Covariant procedures for perturbative non-linear deformation of duality-invariant theories
aa r X i v : . [ h e p - t h ] M a r SU-ITP-11/41
Covariant procedures for perturbative non-linear deformationof duality-invariant theories
John Joseph M. Carrasco a , Renata Kallosh a , and Radu Roiban b a Stanford Institute for Theoretical Physics and Department of Physics,Stanford University, Stanford, CA 94305-4060, USA b Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
We analyze a recent conjecture regarding the perturbative construction of non-linear deformationsof all classically duality invariant theories, including N = 8 supergravity. Starting with an initialquartic deformation, we engineer a procedure that generates a particular non-linear deformation(Born-Infeld) of the Maxwell theory. This procedure requires the introduction of an infinite numberof modifications to a constraint which eliminates degrees of freedom consistent with the dualityand field content of the system. We discuss the extension of this procedure to N = 1 and N = 2supersymmetric theories, and comment on its potential to either construct new supergravity theorieswith non-linear Born-Infeld type duality, or to constrain the finiteness of N = 8 supergravity. PACS numbers: 04.65.+e, 11.15.Bt, 11.30.Pb, 03.50.-z, 03.50.De
I. INTRODUCTION
From our first and most familiar gauge-theory, classical electromagnetism, to the theoretical triumph of maximallysupersymmetric supergravity in four-dimensions, N = 8 supergravity [1], we have at our disposal examples of theorieswhose equations of motion respect a particularly constraining duality invariance: the rotation of the electric field (orits analog) into the magnetic field. Their covariant actions, however, must transform non-trivially for the classicalduality symmetry of the equations of motion to be preserved [2–4]. Introducing deformations of the action must beundertaken with a certain amount of care if one wishes to maintain this invariance. If one is able to consistently includesuch deformations, exciting generalizations of known theories are possible. Additionally, one would have the ability tointroduce counterterms that might otherwise seem to conflict with the known symmetries of duality-invariant theories.In this note we will discuss procedures which, starting from a classical action and quantum generated counterterms,allow us to construct a covariant effective action whose equations of motion are invariant under the same dualitytransformations as the classical action.Linear duality has been an integral part of supergravity theories since their beginning [5, 6]. Non-linear dualitymodels, where the action depends on quartic and higher order powers of vector fields, are well known for gaugetheories: these models are generalized Born-Infeld theories discovered in refs. [7, 8], with a supersymmetric versionlater constructed in [9]. Studied extensively in [2–4, 10–15], they have natural supersymmetric generalizations, asreviewed in refs. [3, 4]. Some attempts to construct the supergravity analog of the Born-Infeld models of non-linearduality have been made in N = 1 supergravity, see e.g. refs. [14, 15] but, as of yet, no models with non-linear dualityhave ever been constructed for N ≥ N = 8 supergravity in D = 4 are believed to be related, at least in part, to itsduality property, i.e. the symmetry of its equations of motion and Bianchi identities under E transformations. TheUV properties of N = 8 supergravity in D = 4 have long been studied, starting with the construction of candidate L -loop order counterterms for L ≥ R +( ∂F ) + R ( ∂F ) + · · · candidate counterterm [16, 18] was shown by explicit computations [19] to be absent. One set of explanations forthis is based on E symmetry [20–22]. E -invariant non-BPS candidate on-shell counterterms with non-linearsupersymmetry appear starting at the 8-loop order [16, 17] and a 1/8 BPS E candidate counterterm is availableat the 7-loop order [23].From a different perspective, it has been argued [24] that locality forbids all counterterms in the real light-conesuperspace; this provides an alternative explanation of the result of the three-loop computation and an argument infavor of all loop finiteness of N = 8 supergravity. Through a pure spinor worldline formalism, manifest maximalupersymmetry gives another explanation of the three-loop UV finiteness, but suggests a 7-loop four-dimensionaldivergence [25], similar to its string theory counterpart [26].Recently an argument for the all-loop order UV finiteness of perturbative N = 8 supergravity, in explanation ofobserved cancellations [19, 27, 28], was presented in ref. [29] based on the conservation of the Noether-Gaillard-Zumino(NGZ) E duality current [2]. As we will review in later sections, conservation of the duality current requires theaction to transform in a specific way. The argument of ref. [29] is based on the observation that a deformation of theclassical N = 8 supergravity action by an E -invariant counterterm leads to an action with different transformationproperties and thus to a violation of the E NGZ current conservation.It was suggested, however, by Bossard and Nicolai [30], based on previous work on dualities [31, 34], that there existprocedures which always allow a duality-consistent perturbative non-linear deformation of general theories – including N = 8 supergravity – which exhibit duality-invariant classical equations of motion. An elegant covariant procedureis described that allows a nonlinear deformation of classical electromagnetism through a modification of the linearvector field self-duality constraint. This constraint exists to eliminate degrees of freedom to comply with the fieldcontent of the theory and to avoid a double counting of vector fields. We find that this procedure, at least unmodified,does not reproduce another simple nonlinear deformation of classical electromagnetism: the Born-Infeld theory [7, 8].By actively expanding the known Born-Infeld deformation, we are able to a posteriori derive a procedure that doesreproduce it. We formulate a procedure general enough to find such deformations. For U (1) theories the deformationis external – i.e. it may be generated by interactions outside Maxwell’s theory. In interacting theories it is generatedby the interactions of the fields of the theory and may either be the result of finite or divergent counterterms. Theprocedure we propose has the potential to exclude counterterms that are incompatible with various expectations ofthe form of the final action.Extensive analysis suggests that manifestly duality-invariant local actions are not available in the presence of Lorentzinvariance . Manifestly duality-invariant actions with hidden Lorentz invariance were initially constructed for two-dimensional scalar fields in [35, 36] based on ideas described in [37] . The generalization of duality-symmetric actionsfor vector fields in four dimensions (as well as m -forms in d -dimentions) was explicitly discussed in [39]. While Lorentzinvariance of the manifestly duality-invariant actions is hidden, it emerges on shell at the classical level and, assumingabsence of anomalies, will also be visible at the level of the quantum scattering matrix. Thus, in such a formulation,the scattering matrix may be expected to be constrained by both manifest Lorentz and duality invariance. Analyzingthe duality invariance of the effective equations of motion of a covariant formulation of these theories, as we will doin this paper, may be interpreted as an intermediate step towards an analysis of the scattering matrix.Ref. [30] also proposes an explicit non-covariant construction of duality-invariant theories using the Henneaux-Teitelboim formulation [34, 38]. In our paper, for the examples limited to the non-linear deformations of the Maxwelltheory, we will also discuss the Hamiltonian approach to the problem which has a simple relation to the covariantsolution.We should spend a few words on terminology. Maxwell theories have no interaction, so the introduction of a non-linear deformation is, of course, a choice. In supergravity theories, on the other hand, “experimentally” identifiedcounterterms (i.e. counterterms arising from explicit calculations) may force deformations upon us. We will usethe word counterterm to specifically mean changes to the action necessitated by explicit calculation (or conjecturedexplicit calculation). In general the form of a given counterterm will not alone be sufficient to deform the action ina way consistent with the duality. The procedures discussed in this paper will generate from these counterterms afinal deformed action compatible with duality symmetries. In Maxwell theories, the role of supergravity countertermsis taken by initial deformation sources generated by external interactions. Analogously to supergravity theories, theprocedures discussed in later sections will take these initial sources and generate final deformed actions.The paper is organized as follows. In section II we introduce the simplest examples of duality invariant theories,Maxwell’s electromagnetism and two of its non-linear deformations. In section III , we introduce constraints designedto help make duality symmetry manifest, and which allow a framework for introducing deformation. In section IV However, Pasti-Soroki-Tonin actions [32] are available, which are Lorentz covariant and duality invariant due to a special choice of gaugesymmetries and a non-polynomial (e.g. inverse powers) dependence on auxiliary fields. In particular, there is an action of this kind withmanifest duality for maximally supersymmetric D=6 supergravity [33]. The ideas of [37] have also been used in [38] for the construction of actions for self-dual form fields in 2 mod 4 dimensions. In the context of the N = 8 supergravity, certain aspects of the E duality may be probed at the level of the scattering matrixthrough soft scalar limits [42].
2e introduce the necessary generalization to supergravity, and reproduce the procedure of ref. [30], for generatingnon-linear deformations but in notation we will find it easier to generalize from. In section V we derive the procedurerequired to introduce the Born-Infeld deformation. In section VI we discuss the applicability of these procedures ina supersymmetric context. We conclude in section VII. In appendix A we discuss duality in supergravity and inappendix B we present the Hamiltonian solutions of the duality invariant BN and BI models.
II. MAXWELL DUALITY-INVARIANT THEORIES
For an excellent review of duality rotations in non-linear electrodynamics, which in this section we follow closely,please see ref. [4]. We begin by considering perhaps the most familiar duality-invariant theory, classical electromag-netism in a vacuum. Maxwell’s equations are given ∂ t B = −∇ × E , ∇ · B = 0 (2.1) ∂ t D = ∇ × H , ∇ · D = 0in addition to relations between the electric field E , the magnetic field H , the electric displacement D , and themagnetic induction B . In a vacuum, D = E , and H = B . The Hamiltonian H = ( E + B ) and the equations ofmotion are invariant under rotations (cid:18) EB (cid:19) (cid:18) cos α − sin α sin α cos α (cid:19) (cid:18) EB (cid:19) . (2.2)Note that the Lagrangian, however, L = ( E − B ) is not invariant, for small rotations α one finds that it transformsas δ L = − α EB . (2.3)This suggests that non-linear deformations of L will require modifications which are also non-invariant. Indeed themost straightforward non-linear modification is the introduction of a chargeless medium. In such a medium we willnow have non-linear relations: D = D ( E , B ) H = H ( E , B ) . (2.4)It is convenient to continue the discussion more covariantly through the introduction of four-component notation.Quite generally, duality transformations may be realized in the path integral as a Legendre transform (see also,e.g. [11]). Given some Lagrangian L ( F ) depending only on the field strength of a vector field, one constructs e L ( F, G ) = L ( F ) − ǫ µνρσ F µν ∂ ρ ˜ A σ , (2.5)in which F is treated as a fundamental field. On the one hand, integrating out ˜ A σ one finds that F should obeythe Bianchi identity ǫ µνρσ ∂ ν F ρσ = 0, i.e. that F may be expressed in terms of a vector potential in the usual way.Plugging this into e L ( F, G ) one finds that it reduces to the original Lagrangian L ( F ). On the other hand, the classicalequations of motion for F require that G µν = ∂ µ ˜ A ν − ∂ ν ˜ A µ is related to F by˜ G µν = 2 ∂ L ( F ) ∂F µν , (2.6)through G µν = − ǫ µνρσ ˜ G ρσ , ˜ G µν = 12 ǫ µνρσ G ρσ . (2.7)The Lagrangian L D ( G ), dual to L ( F ), is obtained by eliminating F between equations (2.6) and (2.5). Regardlessof the form of the original Lagrangian, the Bianchi identity and the equations of motion of the original Lagrangian,expressed in terms of F and G , are ∂ µ ˜ F µν = 0 , ∂ µ ˜ G µν = 0 , (2.8)3nd are formally mapped into linear combinations of themselves by a GL (2) transformation. Further requiring thatthe transformed G may be obtained from the action evaluated on the transformed F though eq. (2.6) and that theresulting action is a deformation of Maxwell’s theory L = − F + O ( F ) restricts [4] the possible transformations to δ (cid:18) FG (cid:19) = (cid:18) B − B (cid:19) (cid:18) FG (cid:19) . (2.9)In other words, the duality transformation exchanges the Bianchi identity and the equations of motion of the originalLagrangian. The original Lagrangian is self-dual if L and L D have the same functional form. It is easy to check thatMaxwell’s theory, with L ( F ) = − F , is such a theory.In the derivation above, the dual field strength is determined by eq. (2.6) and is not an independent field. Sinceduality transformations (2.9) mix the field strength and its dual, it is convenient to interpret G as an independentfield and relate it to F by introducing constraint equations as we discuss in section III.For theories with n v vector fields the strategy for constructing the dual Lagrangian is unchanged. The equations ofmotion and the Bianchi identities remain of the form (2.8) but are invariant under a much larger set of transformations: δ (cid:18) FG (cid:19) = (cid:18) A BC D (cid:19) (cid:18) FG (cid:19) , (2.10) A T = − D B T = B C T = C (2.11)Here A, B, C, D are the infinitesimal parameters of the transformations, arbitrary real n × n matrices and the trans-formations (2.10) generate the Sp (2 n v , R ) algebra. For more general theories, when scalar fields are present, we wouldalso include a δφ ( A, B, C, D ).Consistency of the duality constraint can be expressed as requiring that the Lagrangian must transform underduality in a particular way, defined by the Noether-Gaillard-Zumino (NGZ) identity [2]. The NGZ current conservationrequires universally that for any duality group embeddable into Sp (2 n v , R ) δ L = 14 ( ˜ GBG + ˜
F CF ) . (2.12)This leads to the NGZ identity since the variation δ L ( F, φ ) can be computed independently using the chain rule andthe information about δF and δφ .For example, in the case of a U(1) duality (2.9), A = D = 0 , C = − B , (2.13)we see that eq. (2.12) reduces to δ L = ( ˜ GBG − ˜ F BF ). Taking into account that in the absence of scalars δ L ( F ) = ∂ L ( F ) ∂F µν δF µν = 12 ˜ GBG , (2.14)the NGZ identity which follows from (2.12) requires that12 ˜
GBG = 14 ( ˜
GBG − ˜ F BF ) . (2.15)In this case the NGZ identity simplifies to the following relation F ˜ F + G ˜ G = 0 . (2.16) Here we discuss theories with actions depending on the field strength F but not on its derivatives. When derivatives are present, ananalogous relation is given by a functional derivative over F of the action, see Appendix A. L ( F )
7→ L ( F, φ ) written in terms of the dual field strength e L ( F, φ ) = L ( F, φ ) − F ˜ G . Nowwe consider its invariance under duality transformations (2.10) and δφ . Annotating the transformed F, ˜ G as F ′ , ˜ G ′ ,and the transformed φ as a φ ′ , the invariance of this action implies that Z e L ( F, φ ) = S inv = S [ F ′ , φ ′ ] − Z F ′ ˜ G ′ = S [ F, φ ] − Z F ˜ G . (2.17)According to (2.10), (2.11) δ ( F ˜ G ) = ( AF + BG ) ˜ G + F ( C ˜ F + D ˜ G ) = ˜ GBG + ˜
F CF , (2.18)implying that S inv is invariant under the transformations (2.10), provided that (2.12) is satisfied.We may also present the NGZ identity as follows˜ G − F δ ˜ GδF = 4 δS inv δF , (2.19)which is just the derivative of the defining relation of S inv with respect to F under the assumption that there is somerelation between F and G . We can call it a “reconstruction identity” since it follows from the form of the action S = 14 Z F ˜ G + S inv (2.20)reconstructed using the duality symmetry. When the theory only has linear duality (e.g. only F terms in the action) δS inv /δF vanishes. So eqs. (2.20) and (2.19) tell us that any higher order dependence ( F , F etc.) must be part of S inv .The NGZ identity, in conjunction with eq. (2.6) can be solved to find G ( F ) and various Lagrangians providinga duality symmetry between equations of motion and Bianchi identities. We will discuss two cases of non-lineardeformations of the Maxwell theory for models depending only on F ’s without derivatives. A. Born-Infeld Lagrangian
The Born-Infeld Lagrangian, perhaps the most venerable non-linear deformation of Maxwell’s theory, is L BI = g − (1 − √ ∆) = − F + 132 g (cid:16) ( F ) + ( F ˜ F ) (cid:17) + · · · , (2.21)where g is the coupling constant, and ∆ = 1+2 g ( F / − g ( F ˜ F / . Using eq. (2.6), we find the following expressionfor G , G µν = − ǫ µνρσ ∂ L ( F ) ∂F ρσ (2.22)= 1 √ ∆ ( ˜ F µν + g
14 ( F ˜ F ) F µν ) . (2.23)A little algebra shows that the NGZ identity eq. (2.16) is readily verified and that the dual Lagrangian constructed asdescribed above has the same functional form as L BI . It is worth noting that classical electromagnetism correspondsto g → t = 14 F , z = 14 F ˜ F . (2.24)With these field variables, one can rewrite the Born-Infeld Lagrangian simply as L BI = g − (1 − p g t − g z ) , (2.25)5nd expand as L BI = − t + 12 g (cid:0) t + z (cid:1) − g t (cid:0) t + z (cid:1) + 18 g (cid:0) t + z (cid:1) (cid:0) t + z (cid:1) − g t (cid:0) t + z (cid:1) (cid:0) t + 3 z (cid:1) + · · · . (2.26)We continue the discussion of the BI case soon, but first we will discuss a distinct non-linear deformation ofelectromagnetism. While superficially complicated, this next deformation is, in fact, much easier to generate frompure Maxwell electrodynamics. Indeed we will see a tradeoff between the relative simplicity of the deformed action inthe BI case and the complicated initial deformation source required to generate it and the relative simplicity of theinitial deformation source which results in the superficially complicated action we will now present. B. Bossard-Nicolai Model
With the same variables t and z , one can write the following NGZ-consistent Lagrangian L BN = − t + 12 g (cid:0) t + z (cid:1) − g t (cid:0) t + z (cid:1) + 14 g (cid:0) t + z (cid:1) (cid:0) t + z (cid:1) − g t (cid:0) t + z (cid:1) (cid:0) t + 7 z (cid:1) + 132 g (cid:0) t + z (cid:1) (cid:0) t + 86 t z + 11 z (cid:1) − g t (cid:0) t + z (cid:1) (cid:0) t + 64 t z + 17 z (cid:1) + 164 g (cid:0) t + z (cid:1) (cid:0) t + 1517 t z + 623 t z + 43 z (cid:1) + · · · . (2.27)One simply keeps adding terms necessary so as to maintain the consistency eq. (2.16) order by order, specifically viaa procedure we will discuss in section III C. Unlike the Born-Infeld action, we do not know if this has a closed-formexpression. Note that this Lagrangian differs from L BI starting at O ( g ).It is not difficult to verify that eq. (2.16) is maintained order by order. Using, ˜ G = 2 ∂L∂F = ( ∂ t L ) F + ( ∂ z L ) ˜ F and G = − ( ∂ t L ) ˜ F + ( ∂ z L ) F , we can rewrite the NGZ identity as, (cid:0) ( ∂ t L ) − ( ∂ z L ) − (cid:1) z − (cid:0) ∂ z L )( ∂ t L ) (cid:1) t = 0 (2.28)Although the explicit Lagrangian eq. (2.27) is not provided in ref. [30], it is indeed the non-linear deformation ofclassical electrodynamics that is produced order by order as we will describe shortly. III. TWISTED SELF-DUALITY CONSTRAINTS
While the duality constraints are readily checked in the two above examples, BI and BN, note that, by hand, weforced a functional form of G in terms of F through eq. (2.6). The very act of doing so, prioritizing the primacy ofone over the other, makes the duality between F and G no longer manifest. We can avoid this by introducing whathas been called a “twisted self-duality” constraint – a constraint that guarantees that only one vector field from theduality doublet will ever be independent, but without establishing priority for one over the other. This constraintgeneralizes the equation (2.6), in that it can be considered more fundamental than the Lagrangian L which it, in fact,determines. The symmetry between F and G will only be broken by the solution to this constraint. A. Schr¨odinger’s BI Solution
In the Born-Infeld example, such a constraint was first found by Schr¨odinger in 1935 [8]. To describe Schr¨odinger’sconstruction in the form given in [11] it is useful to consider the duality symmetry in a complex basis where T = F − iG , T ∗ = F + iG , (3.1) Strictly speaking ref. [30] presents this model with negative g so as to generate a positive Hamiltonian, as discussed in appendix B. U (1) duality symmetry is δ (cid:18) F − iGF + iG (cid:19) = (cid:18) iB − iB (cid:19) (cid:18) F − iGF + iG (cid:19) . (3.2)Schr¨odinger suggested the following exact duality covariant cubic self-duality constraint T µν ( T ˜ T ) − ˜ T µν T = g T ∗ µν ( T ˜ T ) . (3.3)It is straightforward to verify that, if this constraint is solved perturbatively, one finds the unique Born-Infeld solutionof the NGZ identity T ˜ T ∗ = F ˜ F + G ˜ G = 0 . (3.4)And, even better, there is an action which is manifestly duality invariant [8, 11], L Sch ( T ) = 4 T ( T ˜ T ) , L Sch = −L ∗ Sch . (3.5)This fascinating Lagrangian is a ratio of two duality invariants T = ( F − iG ) = F − iF G − G , (3.6) T ˜ T = ( F − iG )( ˜ F − i ˜ G ) (3.7)= F ˜ F − iF ˜ G − G ˜ G .
The cubic constraint (3.3) is equivalent to the requirement that the derivative of the Schr¨odinger action L Sch ( T ) over T defines the conjugate ˜ T ∗ : ˜ T ∗ µν ≡ g − ∂ L Sch ∂T µν . (3.8)It follows that ∂ L Sch ∂T µν = 8 (cid:16) T µν T ˜ T ) − ˜ T µν T ( T ˜ T ) (cid:17) = g ˜ T ∗ µν . (3.9)Contraction with T µν demonstrates that (3.4) holds.To make contact with the supergravity formalism and the discussion in Appendix A, we introduce self-dual notation, T ± = ( T ± i ˜ T ) (3.10)such that T + µν T − µν = 0 and T ∗ = ( T ∗ ) + + ( T ∗ ) − ( T ∗ ) ± = ( T ∗ ± i ˜ T ∗ ) . (3.11)Recalling that ( ˜ T ) = − T , we have T − i ( T ˜ T ) = T ( T − i ˜ T ) = 2 T T − = 2( T − ) . (3.12)We can now rewrite the cubic self-duality constraint eq. (3.8) as T + µν ( T − ) + g
16 ( T ∗ ) + µν ( T ˜ T ) = 0 , (3.13)or T + µν ( T − ) − g T ∗ + µν (cid:16) ( T + ) − ( T − ) (cid:17) = 0 , (3.14)and the NGZ identity (2.16) is T ∗ + T + − T ∗− T − = 0 . (3.15)This formulation of the NGZ identity will be useful in later sections.7 . Maxwell Case Note that in the Maxwell case with g = 0 there is a particularly simple duality covariant linear twisted self-dualityconstraint G = ˜ F and F = − ˜ G , which in self-dual notation is T + = F + − iG + = 0 (3.16)and does indeed follow from the g → T + ) ∗ = F − + iG − = 0. Itshould be noted, however, that eq. (3.14) cannot be interpreted as a local perturbative deformation of (3.16). C. BN Case
In contrast, the model in eq. (2.27) which is consistent with NGZ identity satisfies a local deformation of (3.16), inwhich the right-hand side is modified as T + µν = g T ∗ + µν ( T − ) . (3.17)Using eqns. (2.6), (3.1), (3.10), and G + = ( G + i ˜ G )= ( F + i ˜ F )( ∂ z L + i∂ t L )= F + ( ∂ z L + i∂ t L ) , (3.18)we can translate eq. (3.17) back into constraints on derivatives of the action,0 = (1 + ∂ t L − i ∂ z L ) − g t − iz )(1 − ∂ t L − i ∂ z L ) (1 − ∂ t L + i ∂ z L ) . (3.19)Foreshadowing slightly – requiring analyticity of L for small values of F – one may introduce an ansatz in terms ofmonomials in g , t = F /
4, and z = F ˜ F / L = (cid:16) g − X m =0 ,p =0 g p +2 m ) c ( p, m ) t p z m (cid:17) − c (0 , g − , (3.20)and solve eq. (3.19) algebraically, order by order in g , fixing the constant coefficients c ( i,j ) . Doing so results in aLagrangian which satisfies the NGZ equation, and reproduces eq. (2.27).Indeed, as we will see, the covariant procedure proposed in ref. [30] is to modify the linear twisted self-dualityconstraint to a non-linear duality constraint by the introduction of a single deformation (or counterterm) as we justdid to go from eq. (3.16) to eq. (3.17). It so happens that in the cases studied in ref. [30], as with eq. (3.17), a singlesuch deformation was sufficient. We can see already, given the cubic nature of the BI constraint, that in general wewill require a procedure which introduces an infinite number of such deformations to the linear twisted self-dualityconstraint. Indeed the non-covariant procedure of Floreanini, Jackiw, Henneaux and Teitelboim [37, 38], discussed inref. [30] has the potential to allow an infinite amount of information. Ref. [30] seemed to constrain its constants ofintegration to explicitly reproduce the covariant procedure described above and more generally in section IV A. Thisneed not be so. The generalization of the covariant procedure discussed in section V can be arrived at non-covariantlyby allowing arbitrary constants of integration that satisfy the relevant NGZ relation. We have in fact verified thatthe Born-Infeld Hamiltonian can be obtained in this approach, see Appendix B. IV. BOSSARD-NICOLAI (BN) PROPOSAL
We start by explicitly providing an algorithm for the covariant procedure introduced in ref. [30]. We subsequentlyreview the provided supporting examples. 8 . Covariant BN procedure
Bossard and Nicolai posit [30] the existence of procedures which would allow the deformation of all classicallyduality invariant theories, including N = 8 supergravity. This proposal was worked out on three examples in ref. [30],and here we reconstruct the covariant procedure in detail.A convenient language for extended supergravities comes from the fact that any candidate counterterm woulddepend on the graviphoton . More specifically the counterterm would depend on the conjugate self-dual field strength T + AB and the anti-self-dual field strength T − AB . In the G/H coset space, AB are the indices of the antisymmetricrepresentation of the group H . For example, for N = 8 supergravity these would be SU (8) indices (in the 28-dimensional representation) and G/H is E /SU (8). For U (1) the deformation source depends on T ∗ + and T − . Inthis procedure, as with the generalized procedure we present in section V B, we will include the H -symmetry indices.The same procedures work for U (1) with the indices elided.One starts with an initial action S init with a conserved duality current and a manifestly duality-invariant countert-erm, or deformation, ∆ S . It is assumed that ∆ S can be expressed as a manifestly duality invariant function of F and G or, equivalently, on T + AB and T − AB . Classically T + AB = 0 is the linearized twisted self-duality constraint, whichwe will be deforming. The goal is to construct a Lagrangian L final that incorporates the counterterm/deformationyet still conserves the duality current. For the general case this means satisfying NGZ identity given in eqs. (A6)and (A7), and the simpler (2.16) for U (1). Of course, one should also require that it possesses the field content andother relevant symmetries of S init . The construction proceeds as follows:1. Take the variation of the counterterm with respect to the field-strength, and express as a function of T − , and T + which we will call the initial deformation source I (1) δ ∆ SδT + AB → δ I (1) ( T − AB , T + AB ) δT + AB (4.1)2. Constrain the self-dual field strength to the variation of this initial source: T + AB = δ I (1) ( T − AB , T + AB ) δT + AB (4.2)This is a modification of the linear twisted self-duality constraint T + AB =0.
3. Translate eq. (4.2) to a differential constraint on S final , c.f. section III C for the U (1) case.4. Introduce an ansatz for L final in terms of the Lorentz invariants, c.f. eq. (3.20), again for the U (1) case. Thiswill be more complicated, of course, for the generic case.5. Solve for the ansatz order by order in the coupling constant, at each step verifying the consistency of the relevantNGZ relation, the presence of additional desired symmetries of the target Lagrangian and enlarging the ansatzif one runs into an inconsistency.In contrast to ref. [30] we do not call I (1) the “initial deformation.” As we will see in the generalized procedure inorder to even recover the Born-Infeld action we will need to include an infinite number of terms to modify the covarianttwisted self-duality constraint. One can integrate those infinite deformations to achieve a final I BI , but this will notbe the final deformation of the action L Max − L BI , rather it is simply the complete source of the deformations to thelinear twisted self-duality constraint required to generate the BI deformation of the action through the generalizedprocedure. For consistency, then, we refer to I (1) as the initial deformation source. See eq. (A4) for definition of this particular combination of F and G and scalars for supergravities with scalars in the G/H coset space. When I (1) has only terms quadratic in T (as in U (1) and the toy model N = 8 examples of sec. 2 in ref. [30]), the right-hand side ofeq. (4.2) remains linear in T so the deformation of the linear constraint remains linear. . Three BN examples Two examples of the deformation of the linear twisted self-duality condition discussed in ref. [30] relate to Maxwellelectrodynamics and one to a toy model of N = 8 supergravity.The first example, from sec. 2 of ref. [30], is a Maxwell deformation analogous to an N = 8 supergravity counterterm.The deformation is quadratic in F , with derivatives of the Maxwell field, I (1) ∼ C ( dF ) . The dependence onderivatives necessitates the following deformed twisted self-duality constraint [41] δδF ( y ) Z d x ( ˜ GBG + ˜
F BF ) = 0 . (4.3)In this case G is linear in F and the action remains quadratic in F . The reconstruction is based on NGZ identity inthe form S = F ˜ G which is valid only for the actions quadratic in F when S inv = 0 in eqs. (2.17) and (2.19). As theresult of the deformation (4.2) the reconstructed action S ( F ) has some non-polynomial non-local terms required tocomplete the deformation in the action. This example, however, has linear duality since G remains a linear functionof F even with the deformation caused by I (1) ∼ C ( dF ) .A closely related example in sec. 2 is a toy model of an N = 8 supergravity deformation caused by the part ofthe three-loop counterterm which is quadratic in F and quadratic in Weyl curvatures. The quartic in F terms ( ∂F ) present in the N = 8 three-loop counterterm, C + ( ∂F ) + C ( ∂F ) + · · · , are not taken into account in this example.This example, therefore is also of the type given in eqs. (2.17) and (2.19) where S = R F Λ ˜ G Λ + S inv and δS inv δF = 0.In the toy model ˜ G remains a linear function of F , in absence of contribution to the right-hand side of eq. (2.19) from δS inv δF = 0, and therefore the linear duality of the classical action is preserved by deformation. Note that in the caseof linear duality the action is easily reconstructed, all dependence on vectors is in S vect = R F Λ ˜ G Λ and it satisfiesNGZ identity as explained in (2.18). Thus, this example also does not immediately shed light on cases of non-linearduality when the vector dependent part of S inv is present and contains ( ∂F ) terms which require the presence of allincreasing powers of F .In both examples of sec. 2 in ref. [30] a Lorentz covariant single term deformation of the undeformed constraint isemployed as shown in eq. (4.2).The third example is the deformation we discussed as the BN model earlier in section III C. Without derivativesin F , the manifestly U (1) invariant ‘initial’ deformation source, quartic in F , is used in the Lorentz covariant cubicdeformation of the linear constraint (4.2), and its equivalent Hamiltonian formulation. The proposed procedure isequivalent to the one worked out earlier: introduce the initial source, and then solve the twisted self-duality constraintfor a Lagrangian order by order by introducing an ansatz polynomial in the available Lorentz invariants.Any procedure must require that the deformed action, reconstructed using the deformed twisted self-duality con-straint (4.2), satisfies the relevant NGZ constraints (2.16). All examples considered in [30] have the nice property thatthe only input into the right-hand side of (4.2) is a term I (1) quadratic or quartic in field strengths, and they indeedsatisfy the relevant NGZ constraints: (4.3) in the case with derivatives and (2.16) in models without derivatives on F . No allowance is made, however, for cases when the solution of eq. (4.2) is inconsistent with direct higher-loopcalculations, as neither of the examples indicated the need for such a possibility.We will see that the Born-Infeld model requires the presence of an infinite set of deformations of the linear constraint(3.16). Instead of eq. (4.2), we will find that a general procedure will impose, T + AB = δ I (1) δT + AB + · · · + δ I ( n ) δT + AB + · · · = δ I ( T − AB , T + AB , g ) δT + AB , (4.4)where the various terms need not be related to the initial I (1) . In the following section we present a procedure thatsuccessfully reproduces the Born-Infield deformation. V. GENERALIZED COVARIANT PROCEDURE
First we present the procedure that we use to recover the Born-Infeld deformation in the BN framework, and seethat it does, indeed, require an infinite number of modifications to the linear twisted self-duality constraint. Learningfrom this example we modify the procedure of section IV A so as to handle the more general case.10 . Finding the Born-Infeld Deformation
We can begin by introducing an ansatz for the deformation source I ( T − , T ∗ + , g ) in terms of a series expansion, i.e. T + µν = g T ∗ + µν ( T − ) h X n =0 d n (cid:16) g ( T ∗ + ) ( T − ) (cid:17) n i , (5.1)where d n are the real parameters to be constrained so as to reproduce the Born-Infeld deformation. Since we arelooking to reproduce the BI Lagrangian, and we know it ahead of time, we may simply set L to eq. (2.21). It is notdifficult to check (by multiplying with T + and subtracting from the result the product between T − and the conjugateof (5.1)) that there exist solutions obeying the NGZ identity (3.15).As in section III C, we can translate eq. (5.1) into constraints on derivatives of the BI action using G + = F + ( ∂ z L + i∂ t L ),0 = (1 + ∂ t L − i ∂ z L ) + g t − iz )(1 − ∂ t L − i ∂ z L ) (1 − ∂ t L + i ∂ z L ) h X n =0 d n (cid:16) g ( t − iz )(1 − ∂ t L − i ∂ z L ) ( t + iz )(1 − ∂ t L + i ∂ z L ) (cid:17) n i . (5.2)We expand in a series of the coupling constant and solve for d n order by order. We indeed find an infinite serieswhich we can express as a generalized hypergeometric function so the BI twisted self-duality constraint can be given, T + µν = g T + µν ( T − ) F (cid:0) , , ; , ; − g ( T + ) ( T − ) (cid:1) . (5.3)Writing eq. (5.3) as T + µν = δ I ( T − , ¯ T + , g ) δ ¯ T + µν (5.4)we find that the required deformation source takes the following form I ( T − , T + , g ) = 6 g (cid:16) − F ( − , − , ; , ; − g ( T + ) ( T − ) ) (cid:17) (5.5)The procedure then for deforming to BI is to modify eq. (3.16) to eq. (5.3) and then to introduce an ansatz for theLagrangian to be solved for order by order. The resulting Lagrangian should be analytic for small values of the fieldstrength.We have therefore constructed (5.5) a deformation source I ( T − , ¯ T + , g ) which, like Schr¨odinger’s action L Sch ( T ) =4 T ( T ˜ T ) via eq. (3.8), yields a twisted self-duality constraint whose solution is the Born-Infeld action. The differencesbetween the two expressions are striking; moreover, while both are duality invariant, their natural variables and,consequently, the resulting deformed twisted self-duality constraints, (3.9) and (5.3), are different. This opens thepossibility that there may exist other deformations, different from them, which nevertheless generate the same duality-invariant action. It would be interesting to explore this possibility as well as the relation between these actions. B. Generalized Covariant Procedure
Thus, to reproduce a sufficiently general action with a conserved duality current, we must allow the countertermto be a general function of the coupling constant and duality invariants which is analytic for small values of fields.As before, we present this discussion in terms of graviphoton field strengths (see appendix A), but the U (1) examplesfollow by simply dropping the indices.We start with a duality conserving initial action S init , and a duality-invariant counterterm, or deformation, ∆ S .We assume, as BN, that ∆ S can be expressed as a function the conjugate self-dual field-strength T + AB . We wish toarrive at a Lagrangian L final that incorporates the counterterm yet still conserves the duality current. We proceed asfollows: 11. Take the variation of the counterterm with respect to the field-strength, and express as a function of T − , and T + , δ ∆ SδT + AB → δ I ( T − AB , T + AB , g ) δT + AB (5.6)2. Introduce an ansatz for the deformation source I ( T − AB , T + AB , g ). In general, this may be taken to depend onall possible duality invariants .3. Constrain the self-dual field strength to this variation: T + AB = δ I ( T − AB , T + AB , g ) δT + AB (5.7)4. Translate eq. (5.7) to a differential constraint on L final , c.f. section V A for the U (1) case. The differentialconstraint in general is more complicated, see (A6), (A7).5. Introduce an ansatz for L final which is analytic around the origin in terms of the Lorentz invariants. For thecase of U (1), again, this was not so difficult (eq. (3.20)), but in general this is unknown and can depend onother fields (e.g. scalars) in non-trivial ways.6. Solve for both the I ansatz parameters, as well as the Lagrangian ansatz parameters, order by order in thecoupling constant, enforcing the consistency of the relevant NGZ consistency equation (in U (1) case any of theeqs. (2.16), (3.15) or (2.28)), and additional desired symmetries of the target Lagrangian, enlarging the ansatzif one runs into inconsistency.The procedure given in section IV A is recovered by restricting to the lowest order term in the small g expansion of I . We also see that, at least for deformations of Maxwell’s theory, there are an infinite number of classical solutionsrecoverable by this procedure, consistent with the findings of ref. [10, 11] where it was shown that the NGZ identity(2.16) has infinitely many solutions.There exists the possibility that the counterterms generated by iterating on some first counterterm I (1) differ atsome loop level from counterterms discovered by explicit calculation. Unlike the original procedure, if the differ-ence is a duality invariant, our strategy can accommodate it by a suitable modification of δ I ( T − AB , T + AB , g ). Inthe supersymmetric context discussed in the next section this allows for complete supersymmetric invariants to beindependently included starting at some loop order higher than the one at which the first counterterm appears.It is important to note that in the U (1) case without derivatives and scalars, a hermitian deformation and manifestly U (1) invariant deformation I ( T − , T ∗ + , g ) guarantees that the NGZ equation is satisfied. Indeed, using (5.7) it is easyto see that T ∗ + δ I ( T − , T ∗ + , g ) δT ∗ + − T − δ I ( T − , T ∗ + , g ) δT − = T ∗ + T + − T ∗− T − = 0 . (5.8)This was manifestly the case for the deformation ansatz for any real choice of d n in eq. (5.1). This is in contrast tothe NGZ equations relevant for supergravity as we will discuss in appendix A. VI. NONLINEAR U (1) DUALITY AND SUPERSYMMETRY
The NGZ condition for U (1) duality invariance (2.16) has infinitely many solutions which are analytic for sufficientlysmall field strength [10, 11]. As we saw in earlier sections, the BN deformed self-duality constraint selects one such In the case of the non-linear U (1) duality we assumed that I is an analytic function of g ( T + ) ( T − ) . There is, however, in moregeneral theories no reason to forbid higher-order counterterms. In other words, if we have to worry about adding counterterms, wemight as well worry about adding all counterterms allowed by the known symmetries. E.g. for N = 8 supergravity we should at leastinclude in the ansatz all E invariants. A. N = 1 supersymmetric nonlinear electrodynamics Models with nonlinear U (1) duality and N = 1 supersymmetry are constructible in superspace, see [9] and [3, 4].The action is constructed from the standard (anti)chiral field-strength superfields W α = − D D α V , W ˙ α = − D D ˙ α V , (6.1)defined in terms of a real unconstrained prepotential V . The Bianchi identities D α W α = D ˙ α W ˙ α (6.2)are automatically satisfied. Similarly to the bosonic case, the dual (anti)chiral field strengths, M ˙ α and M α , are definedfrom the action S [ W, W ] as followsi M α [ W ] ≡ δδW α S [ W, W ] , − i M ˙ α [ W ] ≡ δδW ˙ α S [ W, W ] . (6.3)The equations of motion for the vector multiplet may be expressed in terms of M and M as D α M α = D ˙ α M ˙ α . (6.4)The superysmmetric generalization of the NGZ relation requires thatIm Z d xd θ (cid:16) W α W α + M α M α (cid:17) = 0 . (6.5)One may understand the structure of this relation by recalling that the bosonic NGZ relation is quadratic in fieldstrengths in addition to being invariant under the infinitesimal duality rotation δF = λG , δG = − λF . (6.6)The Bianchi identities (6.2) and the equations of motion (6.4) are therefore invariant under a similar transformationacting on W and M . Moreover, the supersymmetric NGZ identity eq. (6.5) is also invariant under this transformation.It is worth noting that this equation reduces to the bosonic NGZ relation eq. (2.16) upon setting the fermion andauxiliary fields to zero.The N = 1 Maxwell theory is a solution of eq. (6.5). To construct interacting theories which solve the supersym-metric NGZ relation one may start, following ref. [3], with a general action S = 14 Z d z W + 14 Z d ¯ z ¯ W + 14 Z d z W W Λ (cid:16) D W ,
18 ¯ D W (cid:17) (6.7)13arametrized by the real analytic function of one complex variable Λ( u, ¯ u ). Constructing the dual super-field strengths(6.3) it is not difficult to find that the NGZ constraint requires that Λ be a solution ofIm n ∂ u ( u Λ) − ¯ u ( ∂ u ( u Λ)) o = 0 . (6.8)This partial differential equation has infinitely many solutions, parametrized e.g. by the coefficients of the terms( u ¯ u ) n with n ≥ u = 0 (as well as the coefficient of u ¯ u ). This freedom is sufficient toaccommodate all the solutions of the bosonic deformed self-duality constraints discussed in earlier sections.Indeed, taking the integral over the fermionic superspace coordinates, and setting the gauginos and auxiliary fields to zero, we find L = −
12 ( u + ¯ u ) + u ¯ u Λ( u , ¯ u ) , u ≡ D W (cid:12)(cid:12) θ =0 ,D =0 ,ψ =0 = 14 F + i F ˜ F ≡ ω . (6.9)It is not difficult to see that it is possible to choose functions Λ such that this Lagrangian reproduces the two solutionsdiscussed explicitly in section II. The choice of Λ for the Born-Infeld Lagrangian, section II A, is well-known [3] L BI = 1 g n − p − det( η ab + gF ab ) o = 1 g h − r g ( ω + ¯ ω ) + 14 g ( ω − ¯ ω ) i , Λ BI = g g ( ω + ¯ ω ) + q g ( ω + ¯ ω ) + g ( ω − ¯ ω ) . (6.10)The Lagrangian obtained with the BN deformation, section II B, may be expressed in terms of ω as L = −
12 ( ω + ¯ ω ) + g ω ¯ ω − g ω ¯ ω ( ω + ¯ ω ) + g ω ¯ ω (( ω + ¯ ω ) + 2 ω ¯ ω ) (6.11) − g ω ¯ ω ( ω + ¯ ω ) (cid:0) ( ω + ¯ ω ) + 7 ω ¯ ω (cid:1) + g ω ¯ ω (cid:0) ( ω + ¯ ω ) + 16 ω ¯ ω ( ω + ¯ ω ) + 11( ω ¯ ω ) (cid:1) + . . . implying that Λ( ω, ¯ ω ) isΛ = 12 − g ω + ¯ ω ) + g ω + ¯ ω ) + 2 ω ¯ ω ) (6.12) − g
16 ( ω + ¯ ω ) (cid:0) ( ω + ¯ ω ) + 7 ω ¯ ω (cid:1) + g (cid:0) ( ω + ¯ ω ) + 16( ω + ¯ ω ) + 11( ω ¯ ω ) (cid:1) + . . . More generally, both the general deformation considered in eq. (5.1) and the function Λ have one free coefficient forevery fourth power of the field strength suggesting that there should exist a one to one map between the two functions.Thus, N = 1 supersymmetry does not seem to rule out any of the solutions with positive energy constructed usingeither section IV A or more generally section V B: for every such model one may easily find Λ (at least perturbatively)and thus construct an action in N = 1 superspace whose bosonic component reproduces the initial bosonic action.This result is not completely surprising; it was shown in [3] that all solutions of the bosonic NGZ equation havean N = 1 supersymmetric completion. Since all relevant solutions of the deformed self-duality constraint (5.7) aresolutions of the NGZ relation, the same conclusion must apply to them as well. B. N = 2 supersymmetric non-linear U (1) duality models While all actions constructed in earlier sections have an N = 1 supersymmetric extension, most of them do nothave a known extended supersymmetric counterpart. It may be also useful to recall here the results of [43, 44], namelythat the Born-Infeld action is unique in that it has 4 linearly realized and 4 nonlinearly realized supercharges. This is consistent, as the auxiliary fields alway appear squared after all supersymmetric covariant derivatives are evaluated in eq. (6.7). N = 2 global superspace is parametrized by Z A = ( x a , θ αi , ¯ θ i ˙ α ), with i = 1 , SU (2) R-symmetryindex. Actions describing the dynamics of N = 2 vector multiplets are written in terms of the (anti) chiral superfieldstrengths W and W which satisfy the Bianchi identities D ij W = D ij W . (6.13)They determine the superfield strength in terms of an unconstrained prepotential V ij . W = D D ij V ij , W = D D ij V ij , (6.14)where D is a chiral projector: D iα D U = 0 for any superfield U .As in the case of N = 1 supersymmetric models one may define, following [3], dual (anti) chiral superfields M and M as i M ≡ δδ W S [ W , W ] , − i M ≡ δδ W S [ W , W ] (6.15)in terms of which the equations of motion are D ij M = D ij M . (6.16)To construct the N = 2 analog of the NGZ relation we note that, similarly to the N = 1 setup, the Bianchiidentities (6.13) and the equations of motion (6.16) have the same functional form and are mapped into each otherby the infinitesimal U(1) duality transformations δ W = λ M , δ M = − λ W . (6.17)Considering the fact that the N = 2 NGZ identity should reduce to the equation (6.5) upon ignoring the fields inthe N = 1 chiral multiplet, we are left with [3] Z d Z (cid:16) W + M (cid:17) = Z d ¯ Z (cid:16) W + M (cid:17) (6.18)as the only possible N = 2 extension of (6.5). Solutions of this equation have not been easy to find. The free N = 2supersymmetric Maxwell action S free = 18 Z d Z W + 18 Z d Z W (6.19)satisfies this constraint. The one other known action obeying the constraint (6.18) was discovered by Ketov in [45].It is S = 14 Z d Z X + 14 Z d Z X , (6.20)where the chiral superfield X is a functional of W and W and is a solution of the constraint X = X D X + 12 W . (6.21)Upon solving the constraint (6.21), the action becomes [3, 14, 45, 46] S N =2 = S free + Z d x d θ W W Y ( D W , ¯ D W ) + O ( ∂ µ W ) (6.22)where Y is a Born-Infeld-type functional which in the N = 0 limit reduces to Λ BI ( ω, ¯ ω ) in eq. (6.10). The derivatives D ij and D ij are defined as D ij = D iα D jα and D ij = D i ˙ α D j ˙ α . N = 2 generalization of the Born-Infeld action. In N = 1language, the N = 2 vector multiplet splits into a vector and a chiral N = 1 multiplets. By truncating away thechiral multiplet the equations above correctly reproduce the system (6.7), (6.8) and (6.10).The extra terms with derivatives ∂ µ W appear to be required for N > N = 2 supersymmetry and are compatible withthe duality condition also have the structure of the BI action but exhibit additional terms containing space-timederivatives . They also share the property that they are associated with the D3-brane actions L D3 − brane =1 − q − det (cid:0) η ab + F ab + ∂ a ¯ ϕ∂ b ϕ (cid:1) . It was shown in [3] that an N = 2 self-dual action is given by S BI = S free + S int (6.23) S int = 18 Z d xd θ W W ( (cid:16) D W + D W (cid:17) (6.24)+ 14 (cid:16) ( D W ) + ( D W ) (cid:17) + 34 ( D W )( D W ) ) + 124 Z d Z ( W (cid:3) W + 12 ( W (cid:3) W ) D W + 12 ( W (cid:3) W ) D W + 148 W (cid:3) W ) + O ( W ) . The unique term with no fermionic or space-time derivatives, W W , yields the known F term of the Born-Infeldaction. The six-order terms, apart from W (cid:3) W terms with space-time derivatives, also correspond to the BI model.This action was confirmed in [14, 46].At the current level of understanding of N = 2 supersymmetric duality symmetric theories it is not clear yetwhat role will be played by the BN proposal to deform the twisted self-duality equation. The terms with space-timederivatives of the superfields are not likely to be generated by the initial deformation of self-duality equation, unlessone allows for deformations which contain derivatives of the field strength . VII. DISCUSSION
The question whether duality symmetries of equations of motion survive quantization and constrain the effectiveaction of the theory is very interesting and with far reaching implications both for gravitational and non-gravitationaltheories. A direct construction based on the classical Lagrangian and some number of (perhaps quantum generated)local counterterms would extend the tally of duality invariant theories and could shed light on the quantum propertiesof the theory. For supergravity theories in general and for N = 8 supergravity in particular it may constrain theexistence of higher-loop counterterms not immediately amenable to explicit calculations. In cases in which only theclassical equations of motion are invariant under duality transformations (while the action is not), the constructionis complicated by the fact that simply adding to the action a duality invariant counterterm leads [29] to duality-non-invariant deformed equations of motion and a non-conserved NGZ duality current.In ref. [30] a procedure, which we have broken into five-steps in section IV A, was suggested such that an actionexhibiting a conserved NGZ duality current is constructed if the procedure can be carried out. This directly followsif the first counterterm/deformation is manifestly duality invariant. The deformations discussed in ref. [30] areassumed to depend on fields transforming linearly under duality transformations; in supergravity theories they arethe vector fields. The action constructed following the BN procedure has infinitely many terms which, in the presence This state of affairs appears to be different from the statement [30] that the extension of the BN construction to a supersymmetricsetup does not encounter any difficulties. It is not clear to us whether this statement refers to minimal or extended supersymmetry. Inour discussion there is a fundamental difference between minimal and extended supersymmetry, the former accommodating indeed anysolution of the deformed self-duality equation. With such a deformation it possible that the resulting action is nonlocal (though perturbatively local), as demonstrated in [30] for thecase of and C ( dF ) deformation of maximal supergravity.
16f derivatives acting on the field strengths, may also be nonlocal though local order by order in a weak couplingexpansion.To understand and test this proposal we studied in detail a simple example – that of nonlinear electrodynamics.We found that, while an action can always be constructed, this action typically does not have desirable propertiesunless one assumes the existence of higher-order deformations of a specific form. In particular, using known resultsof supersymmetric nonlinear actions for abelian vector multiplets, we find that the Bossard-Nicolai action generatedby the first I (1) ∼ F deformation of the linear twisted self-duality constraint, may not have a supersymmetricgeneralization beyond N = 1 supersymmetry. To recover the known N = 2 actions of the BI type, the deformationof the linear twisted self-duality constraint must be modified to include all order terms I ( n ) ∼ F n . The generalizedconstruction, extending that of BN, is detailed in section V B. Moreover, for N > F µν . Therefore it is not clearwhat kind of deformed linear twisted duality constraint will provide the action consistent with N > T − , as follows from (3.14). The existenceof two twisted self-duality relations that yield the Born-Infeld action suggests it may be a general feature of thisconstruction of duality-invariant actions.Part of the motivation behind understanding the construction of actions exhibiting non-linear duality symmetriesis provided by applications to supergravity theories. In maximal four-dimensional supergravity it was shown fromseveral standpoints [16, 17], [22]-[26], that the first E duality-invariant potential counterterm may occur at 7 or8 loops. Supersymmetry considerations as well as the structure of scattering amplitudes of N = 8 supergravityimply that this counterterm necessarily contains terms quartic in vector fields. Assuming that the E dualitysymmetry should survive quantization, one is therefore to attempt to construct non-linear duality models withmaximal supersymmetry and with scalar field dependence which twists nontrivially the classical duality constraint.Such models have never been constructed before. Our generalization of the BN proposal, which accounts for knownmodels of non-linear duality, offers a wide pool of bosonic models among which there may exist one which admits amaximally supersymmetric completion. The nontrivial way in which a supersymmetric Born-Infeld action emergedfrom such an analysis makes it difficult to conclude, however, that such a model must exist and what is its precisestructure and relation to the first counterterm. Further detailed analysis is necessary to unravel this issue; along theway to maximal supersymmetry and supergravity we may find novel models of nonlinear duality which are interestingin their own right. Note added.
When this paper was finalized we were informed by G. Bossard and H. Nicolai that they have alsoworked out the Born-Infeld theory in the (Floreanini-Jackiw)-Henneaux-Teitelboim formulation. Such models are expected to contain arbitrary powers of the vector field strength. Presumably, these terms should be related to termsidentified in the analysis of [22] as required for having vanishing soft-scalar limits for multi-point S-matrix elements. cknowledgments We are grateful to Z. Bern, J. Broedel, G. Bossard, S. Ferrara, D. Freedman, A. Linde, H. Nicolai, D. Sorokin,M. Tonin and A. Tseytlin for stimulating discussions. We would especially like to thank S. Ferrara and A. Tseytlinfor insightful comments on an initial draft and S. Ketov and S. Kuzenko for their help in understanding the issuesof N = 2 supersymmetry and self-duality. This work is supported by the Stanford Institute for Theoretical Physics,NSF grants 0756174, PHY-08-55356, and the A.P. Sloan Foundation. Appendix A: Generalization to supergravity1. Duality and Supergravity
The action of the n vector fields of an N > L vectors = i N ΛΣ F − Λ F − Σ + h.c. , (A1)where N ΛΣ ( φ ) is a scalar field dependent symmetric matrix. The scalar fields φ parametrize a coset G/H with thetheory-specific duality group G and its subgroup H isomorphic to the R-symmetry group. For N = 8 supergravity G = E and H = SU (8). The self-duality constraint derived from (2.6) is twisted by this matrix and may bewritten either as a G covariant constraint G +Λ = N ΛΣ F +Λ , G − Λ = N ΛΣ F − Λ , (A2)or as an H covariant one T + AB = 0 , (A3)where T ± ≡ h Λ AB F ± Λ − f Λ AB G p Λ m (A4)and where the kinetic term matrix N ΛΣ ( φ ) is constructed out of the scalar field-dependent sections of an Sp (2 n v , R )bundle over the G/H coset space h Λ AB and f Λ AB ; they transform in an antisymmetric representation of H – see [2, 4, 40]for details. The equations (A3) are the supergravity analog of eq. (3.16).An infinitesimal Sp (2 n v , R ) transformation acts on a duality vector field doublet in a real representation exactly asgiven in eq. (2.10). Here, as there, A, B, C, D are the infinitesimal parameters of the transformations, arbitrary real n v × n v matrices satisfying (2.10). The vector kinetic matrix transforms projectively under Sp (2 n v , R ) N ′ = ( C + D N )( A + B N ) − . (A5)The case of the graviphoton in the absence of scalars and of additional vector fields, A = D = 0 and B = − C , the U (1) ∼ SO (2), follows the Maxwell discussion of section II identically.In N = 8 supergravity, for E , the NGZ identity requires that the following functional differential equation besatisfied δδF ( y ) (cid:16) δS − Z d x ( ˜ GBG + ˜
F CF ) (cid:17) = 0 , (A6)where δS is the variation of the action under E δS = δSδF δF + δSδφ δφ , (A7)and δF and δφ are the variations of vectors and scalars, respectively, under E . Here the E symmetry transfor-mations in the real basis for the doublet ( F, G ) are defined by an Sp (2 n, R ) embedding (cid:18) A BC D (cid:19) = (cid:18) ReΛ − ReΣ ImΛ + ImΣ − ImΛ + ImΣ ReΛ + ReΣ (cid:19) (A8)Λ are parameters of SU (8) and Σ are the SU (8)-orthogonal parameters of E , which control the familiar infinites-imal shift of scalars δφ = Σ + · · · . 18 . Modification of procedures The modification to the procedures of section IV A and section V B is actually quite minimal in terms of thealgorithms. What grows in complexity, which may be the reason there are no non-linear examples currently workedout in supergravity, is the complexity of the NGZ identity that must be maintained. In the N = 8 supergravity caseit is actually eq. (A6) which must be satisfied order by order. Appendix B: Born-Infeld and Bossard-Nicolai Hamiltonians In U (1) duality invariant models there is a simple relation between the Lagrangian and the Hamiltonian formu-lations [10, 11]. The NGZ constraint discussed above can be expressed as a differential equation with solutions,perturbative in g , codified in an arbitrary function of one real variable.The Lagrangian can be expressed in terms of t = F and on z = F ˜ F . We introduce the following (copious)notation to touch the (equally copious) literature x = p t + z , (B1) y = − z , (B2) Y = x , (B3) X = t . (B4)We can write same Hamiltonian as two different functional forms H ( X, y ) = V ( X, Y ). Similarly we can write thesame Lagrangian as two different functional forms L ( t, z ) = k ( t, x ).The nice relation between U (1) duality-conserving Lagrangians and Hamiltonians is simply L ( t, z ) = k ( t, x ) = − H ( X, y ) = − V ( X, Y ) . (B5)These represent general solutions of the differential equation,( ∂ t k ) − ( ∂ x k ) = 1 (B6)which is simply another way of writing the NGZ constraint, (c.f. eq. (2.28)).For example, for Maxwell and for Born-Infeld the respective functional forms are simply L Max ( t, z ) = − t L BI ( t, z ) = − g − (cid:16)p g t − g z − (cid:17) (B7) H Max ( X, y ) =
X H BI ( X, y ) = g − (cid:16)p g X + 2 g y − (cid:17) (B8) V Max ( X, Y ) =
X V BI ( X, Y ) = g − (cid:16)p g X + g X − g Y − (cid:17) (B9)For the BN model (see sections II B and III C), we have L BN ( t, z, g ) = − t + 12 g (cid:0) t + z (cid:1) − g t (cid:0) t + z (cid:1) + 14 g (cid:0) t + z (cid:1) (cid:0) t + z (cid:1) − g t (cid:0) t + z (cid:1) (cid:0) t + 7 z (cid:1) + 132 g (cid:0) t + z (cid:1) (cid:0) t + 86 t z + 11 z (cid:1) − g t (cid:0) t + z (cid:1) (cid:0) t + 64 t z + 17 z (cid:1) + 164 g (cid:0) t + z (cid:1) (cid:0) t + 1517 t z + 623 t z + 43 z (cid:1) + · · · . (B10)19t follows that V BN ( X, Y, g ) = X − g Y + 12 g X Y − g Y (cid:0) X + Y (cid:1) + 18 g X Y (cid:0) X + 7 Y (cid:1) − g Y (cid:0) X + 64 X Y + 11 Y (cid:1) + 18 g X Y (cid:0) X + 30 X Y + 17 Y (cid:1) − g Y (cid:0) X + 400 X Y + 494 X Y + 43 Y (cid:1) + · · · . (B11)The sign of g can be adjusted in the non-covariant procedure through a suitable choice for the first integrationconstant. Notice that when we make a choice g = −
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