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Eleven-Dimensional Supergravity on a Manifold with Boundary


hep-th/9603142 IASSNS-HEP-96/17 PUPT-1597

arXiv:hep-th/9603142v1 21 Mar 1996

ELEVEN-DIMENSIONAL SUPERGRAVITY ON A MANIFOLD WITH BOUNDARY

Petr Hoˇava? r Joseph Henry Laboratories, Princeton University Jadwin Hall, Princeton, NJ 08544, USA and Edward Witten? School of Natural Sciences, Institute for Advanced Study Olden Lane, Princeton, NJ 08540, USA In this paper, we present a systematic analysis of eleven-dimensional supergravity on a manifold with boundary, which is believed to be relevant to the strong coupling limit of the require a re?nement of the standard Green-Schwarz mechanism for their cancellation. This uniquely determines the gauge group to be a copy of E8 for each boundary component, ?xes the gauge coupling constant in terms of the gravitational constant, and leads to several striking new tests of the hypothesis that there is a consistent quantum M -theory with eleven-dimensional supergravity as its low energy limit. E8 ×E8 heterotic string. Gauge and gravitational anomalies enter at a very early stage, and

March, 1996
? ?

horava@puhep1.princeton.edu. Research supported in part by NSF Grant PHY90-21984. witten@sns.ias.edu. Research supported in part by NSF Grant PHY95-13835.

1. Introduction In a previous paper [1], we proposed that the strong coupling limit of the tenR10 × S1 /Z2 = R10 × I (I is the unit interval), with the gauge ?elds entering via tendimensional E8 × E8 heterotic string is eleven-dimensional M -theory compacti?ed on

dimensional vector multiplets that propagate on the boundary of space-time. This implies

in particular that there must exist a supersymmetric coupling of ten-dimensional vector multiplets on the boundary of an eleven-manifold to the eleven-dimensional supergravity multiplet propagating in the bulk. The purpose of the present paper is to explore this coupling. In doing so, one comes quickly to a puzzle. The supergravity action in bulk is ? 1 2κ2 √ d11 x g R + . . . ,
M 11

(1.1)

with M 11 being the eleven-dimensional space-time, and “. . .” being the terms involving fermions and the bosonic three-form ?eld. The supergauge action on the boundary is ? 1 4λ2 √ d10 x g tr F 2 + . . . ,
M 10

(1.2)

where M 10 is the boundary (or a component of the boundary) of M 11 , and F is the ?eld strength of the gauge ?elds that propagate on M 10 . (For E8 × E8 , “tr” is as usual 1/30 of the trace Tr in the adjoint representation.) In the above formulas, κ and λ are

the gravitational and gauge coupling constants. From those constants one can make a dimensionless number η = λ6 /κ4 . The question is what determines the value of η. Note that there is no dilaton or other scalar whose expectation value controls the value of η. In fact, there is no scalar ?eld at all in the theory, propagating either in bulk or on the boundary; in going to strong coupling, the dilaton of the perturbative heterotic string is reinterpreted as the radius of S1 /Z2 . Since string theory has no adjustable parameter corresponding to η, the strong coupling limit of the E8 × E8 heterotic string, if it does have the eleven-dimensional interpretation proposed in our previous paper, must give a de?nite value for η. In fact, we will argue in this paper that by looking more precisely at gravitational and gauge anomalies (which were already used in the previous paper), one can determine η. We get η = 128π 5 , 1 (1.3)

or equivalently λ2 = 2π(4πκ2 )2/3 . (1.4)

In the remainder of this introduction, we sketch the argument that will be used to determine η, and also sketch the other main qualitative results of this paper. The reason for presenting such a detailed sketch ?rst is that the supergravity calculation that occupies the remainder of the paper is unavoidably rather complicated. We recall that anomalies in ten dimensions are described by a formal twelve-form I12 (R, F1 , F2 ) that is a sixth order homogeneous polynomial in the Riemann tensor R and the ?eld strengths F1 and F2 in the two E8 ’s. It has the general form I12 (R, F1 , F2 ) = A(R) + B(R, F1 ) + B(R, F2 ), (1.5)

where A(R) is the contribution of the supergravity multiplet, and B(R, Fi ), for i = 1, 2, is the contribution of the gluinos of the ith E8 . In [1], we introduced I12 (R, F ) = so that I12 (R, F1 , F2 ) = I12 (R, F1 ) + I12 (R, F2 ). (1.7) 1 A(R) + B(R, F ), 2 (1.6)

The idea here is that from the eleven-dimensional point of view, the gauge and gravitational anomaly is localized on the boundary of space-time, and the two terms on the right of E8 ’s, the one propagating on a given boundary component is naturally the only one that contributes to the anomaly form of that component. Anomaly cancellation for the perturbative heterotic string involves a factorization I12 = I4 I8 , (1.8) (1.7) are the contributions of the two components of the boundary of R10 × I. Of the two

2 2 where I4 = tr R2 ? tr F1 ? tr F2 and I8 is an eight-form given by a lengthy quartic poly-

nomial in R and the Fi . As was explained in [1], I12 has an analogous factorization I12 = I4 I8 , with 1 I4 (R, F ) = tr R2 ? tr F 2 2 1 1 1 I8 (R, F ) = ? I4 (R, F )2 + ? tr R4 + (tr R2 )2 . 4 8 32 2

(1.9)

(1.10)

(This way of writing the formula for I8 , which was also noticed by M. Du? and R. Minasian, has a rationale that will become clear in section three.) It was proposed in [1] that this factorization of I12 would permit an extension of the Green-Schwarz anomaly cancellation mechanism to M -theory on eleven-dimensional manifolds with boundary. The Green-Schwarz mechanism in ten dimensions depends on the existence in string theory of a two-form ?eld B whose gauge-invariant ?eld strength H obeys dH = I4 . (1.11)

Such an equation (with only the tr F 2 term in I4 ) holds even in the minimal ten-dimensional supergravity [2,3]. In addition, there are “Green-Schwarz interaction terms,” present in the string theory but not in the minimal low energy supergravity theory, of the form ?L = B ∧ I8 . (1.12)

The combination of (1.11) and (1.12) gives a classical theory that is not gauge invariant, with an anomaly constructed from the twelve-form I12 = I4 I8 . The minimal classical supergravity theory is gauge invariant because the anomalous fermion loops and the GreenSchwarz terms are both absent, and the string theory is gauge invariant because they are both present and the anomalies cancel between them. Now let us discuss how the story will work in eleven dimensions. In doing so, and in most of this paper, we will use an orbifold approach in which we work on an eleven-manifold M 11 with a Z2 symmetry whose ?xed points are of codimension one; alternatively, one can take the quotient and work on the manifold-with-boundary X = M 11 /Z2 , whose boundary points are the Z2 ?xed points in M 11 . In general, the formulation in terms of a manifold with boundary is convenient intuitively, and the orbifold formulation is convenient for calculation. Rather than a two-form B, the eleven-dimensional supergravity multiplet has a threeform ?eld C (denoted by A(3) in our previous paper [1]), whose ?eld strength is a four-form G.1 In the absence of boundaries, G obeys the usual Bianchi identity dG = 0. The analog of (1.11) will have to be a contribution to dG supported at the Z2 ?xed points. As dG is a ?ve-form, we will have to promote the four-form I4 to a ?ve-form supported on the ?xed
1

Following conventions in [2] which have become standard in eleven-dimensional supergravity,

we de?ne GIJ KL = ?I CJ KL ± 23 terms, though the normalization is somewhat unusual. We also de?ne dGIJ KLM = ?I GJ KLM + cyclic permutations of IJ KLM .

3

point set, so that it can appear as a correction to the Bianchi identity. To write such a ?ve-form, one supposes that the ?xed point set is de?ned locally by an equation x11 = 0, and one multiplies by the closed one-form δ(x11 )dx11 to promote the four-form I to a ?ve-form. Thus, the eleven-dimensional analog of the ten-dimensional equation dH = I4 will be an equation dG = cδ(x11 )dx11 I4 , with some constant c. In section two, we will determine the precise equation to be √ 3 2 κ 2/3 1 dG11 IJKL = ? δ(x11 ) tr F[IJ FKL] ? tr R[IJ RKL] . 2π 4π 2

(1.13)

Here F is of course the ?eld strength of the gauge ?eld propagating at x11 = 0, and directly only the tr F ∧ F term in (1.13); the tr R ∧ R term is a sort of higher order tr F[IJ FKL] = (1/24)tr FIJ FKL ± permutations. Actually, in section two, we will see

correction that we infer because it is needed for anomaly cancellation. (Analogously, in ten dimensions, the tr F ∧ F term is required by supersymmetry, and the tr R ∧ R term is an O(α′ ) stringy correction needed for anomaly cancellation.) At this stage the question is, what are the Green-Schwarz terms? In the familiar tendimensional story, because the Green-Schwarz terms are unconstrained by supersymmetry, the Green-Schwarz mechanism makes no general prediction (independent of anomalies or a detailed string model) about what I8 should be. In eleven dimensions, the story will be quite di?erent because the terms analogous to the Green-Schwarz terms are independently known. One of these terms is simply the supergravity. This term, discovered when the model was ?rst constructed [8], has always seemed enigmatic because the rationale behind its apparently “topological” nature was not clear. We feel that the role of this term in canceling anomalies – we explain in section three how C ∧ G ∧ G comes to play the role of a Green-Schwarz term – removes some of C ∧ G ∧ G interaction of eleven-dimensional

the enigma. The

anomalies, but to cancel also the gravitational anomalies requires an additional interaction. This is an eleven-dimensional interaction C ∧ X8 (R), (1.14)

C ∧ G ∧ G is the only “Green-Schwarz” interaction involved in canceling gauge

M 11

with X8 (R) an eight-form constructed as a quartic polynomial in the Riemann tensor. This interaction is known in two ways. (1) Upon dimensional reduction on S1 , it turns into a B ∧ X8 interaction which can be computed as a one-loop e?ect in Type IIA superstring 4

theory [7]. The one loop calculation is exact since a dilaton dependence of the B ∧ X8

coupling would spoil gauge invariance; because it is exact, it can be extrapolated to eleven dimensions and implies the existence of the interaction written in (1.14). (2) Alternatively, this coupling is needed to cancel one-loop anomalies on the ?ve-brane world-volume and thus permit the existence of ?ve-branes in the theory [4-6]. Happily, the two methods agree, with 1 1 X8 = ? tr R4 + (tr R2 )2 . 8 32 (1.15)

As we will see in section three, it is no coincidence that the combination of tr R4 and (tr R2 )2 that appears here also entered in (1.10). The fact that the terms analogous to Green-Schwarz terms are known independently of any discussion of space-time anomalies means that we get an a priori prediction for I8 . (We have no a priori prediction of I4 , as the coe?cients in (1.13) will essentially be adjusted to make anomaly cancellation possible.) We regard the success of this prediction as a compelling con?rmation that eleven-dimensional supergravity on a manifold with boundary is indeed related to ten-dimensional E8 × E8 heterotic string theory as proposed in [1].

Classical and Quantum Consistency The details that we have just explained of how anomaly cancellation works in eleven dimensions have other implications for the structure of the theory. The fact that, once one works on a manifold with boundary, some of the GreenSchwarz terms are present in the minimal supergravity Lagrangian means that the classical Lagrangian, including the vector supermultiplets on the boundary, is not gauge invariant. Thus, the theory with the supergravity multiplet in bulk and the vector multiplets on the boundary is only consistent as a quantum theory. The situation is rather di?erent from perturbative string theory, where since the Green-Schwarz terms arise at the one loop level, one has gauge invariance either classically (leaving out the anomalous chiral fermion loop diagrams and the e?ects of the Green-Schwarz terms) or quantum mechanically (including both of these). light on this. It means that the gauge kinetic energy, of order 1/λ2 , is a higher order correction, of relative order κ2/3 compared to the gravitational action, which is of order 1/κ2 . If one wants a fully consistent classical theory, one must ignore the gauge ?elds completely. Once one tries to include the supergauge multiplet, gauge invariance will fail 5 The relation λ2 ? κ4/3 between the gauge and gravitational couplings sheds a further

classically (in relative order κ2 ), and quantum anomalies are needed to compensate for this failure. Since the classical theory with the gauge ?elds is not going to be fully consistent, one has to expect peculiarities in constructing it. In our analysis in section four, we certainly ?nd such peculiarities. We will organize our discussion of the boundary interactions as an expansion in powers of κ2/3 . In order κ2/3 , things go smoothly, though the calculations are rather involved, roughly as in standard supergravity theories. Some novelties arise in order κ4/3 . In verifying invariance of the Lagrangian in that order, one has to cancel terms that are formally proportional to δ(0). The cancellation also involves adding to the Lagrangian new interactions (of relative order κ4/3 ) proportional to δ(0). We interpret the occurrence of δ(0) terms in the Lagrangian and the supersymmetry variations of ?elds as a symptom of attempting to treat in classical supergravity what really should be treated in quantum M -theory. In a proper quantum M -theory treatment, there would presumably be a built-in cuto? that would replace δ(0) by a ?nite constant times κ?2/9 . For instance, the cuto? might involve having the gauge ?elds propagate in a boundary layer, with a thickness of order κ2/9 , and not precisely on the boundary of space-time. Though the δ(0) terms formally cancel in order κ4/3 , one must expect further di?culties in higher order since, without knowing the correct way to cut o? the linear divergence that gave the δ(0) terms in order κ4/3 , there is some uncertainty in the determination of the correct structure in that order. One must suppose, by analogy with many other problems in physics, that underneath the cancellation of the linear divergences there might be a ?nite remainder, which could be extracted if one understood the correct cuto?. Without understanding the ?nite remainder, one should expect di?culty in proceeding to the next order. In any event, one of the things that happens in the next order – relative order κ2 – has already been explained. One runs into a failure of classical gauge invariance which must be canceled by quantum one-loop anomalies (which are also of relative order κ2 ). It is hard to believe that the classical discussion can usefully be continued to higher order, once the classical gauge invariance has failed and one has begun to run into conventional quantum loops. An attempt to continue the classical discussion would almost undoubtedly soon run into higher order divergences than the δ(0) that we described two paragraphs ago; for instance, one would very likely ?nd δ(0) terms in the supergravity transformation laws and δ(0)2 terms in the Lagrangian. 6

Despite the in?nities that arise in the construction, we hope and expect that the analysis of the anomalies is reliable. This should be analogous to the fact that anomalous loop diagrams can be reliably computed even in unrenormalizable e?ective theories, because the anomalies can be construed as an infrared e?ect and are independent of what cuto? one introduces. Summary To summarize, then, the lessons from our investigation, we will ?nd that anomaly cancellation of the ten-dimensional heterotic string has an elegant eleven-dimensional interpretation that sheds light on properties of the anomaly twelve-form that were not needed before. This sharpens the eleven-dimensional interpretation of the strongly coupled E8 ×E8 heterotic string, ?xing an otherwise unknown dimensionless parameter and adding to our con?dence that the eleven-dimensional description is correct. The gauge anomalies that arise in the classical discussion also give an indication – and not the only one – that the theory only really makes sense at the quantum level.

2. Correction to the Bianchi Identity Our eleven-dimensional conventions are as in [2]. We work with Lorentz signature ?+

+ . . . +. Vector indices will be written as I, J, K, and spinor indices as α, β, γ. We introduce

a frame ?eld eI m with the metric being gIJ = ηmn eI m eJ n . The gamma matrices are 32 × 32 real matrices obeying {ΓI , ΓJ } = 2gIJ . One also de?nes ΓI1 I2 ...In = Γ[I1 . . . ΓIn ] ≡ (1/n!)ΓI1 ΓI2 . . . ΓIn ± permutations. Spinor indices are raised and lowered with a real index in the gamma matrix Γα β one gets a symmetric tensor ΓIαβ = ΓIβα . All spinors I will be Majorana spinors; the symbol ψ α is simply de?ned by ψ α = Cαβ ψ β . The supergravity multiplet consists of the metric g, the gravitino ψIα , and a threeα antisymmetric tensor C obeying Cαβ = ?Cβα , C αβ Cβγ = δγ . In particular, by lowering an

form C (with ?eld strength G, normalized as in a previous footnote). The supergravity Lagrangian, up to terms quartic in the gravitino (which we will not need), is [8] LS = 1 κ2 1 1 1 √ d11 x g ? R ? ψ I ΓIJK DJ ψK ? GIJKL GIJKL 2 2 48 M 11 √ 2 J ? ψ I ΓIJKLM N ψN + 12ψ ΓKL ψ M GJKLM 192 √ 2 I1 I2 ...I11 ? CI1 I2 I3 GI4 ...I7 GI8 ...I11 . ? 3456 7

(2.1)

We work in 1.5 order formalism: the spin connection ? is formally regarded as an independent variable, and eventually set equal to the solution of the bulk equations of motion. The Riemann tensor is the ?eld strength constructed from ?. The transformation laws of local supersymmetry read 1 m ηΓ ψI 2√ 2 δCIJK = ? ηΓ[IJ ψK] 8 √ 2 J ΓI JKLM ? 8δI ΓKLM ηGJKLM + . . . . δψI = DI η + 288 δeI m =

(2.2)

(The . . . are three fermi terms in the transformation law of ψ, often absorbed in a de?nition of “supercovariant” objects; we will not need them.) We suppose that there is a Z2 symmetry acting on M 11 , with codimension one ?xed points. We let M 10 be a component of the ?xed point set; we will study the physics near M 10 . We suppose that the ?elds are required to be invariant under the Z2 ; this means that we could pass to the manifold-with-boundary X = M 11 /Z2 (with boundary M 10 ), but that will not be particularly convenient. If M 10 is de?ned locally by an equation an appropriate lifting of the Z2 action to spinors and the three-form) the supersymmetries that commute with the Z2 action are generated by spinor ?elds η on M 11 that obey Γ11 η = η at x11 = 0. Z2 invariance of the gravitino means that Γ11 ψA = ψA , Γ11 ψ11 = ?ψ11 . A = 1, . . . , 10 (2.4) (2.3) x11 = 0, x11 being one of the coordinates (and the Z2 acting by x11 → ?x11 ), then (with

As in (2.4), we will use A, B, C, D = 1, . . . , 10 for indices tangent to M 10 . For the threeform C, because it is odd under parity (this follows from the CGG interaction in (2.1)), Z2 invariance means that CBCD = 0 at x11 = 0. A gauge-invariant statement that follows from this is that GABCD = 0 at x11 = 0, (2.5)

or in other words, the pull-back of the di?erential form G to M 10 vanishes. We will eventually ?nd a sort of modi?cation of this statement in order κ2/3 . 8

The vector supermultiplets, which propagate on M 10 , consist of the E8 gauge ?eld A (with ?eld strength FCD = ?C AD ? ?D AC + [AC , AD ]) and fermions (gluinos) χ in the adjoint representation, obeying Γ11 χ = χ. The minimal Yang-Mills Lagrangian is LY M = ? 1 λ2 √ d10 x g tr
M 10

1 1 FAB F AB + χΓA DA χ . 4 2

(2.6)

√ (Here d10 x g is understood as the Riemannian measure of M 10 , using the restriction to M 10 of the metric on M 11 .) In (2.6), λ is the gauge coupling constant. We will ultimately see that λ ? κ2/3 , so that LY M is of order κ2/3 relative to LS . The supersymmetry transformation laws are 1 ηΓA χa 2 (2.7) 1 AB a a δχ = ? Γ FAB η. 4 We have here made explicit an index a = 1, . . . , 248 labeling the adjoint representation of δAa = A

E8 . We de?ne an inner product by X a X a = tr X 2 = (1/30)TrX 2 , with Tr the trace in the adjoint representation. We wish to add additional interactions to the above and modify the supersymmetry transformation laws so that LS + LY M + . . . will be locally supersymmetric. The ?rst steps the supergauge multiplet. As in any coupling of matter to supergravity, the variation of
A LY M under local supersymmetry contains terms DA ηSY M , re?ecting the fact that LY M

are as follows. Let TY M and SY M be the energy-momentum tensor and supercurrent of

is only invariant under (2.7) if η is covariantly constant, and ηψTY M , coming from the variation of LY M under a local supersymmetry transformation of the metric. To cancel these variations, it is necessary – as usual in supergravity – to add an interaction ψSY M . In the case at hand, this interaction is L1 = ? 1 4λ2 √ a d10 x g ψ A ΓBC ΓA FBC χa . (2.8)

M 10

A small calculation shows that the variation of L1 under local supersymmetry cancels the DηχF and ηψχDχ variations of LY M , and also cancels part of the ηψF 2 variation. The uncanceled variation turns out, after some gamma matrix gymnastics, to be ?= 1 16λ2 √ a a d10 x g ψ A ΓABCDE FBC FDE η. (2.9)

M 10

Rather as in the coupling of the ten-dimensional vector multiplet to ten-dimensional supergravity [2], there is no way to cancel this variation by adding to the Lagrangian 9

additional matter couplings. A peculiar mixing of the supergravity and matter multiplets is needed. When one veri?es the local supersymmetry of the eleven-dimensional supergravity Lagrangian LS , it is necessary among other things to check the cancellation of the ηGDψ and DηGψ terms. In this veri?cation, it is necessary to integrate by parts and use the Bianchi identity dG = 0.2 To cancel ?, one must modify the Bianchi identity to read √ κ2 a a dG11 ABCD = ?3 2 2 δ(x11 )F[AB FCD] . λ (2.10)

This correction to the Bianchi identity adds an extra variation of LS that precisely cancels ?. Much as in the analogous story in ten dimensions, (2.10) implies that the three-form C is not invariant under Yang-Mills gauge transformations. To determine the gauge transformation law of C, it is convenient to solve the modi?ed Bianchi identity by introducing 2 ωBCD = tr AB (?C AD ? ?D AC ) + AB [AC , AD ] + cyclic permutations of B, C, D . 3 (2.11) Thus ?C ωBCD + cyclic permutations = 6 tr F[AB FCD] . (2.12)

The Bianchi identity can then be solved by modifying the de?nition of G11 ABC , the new de?nition being κ2 δ(x11 )ωABC . G11 ABC = (?11 CABC ± 23 permutations) + √ 2 2λ Under an in?nitesimal gauge transformation δAa = ?DA ?a , ω transforms by A δωABC = ?A (tr ?FBC ) + cyclic permutations of A, B, C, (2.14) (2.13)

so gauge invariance of G11 ABC holds precisely if the three-form C transforms under gauge transformations by κ2 δC11 AB = ? √ δ(x11 ) tr ?FAB . 6 2λ2 (2.15)

2

This occurs when one varies the interaction ψ I ΓIJ KLMN ψN GJ KLM with δψ I ? DI η. To

cancel other variations, one must integrate by parts so that the DI acts on ψN instead of on η. The integration by parts gives a term proportional to dGIJ KLM .

10

A correction to the supersymmetry transformation law of C11 BC is also necessary. It can be determined by requiring that the supersymmetry variation of G11 ABC be gaugeinvariant (otherwise this variation gives gauge non-invariant, uncancellable terms in the supersymmetry variation of the Lagrangian) and is κ2 tr (AB δAC ? AC δAB ) , δC11 BC = ? √ 6 2λ2 (2.16)

where on the right δA is the standard supergravity transformation law given in (2.7). With this correction to δC, the correction to δG is κ2 a δ(x11 ) (ηΓA χa FBC + cyclic permutations of A, B, C) . δG11 ABC = √ 2 2λ Boundary Behavior There is another sense in which we can “solve the Bianchi identity.” We can ask, compatibly with the equation of motion, what can be the behavior near x11 = 0 of a G ?eld that obeys the corrected Bianchi identity found above, which was √ κ2 a a dG11 ABCD = ?3 2 2 δ(x11 )F[AB FCD] . λ (2.18) (2.17)

How can dG acquire such a delta function? G itself cannot have a delta function at x11 = 0, as that would not be compatible with the equations of motion. However, as G is odd under x11 → ?x11 , it is natural for GABCD to have a step function discontinuity at x11 = 0, giving a delta function in dG. In fact, GABCD must have a jump at x11 = 0 given precisely by 3 κ2 a a GABCD = ? √ 2 ?(x11 )F[AB FCD] + . . . . λ 2

(2.19)

Here ?(x11 ) is 1 for x11 > 0 and ?1 for x11 < 0; the . . . are terms that are regular near behavior required by the modi?ed equations of motion and Bianchi identity.

x11 = 0 and therefore (since G is odd under x11 → ?x11 ) vanish at x11 = 0. This is the This discontinuity means that GABCD does not have a well-de?ned limiting value as

of the gauge ?elds at x11 = 0.

x11 → 0. However, G2 has such a limit, which moreover is determined by (2.19) in terms There is another interesting way to think about (2.19). In this paper we are working

“upstairs” on a smooth eleven-manifold M 11 , and requiring Z2 invariance. It is natural conceptually (though sometimes less convenient computationally) to work “downstairs” 11

on the manifold-with-boundary X = M 11 /Z2 . In that case, it is not natural to add a correction to dG supported at the boundary of X (that is, at x11 = 0). More natural is to impose a boundary condition that has the same e?ect. Assuming that one identi?es X with the portion of M 11 with x11 > 0, the requisite boundary condition is simply 3 κ2 a a GABCD |x11 =0 = ? √ 2 F[AB FCD] . 2λ (2.20)

(If one identi?es X with the x11 < 0 portion of M 11 , one would want the opposite sign in (2.20).) The idea here is that, since dG = 0, the integration by parts explained in the footnote just before (2.10) no longer picks up a delta function term, but (since there is now a boundary) it does pick up a boundary term that has the same e?ect.3 Thus, in working downstairs on X, G has a well-de?ned boundary value given by (2.20) (or the same expression with opposite sign if one picks orientations opppositely). In working upstairs on M , G does not quite have a well-de?ned value at x11 = 0, but G2 does.

3. Analysis of Anomalies The most important conclusions of the last section are the gauge transformation law (2.15) for the three-form C, and the formula (2.19) for the behavior of G near M 10 . We will now put these together to get an eleven-dimensional view of gauge and gravitational anomalies. for the two-form B of string theory, and (2.19) will turn the “Chern-Simons interaction” C ∧ G ∧ G of eleven-dimensional supergravity into a Green-Schwarz term. We recall that the CGG interaction is, to be precise, a term √ 2 W =? ?M1 M2 ...M11 CM1 M2 M3 GM4 ...M7 GM8 ...M11 . 3456κ2 M 11 The idea is that (2.15) is analogous to the gauge transformation law δB ? tr ?F

(3.1)

The variation of W under an arbitrary variation of C is therefore √ 2 δW = ? ?M1 M2 ...M11 δCM1 M2 M3 GM4 ...M7 GM8 ...M11 . 1152κ2 M 11
3

(3.2)

In working on X = M 11 /Z2 instead of M 11 , one should replace the 1/κ2 in (2.1) by 2/κ2 ,

because one is integrating (2.1) over a space of half the volume. This factor of 2 goes into verifying the normalization of (2.20).

12

Given that C is not invariant under gauge transformations, neither is W . Using (2.15) for the gauge variation of C, we get for the gauge variation of W δW = ? 1 2304λ2
M 10 a ?M1 M2 ...M10 ?a FM1 M2 GM3 ...M6 GM7 ...M10 .

(3.3)

To proceed further, we need the value of G2 at x11 = 0. This is given by (2.19) in the orbifold approach or equivalently by the boundary condition (2.20) if one works “downstairs.” Either way, one gets δW = ? κ4 128λ6
a b b c c ?M1 M2 ...M10 ?a FM1 M2 FM3 M4 FM5 M6 FM7 M8 FM9 M10 .

(3.4)

M 10

So, as promised, the classical theory is not gauge invariant. There is no way to cure this at the classical level. The only recourse is to quantum anomalies. The anomalous variation of the e?ective action Γ for ten-dimensional Majorana-Weyl fermions in an arbitrary representation of a simple gauge group is δΓ = 1 1 2 (4π)5 5! ?M1 M2 ...M10 Tr (?FM1 M2 FM3 M4 . . . FM9 M10 ) ,
M 10

(3.5)

with Tr being the trace in the fermion representation. The case that we are interested in is that the gauge group is E8 and the fermions are in the adjoint representation. In that case, one has the wonderful and unique (to E8 ) identity TrW 6 = (TrW 2 )3 /7200 (and likewise Tr?F 5 = Tr?F (TrF 2 )2 /7200). If furthermore we write, as is customary, tr W 2 = TrW 2 /30, then TrW 6 = (15/4)(tr W 2 )3 . In this case, therefore, the quantum anomaly (3.5) can be written δΓ = 15 8(4π)5 5! ?M1 M2 ...M10 tr (?FM1 M2 ) tr (FM3 M4 FM5 M6 ) tr (FM7 M8 FM9 M10 ). (3.6)
M 10

It therefore has the right structure to cancel (3.4) (recall that the metric on the Lie algebra was de?ned by ?a F a = tr ?F ). Implementing this cancellation, we learn ?nally that, as promised in the introduction, the gauge coupling is related to the gravitational coupling by λ2 = 2π 4πκ2
2/3

.

(3.7)

One might have expected that the analogs of the Green-Schwarz terms in the present discussion would be boundary interactions, that is interactions supported at x11 = 0. This is not the case, as we have seen. In fact, given a gauge variation of C proportional to δ(x11 ), the possible resulting gauge variation of a boundary interaction would necessarily be proportional to δ(0). Thus, the “Green-Schwarz terms” must be bulk interactions; this goes for the “Chern-Simons” CGG term and other terms discussed below. 13

3.1. Extension to Gravitational Anomalies We determined the gauge coupling by canceling the purely gauge anomalies at the boundary of the eleven-dimensional world. We would now like to include also the gravitational and mixed anomalies. From the above discussion, the anomaly four-form I4 is the four-form that appears (multiplied by δ(x11 )dx11 ) in the Bianchi identity for G. In our work so far, we have seen only a tr F 2 term in I4 , but in view of the known form of the ten-dimensional anomalies, the actual structure must be tr F 2 ? (1/2)tr R2 . Thus, the modi?ed Bianchi identity (2.10) should be replaced by √ κ2 1 a a dG11 ABCD = ?3 2 2 δ(x11 ) F[AB FCD] ? tr R[AB RCD] , λ 2

(3.8)

and the formula (2.19) for the behavior near x11 = 0 should correspondingly be replaced by 3 κ2 1 a a GABCD = ? √ 2 ?(x11 ) F[AB FCD] ? tr R[AB RCD] + . . . . 2 2λ (3.9)

There is also a corresponding local Lorentz transformation law of (3.9), analogous to the E8 gauge transformation law (2.15). The tr R2 terms in these formulas are not required by the low energy supergravity, but (since they are needed for anomaly cancellation, given the structure of the one-loop chiral anomalies), they must be present in the full M -theory. The situation is presumably analogous to what is seen for the perturbative heterotic string, where the tr R2 terms in the analogous formulas arise as corrections of order α′ . Note that the tr R2 correction will appear in (3.8) and (3.9) with the same coe?cient, since (3.9) is deduced from (3.8) by reasoning that was explained above. Having understood how I4 enters in eleven dimensions, we would like now to understand the origin of I8 , or equivalently to complete our understanding of the Green-Schwarz terms. We have already found one of the Green-Schwarz terms above – the long-familiar “Chern-Simons” interaction of eleven-dimensional supergravity. This particular interac2 tion gives a contribution to I8 that is a multiple of I4 , since the boundary behavior of G

tions, as explained at the end of the last subsection, and more precisely will have to be interactions of the form I= C ∧ a tr R4 + b(tr R2 )2 , 14 (3.10)

is G ? I4 , as we have seen. The other Green-Schwarz terms will have to be bulk interac-

M 11

these being the terms that have the right sort of gauge and local Lorentz variations to cancel chiral anomalies. Note that it is impossible to add to (3.10) terms that directly involve F , since the gauge ?elds propagate only on M 10 . It is also impossible for F -dependence to arise indirectly from the behavior of G near M 10 , since (3.10) is independent of G; the C ∧ G ∧ G has already been taken into account (with a coe?cient known from low energy symmetry of M -theory. supergravity), and a term C ∧ G ∧ tr R2 is not possible, as it would violate the parity I will contribute to I8 a term that involves R only, so we get the striking prediction

2 that I8 is a multiple of I4 plus an eight-form constructed only from R. At this point, it is

helpful to note that I8 = ? 1 4 1 tr F 2 ? tr R2 2
2

1 1 + ? tr R4 + (tr R2 )2 , 8 32

(3.11)

and thus has the expected form. This is a satisfying test of M -theory, as this structure of I8 has no known rationale in perturbative string theory. Actually, we can be more precise, since the interaction (3.10) is known (at least up to an overall multiplicative constant; ?xing this constant requires a more precise comparison of the normalizations of string theory and M -theory or a precise knowledge of the two-brane and ?ve-brane tensions in M -theory). As we explained in the introduction, the interaction (3.10) is known up to a constant multiple either from comparison to a one-loop calculation for Type IIA superstrings [7] or from anomaly cancellation for eleven-dimensional ?vebranes [4-6]. Either way, one ?nds that (3.10) is a multiple of √ 2 1 1 C ∧ ? tr R4 + (tr R2 )2 . 8 32 M 11 (3.12)

(4π)3 (4πκ2 )1/3

Thus, at least the relative coe?cient of tr R4 and (tr R2 )2 agrees with the “experimental” structure of I8 . This is again a real test of M -theory, since there is no perturbative string theory reason for this to work. The structure of (3.10) is deduced either via Type IIA perturbation theory or anomaly cancellation for eleven-dimensional ?ve-branes, and any known rationale for comparing the anomaly polynomial of the perturbative heterotic string to either of these involves M -theory. could be shifted by a scaling I4 → uI4 , I8 → u?1 I8 , without a?ecting the factorization I12 = I4 I8 . Modulo this imprecision in the de?nition of I8 , we have from M -theory a 15
2 Notice that the coe?cient ?1/4 of the I4 term in I8 is a matter of convention; it

complete a priori prediction for I8 , which amounts to a prediction for three numbers (I8 is a linear combination of four monomials (tr F 2 )2 , tr F 2 tr R2 , (tr R2 )2 , and tr R4 , but the coe?cient of one monomial can be scaled out as just explained). There are therefore three predictions, of which we have here veri?ed two; veri?cation of the last prediction requires a more precise comparison of di?erent conventions, as noted in the last paragraph.

4. Construction of the Lagrangian In this section, we will proceed with additional steps in the construction of the locally supersymmetric Lagrangian. The formula λ2 = 2π(4πκ2 )2/3 obtained in the last section is of some conceptual interest in organizing the computation. It shows that the theory has only one natural length scale, given by κ2/9 . Moreover, on dimensional grounds, the decomposition of the boundary interactions in terms with more and more matter ?elds is an expansion in powers of κ. The leading boundary interactions (the minimal Lagrangian LY M of the gauge multiplet and terms related to it by supersymmetry) are of order κ2/3 relative to the gravitational action. Formally, the construction of the locally supersymmetric classical action appears to be an expansion in integral powers of κ2/3 . Other exponents must arise in the actual quantum M -theory, since we will run into in?nities which, when cut o? in the quantum theory, must on dimensional grounds give anomalous powers of κ. There are two principal goals of the rather complicated computation performed in this section: (1) To add to our con?dence that the supersymmetric coupling of the vector multiplet on M 10 to the supergravity multiplet on M 11 does exist, by working out the classical construction of this coupling to the extent that it makes sense. (2) To exhibit the limits of the classical construction (beyond what is evident from the discussion of anomalies in section three) by showing how in?nities arise in order κ4/3 . In the computation, one can be guided to a certain extent by the ten-dimensional coupling of the vector and supergravity multiplets [2,3], to which our discussion must reduce at low energies in the appropriate limit. This gives clues to many of the terms that must be added to the Lagrangian and transformation laws. On the other hand, in the computation one de?nitely meets terms (involving D11 η, for instance) that vanish upon dimensional reduction to ten dimensions but must be canceled to achieve local supersymmetry in eleven 16

dimensions. Thus, the existence of the coupling we are constructed (and again, we believe that it only exists in full at the quantum level) goes well beyond ten-dimensional considerations. We will carry out the computation in three stages: (i) ?rst we complete the construction of the Lagrangian in order κ2/3 ; (ii) then we look at some terms in order κ4/3 ; (iii) ?nally we look systematically at all four-fermi variations in order κ2/3 . It might seem illogical to put (ii) before (iii). We have done this because (ii) is much simpler than (iii), and also gives an easy way to determine some of the transformation laws that are needed in (iii). Of course, we cannot hope for a full determination of the structure. Apart from requiring a much fuller knowledge of the quantum mechanics of M -theory than one has, the full structure is presumably non-polynomial, like the α′ expansion of perturbative string theory. Once one reaches a su?ciently high order in κ, one would require among other things a more complete knowledge of the low energy expansion of M -theory in bulk (including higher derivative interactions) in order to proceed. 4.1. Some New Interactions The boundary interactions (that is, interactions supported at x11 = 0) that we discussed in section two are the minimal super Yang-Mills action and the supercurrent coupling L0 = ? 1 2π(4πκ2 )2/3 √ d10 x g
M 10

1 1 1 a tr FAB F AB + tr χΓA DA χ + ψ A ΓBC ΓA FBC χa . 4 2 4 (4.1)

These terms are all of order κ2/3 compared to the supergravity action. There is precisely one more boundary interaction of the same order. To ?nd it, one can look at the terms of order F χGη in the supersymmetry variation of the Lagrangian. One source of such terms source comes as follows. We found in section two a correction (2.17) to the supersymmetry comes from the variation of the supercurrent interaction in (4.1) with δψ ? Gη. Another

variation of GABC 11 . The G2 ABC 11 term in the bulk supergravity action therefore picks up a new variation supported at x11 = 0; this term is again proportional to F χGη. These terms by themselves do not cancel. After a moderately lengthy computation, one ?nds that to cancel them one must add a new boundary interaction, √ 2 √ d10 x g χa ΓABC χa GABC 11 . L1 = 2 )2/3 96π(4πκ M 10 17

(4.2)

This term – and the veri?cation that the F χGη terms cancel – is quite similar to an analogous term and veri?cation in the ten-dimensional supergravity/Yang-Mills coupling. This actually completes the construction of the Lagrangian in order κ2/3 and veri?cation of local supersymmetry up to four-fermi terms, whose analysis we postpone to the next subsection. Instead we turn to something that is of conceptual interest and still relatively simple. In (4.2), we see an interaction in which GABC 11 is evaluated on M 10 , that is at x11 = 0. On the other hand, in (2.17), we found a term in the supersymmetry variation of GABC 11 that is proportional to δ(x11 ). If we combine the two, that is if we vary L1 according to (2.17), we get a result proportional to δ(0). This presumably should be interpreted as a linear divergence that is cut o? somehow in the quantum M -theory. For our present purposes, though, we will be pragmatic, and without worrying about precisely what δ(0) means, we will attempt to formally cancel the δ(0) terms. Obviously, to do this we need more sources of δ(0) terms. Since the term we want to cancel is proportional to χχχF η, there are two sources of terms that might cancel it. We could add to the gravitino variation an extra term δψA ? δ(x11 )χχη. When combined with the χF ψ interaction in L0 , it gives another term of the general form δ(0)χ3 F η. Finally, desired form. After another moderately long calculation, one ?nds that the new terms required in the gravitino variation are4 δψA = ? 1 576π κ 4π
2/3 B δ(x11 ) χa ΓBCD χa ΓA BCD η ? 6δA ΓCD η ,

one could add to the Lagrangian a term δ(0)χ4 , which will again have a variation of the

(4.3)

and that the new interaction required is Lχ = ? δ(0) 96(4π)10/3κ2/3 √ d10 x g χa ΓABC χa χb ΓABC χb .
M 10

(4.4)

The δ(0) presumably means that in the quantum theory this interaction is really of order κ?8/9 , that is, of order κ10/9 relative to the original supergravity action. We focussed here on a particular four-fermi term of relative order κ4/3 because it enabled us to exhibit in a simple fashion the “divergences” that appear in trying to construct the classical Lagrangian. In the next section, we look systematically at the four-fermi terms of order κ2/3 .
4

Here δ(x11 ) is understood as the delta function that transforms as a scalar under di?eomor-

phisms; this involves an implicit power of the 11-11 component of the frame ?eld e11 11 , and is required to match the transformation properties of δψA .

18

4.2. Four-Fermi Terms in Order κ2/3 To this order, the structure of the Lagrangian and the supersymmetry variation of the ?elds can be determined by canceling terms ? χχψη with one covariant derivative acting on one of the fermi ?elds. There are two natural classes of such terms, depending on whether the derivative is normal to the boundary, or acts along the boundary. First consider the class of terms containing the normal derivative, i.e. terms proportional to D11 η or D11 ψA . Such terms clearly vanish upon the dimensional reduction to ten dimensions: from the point of view of a ten-dimensional low-energy observer, they only contain contributions from massive Kaluza-Klein modes of ψA and η that decouple from the low-energy modes as the radius of the eleventh dimension goes to zero. In the present case, however, the cancellation of such terms is not automatic, and will help us determine some new additions to the Lagrangian. What are the possible sources of terms proportional to D11 ψA or D11 η? One source of such terms is generated by the correction (4.3) to the variation of ψA determined in the previous subsection. Since this correction is proportional to the delta function localized at the boundary, it will generate boundary four-fermi terms proportional to D11 ψA when applied to the bulk kinetic term of the gravitino. Given (4.3), this variation generates just one term, equal to ? 1 64π(4πκ2 )2/3 √ d10 x g χa ΓABC χa ηΓAB D11 ψC .
M 10

(4.5)

This term cancels exactly against a similar term that comes from the bulk variation of GABC 11 in the interaction term L1 . No other terms with D11 ψA appear in the supersymmetry variation of the Lagrangian at this order. As to the terms with D11 η, they have two sources among the terms already present in the Lagrangian. First of all, the bulk variation of GABC 11 in L1 produces a term proportional to χΓABC χ ψ A ΓBC D11 η. This term can only be canceled if we introduce a new interaction, L2 = 1 64π(4πκ2 )2/3 √ d10 x g χa ΓABC χa ψ A ΓBC ψ11 . (4.6)

M 10

(When L2 is varied, δψA ? DA η leads to a χχψ11 DA η term; this term cancels against the variations of the bulk supergravity action LS and of L1 give terms of the form χχηDA ψ11 ; happily, these terms cancel against each other.) 19

term of the same form that comes from the variation of GABC 11 in L1 . In addition, the

Another source of terms proportional to D11 η is the variation of the spin connection in the gaugino kinetic term. This point requires a further explanation. In our treatment of eleven-dimensional supergravity in bulk, we have adopted the 1.5 order formalism, which means that the spin connection is ?rst treated as an independent variable and set equal to the solution of its bulk equations of motion at the end of the calculation. One then need not worry about the supersymmetry variation of the spin connection, which vanishes by the equations of motion. In including the boundary interactions, we prefer to continue to use the “bulk” formula for the spin connection. This means that when the spin connection appears in boundary interactions, its supersymmetry variation must be included. (One could avoid this by extending the 1.5 order formalism to incorporate boundary corrections to the spin connection determined by the equations of motion, but we did not ?nd that approach simpler.) In practice, to the order we will calculate, the spin connection only appears in the gravitino kinetic term. Its variation produces an additional term proportional to D11 η (plus other terms we will consider later). Canceling this D11 η term requires a new term in the Lagrangian, L3 = 1 64π(4πκ2 )2/3 √ d10 x g χa ΓABC χa ψ D ΓDABC ψ11 . (4.7)

M 10

Now we will show that no other ψ11 dependent four-fermi terms are generated in the Lagrangian at this order in κ, beyond those given by (4.6) and (4.7). To see this, we will proceed as follows. Supersymmetry variation of such additional four-fermi terms would produce additional terms proportional to D11 η. Notice that these D11 η terms could only be canceled if there is a three-fermi correction to the variation of the gravitino, δ ′ χ ? χψ11 η, and the gaugino kinetic term is varied. We will prove our claim that no new ψ11 -dependent four-fermi terms arise in the Lagrangian at this order, by showing that there are no χψ11 η corrections to the supersymmetry variation of χ. Upon variation of the gaugino kinetic term, such corrections would produce terms of the form χΓA DB χ ψ 11 . . . η and χΓABCDE DF χ ψ 11 . . . η. (4.8)

(Here . . . denotes all possible combinations of Γ matrices.) There is no other possible source of such terms; a simple calculation shows that their cancellation requires the supersymmetry variation of χa to be independent of ψ11 , thus completing our argument. 20

Having canceled all terms with D11 ψ and D11 η, we can determine the rest of the structure at this order in κ by looking at cancellations of χχηψ terms where now the ten-dimensional derivative DA acts on one of the four fermions. First we determine the correction to the supersymmetry variation of χa , by canceling terms of the form χΓA DB χ η . . . ψC and χΓABCDE DF χ η . . . ψG . (4.9)

Terms of this structure must cancel by themselves, since chirality and fermi statistics do not allow one to use integration by parts to move the derivative away from the gauginos. There are two obvious sources of such terms: the variation of eA m in the gaugino kinetic
a term, and the variation of FAB in the supercurrent coupling of L0 . As these do not cancel,

the supersymmetry variation of the gauginos. This correction will produce terms of the required form (4.9) from the variation of the gaugino kinetic term, and the precise form of δ ′ χ will be determined from the cancellation of these terms.5 After a tedious calculation, one obtains δ ′ χa = 1 64 7 ψ A ΓB η ΓAB χa + 9 ψ A ΓA η χa ? 1 ψ A ΓBCD η ΓABCD χa 2

one has to look for another source of such terms. We can add a correction, δ ′ χ ? χψη, to

5 1 A A ψ ΓABC η ΓBC χa + ψ ΓABCDE η ΓBCDE χa . ? 2 24

(4.10)

The correction (4.10) to the supersymmetry variation of χa can be simpli?ed considerably by the Fierz rearrangement formula, leading to δ ′ χa = 1 ψ ΓB χa ΓAB η. 4 A (4.11)

This is exactly what one would have expected from the requirement that the total supersymmetry transformation of χa be “supercovariant.” This also explains why no ψ11 dependent corrections to the supersymmetry variation of χa arise – when varied, such
5

In this and some of the following calculations, we need a Fierz rearrangement formula for

chiral ten-dimensional fermions. All rules follow from the expansion of the product of two fermions ξ and ζ on M 10 that obey Γ11 ξ = ξ and Γ11 ζ = ζ: ζ α ξβ = ? 1 1 1 ξΓABC ζ ΓABC α β + ξΓABCDE ζ ΓABCDE α β . 2 ξΓA ζ ΓA α β ? 32 3 120

21

terms would produce terms with D11 η, and supercovariance of the total supersymmetry variation of the gauginos would be spoiled. Given the correction (4.10) to the supersymmetry variation of the gauginos, the terms that remain to be determined at this order in κ are: (1) The correction to the supersymmetry variation of ψ11 ; on the basis of chirality and fermi statistics, this correction can only be proportional to (χa ΓABC χa ) ΓABC η. (2) Coe?cients of all possible χχψA ψB terms in the Lagrangian; there are exactly four possible inequivalent terms of this structure. We will see momentarily that these additional four-fermi terms do appear in the Lagrangian. If the derivative is on one of the gauginos, we can now use integration by parts to move it to either ψA or η. This leaves us with two classes of terms to cancel – one with DA ψB , and one with DA η. The χχηDψ terms do not get any contribution from the so far undetermined χχψA ψB terms in the Lagrangian, since at this order those will only contribute to χχDηψ terms. Hence, we can use cancellation of the χχηDψ terms to determine the correction to the supersymmetry variation of ψ11 ; another lengthy calculation leads to6 δ ′ ψ11 = 1 576π κ 4π
2/3

We start by canceling terms ? χΓABC χ ψ D . . . η with DE on one of the four fermions.

δ(x11 ) χa ΓABC χa ΓABC η.

(4.12)

Once δ ′ ψ11 has been determined, we can go on and calculate the χΓABC χa ψD . . . DE η terms; their cancellation will determine the coe?cients of the remaining four-fermi terms in the Lagrangian. (As in the case of the χχηDψ terms, there will be a non-zero contribution from the variation of the spin connection in the gaugino kinetic term.) After some additional algebra, one obtains L4 = 1 256π(4πκ2 )2/3 √ d10 x g χa ΓABC χa 3ψ A ΓB ψC ? ψ A ΓBCD ψ D

M 10

13 D 1 ? ψ D ΓABC ψ D ? ψ ΓDABCE ψ E . 2 6

(4.13)

This completes the construction of the boundary Lagrangian to order κ2/3 , which is thus equal to the sum L = L0 + L1 + L2 + L3 + L4 , with the individual terms given by (4.1), (4.2), (4.6), (4.7) and (4.13).
6

The only subtlety here is related to the cancellation of terms χa ΓABC χa ηΓA DB ψC , which

gets a contribution from the variation of the spin connection in the gaugino kinetic term; recall the discussion of the 1.5 order formalism above.

22

We could stop our discussion here; instead, however, one simple point seems worth making. It turns out that the four-fermi terms that we found at order κ2/3 are exactly those implied by supercovariance to this order in κ, and can therefore be absorbed into the de?nition of supercovariant objects. This allows us to summarize the structure of all boundary terms in the Lagrangian at order κ2/3 as constructed in this section, in the following succinct formula: L= 1 2π(4πκ2 )2/3 √ d10 x g
M 10

1 a + ψ A ΓBC ΓA (FBC 8

1 1 tr FAB F AB + tr χΓA DA (?)χ 4 2 √ 2 a ABC a a + FBC )χa + χ Γ χ GABC 11 . 48

(4.14)

a Here the supercovariant spin connection ?mn , Yang-Mills ?eld strength FAB , and ?eld A

strength GABC 11 are given by 1 D 1 D ?A BC = ?A BC + ψ ΓDABCE ψ E ? ψ ΓDABC ψ11 , 8 4 a a a FAB = FAB ? ψ [A ΓB] χ , √ 3 2 ψ [A ΓBC] ψ11 ? ψ [A ΓB ψC] . GABC 11 = GABC 11 + 4

(4.15)

(In accord with the version of the 1.5 order formalism used in this paper, the spin connecequations of motion.) tion ?A BC ≡ eBm eCn ?mn in (4.15) is a composite of eA m and ψA , and solves the bulk A Hence, we see that – just as in the case of pure eleven-dimensional supergravity [8] – no further four-fermi terms are generated at order κ2/3 beyond those required by elevendimensional supercovariance. Of course, at higher orders in κ we encounter additional four-fermi terms that are not explained in this way – the ?rst example of such terms is the term Lχ of (4.4), which is quartic in the gauginos and appears at relative order κ4/3 .

23

References [1] P. Hoˇava and E. Witten, “Heterotic and Type I String Dynamics from Eleven Dir mensions,” Nucl. Phys. B460 (1996) 506, hep-th/9510209. [2] E. Bergshoe?, M. de Roo, B. de Wit and P. van Nieuwenhuisen, “Ten-Dimensional Maxwell-Einstein Supergravity, its Currents and the Issue of its Auxiliary Fields,” Nucl. Phys. B195 (1982) 97. [3] G. Chapline and N.S. Manton, “Uni?cation of Yang-Mills Theory and Supergravity in Ten Dimensions,” Phys. Lett. B120 (1983) 105. [4] M.J. Du?, J.T. Liu, and R. Minasian, “Eleven-Dimensional Origin of String/String Duality: A One Loop Test,” Nucl. Phys. B452 (1995) 261, hep-th/9506126. [5] J.A. Harvey and J. Blum, unpublished. [6] E. Witten, “Five-Branes and M -Theory on an Orbifold,” Nucl. Phys. B463 (1996), hep-th/9512219. [7] C. Vafa and E. Witten, “A One-Loop Test of String Duality,” Nucl. Phys. B447 (1995) 261, hep-th/9505053. [8] E. Cremmer, B. Julia, and J. Scherk, “Supergravity Theory in 11 Dimensions,” Phys. Lett. B76 (1978) 409.

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