The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity
aa r X i v : . [ h e p - t h ] D ec Preprint typeset in JHEP style - HYPER VERSION
The action principle and the supersymmetrisation ofChern-Simons terms in eleven-dimensional supergravity.
Bertrand Sou`eres and Dimitrios Tsimpis
Universit´e Claude Bernard (Lyon 1)UMR 5822, CNRS/IN2P3, Institut de Physique Nucl´eaire de Lyon4 rue Enrico Fermi, F-69622 Villeurbanne Cedex, FranceE-mail: [email protected] , [email protected] Abstract:
We develop computational tools for calculating supersymmetric higher-order deriva-tive corrections to eleven-dimensional supergravity using the action principle approach. We showthat, provided the superspace Bianchi identities admit a perturbative solution in the derivativeexpansion, there are at least two independent superinvariants at the eight-derivative order ofeleven-dimensional supergravity. Assuming the twelve-superforms associated to certain anoma-lous Chern-Simons terms are Weil-trivial, there will be a third independent superinvariant atthis order. Under certain conditions, at least two superinvariants will survive to all orders inthe derivative expansion. However only one of them will be present in the quantum theory: thesupersymmetrization of the Chern-Simons terms of eleven-dimensional supergravity required forthe cancellation of the M5-brane gravitational anomaly by inflow. This superinvariant can beshown to be unique at the eight-derivative order, assuming it is quartic in the fields. On theother hand, a necessary condition for the superinvariant to be quartic is the exactness, in τ -cohomology, of X , –the purely spinorial component of the eight-superform related by descentto the M5-brane anomaly polynomial. In that case it can also be shown that the solution ofthe Weil-triviality condition of the corresponding twelve-form, which is a prerequisite for theexplicit construction of the superinvariant, is guaranteed to exist. We prove that certain highlynon-trivial necessary conditions for the τ -exactness of X , are satisfied. Moreover any potentialsuperinvariant associated to anomalous Chern-Simons terms at the eight-derivative order mustnecessarily contain terms cubic or lower in the fields. ontents
1. Introduction 22. Cohomology in superspace 5 τ -cohomology 72.3 Spinorial cohomology 82.4 Pure-spinor cohomology 9
3. The action principle 10 O ( l ) correction (five derivatives) 12
4. The O ( l ) correction (eight derivatives) 13 τ -exactness of X
5. Discussion 19A. Weil triviality at l l l γ -matrices 36E. Eleven-dimensional superspace 38F. Tensor representation of a Young diagram 39 – 1 – . Introduction Eleven-dimensional supergravity [1] is believed to be the low-energy limit of M-theory [2], theconjectured nonperturbative completion of string theory. As such it is expected to receive aninfinite tower of higher-order corrections in an expansion in the Planck length or, equivalently,in the derivative expansion. At present such higher-order corrections cannot be systematicallyconstructed within M-theory, so one must resort to indirect approaches.One such approach is to calculate the higher-order corrections within perturbative string theory,in particular type IIA in ten dimensions, which is related to eleven-dimensional supergravity bydimensional reduction. The effective action of string theory can be systematically constructedperturbatively in a loop expansion in the string coupling, S eff = ∞ X g =0 g g − s Z d x √ G L g , (1.1)where g is the loop order (equivalently, the genus of the Riemann surface), g s is the stringcoupling constant, G is the spacetime metric and L g is the effective action at order g . Each L g admits a perturbative expansion in an infinite series of higher-order derivative terms. Moreover itis expected that each L g should correspond to an independent superinvariant in ten dimensions,see e.g. [3].The bosonic part of the tree-level effective action takes schematically the following form, L = L IIA + α ′ (cid:0) I ( R ) − I ( R ) + · · · (cid:1) + O ( α ′ ) , (1.2)where L IIA is the (two-derivative) Lagrangian of ten-dimensional IIA supergravity, and the el-lipses stand for terms which have not been completely determined yet. Unlike the case of N = 1superstrings, the first higher-derivative correction starts at order α ′ (eight derivatives). The I , I in (1.2) are defined as follows, I ( R ) = t t R + 12 ε t BR I ( R ) = − ε ε R + 4 ε t BR . (1.3)These were constructed in [4], to which we refer for further details, by directly checking invarianceunder part of the supersymmetry transformations. The terms in (1.3) linear in B are, up to anumerical coefficient, Hodge-dual to the Chern-Simons term B ∧ X [5, 6]. The eight-form X , see (4.2) below, is related by descent to the M5-brane anomaly polynomial and is a linearcombination of (tr R ) and tr R . Note that the Chern-Simons term drops out of (1.2).The R part of the tree-level effective action was determined in [7, 8] via four-graviton scatteringamplitudes and in [9, 10, 11] from the vanishing of the worldsheet beta-function at four loops.The NSNS sector of the four-field part of the effective action (common to all superstring theoriesin ten dimensions) was determined in [12]: it is captured by the simple replacement R → ˆ R ,– 2 –here ˆ R is a modified Riemann tensor with torsion which includes the NSNS three-form and thedilaton. The ε ε R term does not contribute to tree-level four-point scattering amplitudes,but gives a nonvanishing contribution to the five-graviton scattering amplitude. The completetree-level four-point effective action for type II superstrings was first determined in [13] and, inaddition to the NSNS sector, consists of terms of the form ( ∂F ) ˆ R and ∂ F , where F standsfor all RR flux.The superinvariant I can be further decomposed into two separate N = 1 superinvariants inten dimensions [4], I = − I a + 24 I b , where, I a = ( t + 12 ε B )(tr R ) + · · · I b = ( t + 12 ε B )tr R + · · · , (1.4)correspond to the supersymmetrization of the B ∧ (tr R ) and B ∧ tr R Chern-Simons termsrespectively. As we show in the following, if the uplift of I a , I b gives rise to two separatesuperinvariants in eleven dimensions, they will necessarily have to be cubic or lower in the fields.The one-loop effective action takes the following form [7, 14], L = α ′ (cid:0) I ( R ) + 18 I ( R ) + · · · (cid:1) + O ( α ′ ) . (1.5)In particular we see that in this case the Chern-Simons term does not drop out, cf. (1.3). Theellipses above indicate terms which are not completely known, although partial results existthanks to five- and six-point amplitude computations [15, 16, 17, 18]. Contrary to the tree-levelsuperinvariant L which is suppressed at strong coupling, the uplift of the one-loop superinvariant L is expected to survive in eleven dimensions, and thus to be promoted to an eleven-dimensionalsuperinvariant. We will refer to the latter as the supersymmetrization of the Chern-Simons term C ∧ X , the uplift of the ten-dimensional Chern-Simons term, where C is the three-form potentialof eleven-dimensional supergravity.An argument of [19], which we review in the following, guarantees that if the supersymmetrizationof the Chern-Simons term is quartic or higher in the fields, then it is unique at the eight-derivativeorder . The uniqueness of this superinvariant is also supported by the results of [20, 21, 22] whichuses the Noether procedure to implement part of the supersymmetry transformations of eleven-dimensional supergravity. The results of these references constrain the supersymmetrization ofthe Chern-Simons term to be of the form,∆ L = l (cid:0) t t R − ε ε R − ε t CR + R G + · · · (cid:1) + O ( l ) , (1.6) Note that [12] contains an error that has unfortunately caused some confusion in the literature: the expansionof the t t R terms of eq. (2.11) in that reference indeed has the form of the term in the square brackets on theright-hand side of eq. (2.13) therein. However, if one replaces R by the modified Riemann tensor ˆ R , given ineq. (2.12) therein, eq. (2.13) no longer gives the correct expansion of t t ˆ R . The existence of independent superinvariants starting at order higher than eight in the derivative expansionwill of course spoil the uniqueness of the superinvariant at higher orders. – 3 –here l is the Planck length. The ellipses indicate terms which were not determined by theanalysis of [20, 21, 22], while the R G terms were only partially determined. The reductionof the above to ten dimensions is consistent, as expected, with the one-loop IIA superinvariant(1.5). In addition the quartic interactions R ( ∂G ) and ( ∂G ) were determined in [23] by eleven-dimensional superparticle one-loop computations in the light cone, and in [24, 25, 26] by adifferent method which uses tree amplitudes instead. The t t R terms have also been obtainedby four-graviton one-loop amplitudes in eleven dimensions [27, 28], while it can be shown [29]that higher loops do not contribute to the superinvariant (1.6).In the present paper we reexamine the problem of calculating supersymmetric higher-orderderivative corrections to eleven-dimensional supergravity from the point of view of the actionprinciple approach. This method relies on the superspace formulation of the theory and isparticularly well suited to the supersymmetrization of Chern-Simons terms. Given an eleven-dimensional Chern-Simons term there is an associated gauge-invariant twelve-superform obtainedby exterior differentiation. The action principle approach can be carried out provided the twelve-form is Weil-trivial, i.e. exact on the space of on-shell superfields. Computing the superinvariantthen boils down to explicitly solving the Weil-triviality condition for the twelve-form.We show that, provided the superspace Bianchi identities admit a perturbative solution inthe derivative expansion, there will be at least two independent superinvariants at the eight-derivative order. If we also assume that the twelve-superforms associated to the anomalous (inthe presence of an M5-brane) Chern-Simons terms, C ∧ (Tr R ) and C ∧ Tr R , are separatelyWeil-trivial, there will be a third independent superinvariant at this order. Moreover we arguethat, under certain conditions, at least two of the superinvariants should be expected to surviveto all orders in the derivative expansion. However only one of those would correspond to thesupersymmetrization of C ∧ X , cf. (1.6).As already noted this superinvariant can be shown to be unique, assuming it is quartic in thefields. On the other hand, a necessary condition for the superinvariant to be quartic is theexactness, in the so-called τ -cohomology, of X , –the purely spinorial component of X . In thatcase we also show that the solution of the Weil-triviality condition of the corresponding twelve-form is guaranteed to exist. Proving the τ -exactness of X , is the first and arguably mostdifficult step in obtaining the explicit solution to the Weil-triviality condition of the twelveform,and therefore constructing the superinvariant using the action principle.To tackle this computationally intensive problem we have built on the computer program [30],to supplement it, among other things, with functionalities related to Young tableaux [31]. Bya combination of calculational techniques involving the implementation of Fierz identities andYoung tableaux projections we prove that certain highly non-trivial necessary conditions forthe τ -exactness of X , are satisfied. As a corollary of our work, it follows that any potentialsuperinvariant associated to the anomalous Chern-Simons terms, C ∧ (Tr R ) and C ∧ Tr R ,must necessarily contain terms cubic or lower in the fields. There is disagreement between [23] and [25] concerning part of the ( ∂G ) terms. – 4 –he plan of the rest of the paper is a follows. In section 2 we review the different superspacecohomologies that will be useful in the following. In section 3 we introduce the action principleapproach and in section 3.1 we show how to obtain the eleven-dimensional supergravity of [1]in this framework. In section 3.2 we apply the action principle to derive the five-derivativecorrection. Section 4 considers the eight-derivative correction. In section 4.1 we examine thenumber of independent superinvariants at the eight-derivative order. Section 4.2 addresses theproblem of the τ -exactness of X , . In section 4.3 we discuss the conditions for the existenceof the superinvariants to all orders in the perturbative expansion. We conclude in section 5.Further technical details are included in the appendices.
2. Cohomology in superspace
In this section we review the various superspace cohomology groups that will be useful in thefollowing. This is not new material, but we are including it here to make the paper self-containedand for the benefit of the readers who may not be familiar with the relevant literature.Let us start by explaining our conventions: Eleven-dimensional superspace [32, 33] consistsof eleven even (bosonic) and thirty-two odd (fermionic) dimensions, with structure group theeleven-dimensional spin group. Let A = ( a, α ) be flat tangent superindices, where a = 0 , . . . α = 1 , . . .
32 is a Majorana spinor index. Curved superindiceswill be denoted by M = ( m, µ ), with the corresponding supercoordinates denoted by Z M =( x m , θ µ ). The supercoframe is denoted by E A = ( E a , E α ) while the dual superframe is denotedby E A = ( E a , E α ). We can pass from the coframe to the coordinate basis using the supervielbein, E A = d z M E M A .We shall assume the existence of a connection one-form Ω AB with values in the Lie algebra ofthe Lorentz group. In particular this implies that,Ω ( ac η b ) c = 0 , Ω αβ = 14 ( γ ab ) αβ Ω ab , Ω aβ = 0 = Ω αb . (2.1)The associated supertorsion and supercurvature tensors are then given by: T A = DE A := d E A + E B ∧ Ω BA = 12 E C ∧ E B T BC A R AB = dΩ AB + Ω AC ∧ Ω C B = 12 E D ∧ E C R CDAB , (2.2)where the exterior derivative is given by d = d z M ∂ M . The assumption of a Lorentzian structuregroup implies that the components of the curvature two-form obey a set of equations analogousto (2.1). The super-Bianchi identities (BI) for the torsion and the curvature, DT A = E B ∧ R BA ,DR AB = 0 , (2.3)– 5 –ollow from the definitions (2.2). Moreover, a theorem due to Dragon [34] ensures that for aLorentz structure group the second BI above follows from the first and need not be consideredseparately. Once constraints are imposed the BI cease to be automatically satisfied. As wasshown in [33], by imposing the conventional constraint T cαβ = iγ cαβ , (2.4)and solving the torsion BI, one recovers ordinary eleven-dimensional supergravity. In particularone determines in this way all components of the torsion. In addition one can construct a closedsuper four-form G and a super seven-form G obeying [35, 36], d G = 0 , d G + 12 G ∧ G = 0 , (2.5)whose bosonic components correspond to the eleven-dimensional supergravity four-form and itsHodge-dual, respectively: G m ...m = ( ⋆G ) m ...m . (2.6)The solution of the eleven-dimensional superspace BI is reviewed in appendix E. Let Ω n be the space of n -superforms. Thanks to the nilpotency of the exterior superderivative,one can define de Rham cohomology groups in superspace in the same way as in the case ofbosonic space, H n = { ω ∈ Ω n | d ω = 0 } / { ω ∼ ω + d λ, λ ∈ Ω n − } . (2.7)The fact that the topology of the odd directions is trivial means that the de Rham cohomologyof a supermanifold coincides with the de Rham cohomology of its underlying bosonic manifold,also known as the body of the supermanifold. In the remainder of the paper we shall assumethat the body has trivial topology. This is the simplest type of supermanifold, sometimes calleda graded manifold . It implies in particular that every d-closed superform is d-exact.There is an important caveat to the previous statement: it is only valid when the cohomologyis computed on the space of unconstrained superfields. Once constraints are imposed it ceasesto be automatically satisfied. Adopting the terminology of [37], we shall call Weil-trivial thosed-closed superforms which are also d-exact on the space of constrained (also referred to as “on-shell”, or “physical”) superfields. The cohomology groups computed on the space of constrainedsuperfields will be denoted by H n (phys), as in [19]. As already emphasized, there is no a priorireason why H n (phys) should coincide with the cohomology of the body of the supermanifold. The G BI receives a correction at the eight-derivative order, cf. (4.3) below. – 6 – .2 τ -cohomology The space of superforms can be further graded according to the even, odd degrees of the forms.We denote the space of forms with p even and q odd components by Ω p,q so that,Ω n = ⊕ X p + q = n Ω p,q . (2.8)A ( p, q )-superform ω ∈ Ω p,q can be expanded as follows, ω = 1 p ! q ! E β q . . . E β E a p . . . E a ω a ...a p β ...β q . (2.9)In the following we will use the notation Φ ( p,q ) ∈ Ω p,q for the projection of a superform Φ ∈ Ω n onto its ( p, q ) component.The exterior superderivative, d : Ω p,q → Ω p +1 ,q + Ω p,q +1 + Ω p − ,q +2 + Ω p +2 ,q − , when written outin this basis will give rise to components of the torsion as it acts on the coframe. Following [38]we split d into its various components with respect to the bigrading,d = d b + d f + τ + t , (2.10)where d b , d f are even, odd derivatives respectively, such that d b : Ω p,q → Ω p +1 ,q , d f : Ω p,q → Ω p,q +1 . The operators τ and t are purely algebraic and can be expressed in terms of the torsion.Explicitly, for any ω ∈ Ω p,q we have,(d b ω ) a ...a p +1 β ...β q = ( p + 1) (cid:16) D [ a ω a ...a p +1 ] β ...β q + p T [ a a | c ω c | a ...a p +1 ] β ...β q + q ( − p T [ a | ( β | γ ω | a ...a p +1 ] γ | β ...β q ) (cid:17) (d f ω ) a ...a p β ...β q +1 = ( q + 1) (cid:16) ( − p D ( β | ω a ...a p | β ...β q +1 ) + q T ( β β | γ ω a ...a p γ | β ...β q +1 ) + p ( − p T ( β | [ a | c ω c | a ...a p ] | β ...β q +1 ) (cid:17) ( τ ω ) a ...a p − β ...β q +2 = 12 ( q + 1)( q + 2) T ( β β | c ω ca ...a p − | β ...β q +2 ) ( tω ) a ...a p +2 β ...β q − = 12 ( p + 1)( p + 2) T [ a a γ ω a ...a p +2 ] γβ ...β q − . (2.11)– 7 –he nilpotency of the exterior derivative, d = 0, implies the following identities: τ = 0d f τ + τ d f = 0d f + d b τ + τ d b = 0d b d f + d f d b + τ t + tτ = 0d b + d f t + t d f = 0d b t + t d b = 0 t = 0 . (2.12)The first and the last of these equations are algebraic identities and are always satisfied. On theother hand, as a consequence of the splitting of the tangent bundle into even and odd directions,the remaining identities are only satisfied provided the torsion tensor obeys its Bianchi identity.The first of the equations in (2.12), the nilpotency of the τ operator, implies that we can considerthe cohomology of τ , as first noted in [38] (see also [35] for some related concepts). Explicitlywe set, H p,qτ = { ω ∈ Ω p,q | τ ω = 0 } / { ω ∼ ω + τ λ, λ ∈ Ω p +1 ,q − } . (2.13)As in the case of de Rham cohomology, one could make a distinction between cohomology groupscomputed on the space of unconstrained superfields and those computed on the space of physicalfields.Suppose now that the conventional constraint (2.4) is imposed so that τ reduces to a gammamatrix. It was conjectured in [19], consistently with the principle of maximal propagation of[39], that in this case the only potentially nontrivial τ -cohomology appears as a result of theso-called M2-brane identity, ( γ a ) ( α α ( γ ab ) α α ) = 0 . (2.14)Explicitly, for p = 0 , ,
2, one may form the following τ -closed ( p, q )-superforms, ω α ...α q = S α ...α q ; ω aα ...α q = ( γ ab ) ( α α P bα ...α q ) ; ω abα ...α q = ( γ ab ) ( α α U α ...α q ) , (2.15)with S , P , U , arbitrary superfields. It can be seen using (2.14) that the forms ω above cor-respond to nontrivial elements of H p,qτ with p = 0 , ,
2. The conjecture of [19] means that allnontrivial cohomology is thus generated, and that all H p,qτ groups are trivial for p ≥
3. This wassubsequently proven in [40] and [41, 42, 43, 44].
Following [19], let us now define a spinorial derivative d s which acts on elements of τ -cohomology,d s : H p,qτ → H p,q +1 τ . For any ω ∈ [ ω ] ∈ H p,qτ we set,d s [ ω ] := [d f ω ] . (2.16)– 8 –o check that this is well-defined, one first shows that d f ω is τ -closed, τ d f ω = − d f τ ω = 0 , (2.17)where we used the second equation in (2.12). Morever d s [ ω ] is independent of the choice ofrepresentative, [d f ( ω + τ λ )] = [d f ω − τ d f λ ] = [d f ω ] . (2.18)Furthermore it is simple to check that d s = 0,d s [ ω ] = d s [d f ω ] = [d f ω ] = − [(d b τ + τ d b ) ω ] = 0 , (2.19)where we took into account the third equation in (2.12). We can therefore define the correspond-ing spinorial cohomology groups H p,qs as follows, H p,qs = { ω ∈ H p,qτ | d s ω = 0 } / { ω ∼ ω + d s λ, λ ∈ H p,q − τ } . (2.20)The notion of spinorial cohomology was originally introduced in [45, 39] and applied in a seriesof papers with the aim of computing higher-order corrections to supersymmetric theories [46,47, 48, 49, 50], and more recently in [51, 52, 53]. The spinorial cohomology as presented abovewas introduced in [19] and is independent of the value of the dimension-zero torsion. It reducesto the spinorial cohomology of [45, 39] upon imposing the conventional constraint (2.4). It was first pointed out by P. Howe [54] and subsequently elaborated in [19], that in the casewhere the dimension-zero torsion is flat, cf. (2.4), the cohomology groups H ,qs are isomorphic toBerkovits’s pure-spinor cohomology groups [55]. Therefore, in view of what was said in section2.3, the latter are also isomorphic to the spinorial cohomology groups that had been computeda few months earlier in [39]. In the following we briefly explain the equivalence between the twoformulations.The pure spinor cohomology groups are defined as follows. Consider an eleven-dimensional purespinor, λ α , `a la Berkovits, i.e. such that it obeys, λ α γ aαβ λ β = 0 . (2.21)The pure spinor λ α is assigned ghost number one. Furthermore we define a form of ghost number q as a multi-pure spinor, ω = λ α . . . λ α q ω α ...α q . (2.22)Note that the above definition implies that ω ∈ [ ω ] ∈ H ,qτ : indeed shifting ω α ...α q by a τ -exactterm would drop out of the right-hand side above due to the contractions with the pure spinors;moreover ω α ...α q is trivially τ -closed. This definition is different from an eleven-dimensional pure spinor `a la Cartan, used in [56], which obeys λ α γ abαβ λ β = 0 in addition to (2.21). – 9 –he pure-spinor BRST operator is defined as follows, Q := λ α D α , (2.23)where D α is the spinor component of the covariant derivative defined in flat superspace. Thereforethe action of Q on omega, Qω = λ α . . . λ α q λ α q +1 D α q +1 ω α ...α q , (2.24)corresponds precisely to the action of d s defined in (2.16). Indeed, for flat superspace the torsionterms drop out and d f reduces to D α , cf. the second line of eq. (2.11). Moreover the contractionwith the pure spinors on the right-hand side above implies that Qω ∈ [ Qω ] ∈ H ,q +1 τ , for the samereasons noted below (2.22). In other words, in the linearized limit the pure-spinor cohomologygroups of ghost number q are isomorphic to the spinorial cohomology groups H ,qs . For anextended review of pure-spinor superfields, see [57].
3. The action principle
The action principle , also known as ectoplasmic integration , is a way of constructing superin-variants in D spacetime dimensions as integrals of closed D -superforms [58, 59]. Indeed if L isa closed D -superform, the following action is invariant under supersymmetry, S = 1 D ! Z d D x ε m ...m D L m ...m D | , (3.1)where a vertical bar denotes the evaluation of a superfield at θ µ = 0. This can be seen asfollows. Consider an infinitesimal super-diffeomorphism generated by a super-vector field ξ . Thecorresponding transformation of the action reads, δL = L ξ L = (d i ξ + i ξ d) L = d i ξ L , (3.2)where we took into account that L is closed. On the other hand, local supersymmetry transfor-mations and spacetime diffeomorphisms are generated by ξ | and, in view of (3.2), the integrandin (3.1) transforms as a total derivative under such transformations. The action is thus invariantassuming boundary terms can be neglected.This method is particularly well-suited to actions with Chern-Simons (CS) terms and indeedhas been used to construct all Green-Schwarz brane actions [60, 61], see [62, 63] for more recentapplications to other theories and [64] for applications to higher-order corrections. The idea isas follows: let Z D be the CS term and W D +1 = d Z D its exterior derivative. Obviously W D +1 isa closed form. On the other hand one might be led to conclude that the de Rham cohomologygroup of rank D +1 must be trivial on a supermanifold whose body is D -dimensional, hence W D +1 must also be exact. This means that it can be written as W D +1 = d K D where now, contrary to Z D , K D is a globally-defined (gauge-invariant) superform. It follows that L D := Z D − K D is aclosed superform, and can therefore be used to construct a supersymmetric action as in (3.1).– 10 –leven-dimensional supergravity is another example of an action with Chern-Simons terms,and we turn to the application of the action principle to this case in the following sections.Unfortunately there is a caveat to the previous argument that W D +1 is exact. As already notedin section 2.1, this argument can be applied only in the case where the cohomology is computedon the space of unconstrained superfields, but is not a priori true on the space of physical (on-shell) superfields. Interestingly it does turn out to be true in all known cases. As we will seein the following this includes the case of ordinary eleven-dimensional supergravity as well asits supersymmetric corrections with five derivatives. In section 4.2 we show that a sufficientcondition for the Weil triviality of the eight-derivative correction is the τ -exactness of X , .We shall parameterize the derivative expansion in terms of the Planck length l , so that theCremmer-Julia-Scherk two-derivative action (CJS) corresponds to zeroth order in l . In section 4we show that, provided the four- and seven-form BI are satisfied at order O ( l ), cf. (4.3), thereare at least two Weil-trivial twelveforms W and hence at least two independent supersymmetricactions with eight derivatives. Provided the twelve-forms associated to certain anomalous CSterms are Weil-trivial, cf. (4.16) below, there will be a third independent superinvariant at thisorder. We argue that at least two of those superinvariants will exist to all orders in the derivativeexpansion.As we will see in detail in the following, in practice one solves for the flat components of theclosed superform L D in a stepwise fashion in increasing engineering dimension. Once all flatcomponents of L D have been determined in this way, the explicit form of the action (3.1) canbe extracted using the formula, L m ...m D (cid:12)(cid:12) = e m D a D · · · e m a L a ...a D (cid:12)(cid:12) + D e m D a D · · · e m a ψ m α L α a ...a D (cid:12)(cid:12) + · · ·· · · + ψ m D α D · · · ψ m α L α ...α D (cid:12)(cid:12) , (3.3)where ψ αm := E mα | and e ma := E ma | are identified as the gravitino and the vielbein of (bosonic)spacetime respectively. In particular the bosonic terms of the Lagrangian can be read off imme-diately from L a ...a D . The eleven-dimensional supergravity action reads [1], S = Z (cid:0)
R ⋆ − G ∧ ⋆ G − C ∧ G ∧ G (cid:1)(cid:12)(cid:12) , (3.4)where d C = G is the threeform potential; it is understood that only the bosonic (11 , W = − G ∧ G ∧ G = d Z ; Z = − C ∧ G ∧ G . (3.5)– 11 –sing the BI (2.5) this can also be written in a manifestly Weil-trivial form, W = d K ; K = 13 G ∧ G . (3.6)Taking L = Z − K we obtain that the following action is invariant under supersymmetry, S = Z (cid:0) − G ∧ G − C ∧ G ∧ G (cid:1)(cid:12)(cid:12) . (3.7)This can then be put in the form (3.4) by using the on-shell conditions ⋆G = G and G ∧ ⋆ G =6 R ⋆
1, cf. appendix A. Therein we also give the details of the solution of the superspace equation W = d K and we show, as a byproduct, that the solution for K given in (3.6) is unique upto exact terms. O ( l ) correction (five derivatives) It was shown in [49], by directly computing the relevant spinorial cohomology group, that there isa unique superinvariant at the five derivative level (order l in the Planck length). The modifiedeleven-dimensional action to order l reads, S = Z (cid:16)
R ⋆ − G ∧ ⋆ G − C ∧ G ∧ G + l (cid:0) C ∧ G ∧ tr R + 2 tr R ∧ ⋆ G (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) , (3.8)where an arbitrary numerical coefficient has been absorbed in the definition of l and tr R := R ab ∧ R ba ; it is understood that only the bosonic (11 ,
0) components of the forms enter the formulaabove. This action can also be easily understood from the point of view of the action principleas follows. Consider the twelve-form corresponding to the CS term at order l , W = G ∧ G ∧ tr R = d Z ; Z = C ∧ G ∧ tr R . (3.9)Using the BI (2.3), (2.5) this can also be written in a manifestly Weil-trivial form, W = d K ; K = − G ∧ tr R . (3.10)Taking L = Z − K we obtain the following superinvariant at order l ,∆ S = Z (cid:0) C ∧ G ∧ tr R + 2 G ∧ tr R (cid:1)(cid:12)(cid:12) . (3.11)This can be seen, using the Hodge duality relation G = ⋆G , to precisely correspond to theorder- l terms in (3.8). In appendix B we work out in detail the superspace equation W = d K and confirm that thesolution (3.10) for K is unique up to exact terms, in accordance with the spinorial cohomologyresult of [49]. As explained in [49], on a topologically trivial spacetime manifold this superinvariant can be removed by anappropriate field redefinition of the threeform superpotential. However on a spacetime with nonvanishing firstPontryagin class the superinvariant cannot be redefined away without changing the quantization condition of thefourform field strength. The Hodge duality relation between G and G is expected to receive higher-order corrections (see below4.11). These can be neglected here since ∆ S is already a higher-order correction. – 12 – . The O ( l ) correction (eight derivatives) As was shown in [6, 65], the requirement that the M5-brane gravitational anomaly is cancelledby inflow from eleven dimensions implies the existence of certain CS terms Z at the eight-derivative order in the eleven-dimensional theory. The corresponding twelve-form reads, W = l G ∧ X = d Z ; Z = l C ∧ X , (4.1)where X is related to the M5-brane anomaly polynomial by descent, X = tr R −
14 (tr R ) , (4.2)and we have set (tr R ) := tr R ∧ tr R , tr R := R ab ∧ R bc ∧ R cd ∧ R da . At eight derivatives themodified four- and seven-form BI read,d G = 0 ; d G + 12 G ∧ G = l X , (4.3)where a numerical coefficient has been absorbed in the definition of l . We expand the formsperturbatively in l , G = G (0)4 + l G (1)4 + · · · ; G = G (0)7 + l G (1)7 + · · · , (4.4)and similarly for the supercurvature R AB . Note that in the expansion above the bosonic com-ponents of the lowest-order fields, G (0) m ...m etc, are identified with the fieldstrengths of thesupergravity multiplet, while the higher-order fields G (1)4 etc, are composite higher-derivativefields which are polynomial in the fieldstrengths of the supergravity fields.Solving perturbatively the BI at each order in l , taking into account that the exterior superderiva-tive d = d z M ∂ M is zeroth-order in l , implies,d G (0)4 = 0 ; d G (1)4 = 0 ;d G (0)7 + 12 G (0)4 ∧ G (0)4 = 0 ; d G (1)7 + G (1)4 ∧ G (0)4 = X (1)8 , (4.5)where we have set l X = l X (1)8 + · · · . Note that X (1)8 only involves the lowest-order curvature R (0) . Let us expand the twelve-form W perturbatively in l , W = l W (1)12 + · · · , so that, W (1)12 = X (1)8 ∧ G (0)4 = d Z ; Z = X (1)8 ∧ C (0)3 . (4.6)It then follows from (4.5) that this can also be written in a manifestly Weil-trivial form as follows, W (1)12 = d K ; K = G (1)7 ∧ G (0)4 − G (0)7 ∧ G (1)4 . (4.7)In particular we see that it suffices to solve the four- and seven-form BI in order to determinethe order- l superinvariant corresponding to L = Z − K ,∆ S = l Z (cid:16) X (1)8 ∧ C (0)3 − G (1)7 ∧ G (0)4 + 2 G (0)7 ∧ G (1)4 (cid:17)(cid:12)(cid:12)(cid:12) , (4.8)– 13 –here it is understood that only the bosonic (11 ,
0) components of the forms enter. This isthe superinvariant corresponding to the supersymmetrization of the CS term (4.1). The actionwould then read to this order, S = Z (cid:16) R (0) ⋆ − G (0)4 ∧ ⋆ G (0)4 − C (0)3 ∧ G (0)4 ∧ G (0)4 + l (cid:0) X (1)8 ∧ C (0)3 − G (1)7 ∧ G (0)4 + 2 G (0)7 ∧ G (1)4 (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) , (4.9)where R (0) , G (0) are identified with the fieldstrengths of the physical fields in the supergravitymultiplet, while the first-order fields R (1) , G (1) should be thought of as gauge-invariant functionsof the physical fields. We see that the action above is in agreement with the expectation thatthe bosonic part of the derivative-corrected supergravity action should be of the form, S = Z (cid:16)
R ⋆ − G ∧ ⋆ G − C ∧ G ∧ G + l (cid:0) X ∧ C + ∆ L ⋆ (cid:1)(cid:17) , (4.10)with ∆ L a function of R , G and their derivatives. Since ∆ L is gauge invariant, we see inparticular that the CS terms do not receive higher-order corrections beyond eight derivatives.Varying (4.10) with respect to C implies, d ⋆ G + 12 G ∧ G = X + δδC (∆ L ⋆ . (4.11)It is straightforward to see that the second term on the right-hand side above is exact by virtue ofthe fact that ∆ L is gauge invariant and thus only depends on C through G . Indeed the variationof the C -dependent terms in the ∆ L part of the action (4.10) can be written (possibly up tointegration by parts) in the form R Φ ∧ d δC , for some seven-form Φ . Therefore by appropriatelycorrecting the lowest-order duality relation by higher-derivative terms, G = ⋆G + O ( l ), onearrives at the modified BI (4.3). We have seen that provided the modified BI (4.3) are satisfied, there will be at least one superin-variant at eight derivatives, cf. (4.8). A second independent superinvariant can also be similarlyconstructed as follows. Consider the twelve-form, W ′ = 16 G ∧ G ∧ G . (4.12)Expanding perturbatively to order l we obtain, W ′ (1)12 = 12 G (0)4 ∧ G (0)4 ∧ G (1)4 = d Z ; Z = 12 G (1)4 ∧ G (0)4 ∧ C (0)3 , (4.13)The above can also be written in a manifestly Weil-trivial form using (4.5), W ′ (1)12 = d K ; K = − G (0)7 ∧ G (1)4 . (4.14)– 14 –he order- l superinvariant corresponding to Z − K then reads,∆ S ′ = l Z G (1)4 ∧ (cid:16) G (0)4 ∧ C (0)3 + G (0)7 (cid:17) , (4.15)where it is understood that only the bosonic (11 ,
0) components of the forms enter. The abovesuperinvariant does not contain the correct CS terms required by anomaly cancelation, cf. (4.10),and should therefore be excluded by the requirement of quantum consistency of the theory.However if one is only interested in counting superinvariants at order l in the classical theory,the above superinvariant is perfectly acceptable and its existence is guaranteed provided the BIare obeyed to order l .Dropping the requirement of quantum consistency, relying on classical supersymmetry alone,one may also consider the following two twelveforms, U = l G ∧ tr R ; V = l G ∧ (tr R ) , (4.16)so that U − V is the Weil-trivial twelveform corresponding to the CS terms of eleven-dimensionalsupergravity required for anomaly cancellation, cf. (4.1). It follows that either U , V are bothWeil-trivial, or neither U nor V is Weil-trivial. If the former is true, there would exist gauge-invariant elevenforms K U , K V so that at order l we have U (1) = d K U , V (1) = d K V . One canthen construct two corresponding superinvariants using the action principle,∆ S U = l Z (cid:16) tr R ∧ C (0)3 − K U (cid:17) ; ∆ S V = l Z (cid:16) (tr R ) ∧ C (0)3 − K V (cid:17) . (4.17)By the argument at the end of the last section, ∆ S U , ∆ S V should correspond to a modifiedBI obtained by replacing the right-hand side of the second equation in (4.3) by tr R , (tr R ) respectively. Then K U , K V would still be given by (4.7) but with G (1)4 , G (1)7 solutions of the newmodified BI.Together with the superinvariant ∆ S ′ of (4.15), we would then have a total of at least three inde-pendent superinvariants at the eight-derivative order, with only one linear combination thereof,∆ S of (4.8), corresponding to the quantum-mechanically consistent eight-derivative correction.As we will see in section (4.2), if ∆ S U , ∆ S V exist they must necessarily be cubic or lower in thefields. τ -exactness of X Based on what is known about superinvariants in
D <
11 dimensions [66], it is plausible toassume that the superinvariant (4.8) corresponding to the supersymmetrization of the CS term(4.1) should be quartic or higher in the fields. As pointed out in [19], a necessary condition forthe superinvariant to be quartic is that the order- l sevenform should be quartic or higher inthe fields. Since G (1)0 , cannot be quartic or higher in the fields, as can be seen by dimensional– 15 –nalysis, the order- l seven-form BI (4.5) must be solved for G (1)0 , = 0. It then follows that thepurely spinorial component of the M5-brane anomaly eightform X , is τ -exact. Explicitly, thefirst nontrivial component (at dimension four) of the seven-form BI then reads, γ f ( α α | G (1) f | α ...α ) = X (1) α ...α . (4.18)As explained in detail in appendix E, taking the form of G (0) into account, cf. (E.1), it followsthat the Weil-triviality condition, W = d K , (4.19)is solved up to dimension 7/2 for K , = K , = K , = 0. At dimension four, condition (4.19)then takes the form, γ f ( α α | K fab | α ...α ) = W (1) abα ...α . (4.20)From this it follows that (4.19) is solved, up to τ -exact terms, for K , given in terms of G (1)1 , ,cf. (4.18), K abcα ...α = 3 (cid:0) γ ab (cid:1) α α G (1) cα ...α , (4.21)where it is understood that all bosonic (spinor) indices are antisymmetrized (symmetrized). Notethat the solution for K , above relies on the M2-brane identity (2.14).Moreover, it can be shown that all higher components of K solving (4.19) are automaticallyguaranteed to exist. To see this, let us define the twelve-form, I := W − d K , (4.22)which is closed by construction,0 = (d I ) p, − p = τ I p +1 , − p + d f I p, − p + d b I p − , − p + tI p − , − p . (4.23)On the other hand, as we saw above, provided (4.18) holds, condition (4.19) is solved up todimension four, i.e. I p, − p = 0 for p = 0 , ,
2. Setting p = 2 in (4.23) then gives τ I , = 0, whichimplies I , = 0 up to a τ -exact piece that can be absorbed in K , , since all τ -cohomology groups H p, − pτ are trivial for p ≥
3, cf. section 2.2. By induction we easily see that I p, − p = 0, for all p ≥
3. In other words, provided (4.18) holds, the Weil-triviality condition (4.19) is guaranteedto admit a solution.In the present paper we provide highly nontrivial evidence corroborating (4.18). We will givethe outline of the argument here, relegating the technical details to appendix C. The component X , of the anomaly polynomial in (4.18) contains a large number of terms of the form G , whichcan be organized in terms of irreducible representations of B . Using certain Fierz identities,cf. appendix D, we have been able to show that almost all of these terms are indeed τ -exact. Thereare only three irreducible representations of B corresponding to terms which can potentially bepresent in X , and are not τ -exact. These are: (04000), (03002) and (02004), where we use theDynkin notation for B , see e.g. appendix C of [19].– 16 –n the other hand we show that, after Fierzing, X , can be put in the form, X , = ( γ a a )( γ a a )( γ a a )( γ a a ) G a a ; a a ; a a ; a a + ( γ a a )( γ a a )( γ a a )( γ a ...a ) G a a ; a a ; a a ; a ...a + ( γ a a )( γ a a )( γ a ...a )( γ a ...a ) G a a ; a a ; a ...a ; a ...a , (4.24)where G a a ; ... ; a a , G a a ; ... ; a ...a , G a a ; ... ; a ...a denote certain sums of G terms with 8,4,2indices contracted respectively, cf. (C.6), and we have supressed spinorial indices for simplicityof notation. Furthermore we show that (04000) can only be potentially present in the projectionof G a a ; ... ; a a onto the Young diagram associated to the partition [4 , G a a ; ... ; a ...a onto the Young diagram [4 , , , , X , to be τ -exact is that the two aforementioned projectionsshould vanish identically up to τ -exact terms,Π G a a ; ... ; a a ≈ G a a ; ... ; a ...a ≈ . (4.25)In the above ≈ denotes equality up to τ -exact terms. These two constraints are highly nontrivial,involving seemingly miraculous cancellations between hundreds of terms. We have shown that,remarkably, (4.25) are indeed identically satisfied.Furthermore we show that the required cancellations for (4.25) to hold, crucially rely on therelative coefficient between tr R and (tr R ) in X . In other words we show that the purelyspinorial components of tr R , (tr R ) are not separately τ -exact. Consequently, if the twelve-forms U , V are Weil-trivial, the corresponding modified order- l BI will be solved for some G (1)7 which are cubic or lower in the fields. (Indeed if G (1)7 were quartic or higher, G (1)0 , would vanishand the purely spinorial components of tr R , (tr R ) would be τ -exact.) It then follows from(4.7) that also K U , K V will be cubic or lower, and similarly for ∆ S U , ∆ S V , cf. (4.17). The perturbative expansion of the curved components following from (4.4) reads, G M ...M = G (0) M ...M + l G (1) M ...M + · · · , (4.26)and similarly for G and R AB . Note that in terms of flat components there is a mixing betweenzeroth order and order l due to,Φ = E A Φ A = E (0) A Φ (0) A + l ( E (0) A Φ (1) A + E (1) A Φ (0) A ) + · · · , (4.27)where we have expanded the coframe, E A = E (0) A + l E (1) A + · · · , and we have considered anarbitrary one-form Φ for simplicity. However, if one restricts to the top bosonic component of asuperform at θ = 0 as in (3.1), then there is no mixing:Φ (0) m | = e ma Φ (0) a | + ψ αm Φ (0) α | ; Φ (1) m | = e ma Φ (1) a | + ψ αm Φ (1) α | , (4.28)– 17 –here e ma , ψ αm were defined below (3.3). Indeed the O ( l ) corrections to the coframe E A onlystart at higher orders in the θ -expansion and could be systematically determined as in e.g. [67]once the O ( l ) corrections to the torsion components have been determined.In practice the BI are solved for the flat components of the superforms involved, G (0) A ...A , G (1) A ...A etc, at each order in l . Consequently the corresponding BI, d G = 0 etc, are only shown to besatisfied up to terms of the next order in l . In principle there may be an integrability obstructionto the solution of the BI at next-to-leading order in the derivative corrections, although thatwould most probably be prohibitively difficult to check in practice. In the following we shall seethat the integrability of a certain superinvariant is guaranteed provided the BI admit solutionsto all orders in l . Note however that all-order integrability need not be a consequence of the BI.The phenomenon of inducing a higher-order correction at next-to-leading order is also well un-derstood at the level of the component action, S = S (0) + l S (1) + · · · . The condition of invarianceof the action under supersymmetry transformations δ = δ (0) + l δ (1) + · · · reads, δ (0) S (0) = 0 ; δ (0) S (1) + δ (1) S (0) = 0 , (4.29)and similarly at higher orders. The term δ (1) S (0) in the second equation above is proportionalto the lowest-order equations of motion. Therefore in constructing S (1) we only need to checkits invariance with repsect to the lowest-order supersymmetry transformations δ (0) and only upto terms which vanish by virtue of the lowest-order equations of motion. This corresponds,in the superspace approach, to the fact that in solving the first-order BI one uses the zeroth-order equations for the various superfields. Once S (1) is thus constructed, the correction δ (1) to the supersymmetry transformations can be read off. Since δ (1) S (1) = 0 in general, thisinduces a correction S (2) to the action and a corresponding correction δ (2) to the supersymmetrytransformations, and so on.The existence of an intergrability obstruction can also be understood in the context of theNoether procedure. Indeed at next-to-leading order we have, δ (2) S (0) + δ (1) S (1) + δ (0) S (2) = 0 . (4.30)Therefore there must exist an action S (2) such that its variation with respect to lowest-ordersupersymmetry transformations is equal to − δ (1) S (1) , up to terms that vanish by virtue of thelowest-order equations of motion. This condition will not be automatically satisfied for every S (1) .In particular one would like to know how many of the independent superinvariants at order l presented in section 4.1 survive to all orders in the derivative expansion. Assuming M-theoryis a non-pertubatively consistent theory, we expect the superinvariant (4.8), corresponding tothe supersymmetrization of the CS term required for anomaly cancellation, to be integrable toall orders. Moreover, assuming this superinvariant is at least quartic in the fields, a similarargument as the one detailed below (4.23) shows that it must be unique at order l [19].– 18 –n addition, if one assumes that the BI admit a solution to all orders in a perturbative expansionin l , then there is one linear combination of the superinvariants presented in section 4.1 that isguaranteed to exist to all orders in l . Indeed in that case the twelve-form, W = (cid:0) l X − G ∧ G (cid:1) ∧ G = d (cid:0) G ∧ G (cid:1) , (4.31)is Weil-trivial by virtue of (4.3), which should now be considered valid to all orders in l . Howeverthis is not the superinvariant which corresponds to the supersymmetrization of the anomaly term,cf. (4.8). Indeed by the usual action principle procedure the twelve-form above would give riseto the superinvariant, ∆ S = Z (cid:18) l X ∧ C − G ∧ G ∧ C − G ∧ G (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (4.32)Expanding to order l and assuming G receives a nonvanishing correction at this order, we seethat (4.32) does not coincide with (4.8) and the corresponding l -corrected action is differentfrom (4.9).In conclusion, under the aforementioned assumptions, we would then expect (at least) two in-dependent superinvariants to exist to all orders in a perturbative expansion in l . Only oneof these, the one corresponding to the supersymmetrization of the CS anomaly term, will bequantum-mechanically consistent.
5. Discussion
We have shown that the highly nontrivial constraints (4.25) are satisfied, corroborating theexpectation that the purely spinorial component of X is τ -exact. Furthermore we have seenthat the τ -exactness of X , suffices for the existence of the superinvariant at order l . Solvingthe τ -exactness of X , is the first step, and arguably the most difficult, towards the explicitconstruction, via the action principle approach, of the supersymmetrization of the Chern-Simonsterm C ∧ X of eleven-dimensional supergravity required for the quantum consistency of thetheory.Conclusively proving the τ -exactness of X , would in addition require checking that the repre-sentation (03002) is absent from X , . This representation is potentially present in two differentYoung diagrams. As a consequence, showing the cancellation would involve, after projectingonto the appropriate Young diagram, Fierzing hundreds of four- γ terms. This is equivalent toeight-spinor Fierzing, as opposed to the four-spinor Fierzing which is sufficient in order to showthe absence of the (04000) and (02004) representations. At present, this seems prohibitivelydifficult even with the help of a computer.As a corollary of this work, we have shown that if the anomalous Chern-Simons terms C ∧ (Tr R ) and C ∧ Tr R can be supersymmetrized independently, the corresponding superinvariantsmust necessarily contain terms cubic or lower in the fields. The existence of eleven-dimensional– 19 –ubic superinvariants at the eight-derivative order has not been examined in the past. Theirexistence would presumably imply, by dimensional reduction, the presence of cubic terms inthe ten-dimensional superinvariants I a , I b mentioned in the introduction. This would not beinconsistent with the results of [4] who have excluded from the outset such terms in their analysis.This is an interesting open question to which we hope to return in the future. Acknowledgements
We would like to thank Paul Howe for valuable discussions. D.T. would like to thank the GalileoGalilei Institute for Theoretical Physics for hospitality and the INFN for partial support duringthe completion of this work.
A. Weil triviality at l In this section we give the details of the solution of the superspace equation W = d K atlowest order in the Planck length. As a byproduct we will see that the solution for K given in(3.6) is unique up to exact terms. We will look for the solution to,d K = − G ∧ G ∧ G , (A.1)for K gauge-invariant, i.e. function of the fieldstrengths of the physical fields. The explicitconstruction of K in flat components proceeds by solving the BI at each engineering dimensionin a stepwise fashion, from dimension − K α ...α to K a ...a ). In componentsthe BI (A.1) reads, D [ A K A ...A ) + 112 T F [ A A | K F | A ...A ) = − G [ A ...A G A ...A G A ...A ) , (A.2)where the torsion term arises from the the action of the exterior derivative on the supervielbein.The [ ABC ) notation stands for symmetrization or antisymmetrization, depending on the bosonicor fermionic nature of the indices.
In the following, antisymmetrisation of the indices a i andsymmetrisation of the indices α i is always implied. The engineering (mass) dimensions of the physical fields which will be involved in the construc-tion of K are, [ D a ] = 1[ D α ] = 1 / T a a α ] = 3 / h T aαβ i = 1 [ G abαβ ] = [ T αβa ] = 0[ G abcd ] = 1 From dimension -3 to -1/2
From dimension − − / K is K α ...α a ...a , appearing for the first time in the 0-dimensional equation (6 fermionic indices and 6 bosonic indices). For example, the equation(A.2) at dimension − / D α K α ...α a ...a − D a K a ...a α ...α +112 (cid:18) T α α f K fα ...α a ...a − T a a γ K γa ...a α ...α − T a α γ K γα ...α a ...a (cid:19) = 0 , and involves K ( − / α ...α a ...a , K ( − α ...α a ...a , K ( − / α ...α a ...a and K ( − α ...α a ...a , which cannot be ex-pressed in terms of the physical fields: the equation is thus trivially satisfied. Dimension 0 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 0, eq. (A.2) reads:12 D α z }| { K α ...α a ...a + 12 D a z }| { K a ...a α ...α +112 (cid:18) T α α f K fα ...α a ...a + 522 T a a γ K γa ...a α ...α | {z } + 1222 T a α γ K γα ...α a ...a | {z } (cid:19) = − G a a α α G a a α α G a a α α . Most terms vanish and the equation simplifies as follows,( γ f ) α α K fa ...a α ...α = 90 ( γ a a ) α α ( γ a a ) α α ( γ a a ) α α . Using the M2-brane identity as well as the so-called M5-brane identity,( γ e ) α α ( γ ea ...a ) α α = 3 ( γ a a ) α α ( γ a a ) α α , (A.3)it is easy to check that the solution is given by, K a ...a α ...α = 42 ( γ a ...a ) α α ( γ a a ) α α . (A.4) Dimension 1/2 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 1 /
2, eq. (A.2) reads,512 D α ( γ (5) γ (2) ) =0 z }| { D α K α ...α a ...a − D a z }| { K a ...a α ...α +112 (cid:18) T α α f K fα ...α a ...a − T a a γ K γa ...a α ...α | {z } − T a α γ K γα ...α a ...a | {z } (cid:19) = 0 , In the following we will use superscripts to indicate the dimension. This should not be confused with thenotation in the main text, e.g. (4.4) where the superscript denotes the order in the derivative expansion. – 21 –hich simplifies to, ( γ f ) α α K fa ...a α ...α = 0 . Since [ K fa ...a α ...α ] = 1 / K a ...a α ...α = 0 . (A.5) Dimension 1 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 1, eq. (A.2) reads,412 D α z }| { K α ...α a ...a + 812 d a ( γ (5) γ (2) ) =0 z }| { D a K a ...a α ...α +112 (cid:18) T α α f K fα α a ...a + 1433 T a a γ K γa ...a α ...α | {z } + 1633 T a α γ K γα ...α a ...a (cid:19) = − G a a a a G a a α α G a a α α , which becomes, using (E.1),( γ f ) α α K fa ...a α α = − i G a a a f ( γ f ) α α ( γ a ...a ) α α + 718 i G fghi ( γ a ...a fghi ) α α ( γ a a ) α α + 70 i ( γ a a ) α α ( γ a a ) α α G a ...a . (A.6)The last term above can be expanded as,70 i ( γ a a ) α α ( γ a a ) α α G a ...a = 703 i ( γ f ) α α ( γ fa ...a ) α α G a ...a = 42 i ( γ f ) α α ( γ [ fa ...a | ) α α G | a ...a ] − i ( γ f ) α α ( γ [ a ...a | ) α α G | a a a ] f . Similarly, the second term on the right-hand side of (A.6) can be written in a manifestly τ -exactform,718 ( γ a ...a fghi ) α α ( γ a a ) α α = − ǫ ja ...a fghi ( γ j ) α α ( γ a a ) α α = − ǫ [ ja ...a | fghi ( γ j ) α α ( γ | a a ] ) α α + 19 ǫ a ...a fghi ( γ j ) α α ( γ a j ) α α | {z } . Then eq. (A.6) takes the following form,( γ j ) α α K ja ...a α α = ( γ j ) α α (cid:16) i ( γ [ ja ...a ) α α G a ...a ] − i ǫ [ ja ...a | i ...i ( γ | a a ] ) α α G i ...i (cid:17) . Since the cohomology group H , τ is trivial, the solution to the above equation reads, K a ...a α α = 42 i ( γ a ...a ) α α G a ...a − i ǫ a ...a i ...i ( γ a a ) α α G i ...i , up to τ -exact terms. – 22 – imension 3/2 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 3 /
2, eq. (A.2) reads,312 D α K α α a ...a − D a z }| { K a ...a α ...α − (cid:18) T α α f K fa ...a α + 611 T a a γ K γa ...a α ...α + 922 T a α γ K γα α a ...a | {z } (cid:19) = 0 , which becomes, using (E.1),( γ f ) α α K fa ...a α = + 252 ( γ a ...a ) α α ( γ a a ) α γ T a a γ − ǫ a ...a i ...i ( γ a a ) α α ( γ i i ) α γ T i i γ + 504 ( γ a a ) ( α α | ( γ a ...a ) | α γ ) T a a γ . (A.7)The decomposition of K fa ...a α in irreducible components is given by(10000) ⊗ (00001) = (10001) ⊕ (00001) , whereas T abα is in the representation (01001). It follows that, K a ...a α = 0 , (A.8)and moreover the right-hand side of (A.7) must vanish identically. This can be verified by e.g.taking the Hodge dual of ( γ i i ) α γ in the second term of (A.7), and using the γ -tracelessness of T abγ , cf. (E.3). Dimension 2 - ( A A → α α , A . . . A → a . . . a ) At dimension 2, eq. (A.2) reads,212 D α K α a ...a + 1012 D a K a ...a α α +112 (cid:18) T α α f K fa ...a + 1033 T a a γ K γa ...a α α + 1522 T a α γ K γα a ...a (cid:19) = − G a a a a G a a a a G a a α α , which becomes, using (E.1),( γ f ) α α K fa ...a = − i (cid:18) i ( γ a ...a ) α α d a G a ...a − i ǫ a ...a i ...i ( γ a a ) α α d a G i ...i (cid:19) − i T a α ǫ (cid:16) i ( γ a ...a ) ǫα G a ...a − i ǫ a ...a i ...i ( γ a a ) ǫα G i ...i (cid:17) − G a ...a G a ...a ( γ a a ) α α . (A.9)– 23 –uliplying by γ (1) and taking the trace leads to, K a ...a = 172 ǫ a ...a G d ... d G d ... d . (A.10)On the other hand contracting (A.9) with γ (2) or γ (5) imposes that the contraction of the right-hand side must be identically zero. This can indeed be straightforwardly verified using (E.3). Dimension 5/2 - ( A → α , A . . . A → a . . . a ) The equation at dimension 5 / K a ...a . It reads,112 D α K a ...a − D a K a ...a α − (cid:18) T a a γ K γa ...a α − T a α γ K γa ...a (cid:19) = 0 , which becomes, using (E.1) and (A.10),172 ǫ a ...a D α G abcd G abcd − i G a a gh ( γ gha ...a T a a ) α + 2310 i G a ...a ( γ a ...a T a a ) α . Using (E.2), (D.1) we then obtain the constraint,0 = ǫ a ...a T d d δ ( γ d d ) δα G d ...d + 774 ǫ a ...a b ...b ( γ b ...b T a a ) α G a ...a − ǫ a ...a b b gh ( γ b b T a a ) α G a a gh , (A.11)which can be seen to be automatically satisfied by contracting (A.11) with ǫ a ...a . The nextequation (of dimension 3) is trivially satisfied, since the purely bosonic component of a twelveformvanishes automatically in eleven dimensions. Action at O ( l )We have thus constructed the explicit expression of all components of K and have seen thatit is unique up to exact terms. Its purely bosonic component in particular takes the followingform, K (2) = 172 · ǫ a ...a G d ...d G d ...d dx a ∧ . . . ∧ dx a = 13 G ∧ ⋆ G (A.12)= − R ⋆ G ∧ ⋆ G , (A.13)where in this subsection we have reverted to bosonic conventions for bosonic forms. Using theaction principle then leads to the CJS action of section 3.1.The last two equalities in (A.12) above can be seen as follows. The volume element is definedas, d V = ⋆ ǫ a ...a dx a ∧ . . . ∧ dx a , – 24 –rom which it follows that, G ∧ ⋆ G = 17! (4!) G a ...a ǫ a ...a b ...b G b ...b − ǫ a ...a d V z }| { dx a ∧ . . . ∧ dx a = 14! G d ...d G d ...d d V .
Moreover, taking the trace of the third relation of (E.3) gives,
R ⋆ G d ...d G d ...d d V = 16 G ∧ ⋆ G . B. Weil triviality at l In this section we are looking for the solution to the equation,d K = G ∧ G ∧ R ab ∧ R ba . (B.1)We will construct all components of K explicitly and confirm that the solution of section 3.2is unique up to exact terms. In components the equation above takes the following form, D [ A K A ...A ) + 112 T F [ A A | K F | A ...A ) = 11!4(4!) R [ A A | c c R | A A | c c G | A ...A G A ...A ) . (B.2)The dimensions of the physical fields are the same as before, with the addition of [ R abcd ] = 2.The dimensions of the various components of K range from − / K α ...α ) to 5 ( K a ...a ). Dimension 0 to 3/2
Since the dimension of K α ...α is − /
2, it must be set to zero as it cannot be expressed in termsof the physical fields. The equation of dimension 0 then takes the form, D α K α ...α | {z } + 112 T α α f K fα ...α = 11!4(4!) R α α c c R α α c c G α ...α G α ...α | {z } , which simplifies to, ( γ f ) α α K fα ...α = 0 . (B.3)Since [ K fα ...α ] = 0 and H , τ is nontrivial, a τ -nonexact solution involving only γ -matricescould exist. In that case K fα ...α would necessary transform as a scalar, since the only availablegauge-invariant superfield of zero dimension is a constant. On the other hand,(10000) ⊗ (00001) ⊗ S = 1 × (00000) + · · · , i.e. the decomposition of K fα ...α contains a unique scalar combination. It follows that, K fα ...α ∝ ( γ f ) α α ( γ a ) α α ( γ a ) α α ( γ b ) α α ( γ b ) α α . (B.4)– 25 –owever it can be verified that this expression does not satisfy eq. (B.3), unless K a α ...α = 0.The right-hand side of eq. (B.2) vanishes from dimension 0 to dimension 3 /
2, and the equationsto solve are all similar to (B.3): The component K (1 / a a α ...α will be set to zero because thereis no gauge-invariant field of dimension 1 /
2. The components K (1) a a a α ...α , K (3 / a a a a α ...α willbe set to zero, up to exact terms, as a consequence of the triviality of H , τ , H , τ . Dimension 2 - ( A . . . A → α . . . α , A . . . A → a . . . a ) This is the first equation with a non-zero right-hand side,812 D α z }| { K α ...α a ...a + 412 D a z }| { K a ...a α ...α + 112 (cid:18) T α α f K fa ...a α ...α + 111 T a a γ K γα ...α a a | {z } + 1633 T a α γ K γα ...α a ...a | {z } (cid:19) =3 455 11!4(4!) R α α c c R α α c c G a a α α G a a α α , which becomes, using (E.1),( γ f ) α α K fa ...a α ...α = − i ( γ f ) α α ( γ fa ...a ) α α R α α c c R α α c c . Since H , τ is trivial, the solution reads, K (2) a ...a α ...α = − i ( γ a ...a ) α α R α α c c R α α c c , up to τ -exact terms. Dimension 5/2 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 5 /
2, eq. (B.2) reads,712 D α K α ...α a ...a − D a z }| { K a ...a α ...α − (cid:16) T α α f K fa ...a α ...α + 533 T a a γ K γα ...α a ...a | {z } + 3566 T a α γ K γα ...α a ...a | {z } (cid:17) = 11!4(4!) (cid:16) R α α c c R α a c c G a a α α G a a α α (cid:17) , which becomes, using (E.1),( γ f ) α α K fa ...a α ...α = − i ( γ a ...a ) α α R α α c c (cid:18) ( γ e e ) α α ( γ [ c c T e e ] ) α + 124 ( γ c c e ...e ) α α ( γ e e T e e ) α (cid:19) + 1800 i ( γ a a ) α α ( γ a a ) α α R α α c c R α a c c . (B.5)– 26 –he second term in (B.5) can be written,1800 i ( γ [ a a | ) α α ( γ | a a | ) α α R α α c c R α | a ] c c (B.6)= 600 i ( γ g ) α α ( γ g [ a a a a | ) α α R α α c c R α | a ] c c = 600 i ( γ g ) α α (cid:18)
65 ( γ [ ga a a a | ) α α R α α c c R α | a ] c c + 15 ( γ a a a a a ) α α R α α c c R α gc c (cid:19) . One can then verify that the second term on the right-hand side of (B.6) cancels with the firstterm on the right-hand side of (B.5). Since the first term on the right-hand side of (B.6) is in a τ -exact form and H , τ is trivial, the solution reads, K (5 / a ...a α ...α = 720 i ( γ a ...a ) α α R α α c c R α a c c , up to τ -exact terms. Dimension 3 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 3, eq. (B.2) reads,12 D α K α ...α a ...a + 12 D a z }| { K a ...a α ...α + 112 T α α f K fa ...a α ...α + 522 T a a γ K γα ...α a ...a | {z } + 611 T a α γ K γα ...α a ...a | {z } = 11!4(4!) (cid:18) − R α a c c R α a c c G a a α α G a a α α + 2 1154 R α α c c R α α c c G a a α α G a ...a + 2 377 R a a c c R α α c c G a a α α G a a α α (cid:19) , which becomes, using (E.1), − i ( γ f ) α α K fa ...a α ...α = − D α K α ...α a ...a −
12 d a K a ...a α ...α − T ǫa α K ǫα ...α a ...a − R α a c c R α a c c G a a α α G a a α α + 225 R α α c c R α α c c G a a α α G a ...a + 1350 R a a c c R α α c c G a a α α G a a α α . – 27 –et us now examine separately each group of terms in the equation above with the same type offield content. There are four G terms which read, − i G a ...a ( γ a a ) α α R α α c c R α α c c −
360 ( γ a ...a ) α α R α α c c T c α ǫ T c ǫβ ( γ a ) βα + 720 ( γ a ...a ) α α R α α c c T c α ǫ T a ǫβ ( γ c ) βα − i ( γ a ...a ) ( α ǫ | T a α ǫ R | α α | c c R | α α ) c c . (B.7)The last term in (B.7) can be split in two parts, − i (cid:18)
26 ( γ a ...a ) α ǫ T a α ǫ R α α c c R α α c c + 46 ( γ a ...a ) α α T a α ǫ R ǫα c c R α α c c (cid:19) . The first one leads to,( γ g ) α α (cid:18) i ǫ ga ...a b ...b G b ...b R α α c c R α α c c (cid:19) + 225 i G a ...a ( γ a a ) α α R α α c c R α α c c , where the first term is τ -exact, and the second term cancels with the first one in (B.7). It canthen be verified that the three remaining G terms cancel out.Moreover there are three terms of the schematic form G ( DG ), − i ( γ a ...a ) α α R α α c c d c T c α β ( γ a ) βα − i ( γ a ...a ) α α R α α c c d c T a α β ( γ c ) βα − i ( γ a ...a ) α α R α α c c d a R α α c c , (B.8)which cancel out.There are two RG terms which read, − γ a a ) α α ( γ a a ) α α R a a c c R α α c c + 45 ( γ a ...a ) α α R α α c c (cid:16) ( γ e e γ a ) α α R c c e e − γ e e γ c ) α α R c a e e (cid:17) . (B.9)The first term of (B.9)can be put in a τ -exact form, − γ a a ) α α ( γ a a ) α α R a a c c R α α c c =( γ g ) α α (cid:16) − i ( γ [ ga ...a | ) α α R | a a ] c c R α α c c + 180 ( γ a ...a ) α α R a gc c R α α c c (cid:17) , while the remaining RG terms cancel out.There are two T terms which read,+ 2700 ( γ a a ) α α ( γ a a ) α α R α a c c R α a c c (B.10)+ 1080 i ( γ a ...a ) α α (cid:16) ( γ e e ) α α ( γ [ c c T e e ] ) α + 124 ( γ c c e ...e ) α α ( γ e e T e e ) α (cid:17) R α a c c . – 28 –he first term can be put in a τ -exact form,2700 ( γ a a ) α α ( γ a a ) α α R α a c c R α a c c =13 2700 ( γ g ) α α (cid:18)
75 ( γ ga ...a ) α α R α a c c R α a c c + 25 ( γ a ...a ) α α R α a c c R α gc c (cid:19) , while the remaining T T terms cancel out. Taking the triviality of H , τ into account, the non-vanishing terms extracted from the RG , T , and G terms lead to the solution, K (3) a ...a α ...α = 504 i ( γ a ...a ) α α (cid:16) − R a a c c R α α c c + 2 R α a c c R α a c c (cid:17) − i ǫ a ...a b ...b G b ...b R α α c c R α α c c , (B.11)up to τ -exact terms. Dimensions 7/2 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 7 /
2, eq. (B.2) reads,512 D α K α ...α a ...a − D a K a ...a α ...α − (cid:18) T α α f K fa ...a α ...α − T a a γ K γα ...α a ...a + 3566 T a α γ K γα ...α a ...a (cid:19) = 11!4(4!) (cid:18) R α a c c R a a c c G a a α α G a a α α + 4 166 R α a c c R α α c c G a a α α G a ...a (cid:19) . The right-hand side of the equation above contains terms of the form G ( DT ), T ( DG ), T R , and
T G . The first two groups of terms simply vanish (without the use of any equations of motionor BI). Two τ -exact terms can be extracted from RT and T G , and the remaining terms cancelout. This leads to the solution, K (7 / a ...a α ...α = 2016 i ( γ a ...a ) α α R a a c c R α a c c + 4 ǫ a ...a b ...b G b ...b R α α c c R α a c c , up to τ -exact terms. – 29 – imensions 4 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 4, eq. (B.2) reads,412 D α K α ...α a ...a + 812 D a K a ...a α ...α + 112 (cid:18) T α α f K fa ...a α α + 1433 T a a γ K γα ...α a ...a + 1633 T a α γ K γα ...α a ...a (cid:19) = 11!4(4!) (cid:18) R a a c c R a a c c G a a α α G a a α α + 35 R α α c c R α α c c G a a a a G a ...a + 4 · R α α c c R a a c c G a a a a G a a α α − · R α a c c R α a c c G a a α α G a a a a (cid:19) . The terms in the equation above can be cast in eight groups: R , RG , R ( DG ), G , G ( DG ), GT , T ( DT ), and G ( DR ). Parts of the terms of the form R , G R , and GT can be put in a τ -exact form, while the remaining terms cancel out. Taking into account the BI, D a R a a c c = − T a a γ R γa c c , (B.12)we see that the term G ( DR ) cancel against a term from GT . Taking into account the equationof motion of G we see that a term from G ( DG ) cancels against a term in G , ǫ a ...a b ...b D a G b ...b = 12 ǫ a ...a b ...b D c G cb ...b = 105 G a ...a G a ...a . (B.13)We are thus led to the solution, K (4) a ...a α ...α = − i ( γ a ...a ) α α R a a c c R a a c c − ǫ a ...a b ...b G b ...b R α α c c R a a c c + 12 ǫ a ...a b ...b G b ...b R α a c c R α a c c , up to τ -exact terms. Dimensions 9/2 - ( A . . . A → α . . . α , A . . . A → a . . . a ) At dimension 9 /
2, eq. (B.2) reads,312 D α K α α a ...a − D a K a ...a α ...α − (cid:18) T α α f K fa ...a α + 611 T a a γ K γα ...α a ...a + 922 T a α γ K γα α a ...a (cid:19) = 11!4(4!) (cid:18) R α α c c R α a c c G a ...a G a ...a + 4 355 R a a c c R α a c c G α α a a G a ...a (cid:19) . – 30 –he terms in the equation above can be cast in seven groups: R ( DT ), RT G , G ( DT ), G T , T , T G ( DG ) and T ( DR ). One term of the form RT G is τ -exact, while all the remaining terms canbe seen to cancel out, using (B.12) and (B.13) to convert a term of the form T ( DR ) to the form T , and a term of the form T G ( DG ) to the form G T . Up to τ -exact terms, the component ofdimension 9 / K (9 / a ...a α α = 60 i ǫ a ...a b ...b G b ...b R a a c c R α a c c . Dimensions 5 - ( A A → α α , A . . . A → a . . . a ) At dimension 5, eq. (B.2) reads,212 D α K α a ...a + 1012 D a K a ...a α α + 112 (cid:18) T α α f K fa ...a + 1522 T a a γ K γα α a ...a + 1033 T a α γ K γα a ...a (cid:19) = 11!4(4!) (cid:18) R α α c c R a a c c G a ...a G a ...a − R α a c c R α a c c G a a a a G a ...a + 2 111 R a a c c R a a c c G a a α α G a a α α (cid:19) . The terms in the equation above can be cast in nine groups: RT , GT ( DT ), G T , GR , GR ( DG ), RG , R ( DR ), G ( DR ), and T ( DG ). One term in GR is τ -exact, while all theremaining terms cancel out, as can be seen using eq. (B.12) and (B.13) to convert a term of theform R ( DR ) to the form RT , a term of the form G ( DR ) to the form G T , and a term of theform T ( DG ) to the form G T . Up to τ -exact terms, the component of dimension 5 then reads, K (5) a ...a = − ǫ a ...a b ...b G b ...b R a a c c R a a c c . (B.14) Dimensions 11/2 - ( A → α , A . . . A → a . . . a ) Since there is no new component of K appearing, this equation should be satisfied automatically,112 D α K a ...a − D a K a ...a α − (cid:18) T a α f K fa ...a − T a a γ K γα a ...a (cid:19) = 11!4(4!) (cid:18) R α a c c R a a c c G a ...a G a ...a (cid:19) . The equation contains six types of terms:
T R , GR ( DT ), G T R , GT ( DR ), RT ( DG ), and T G .As expected all the terms cancel out, as can be seen using (B.12) and (B.13) to convert a termof the form GT ( DR ) to the form T G , and a term of the form RT ( DG ) to the form G T R .– 31 – ction at O ( l )We have constructed the explicit expression of each component of K and showed that it isunique up to exact terms. In particular the top component, given in eq. (B.14), precisely agreeswith (3.10), leading to the superinvariant of section 3.2. C. Weil triviality at l The same method will be used to generate the corrections at l -order, cf. section 4. We will lookfor the solution to the equation d K = G ∧ X (1)8 . In components this reads, D [ A K A ...A ) + 112 T F [ A A | K F | A ...A ) =11!(4!)4 (cid:18) G [ A ...A R | A A | c c R | A A | c c R | A A | c c R | A A ) c c − G [ A ...A R | A A | c c R | A A | c c R | A A | d d R | A A ) d d (cid:19) . (C.1)The dimensions of the various components of K now range from [ K α ...α ] = to [ K a ...a ] = 8. Dimension 3 and 7/2
If we assume that the superivariant at O ( l ) is quartic or higher in fields, the first potentiallynonvanishing component of K appears at dimension 4 (it is of the form G ). We thus obtain, K (5 / α ...α = K (3) a α ...α = K (7 / a a α ...α = 0 . This is consistent with (C.1), whose right-hand side vanishes for dimensions lower than 4.
Dimension 4 - ( A . . . A → α . . . α , A A → a a ) Eq. (C.1) takes the following form,212 D a z }| { K (3) a α ...α + 1012 D α z }| { K (7 / α ...α a a + 112 (cid:18) T α α f K (4) fa a α ...α + 1033 T a α γ z }| { K (3) γα ...α a + 166 T a a γ z }| { K (5 / γα ...α (cid:19) = 11!4!4 G a a α α (cid:18) R α α c c R α α c c R α α c c R α α c c − R α α c c R α α c c R α α d d R α α d d (cid:19) , – 32 –hich simplifies to,( γ f ) α α K (4) fa a α ...α = 2520 ( γ a a ) α α (cid:18) R α α c c R α α c c R α α c c R α α c c − R α α c c R α α c c R α α d d R α α d d (cid:19) = ( γ a a ) α α X (8) α ...α . (C.2)Explicitly, the term (tr R ) reads (omitting the factor − / ( γ u u )( γ u u )( γ u u )( γ u u ) G u u y y G u u y y G u u z z G u u z z · ( γ u u )( γ u u )( γ u u )( γ v ...v ) G u u y y G u u y y G u u v v G v ...v ( γ u u )( γ u u )( γ v ...v x x )( γ w ...w x x ) G u u y y G u u y y G v ...v G w ...w ( γ u u )( γ u u )( γ v ...v )( γ w ...w ) G u u v v G u u w w G v ...v G w ...w ( γ u u )( γ v ...v )( γ w ...w y y )( γ x ...x y y ) G u u v v G v ...v G w ...w G x ...x ( γ u ...u y y )( γ v ...v y y )( γ w ...w z z )( γ x ...x z z ) G u ...u G v ...v G w ...w G x ...x , while the term tr R reads, ( γ u u )( γ u u )( γ u u )( γ u u ) G u u y y G u u y z G u u y z G u u z z · ( γ u u )( γ u u )( γ u u )( γ v ...v ) G u u y y G u u v y G u u v y G v ...v ( γ u u )( γ u u )( γ v ...v )( γ w ...w ) G u u v w G u u v w G v ...v G w ...w ( γ u u )( γ u u )( γ v ...v x )( γ w ...w x ) G u u v y G u u w y G v ...v G w ...w ( γ u u )( γ v ...v y )( γ w ...w y )( γ x ...x y y ) G u u v w G x ...x G v ...v G w ...w ( γ u ...u y y )( γ v ...v y z )( γ w ...w y z )( γ x ...x z z ) G u ...u G v ...v G w ...w G x ...x . Suppose now that the purely femionic component of X can be cast in the τ -exact form ofeq. (4.18). The right-hand side of eq. (C.2) would then take the form,( γ a a ) α α X (8) α ...α = ( γ a a ) α α ( γ f ) α α G fα ...α = ( γ f ) α α (cid:16) γ [ a a | ) α α G | f ] α ...α − γ fa ) α α G a α ...α (cid:17) = ( γ f ) α α (cid:16) γ [ a a | ) α α G | f ] α ...α (cid:17) , which yields, K fa a α ...α = 3 ( γ [ a a | ) α α G | f ] α ...α . In the following we will examine whether X , can be τ -exact. Since (C.2) contains many differenttypes of terms, it is useful to reduce this expression by simplifying every pair of γ -matrices whosebosonic indices contain contractions, using the decompositions in appendix D. When appliedto (tr R ) , this method will give three terms of the form γ (2) γ (2) γ (2) γ (2) , γ (2) γ (2) γ (2) γ (6) and γ (2) γ (2) γ (6) γ (6) , together with several manifestly τ -exact terms. Applied to tr R , this methodwill give several of terms of the form previously encountered, plus some new terms of the form– 33 – (2) γ (2) γ (5) γ (5) , which are equivalent to γ (2) γ (2) γ (6) γ (6) by Hodge duality. In order to compare(tr R ) with tr R , all the γ (6) γ (6) terms must be converted into the form γ (5) γ (5) . This createsnew γ -matrices with contracted bosonic indices, which are simplified as before using appendixD. At the end of this process all the terms have the form γ (2) γ (2) γ (2) γ (2) , γ (2) γ (2) γ (2) γ (6) or γ (2) γ (2) γ (5) γ (5) contracted with G (without any contractions among γ -matrices),( γ a a )( γ a a )( γ a a )( γ a a ) G a a ; a a ; a a ; a a (C.3)( γ a a )( γ a a )( γ a a )( γ a ...a ) G a a ; a a ; a a ; a ...a (C.4)( γ a a )( γ a a )( γ a ...a )( γ a ...a ) G a a ; a a ; a ...a ; a ...a , (C.5)up to manifestly τ -exact terms which we do not need to write out explicitly. In the above, G a a ; ... ; a a , G a a ; ... ; a ...a , G a a ; ... ; a ...a , denote certain sums of G terms with 8,4,2 indicescontracted respectively. More explicitly, G a a ; a a ; a a ; a a = 72 G a efg G a a a e G a a a h G a fgh + · · · G a a ; a a ; a a ; a ...a = 252 G a a fg G a a a f G a a a a G a a a g + · · · G a a ; a a ; a ...a ; a ...a = 12 G a a a f G a a a f G a a a a G a a a a + · · · , (C.6)where the ellipses stand for more than a hundred terms of this form. No obvious cancellationsappear between these three types of terms at this point.Let us further analyse how X , is decomposed into irreducible components. First, the productof four γ -matrices contains a symmetric product of eight spinor indices which can be decomposedas follows in irreps of B ,(00001) ⊗ S = 1(00000) ⊕ · · · ⊕ | {z }
45 terms with multiplicity 1 ⊕ ⊕ ⊕ | {z } . Each irrep on the right-hand side above corresponds to a γ -structure which can be thought ofas a Clebsch-Gordan coefficient: the γ -structure corresponding to (00000) can be thought of asa Clebsch-Gordan coefficient from the scalar to (00001) ⊗ S , etc.Next, the product of four four-forms G can be decomposed as follows in irreps of B ,(00010) ⊗ S = 4(00000) ⊕ · · · ⊕ ⊕ · · · ⊕ | {z }
95 terms, various multiplicites , and all 95 terms except (00006), (00008), (01006), and (10006) can be found in (00001) ⊗ S . Thisanalysis implies that the contraction of four γ -matrices with four fourforms G can be decomposedinto 51 γ -structures, each contracted with (multiple) G terms corresponding to the same irrepof B . – 34 –or example, the term (00000) in the decomposition of (00001) ⊗ S gives rise to a single γ -structure contracted with the four possible G terms giving rise to a scalar. Explicitly we have,( γ e )( γ e )( γ e )( γ e ) (cid:18) α G a ...a G a ...a G b ...b G b ...b + α G a a b b G b b c c G c c d d G d d a a + α G a b c c G c c d f G a d g g G g g b f + α G a b ...b G b ...b c G a d ...d G c d ...d (cid:19) , for some constants α , . . . , α . Similarly, the (00004) gives rise to the following term, (cid:18) β ( γ e )( γ e )( γ a ...a )( γ b ...b ) + β ( γ [ a )( γ a ...a ] )( γ [ b )( γ b ...b ] ) (cid:19) × (cid:18) α G a a b e G a b b e G a ...a e G b ...b e + α G a a b e G a a b e G a b b e G a b b e + α G a ...a G a b b e G a b b e G b b e e + α G a b b e G a ...a G a b e e G b ...b e + α G a ...a G a b e e G a b e e G b ...b + α G a ...a b G a ...a b G b b e e G b b e e (cid:19) , for some constants β , β , α , . . . , α . The 51 γ -structures involved in the decomposition of X (8) can all be found explicitly, and only three of them are not τ -exact: (04000), (03002), and (02004).In other words, except for the structures corresponding to these three irreps all other γ -structuresappearing in X , involve at least one contraction with a γ (1) .Going back to (C.3): the G a a ; ... ; a a term, by virtue of its contraction with the four γ -matrices,transforms in the symmetrized product of four Young diagrams , cf. appendix F. Decomposingin irreducible representations of S , ⊗ S = ⊕ ⊕ ⊕ ⊕ | {z } Y T (5 terms) . (C.7)At the same time G a a ; ... ; a a admits a decomposition into modules of B × S , P R V R × R ,where V R is the plethysm of the module V = (10000) of B with respect to the Young diagram R of S . Moreover only the plethysms corresponding to the right-hand side of (C.7) will appear inthe decomposition of G a a ; ... ; a a under B × S . On the other hand we can compute the module V R corresponding to each R on the right-hand side of (C.7), using [68], with the result thatonly the plethysm corresponding to Y T contains (04000), while neither (02004) nor (03002) iscontained in any of the plethysms corresponding to the Young diagrams on the right-hand sideof (C.7). – 35 –he G a a ; ... ; a ...a term of (C.4) admits the following decomposition in irreps of S , ⊗ S ⊗ = | {z } Y T ⊕ . . . (16 terms) . (C.8)Only the plethysms corresponding to the Young diagrams on the right-hand side of (C.8) willappear in the decomposition of G a a ; ... ; a ...a under B × S . On the other hand it can beshown that only the plethysm corresponding to Y T contains (03002), while neither (04000)nor (02004) is contained in any of the plethysms corresponding to the Young diagrams on theright-hand side of (C.8).Finally, the G a a ; ... ; a ...a term of (C.5) admits the following decomposition in irreps of S , ⊗ S ⊗ ⊗ S = | {z } Y T ⊕ | {z } Y T ⊕ . . . (23 terms) . (C.9)Moreover only the plethysms corresponding to the Young diagrams on the right-hand side of(C.9) will appear in the decomposition of G a a ; ... ; a ...a under B × S . On the other hand itcan be shown that only the plethysm corresponding to Y T contains (02004); only the plethysmcorresponding to Y T contains (03002), while (04000) is not contained in any of the plethysmscorresponding to the Young diagrams on the right-hand side of (C.9).Using the method of appendix F, the γ -matrices in (C.3) and (C.5) can be projected respectivelyonto Y T and Y T . The terms (C.3), (C.5) can thus be shown to vanish. Moreover, it can beseen that the cancellations are sensitive to the relative coefficient between (tr R ) and tr R inside X . In other words, it can be shown that (tr R ) and tr R are not separately τ -exact. D. Eleven-dimensional γ -matrices In this section we give our conventions for the eleven-dimensional γ -matrices, and list a numberof Fierz identities used in the analysis presented in the main text.Hodge duality for γ -matrices is defined as follows, ⋆γ ( n ) = − ( − n ( n − γ (11 − n ) , (D.1)where our definition of the Hodge operator reads,( ⋆S ) a ...a k = 1(11 − k )! ǫ a ...a k b ...b − k S b ...b − k . – 36 –he symmetry properties of the γ -matrices are given by,( γ a ...a n ) αβ = ( − ( n − n − ( γ a ...a n ) βα , where γ (0) is identified with the charge conjugation matrix.The following Fierz identities were used in the analysis. Antisymmetrisation over the a i and b j indices is always understood, as well as symmetrization over all fermionic indices of the γ -matrices (which are suppressed here to avoid cluttering the notation),( γ a ...a e )( γ b ...b e ) =+ 120 δ a ...a b ...b ( γ e )( γ e )+ 1 ( γ a ...a )( γ b ...b ) − δ a ...a b ...b ( γ e )( γ e a a b b )+ 25 δ a b ( γ e )( γ e a ...a b ...b ) −
150 12 (cid:18) δ a b ( γ a a )( γ a a b ...b ) + ( a ↔ b ) (cid:19) + 600 δ a ...a b ...b ( γ a a )( γ b b ) ( γ a ...a e e )( γ b ...b e e ) = −
12 12 (cid:18) ( γ a a )( γ a a b ...b ) + ( a ↔ b ) (cid:19) + 288 δ a a b b ( γ a a )( γ b b ) −
96 12 (cid:18) δ a b ( γ a )( γ a a b ...b ) + ( a ↔ b ) (cid:19) + 192 δ a ...a b ...b ( γ a )( γ b )+ 2 ( γ e )( γ e a ...a b ...b ) − δ a a b b ( γ e )( γ e a a b b )+ 48 δ a ...a b ...b ( γ e )( γ e )( γ a ...a e ...e )( γ b ...b e ...e ) =+ 36 δ a ...a b ...b ( γ e )( γ e ) − δ a b ( γ e )( γ e a a b b )+ 216 δ a b ( γ a a )( γ b b ) −
36 12 (cid:18) ( γ a )( γ a a b ...b ) + ( a ↔ b ) (cid:19) + 324 δ a a b b ( γ a )( γ b ) ( γ a a e ...e )( γ b b e ...e ) =+ 48 δ a a b b ( γ e )( γ e ) −
96 ( γ e )( γ e a a b b )+ 168 ( γ a a )( γ b b )+ 672 δ a b ( γ a )( γ b )( γ a e ...e )( γ b e ...e ) =+ 240 δ a b ( γ e )( γ e )+ 1680 ( γ a )( γ b ) ( γ e ...e )( γ e ...e ) =+ 4320 ( γ e )( γ e )( γ a a e ...e )( γ b b e ...e ) = − δ a a b b ( γ e )( γ e )+ 24 ( γ e )( γ e a a b b ) −
42 ( γ a a )( γ b b )+ 168 δ a b ( γ a )( γ b ) ( γ a e ...e )( γ b e ...e ) = − δ a b ( γ e )( γ e )+ 336 ( γ a )( γ b )( γ e ...e )( γ e ...e ) = −
720 ( γ e )( γ e )– 37 – γ a ...a e )( γ b ...b e ) =+ 6 12 (cid:18) ( γ a a )( γ a a b ...b ) + ( a ↔ b ) (cid:19) − δ a a b b ( γ a a )( γ b b ) −
48 12 (cid:18) δ a b ( γ a )( γ a a b ...b ) + ( a ↔ b ) (cid:19) + 96 δ a ...a b ...b ( γ a )( γ b ) − γ e )( γ e a ...a b ...b )+ 72 δ a a b b ( γ e )( γ e a a b b ) − δ a ...a b ...b ( γ e )( γ e ) ( γ a ...a e e )( γ b ...b e e ) = − δ a ...a b ...b ( γ e )( γ e )+ 36 δ a b ( γ e )( γ e a a b b ) − δ a b ( γ a a )( γ b b ) −
12 12 (cid:18) ( γ a )( γ a a b ...b ) + ( a ↔ b ) (cid:19) + 108 δ a a b b ( γ a )( γ b )( γ a e )( γ b ...b e ) =+ 1 ( γ e )( γ e a b ...b )+ 12 δ a b ( γ b )( γ b b ) ( γ a e )( γ b e ) = + 1 ( γ a )( γ b ) − δ a b ( γ e )( γ e )( γ a ...a e )( γ b ...b e ) = − η a b ( γ a a )( γ a b ...b ) − η a b ( γ a )( γ a a b ...b ) − δ a ...a c ...c η c b η c b η c b ( γ a )( γ b b )+ 240 δ a ...a c ...c η c b η c b η c b ( γ a )( γ b b )+ 140 η a b ( γ [ a )( γ a a b ...b ] ) − δ a a c c η c b η c b ( γ e )( γ e a a b ...b ) ( γ a e )( γ b ...b e ) = − γ [ a )( γ b ...b ] ) − η a b ( γ e )( γ e b ...b )+ 1 ( γ a )( γ b ...b ) E. Eleven-dimensional superspace
In this section we review the properties of on-shell eleven-dimensional superspace at lowest orderin the Planck length [33]. The theory thus obtained is equivalent to CJS supergravity [1].The non-zero superfield components are as follows, G abαβ = − i ( γ ab ) αβ T αβ f = − i ( γ f ) αβ T aαβ = − (cid:18) ( γ bcd ) αβ G abcd + 18 ( γ abcde ) αβ G bcde (cid:19) R αβab = i (cid:18) ( γ gh ) αβ G ghab + 124 ( γ abghij ) αβ G ghij (cid:19) R αabc = i (cid:0) ( γ a T bc ) α − γ [ b T c ] a ) α (cid:1) . (E.1)– 38 –he action of the spinorial derivative on the superfields reads, D α G abcd = 6 i ( γ [ ab | ) αǫ T | cd ] ǫ D α R abcd = d [ a | R α | b ] cd − T abǫ R ǫαcd + 2 T [ a | αǫ R ǫ | b ] cd D α T abβ = 14 R abcd ( γ cd ) αβ − D [ a T b ] αβ − T [ a | αǫ T [ b ] ǫβ . (E.2)The equations of motion for the field-strengths G , R and T are given by, D f G fa a a = − ǫ a a a b ...b c ...c G b ...b G c ...c ( γ a ) αǫ T abǫ = 0 R ab − η ab R = 112 (cid:18) G afgh G bfgh − η ab G fghi G fghi (cid:19) . (E.3) F. Tensor representation of a Young diagram
A Young diagram with n boxes, see [69] for a review, represents an irreducible representationof the symmetric group S n . It is possible to give explicit expressions for Young diagrams in theform of tensors. The method is more easily understood using a specific example. Consider atensor T a a a a without any a priori symmetry properties, and let us construct its projectiononto . Several symmetry operations will have to be applied on the tensor, but the Youngdiagram does not state which indices correspond to its different boxes. First one must determineall the standard tableaux , i.e. all the Young diagrams with numbered boxes, with increasingnumbers in all rows and columns. Different Young tableaux corresponding to the same Youngdiagram give equivalent but distinct representations of the symmetric group. The diagramhas three standard tableaux, , , and , to which correspond three tensors, T (1) , T (2) and T (3) respectively.To obtain the tensor corresponding to a given standard tableau, one must first symmetrize overthe indices indicated in each row, and then antisymmetrize over the indices indicated in eachcolumn. For example, ( T (1) ) a a a a will be obtained by first symmetrizing over the indices a , a and a , (cid:0) T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a (cid:1) , and then antisymmetrizing over a and a ,(Π (1) T ) a a a a = ( T (1) ) a a a a =18 (cid:0) T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a + T a a a a (cid:1) . The overall normalization above can be straightforwardly determined by imposing Π (1) Π (1) T =Π (1) T , where Π (1) T = T (1) is the projection of the tensor T onto the Young tableau .– 39 –or example the tensors T (1) and T (2) , associated with and respectively, obey thefollowing properties,( T (1) ) [ ab | c | d ] = 0 ( T (2) ) [ abc ] d = 0( T (1) ) [ a | bc | d ] = ( T (1) ) abcd ( T (2) ) [ ab ] cd = ( T (2) ) abcd ( T (1) ) a ( bc ) d = ( T (1) ) abcd ( T (2) ) ab ( cd ) = ( T (2) ) abcd . More generally, each T ( i ) has exactly three independent orderings of indices, which can be takento be T (1) a a a a , T (1) a a a a and T (1) a a a a . Any symmetry operation on the indices of T ( i ) can beexpressed as a linear combination of these three orderings, e.g., T (1) a ( a a a ) = T (1) a a a a + 13 T (1) a a a a + 13 T (1) a a a a T (1)[ a a ] a a = 12 T (1) a a a a + 12 T (1) a a a a + 0 T (1) a a a a . A tensor T projected onto a non-standard tableau can be expressed as a linear combinationof the three standard ones. For example it is straightforward (but tedious) to check that theprojection onto the non-standard tableau can be decomposed as,(Π (4) T ) a a a a = T (1) a a a a + T (1) a a a a + 0 T (1) a a a a + 0 T (2) a a a a − T (2) a a a a + T (2) a a a a + T (3) a a a a + 0 T (3) a a a a − T (3) a a a a . (F.1)Every other tableau (corresponding to the same Young diagram ) and any symmetry op-eration on the indices can be expressed as a linear combination of those nine elements. Theautomatization of general decompositions onto Young tablaux, such as the one above, has beenimplemented in the computer program [31].More generally a tensor T a a a a without any a priori symmetry properties can be decomposedinto ten Young tableaux, ⊗ |{z} T = | {z } T S ⊕ | {z } T (1 , , ⊕ | {z } T ′ (1 , ⊕ | {z } T ′′ (1 , , ⊕ |{z} T A , (F.2)where T (1) , T (2) and T (3) are the Young tableaux appearing on the right-hand side of (F.1)above, and correspond to the term 3 . The remaining Young tableaux in the decompositioncan be explicitly constructed using the same method.Consider now a tensor T with a symmetry structure given by, e.g., ⊗ . The previousdecomposition of ⊗ can also be used to decompose T into its irreducible components. Indeed,a tensor with structure ⊗ can be viewed as a particular set of symmetry operationsperformed on the indices of a tensor without any symmetry (i.e. with structure ⊗ ). Therefore T can be expressed as a linear combination of the tensors already used in the decomposition(F.2). – 40 –he following example shows the decomposition of the symmetric product of two threeforms H , T a ...a := H a a a H a a a −→ ⊗ S = ⊕ . There are five standard tableaux corresponding to each of the Young diagrams , . Thetensors corresponding to these Young tableaux can be denoted by T (1) , . . . T (5) and T ′ (1) , . . . T ′ (5) ,respectively. In the particular example above, it can be shown that, H a a a H a a a = T (1) a a a a a a + T ′ (1) a a a a a a + T ′ (1) a a a a a a − T ′ (1) a a a a a a , i.e. only the tensors T (1) and T ′ (1) , corresponding to and respectively, enter the decom-position. – 41 – eferences [1] E. Cremmer, B. Julia, and J. Scherk. Supergravity theory in 11 dimensions. Phys. Lett. ,B76:409–412, 1978.[2] E. Witten. String theory dynamics in various dimensions.
Nucl. Phys. , B443:85–126, 1995.[3] Kasper Peeters, Pierre Vanhove, and Anders Westerberg. Supersymmetric higher-derivative actionsin ten and eleven dimensions, the associated superalgebras and their formulation in superspace.
Class. Quant. Grav. , 18:843–890, 2001.[4] M. de Roo, H. Suelmann, and A. Wiedemann. The supersymmetric effective action of the heteroticstring in ten-dimensions.
Nucl. Phys. , B405:326–366, 1993.[5] Cumrun Vafa and Edward Witten. A one loop test of string duality.
Nucl. Phys. , B447:261–270,1995.[6] M. J. Duff, James T. Liu, and R. Minasian. Eleven-dimensional origin of string / string duality: Aone-loop test.
Nucl. Phys. , B452:261–282, 1995.[7] Michael B. Green and John H. Schwarz. Supersymmetrical Dual String Theory. 2. Vertices andTrees.
Nucl. Phys. , B198:252–268, 1982.[8] David J. Gross and Edward Witten. Superstring Modifications of Einstein’s Equations.
Nucl.Phys. , B277:1, 1986.[9] Marcus T. Grisaru, A. E. M. van de Ven, and D. Zanon. Four Loop beta Function for the N=1 andN=2 Supersymmetric Nonlinear Sigma Model in Two-Dimensions.
Phys. Lett. , B173:423–428, 1986.[10] Marcus T. Grisaru and D. Zanon. σ Model Superstring Corrections to the Einstein-hilbert Action.
Phys. Lett. , B177:347–351, 1986.[11] M. D. Freeman, C. N. Pope, M. F. Sohnius, and K. S. Stelle. Higher Order σ Model Countertermsand the Effective Action for Superstrings.
Phys. Lett. , B178:199–204, 1986.[12] David J. Gross and John H. Sloan. The Quartic Effective Action for the Heterotic String.
Nucl.Phys. , B291:41–89, 1987.[13] Giuseppe Policastro and Dimitrios Tsimpis. R , purified. Class. Quant. Grav. , 23:4753–4780, 2006.[14] Michael B. Green and John H. Schwarz. Supersymmetrical Dual String Theory. 3. Loops andRenormalization.
Nucl. Phys. , B198:441–460, 1982.[15] Kasper Peeters, Pierre Vanhove, and Anders Westerberg. Chiral splitting and world-sheetgravitinos in higher- derivative string amplitudes.
Class. Quant. Grav. , 19:2699–2716, 2002.[16] David M. Richards. The One-Loop Five-Graviton Amplitude and the Effective Action.
JHEP ,10:042, 2008.[17] David M. Richards. The One-Loop H R and H ( ∇ H ) R Terms in the Effective Action.
JHEP ,10:043, 2008.[18] James T. Liu and Ruben Minasian. Higher-derivative couplings in string theory: dualities and theB-field.
Nucl. Phys. , B874:413–470, 2013.[19] P. S. Howe and D. Tsimpis. On higher-order corrections in M theory.
JHEP , 09:038, 2003.[20] Yoshifumi Hyakutake and Sachiko Ogushi. R corrections to eleven dimensional supergravity viasupersymmetry. Phys. Rev. , D74:025022, 2006. – 42 –
21] Yoshifumi Hyakutake and Sachiko Ogushi. Higher derivative corrections to eleven dimensionalsupergravity via local supersymmetry.
JHEP , 02:068, 2006.[22] Yoshifumi Hyakutake. Toward the Determination of R F Terms in M-theory.
Prog. Theor. Phys. ,118:109, 2007.[23] Kasper Peeters, Jan Plefka, and Steffen Stern. Higher-derivative gauge field terms in the M-theoryaction.
JHEP , 08:095, 2005.[24] Stanley Deser and D. Seminara. Counterterms / M theory corrections to D = 11 supergravity.
Phys. Rev. Lett. , 82:2435–2438, 1999.[25] Stanley Deser and D. Seminara. Tree amplitudes and two loop counterterms in D = 11supergravity.
Phys. Rev. , D62:084010, 2000.[26] Stanley Deser and D. Seminara. Graviton-form invariants in D=11 supergravity.
Phys. Rev. ,D72:027701, 2005.[27] Michael B. Green, Michael Gutperle, and Pierre Vanhove. One loop in eleven dimensions.
Phys.Lett. , B409:177–184, 1997.[28] J. G. Russo and A. A. Tseytlin. One-loop four-graviton amplitude in eleven-dimensionalsupergravity.
Nucl. Phys. , B508:245–259, 1997.[29] Michael B. Green, Hwang h. Kwon, and Pierre Vanhove. Two loops in eleven dimensions.
Phys.Rev. , D61:104010, 2000.[30] U. Gran. GAMMA: A Mathematica package for performing Gamma-matrix algebra and Fierztransformations in arbitrary dimensions. 2001.[31] B. Sou`eres. In progress.[32] E. Cremmer and S. Ferrara. Formulation of eleven-dimensional supergravity in superspace.
Phys.Lett. , B91:61, 1980.[33] L. Brink and P. Howe. Eleven-dimensional supergravity on the mass shell in superspace.
Phys.Lett. , B91:384–386, 1980.[34] N. Dragon. Torsion and curvature in extended supergravity.
Z. Phys. , C2:29–32, 1979.[35] R. D’Auria and P. Fre. Geometric Supergravity in d = 11 and Its Hidden Supergroup.
Nucl. Phys. ,B201:101–140, 1982. [Erratum: Nucl. Phys.B206,496(1982)].[36] Antonio Candiello and Kurt Lechner. Duality in supergravity theories.
Nucl. Phys. , B412:479–501,1994.[37] L. Bonora, P. Pasti, and M. Tonin. Chiral Anomalies in Higher Dimensional SupersymmetricTheories.
Nucl. Phys. , B286:150–174, 1987.[38] L. Bonora, K. Lechner, M. Bregola, P. Pasti, and M. Tonin. A Discussion of the constraints in N=1SUGRA-SYM in 10-D.
Int. J. Mod. Phys. , A5:461–477, 1990.[39] Martin Cederwall, Bengt E. W. Nilsson, and Dimitrios Tsimpis. Spinorial cohomology andmaximally supersymmetric theories.
JHEP , 02:009, 2002.[40] M. V. Movshev, A. Schwarz, and Renjun Xu. Homology of Lie algebra of supersymmetries and ofsuper Poincare Lie algebra.
Nucl. Phys. , B854:483–503, 2012.[41] Friedemann Brandt. Supersymmetry algebra cohomology I: Definition and general structure.
J.Math. Phys. , 51:122302, 2010. – 43 –
42] Friedemann Brandt. Supersymmetry Algebra Cohomology: II. Primitive Elements in 2 and 3Dimensions.
J. Math. Phys. , 51:112303, 2010.[43] Friedemann Brandt. Supersymmetry algebra cohomology III: Primitive elements in four and fivedimensions.
J. Math. Phys. , 52:052301, 2011.[44] Friedemann Brandt. Supersymmetry algebra cohomology IV: Primitive elements in all dimensionsfrom D=4 to D=11.
J. Math. Phys. , 54:052302, 2013.[45] Martin Cederwall, Bengt E. W. Nilsson, and Dimitrios Tsimpis. The structure of maximallysupersymmetric Yang-Mills theory: Constraining higher-order corrections.
JHEP , 06:034, 2001.[46] Martin Cederwall, Bengt E. W. Nilsson, and Dimitrios Tsimpis. D = 10 super-Yang-Mills at O ( α ′ ). JHEP , 07:042, 2001.[47] Martin Cederwall, Bengt E. W. Nilsson, and Dimitrios Tsimpis. Spinorial cohomology of abelian d= 10 super-Yang-Mills at O( α ′ ). JHEP , 11:023, 2002.[48] P. S. Howe, S. F. Kerstan, U. Lindstrom, and D. Tsimpis. The deformed M2-brane.
JHEP , 09:013,2003.[49] Dimitrios Tsimpis. 11D supergravity at O ( l ). JHEP , 10:046, 2004.[50] Martin Cederwall, Ulf Gran, Bengt E. W. Nilsson, and Dimitrios Tsimpis. Supersymmetriccorrections to eleven-dimensional supergravity.
JHEP , 05:052, 2005.[51] Renjun Xu, Albert Schwarz, and Michael Movshev. Integral invariants in flat superspace.
Nucl.Phys. , B884:28–43, 2014.[52] Chi-Ming Chang, Ying-Hsuan Lin, Yifan Wang, and Xi Yin. Deformations with MaximalSupersymmetries Part 1: On-shell Formulation. 2014.[53] Chi-Ming Chang, Ying-Hsuan Lin, Yifan Wang, and Xi Yin. Deformations with MaximalSupersymmetries Part 2: Off-shell Formulation.
JHEP , 04:171, 2016.[54] Paul S. Howe. Private communication.[55] Nathan Berkovits. Towards covariant quantization of the supermembrane.
JHEP , 09:051, 2002.[56] P. S. Howe. Pure spinors, function superspaces and supergravity theories in ten-dimensions andeleven-dimensions.
Phys. Lett. , B273:90–94, 1991.[57] Martin Cederwall. Pure spinor superfields – an overview.
Springer Proc. Phys. , 153:61–93, 2014.[58] R. D’Auria, P. Fre, P. K. Townsend, and P. van Nieuwenhuizen. Invariance of Actions, Rheonomyand the New Minimal N = 1 Supergravity in the Group Manifold Approach. Annals Phys. ,155(CERN-TH-3495):423, 1984.[59] S. James Gates, Jr., Marcus T. Grisaru, Marcia E. Knutt-Wehlau, and Warren Siegel. Componentactions from curved superspace: Normal coordinates and ectoplasm.
Phys. Lett. , B421:203–210,1998.[60] Paul S. Howe, O. Raetzel, and E. Sezgin. On brane actions and superembeddings.
JHEP , 08:011,1998.[61] Igor A. Bandos, Dmitri P. Sorokin, and Dmitrii Volkov. On the generalized action principle forsuperstrings and supermembranes.
Phys. Lett. , B352:269–275, 1995.[62] Sergei M. Kuzenko and Gabriele Tartaglino-Mazzucchelli. Conformal supergravities asChern-Simons theories revisited.
JHEP , 03:113, 2013. – 44 –
63] Sergei M. Kuzenko and Joseph Novak. On supersymmetric Chern-Simons-type theories in fivedimensions.
JHEP , 02:096, 2014.[64] N. Berkovits and P. S. Howe. The Cohomology of superspace, pure spinors and invariant integrals.
JHEP , 06:046, 2008.[65] Dan Freed, Jeffrey A. Harvey, Ruben Minasian, and Gregory W. Moore. Gravitational anomalycancellation for M theory five-branes.
Adv. Theor. Math. Phys. , 2:601–618, 1998.[66] Paul S. Howe and U. Lindstrom. Higher Order Invariants in Extended Supergravity.
Nucl. Phys. ,B181:487–501, 1981.[67] Dimitrios Tsimpis. Curved 11D supergeometry.
JHEP , 11:087, 2004.[68] A. M. Cohen, M. van Leeuwen, and B. Lisser.
LiE v.2.2
A computer algebra package for Liegroup computations (1998).[69] S. A. Fulling, Ronald C. King, B. G. Wybourne, and C. J. Cummins. Normal forms for tensorpolynomials. 1: The Riemann tensor.
Class. Quant. Grav. , 9:1151–1197, 1992., 9:1151–1197, 1992.