, making use of manifest duality invariances. Another topic wehave not touched upon are K¨ahler anomalies [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] in N = 1 supergravity or gravitational anomalies. We hope to return to some of these questionsin the future. Acknowledgments.
We are grateful to Bernard de Wit and Henning Samtleben for useful discussions. This workis supported in part by the European Community’s Human Potential Programme undercontract MRTN-CT-2004-005104 ‘Constituents, fundamental forces and symmetries of theuniverse’. The work of J.D.R. and A.V.P. is supported in part by the FWO - Vlaanderen,project G.0235.05 and by the Federal Office for Scientific, Technical and Cultural Affairsthrough the ‘Interuniversity Attraction Poles Programme – Belgian Science Policy’ P6/11-P.The work of J.D.R. has also been supported by a Marie Curie Early Stage Research Train-ing Fellowship of the European Community’s Sixth Framework Programme under contractnumber (MEST-CT-2005-020238-EUROTHEPHY). The work of T.S. and M.Z. is supportedby the German Research Foundation (DFG) within the Emmy-Noether Programme (Grantnumber ZA 279/1-2). 33 eferences [1] M. B. Green and J. H. Schwarz,
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