aa r X i v : . [ h e p - t h ] S e p O P bundles in M-theory Hisham Sati ∗ Department of MathematicsYale UniversityNew Haven, CT 06511USA
Hausdorff Research Institute for Mathematics,Poppelsdorfer Allee 45D-53115 BonnGermany
Abstract
Ramond has observed that the massless multiplet of eleven-dimensional supergravity can be generatedfrom the decomposition of certain representation of the exceptional Lie group F into those of its maximalcompact subgroup Spin(9). The possibility of a topological origin for this observation is investigated bystudying Cayley plane, O P , bundles over eleven-manifolds Y . The lift of the topological terms givesconstraints on the cohomology of Y which are derived. Topological structures and genera on Y arerelated to corresponding ones on the total space M . The latter, being 27-dimensional, might providea candidate for ‘bosonic M-theory’. The discussion leads to a connection with an octonionic version ofKreck-Stolz elliptic homology theory. ∗ E-mail: [email protected] ontents O P Bundles 3 O P Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Genera of O P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 O P bundles over eleven-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Relating Y and M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.1 Topological consequences: the higher structures . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Topological terms in the lifted action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 BO h i -manifolds with fiber O P . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The relation between M-theory and type IIA string theory leads to very interesting connections to K-theory[13, 12] and twisted K-theory [35] [7] [8]. Exceptional groups have also long appeared in physics. In particular,the topological piece of the M-theory action is encoded in part by an E gauge theory in eleven dimensions[64]. This captures the cohomology of the C -field. Models for the M-theory C -field were proposed in [12]with and without using E . The E bundle leads to a loop bundle on the type IIA base of the circle bundle[2] [35]. The role of E and LE was emphasized in [51, 53]. In particular, in [51] an important role for the String orientation was found within the E construction. It is in the case when the base X is String -oriented that the topological action has a WZW-like interpretation and the degree-two component of theeta-form [35] is identified with the Neveu-Schwarz B -field [51].In this paper we study another side of the problem, by including the whole eleven-dimensional supermul-tiplet ( g, C , Ψ), i.e. the metric, the C -field, and the Rarita-Schwinger field, and not just the C -field. Thisturns out to be related to another exceptional Lie group, namely F , the exceptional Lie group of rank 4.Ramond [41] [43] [44] gave evidence for F coming from the following two related observations:1. F appears explicitly [44] in the light-cone formulation of supergravity in eleven dimensions [11]. Thegenerators T µν of the little group S O(9) of the Poincar´e group I S O(1 ,
10) in eleven dimensions andthe spinor generators T a combine to form the 52 operators that generate the exceptional Lie algebra f such that the constants f µνab in the commutation relation[ T µν , T a ] = if µνab T b (1.1)are the structure constants of f . The 36 generators T µν are in the adjoint of S O(9) and the 16 T a generate its spinor representation. This can be viewed as the analog of the construction of E out ofthe generators of S O(16) and of E /S O(16) in [18].2. The identity representation of F , i.e. the one corresponding to Dynkin index [0 , , , F ) −→ (44 , , , (1.2)1he numbers on the right hand side correctly matching the number of degrees of freedoms of the mass-less bosonic content of eleven-dimensional supergravity with the individual summands corresponding,respectively, to the graviton, the gravitino, and the C -field.The purpose of this paper is to expand on Ramond’s observations by investigating the possibility ofhaving an actual O P = F / Spin(9) bundle over Y through which the above observations can be explainedgeometrically and topologically. Since F is the isometry group of the Cayley plane, the O P bundle will bethe bundle associated to a principal F bundle. We analyze some conditions under which this is possible.In physics, the lifting of M-theory via the sixteen-dimensional manifold O P brings us to 27 dimensions.Given a Kaluza-Klein interpretation, this suggests the existence of a theory in 27 dimensions, whose di-mensional reduction over O P leads to M-theory. The higher dimensional theory involves spinors, and it isnatural to ask whether or not the theory can be supersymmetric. In one form we propose this as a candidatefor the ‘bosonic M-theory’ sought after in [24], from gravitational geometric arguments, and in [45], frommatrix model arguments.We consider the point of view of eleven-dimensional manifolds in M-theory with extra topological struc-ture, such as a String structure. Since any Y with a String structure is zero bordant in the
String bordismgroup Ω h i then this raises the question of whether there is an equivalence with a total space of a bundle inwhich Y is a base. For the Spin case, Kreck and Stolz [29] constructed an elliptic homology theory in whicha spin manifold of dimension 4 k is Spin bordant to the total space of an H P bundle over a zero-bordantbase if and only if its elliptic genus Φ ell ∈ Q [ δ, ε ] vanishes, where the generators δ , ε have degree 4 and 8,respectively. The same authors also expected the existence of a homology theory based on O P bundlesfor the String case, i.e. for manifolds such that p = 0, where p is the first Pontrjagin class. So in ourcase, we ask whether there is a manifold M which is an O P bundle over a zero bordant base and whatconsequence that has on the elliptic and the Witten genus.Some aspects of the connection to this putative homology theory are1. The elliptic homology theory requires the fundamental class [ O P ] of O P to be inverted. Thissuggests connecting the lower-dimensional theory, in our case eleven-dimensional M-theory, to a higherdimensional one obtained by increasing the dimension by 16.2. Previous works have used elliptic cohomology. We emphasize that in this paper we make use ofa homology theory. Thus this not only provides further evidence for the relation between elliptic(co)homology and string/M-theory, but it also provides a new angle on such a relationship.In previous work [30] [31] [32] [50] [52] evidence from various angles for a connection between stringtheory and elliptic cohomology was given. These papers relied heavily on analogies with the case in stringtheory, and were thus not intrinsically M-theoretic. In [47] [48] [49] a program was initiated to make therelation directly with M-theory. Thus, from another angle, the general purpose of this paper is two-fold: • to point out further connections between elliptic cohomology and M-theory • to make the connection more M-theoretic, i.e. without reliance on any arguments from string theory. O P is the Cayley, or octonionic projective, plane. For an extensive description see [46] [19] [6]. Thegroup F acts transitively on O P , from which is follows that O P ∼ = F / Spin(9). In fact F is the isometry viewed as a generator. O P . The tangent space to O P at a point is the coset of the corresponding Lie algebras f / so (9),which is O ∼ = R . O P Bundles
The low energy limit of M-theory (cf. [63] [62] [14]) is eleven-dimensional supergravity [11], whose fieldcontent on an eleven-dimensional spin manifold Y with Spin bundle SY is • Two bosonic fields: The metric g and the three-form C . It is often convenient to work with Cartan’smoving frame formalism so that the metric is replaced by the 11-bein e AM such that e AM e BN = g MN η AB ,where η is the flat metric on the tangent space. • One fermionic field: The Rarita-Schwinger vector-spinor Ψ , which is classically a section of SY ⊗ T Y , i.e. a spinor coupled to the tangent bundle.We now give the main theme around which this paper is centered. Main Idea:
We interpret Ramond’s triplets as arising from O P bundles with structure group F over oureleven-dimensional manifold Y , on which M-theory is ‘defined’. One major advantage of the introduction of an O P bundle is that in this picture the bosonic fields ofM-theory, namely the metric and the C -field, can be unified. Theorem 2.1.
The metric and the C -fields are orthogonal components of the positive spinor bundle of O P .Proof. The spinor bundle S + ( O P ) of the Cayley plane is isomorphic to S + ( O P ) = S ( V ) ⊕ Λ ( V ) , (2.1)where V is a nine-dimensional vector space and S denotes the space of traceless symmetric 2-tensors.This follows from an application of proposition 3 in [16] which requires the study the 16-dimensional spinrepresentations ∆ ± as Spin(9)-representations. The element e · · · e belongs to the subgroup g Spin(9) ⊂ Spin(16) and acts on ∆ ± by multiplication by ( ± +16 is an SO (9)-representation,but ∆ − is a Spin(9)-representation [1]. Both representations do not contain non-trivial Spin(9)-invariantelements. Such an element would define a parallel spinor on F / Spin(9) but, since the Ricci tensor of O P is not zero, the spinor must vanish by the Lichnerowicz formula [33] D = ∇ + R scal . Then ∆ +16 as aSpin(9) representation is given by equation (2.1), and ∆ − is the unique irreducible Spin(9)-representationof dimension 128.Thus we have Theorem 2.2.
The massless fields of M-theory are encoded in the spinor bundle of O P . O P Bundles
Having motivated O P bundles in M-theory, we now carry on with our proposal and construct such bundlesin eleven dimensions. We study the properties of the O P bundle as well as the associated F bundle andgive some consistency conditions. As bundles are characterized by characteristic classes and genera, we‘compare’ the structure of the base and that of the total space. For that purpose we start with discussingthe relevant genera of the fiber. 3 .2 Genera of O P A genus is a function on the cobordism ring Ω (see section 3 for cobordism). More precisely, it is a ringhomomorphism ϕ : Ω ⊗ R → R , where R is any integral domain over Q . It could be Z , Z p or Q itself. Generain general have expressions given in terms of characteristic classes. Two important ‘modern’ genera are theelliptic genus Φ ell and the Witten genus Φ W . The first is characterized by two parameters, denoted ε and δ , whose various values give different specializations of Φ ell . Special values of the parameters correspondto more ‘classical’ genera. The values δ = ε = 1 leads to the L -genus L : Ω ⊗ Q → Q , and the values δ = − , ε = 0 leads to the b A -genus b A : Ω ⊗ Q → Q . Depending on the type of cobordism considered,Ω and also R can vary. For instance, when the manifolds are Spin then the b A -genus is an integer and so b A : Ω Spin ⊗ Z → Z . The Witten genus is defined for any topological manifold but it becomes a modular formfor special manifolds, namely ones with a String structure or BO h i -structure, and those are the manifoldsthat satisfy p = 0, where p is the first Pontrjagin class of the tangent bundle. The Witten genus is a mapΦ W : Ω BO h i ⊗ R → M F = R [ E , E ], where M F is the ring of modular forms generated by the Eisensteinseries E and E , and R is usually Q or Z . We describe this more precisely below.It is natural to ask what the values of the elliptic genus and of the Witten genus of O P are. First,however, we consider the classical genera.
1. The classical genera.
We give the following specialization.
Lemma 2.3.
1. The b A -genus of O P is zero, b A [ O P ] = 0 .2. The L -genus of O P is u , where u is the generator of H ( O P ; Z ) .
2. The Witten genus.
Next we consider another genus, the Witten genus, which can be defined in thefollowing way. There is a convenient collection of manifolds { M n } that generate the rational cobordismring Ω ⊗ Q [34]. The advantage of this basis is that each M n has a single nonzero Pontrjagin class, thetop one p n = d n (2 n − m where m generates H n ( M n ). On this basis, Φ W ( M k ) = num k E k for k > W ( M ) = 0, where num n /d n = B n / n is the given numerator, with num n and d n relatively prime,and B n the even Bernoulli numbers. The ring of modular forms for the full modular group is (cf. [4]) M F = Z [ E , E , ∆] / ( E − E − q Q n (1 − q n ) . By inspecting the Bernoulli numberswe can see that the first four terms in d n are 24 , , , Theorem 2.4.
The Witten genus of O P is zero, Φ W ( O P ) = 0 .Proof. O P has positive scalar curvature, so its b A -genus is zero b A ( O P ) = 0. O P is also a String manifold,so its Witten genus Φ W ( O P ) : Ω BO h i = π M O h i → π ∗ eo = M F ∗ must be a modular form for SL (2 , Z )of weight equals to half its dimension [67], i.e. 8. What modular forms do we have? The ring of integralmodular forms is (cf. [4]) M F ∗ = Z [ E , E , ∆] / (2 · ∆ − E + E ) (2.2)where E ∈ M F , E ∈ M F , and ∆ ∈ M F . Thus the only modular form of weight 8 is E . However theform of the Eisenstein series is E = 1+ higher terms, so that the required modular form does not start withzero. Therefore Φ W ( O P ) = 0. 4 . The elliptic genus. Next we consider the elliptic genus Φ ell : Ω BO h i∗ ⊗ Q → Q [ δ, ε ], where thegenerators δ and ε have degrees 4 and 8, respectively. Theorem 2.5.
The elliptic genus of the Cayley plane is Φ ell ( O P ) = ε .Proof. There are several ways to prove this. The first one is to use the idea of cobordism as in the proof ofthe case of the quaternionic projective plane H P . However, we can simply apply a result from [21]. Since O P is a connected homogeneous space of a compact connected Lie group F , and since O P is oriented andadmits a Spin structure, then the normalized elliptic genus Φ norm := Φ ell /ε is a constant modular functionΦ norm ( O P ) = σ ( O P ) . (2.3)Thus we immediately get the result.
4. The Ochanine genus.
We next consider the Ochanine genus [40], which is a generalization of theelliptic genus in such a way that it involves q -expansions. The Ochanine genus is a ring homomorphismΦ och : Ω spin ∗ −→ KO ∗ (pt)[[ q ]] , (2.4)from the Spin cobordism ring to the ring of power series with coefficients in KO ∗ (pt) = Z (cid:2) η, ω, µ, µ − (cid:3) / (cid:0) η, η , ηω, ω − µ (cid:1) , (2.5)where η ∈ KO , ω ∈ KO , and µ ∈ KO are generators of degrees 1, 4, and 8, respectively, and are givenby the normalized Hopf bundles γ R P − γ H P − γ O P − R P = S , H P = S , and O P = S .For a manifold M m of dimension m , corresponding to the projection map π M m : M m → pt there is theGysin map π M m ! : KO ( M m ) → KO m (pt) = KO m (pt). Now consider a real vector bundle E on M m andform the following combination of exterior powers and symmetric powers of E Θ q ( E ) = X i ≥ Θ i ( E ) q i = O n ≥ (cid:0) Λ − q n − ( E ) ⊗ S q n ( E ) (cid:1) , (2.6)which, since it is multiplicative under Whitney sum, can be considered as an exponential map Θ q : KO ( M m ) → KO ( M m )[[ q ]]. Now specialize E to be the reduced tangent bundle ^ T M m , which is T M m − m . Then theOchanine genus is defined to be [40] [29]Φ och ( M m ) := X i ≥ Φ i och ( M m ) q i = X i ≥ π M m ! (cid:16) Θ i ( ^ T M m ) (cid:17) q i = θ ( q ) − m h Θ q ( T M m ) , [ M m ] KO i ∈ KO m (pt)[[ q ]] , (2.7)where [ M m ] KO ∈ KO m ( M m ) denotes the Atiyah-Bott-Shapiro orientation [5] of M m , h , i : KO i ( X ) ⊗ KO j ( X ) → KO j − i is the Kronecker pairing, and θ ( q ) := Θ q (1) = Y n ≥ − q n − − q n = 1 − q + q − q ± · · · ∈ Z [[ q ]] , (2.8)5s the Ochanine genus of the trivial line bundle.The degree zero part Θ ( E ) is a trivial real line bundle, and corresponds to the Atiyah invariantΦ ( M ) = π M m ! (1) = h , [ M m ] KO i = α ( M m ). The cobordism invariant α ∈ KO m [ ? ] can be interpreted asthe index of a family of operators associated to M m parametrized by S m [22]. Thus the α -invariant is theclassical value of the Ochanine genus in the same way that the b A -genus and the L -genus are the classicalvalues of the elliptic genus corresponding, respectively, to δ = b A ( C P ) = − , ε = b A ( H P ) = 0 , and δ = L ( C P ) = 0 , ε = L ( H P ) = 1 . (2.9)The Ochanine genus is related to the restriction Φ ell , int to Ω spin ∗ of the universal elliptic genus Φ ell , uni : Ω SO ∗ → Q [[ q ]], whose parameters are δ = − − X n ≥ X d | n, d odd d q n = −
18 + q − expansion ,ε = X n ≥ X d | n, nd odd d q n = 0 + q − expansion . (2.10)More precisely, Φ ell,int = P h ◦ Φ och : Ω spin ∗ → Z [[ q ]], where P h is the Pontrjagin character
P h : KO ∗ ( X ) −→ ⊗ C K ∗ ( X ) −→ ch H ∗∗ ( X ; Q ) , (2.11)which can be thought of as the analog for real vector bundles of the Chern character for complex vectorbundles.We now check the value of Φ och for O P . Theorem 2.6.
The Ochanine genus of O P is Φ och ( O P ) = ε µ .Proof. The Ochanine genus Φ och ( O P ) is the map Ω spin16 → KO [[ q ]]. Note that Ω spin16 = Z ⊕ Z ⊕ Z and that KO (pt) = Z with generator µ . The image Φ och (Ω spin16 ) is the set of all modular forms of degree 16 andweight 8 over KO = Z . Let M Γ ( KO ) be the graded ring of modular forms over KO for Γ, a subgroupof finite index in SL (2; Z ). For M Γ ∗ ( Z ) = Z [ δ , ε ], where δ = − δ ∈ M Γ2 ( Z ) and δ and ε ∈ M Γ4 ( Z ) are thegenerators in (2.10), we have M Γ ( KO ) ∼ = KO ⊗ M Γ ∗ ( Z )= Z ⊗ Z [ δ , ε ] . (2.12)Then a modular form of degree 16 and weight 8 can be written in a unique way as a polynomial P ( δ , ε ) ofweight 8 with integer coefficients. Still applying the construction in [40], the Ochanine genus in our case isΦ och ( O P ) = (cid:0) a ( O P ) δ + a ( O P ) δ ε + a ( O P ) ε (cid:1) µ , (2.13)with uniquely defined homomorphisms , for i = 1 , , a i · µ : Ω spin16 = Z ⊕ Z ⊕ Z −→ KO = Z . (2.14)6he integers a i can be determined as follows. We have already seen that the lowest coefficient is given bythe Atiyah invariant. Since O P admits a Riemannian metric of positive scalar curvature then, from [22], α ( O P ) = 0, and hence we have determined that a ( O P ) = 0. Another way of seeing this is to noticethat for manifolds of dimension 4 k , the Atiyah invariant is essentially the b A -genus, which, by Lichnerowicztheorem [33], vanishes for a manifold with positive scalar curvature. The highest coefficient, a ( O P ), isgiven by the Ochanine k -invariant, which in this case is just the signature a ( O P ) = σ ( O P ) = 1. Itremains to calculate a . This is given by the first KO-Pontrjagin class Π a ( O P ) = Π ( T O P ) = − Λ ( T O P − , (2.15)which is just − ( T O P − n -dimensionalvector bundle ξ over a space X , Π u ( ξ ) ∈ KO ( X ) are defined by(1 + t ) n ∞ X k =0 t k (1 + t ) k Π k ( ξ ) = ∞ X k =0 t k Λ k ( ξ ) . (2.16)For k = 1 this gives the first KO-Pontrjagin class used in (2.15). Alternatively, we can look at the q -components of Φ och from the first line of equation (2.7) and getΦ ( O P ) = h , [ O P ] KO i = α ( O P )Φ ( O P ) = h− Π ( O P ) , [ O P ] KO i = h− ( T O P − , [( O P )] KO i . (2.17)We still have to calculate a . We use the topological Riemann-Roch theorem (see [58]) which states that for M a closed Spin manifold and x ∈ g KO ∗ ( M ), then P h h x, [ M ] KO i = h b A ( M ) P h ( x ) , [ M ] i H , where h , i H isthe Kronecker pairing on cohomology. Taking M = O P and x = T O P , we get for a h b A ( O P ) P h ( T O P ) , [ O P ] i H , (2.18)which is zero because, as we have seen, b A ( O P ) = 0. O P bundles over eleven-manifolds Consider the fiber bundle E → Y with fiber O P and structure group F . There is a universal bundle ofthis type. O P bundles over Y are pullbacks of the universal bundle O P = F / Spin(9) −→ B Spin(9) −→ BF (2.19)by the classifying map f : Y → BF . In this paper we will consider the diagram O P / / M " " DDDDDDDDDDDDDDDDDD π (cid:15) (cid:15) Y f / / BF . (2.20)Note that the map from M to BF can be f π and this will be useful later in section 3. We first have thefollowing. 7 roposition 2.7. The obstruction to existence of a section of an O P fiber bundle over an eleven-dimensionalmanifold Y lies in H ( Y ; Z ) , H ( Y ; Z ) and H ( Y ; Z ) .Proof. For a fiber bundle F → E → B , the existence to having a section lies in the groups H r ( B ; π r − ( F )) forall nonzero r ∈ N . In our case, O P has π i = 0 for i ≤
7, so that the first obstruction is in H (cid:0) Y ; π ( O P ) (cid:1) ,which is H ( Y ; Z ). The next two nontrivial homotopy groups of O P , both are Z , in dimension 9 and10 so that the obstructions are in H ( Y ; Z ) and H ( Y ; Z ). O P has further nontrivial homotopygroups but that would bring us to H ≥ , which are zero for an eleven-manifold. Remarks1.
The first obstruction H ( Y ; Z ) is called the primary obstruction. Note that the primary obstruction is a Z -class whereas the secondary obstructions are Z -classes.In forming bundles with O P as fibers, we are forming bundles of BO h i -manifolds over Y . We willnext investigate the relation between structures on Y , on the fiber O P , and on the total space M . Y and M We ask the question whether topological conditions on Y , namely having Spin , String , or
Fivebrane struc-ture [55] [56], will lead to (similar) structures on M . The answer to such a question is possible because weknow about the (non-)existence of these structures on O P .The condition λ := p = 0 for lifting the structure group of the tangent bundle to String( n ) is relatedto the condition W = 0 for orientation with respect to either the p = 2 integral Morava K-theory K (2) orLandweber’s elliptic cohomology theory E (2) [30]. The first condition implies the second, but the converse isnot true, a counterexample being X = S × S × C P [30]. Thus if we assume the String orientation, thenwe are already guaranteed the W orientation, and so the discussion and constructions in [30] [31] [32] [50]for ten-dimensional string theory apply. The condition λ = 0 can be extended from ten to eleven dimensionsand vice versa. This is because for Y = X × S the first Pontrjagin classes are related as (using bundlenotation) p ( T X ⊕ T S ) = p ( T X ) + p ( T S ), but for dimensional reasons p ( T S ) = 0 so that we have p ( Y ) = p ( X ). Thus the String condition can be translated from M-theory to string theory and backas desired.There is no cohomology in degree four for O P , so we immediately have Proposition 2.8. O P admits a BO h i -structure. Remark. If Y is a BO h i -manifold, i.e. is M O h i -orientable, then it has an M O h i homology fundamentalclass, [ Y ] MO h i ∈ M O h i ( Y ) . (2.21)Any integral expression will involve this class. This would also enter the construction of the BO h i partitionfunction.We would like to check to what extent we can know the cohomology of the total space M in terms ofthe cohomology of the base Y , given that we know the cohomology of the fiber O P . One way to detectthis is by using the Serre spectral sequence for the bundle E p,q = H p (cid:0) Y , H q ( O P ) (cid:1) ⇒ H p + q ( M ) . (2.22)8onsider the case of a product M = O P × Y , i.e. when the bundle is trivial. In this case, using theK¨unneth theorem and the fact that the cohomology of O P is nonzero only in degrees 8 and 16, we get Proposition 2.9. H n ( O P × Y ; C ) ∼ = H n − ( Y ; C ) ⊕ H n − ( Y ; C ) . (2.23)We next consider the case when the bundle is not trivial. A simplification is made if coefficients are takenso that the cohomology of the fiber is trivial in those coefficients. The torsion (‘bad’) primes for F are 2and 3, so that one might expect that those are the primes that do not cause such a simplification. It willturn out that this is true only for p = 3, as we now show. We first show that p = 3 occurs and then that itis the only one.The cohomology of the classifying spaces of Spin(9) and F with Z p coefficients, p = 2 ,
3, are as follows.The cohomology ring of BF with coefficients in Z is given by the polynomial ring [9] H ∗ ( BF ; Z ) = Z [ x , x , x , x , x ] , (2.24)where x i are polynomial generators of degree i related by the Steenrod square operation Sq i : H n ( BF ; Z ) → H n + i ( BF ; Z ) as x = Sq x , x = Sq x , x = Sq x . (2.25) H ∗ ( BF ; Z ) is generated by x i for i = 4, 8, 9, 20, 21, 25, 26, 36, 48, with the structure of a polynomialalgebra [61]. Considering p = 3, this is H ∗ ( BF ; Z ) ∼ = Z [ x , x ] ⊗ (cid:0) Z [ x , x ] ⊗ (cid:8) , x , x (cid:9) + Λ( x ) ⊗ Z [ x ] ⊗ { , x , x , x } (cid:1) . (2.26)The generators can be chosen to be related by the Steenrod power operations at p = 3, P i : H n ( BF ; Z ) → H n +4 i ( BF ; Z ), as x = P x x = βx = βP x x = P P x x = βP P x x = P βP x x = βP βP x (2.27)and x = P x . If we restrict to degrees ≤
11 then we have the truncated polynomial H ∗ ( BF ; Z ) ∼ = Z [ x , x ] + Λ( x ) . (2.28)The classes coming from B Spin(9) are just the Stiefel-Whitney classes in the Z case and the Pontrjaginclasses (reduced mod 3) in the integral ( Z case). These are actually not much different from the classes of B Spin(11). Explicitly, at p = 2 the cohomology ring of B Spin(9) is given by the polynomial ring [42] H ∗ ( B Spin(9); Z ) = Z [ w , w , w , w , w ′ ] , (2.29)where w i is the restriction of the universal Stiefel-Whitney class, and w ′ is the Stiefel-Whitney class ω (∆ Spin(9) ) of the spin representation ∆
Spin(9) : Spin(9) → O (16). At p = 3, H ∗ ( B Spin(9); Z ) is generatedby the first four Pontrjagin classes [61] H ∗ ( B Spin(9); Z ) = Z [ p , p , p , p ] , deg( p i ) = 4 i . (2.30)Let us look at Z coefficients. From (2.28) and (2.30) we see that H ( B Spin(9); Z ) = 0 while H ( BF ; Z ) =0, which implies that the map H ( BF ; Z ) → H ( B Spin(9); Z ) cannot be injective. Therefore, at p = 39he Serre spectral sequence is not trivial. In the case of Z , the situation is reversed, this time in degreeeight: H ( B Spin(9); Z ) = 0 and H ( BF ; Z ) = 0.Now we proceed with the uniqueness by applying the results in [28]. The cohomology of O P is H ∗ ( O P ; C ) = C [ x ] /x , | x | = deg x = 8, as an algebra. Then, requiring that the Serre fibering O P → M → Y be trivial over C implies for the E -term E = H ∗ ( Y ; C ) ⊗ C C [ x ] /x . (2.31)Now the E term is E | x | +1 = E and the fibering is nontrivial if and only if we have a nonzero differential d (1 ⊗ x ) = 0. If d (1 ⊗ x ) = a ⊗ = 0 then 0 = d (1 ⊗ x ) = 3( a ⊗ x ). Hence the characteristic of C mustnot be relatively prime to 3, the degree of the ideal in the cohomology ring of O P . Therefore, we have Proposition 2.10.
The Serre spectral sequence for the fiber bundle O P → M → Y is nontrivial onlyfor cohomology with Z coefficients. We will make use of this and also say more in section 2.4.2 – see theorem 2.15 and the discussion around it.
Proposition 2.11. If Y admits a String structure then so does M provided that there is no contributionfrom the degree four class from BF .Proof. We have the O P bundle over Y with total space M M
27 ˜ f / / π (cid:15) (cid:15) B Spin(9) Bi (cid:15) (cid:15) Y f / / BF , (2.32)which gives the decomposition T M = π ∗ T Y ⊕ ˜ f ∗ T , and so the tangential Pontrjagin class is p ( M ) = π ∗ (cid:0) p ( Y ) + f ∗ p ( T ) (cid:1) . (2.33)In the case Y is a 3-connected BO h i -manifold, we have that H ( Y ; Z ) is free and π ∗ : H ( Y ; Z ) → H ( M ; Z ) is an isomorphism. Thus M is also a BO h i -manifold if and only if f ∗ x = 0 ∈ H ( Y ; Z ),where x ∈ H ( BF ; Z ) is the generator. Therefore we have shown that M is String if and only if G inM-theory gets no contribution from BF . Remarks1.
The quantization condition for the field strength G in M-theory is known [64]. Since this field does notseem to get a contribution from a class in BF , the condition in Proposition 2.11 seems reasonable. In somesense we could view the presence of such a degree four class as an anomaly which we have just cured. We connect the above discussion back to cobordism groups. While there is no transfer map from Ω h i ( BF )to Ω h i , there is a transfer map after killing x [26]. Denoting by BF h x i the corresponding classifyingspace that fibers over BF , killing x is done by pulling back the path fibration P K ( Z , → K ( Z ,
4) with amap x : BF → K ( Z ,
4) realizing x . The corresponding transfer map is Ω h i ( BF h x i ) → Ω h i .Next, for the higher structures we have This is the analog of the
String group when G = Spin, in the sense that it is the 3-connected cover. roposition 2.12. 1. In order for M to admit a Fivebrane structure, the second Pontrjagin class of Y should be the negative of that of O P , i.e. p ( T Y ) = − p ( T O P ) = − u . b A ( M ) = 0 , irrespective of whether or not the b A -genus of Y is zero. Φ W ( M ) = 0 . Φ ell ( M ) = 0 .Proof. For part (1) note that if Y admits a Fivebrane structure then M does not necessarily admit sucha structure. This is because the obstruction to having a Fivebrane structure is p [56] but we know that p ( O P ) = u = 0. However, we can choose Y appropriately so that it conspires with O P to cancelthe obstruction and lead to a Fivebrane structure for M . Noting that the tangent bundles are relatedas T M = T Y ⊕ T O P , then considering the degree eight part of the formula (see [37]) p ( E ⊕ F ) = P p ( E ) p ( F ) mod 2 − torsion, we get for our spaces p ( T Y ⊕ T O P ) = p ( T Y ) p ( T O P ) + p ( T Y ) + p ( T O P ) mod 2 − torsion . (2.34)Since we have p ( T O P ) = 0, then requiring that p ( T M ) = 0 leads to the constraint that p ( T Y ) + p ( T O P ) = 0 modulo 2-torsion.For part (2) we use the multiplicative property of the b A -genus for Spin fiber bundles to get b A ( M ) = b A ( Y ) b A ( O P ) . (2.35)Since the b A -genus of O P is zero then the result follows.For part (3) we use a result of Ochanine [39]. Taking the total space M and the base Y to be closedoriented manifolds, and since the fiber O P is a Spin manifold and the structure group F of the bundle iscompact, then the multiplicative property of the genus can be appliedΦ W ( M ) = Φ W ( O P )Φ W ( Y ) . (2.36)We proved in Theorem 2.4 that Φ W ( O P ) = 0, so it follows immediately that Φ( M ) is zero regardless ofwhether or not Φ W ( Y ) vanishes. Even more, Φ W ( Y ) is zero because Y is odd-dimensional. For part (4) we use the fact that the fiber is Spin and the structure group F is compact and connectedso we can apply the multiplicative property of the elliptic genus [39]Φ ell ( M ) = Φ ell ( Y )Φ ell ( O P ) . (2.37)In this case the genus for the fiber is not zero (see Proposition 2.5) but the elliptic genus of Y is zero, againbecause of dimension. Therefore Φ ell ( M ) = 0.We next consider the relation between the Ochanine genera of the base and of the total space.Having the Ochanine genera for S and X , we now proceed to determine the corresponding genus forthe eleven-dimensional manifold Y . Proposition 2.13.
Let Y be an eleven-dimensional Spin manifold which is the total space of a circlebundle over a ten-dimensional Spin manifold X . Then the Ochanine genus of Y is Φ och ( Y ) = Φ och ( X ) · α ( S ) . (2.38) However, see the case when Y is a circle bundle at the end of this section. roof. Unlike other genera, the Ochanine genus does not in general enjoy a multiplicative property on fiberbundles. However, in the special case when the fiber is the circle with a U (1) action Φ och does becomemultiplicative on the circle bundle [29]. We simply apply the result for S → Y → X to getΦ och ( Y ) = Φ och ( X ) · Φ och ( S ) . (2.39)With Φ och ( S ) = α ( S ) the degree one generator in KO ∗ (pt), the result follows.Now that we have the Ochanine genus for Y , we go back and consider the original questions of findingthe Ochanine genus of M , given that of Y . Theorem 2.14.
The Ochanine genus of the total space M of an O P bundle over an eleven-dimensionalcompact Spin manifold Y , which is a circle bundle over a ten-dimensional Spin manifold X , is Φ och ( M ) = Φ och ( O P ) · Φ och ( X ) · α ( S ) , (2.40) where Φ och ( O P ) is given in Theorem 2.6 and Φ och ( X ) is given as follows: If k ( X ) = 0 ∈ Z then Φ och ( X ) = α ( X ) , while if k ( X ) = 1 ∈ Z then in KO ⊗ Z we have Φ och ( X ) = α ( X ) + η µ ( q + q + q + · · · ) . (2.41) Proof.
As mentioned in the proof of Proposition 2.13 above, Φ och is not in general multiplicative for fiberbundles. Again, interestingly, we are in a special case where such a property holds [29]. It is so becausethe dimension of the fiber O P is a multiple of 4, the structure group F is a compact connected Lie group,and the base Y is a closed Spin manifold. Applying to the fiber bundle O P → M → Y , and usingproposition 2.13, then gives the formula in the theorem. Remark.
The circle in Theorem 2.14 is the one with the nontrivial/nonbounding/supersymmetric/Ramond-Ramond Spin structure.
Having motivated and then constructed O P bundles in M-theory, we now turn to the discussion of some ofthe consequences. The most obvious question from a physics point of view is to characterize the corresponding‘theory’ in 27 dimensions. We will not be able to achieve that, but we will be able to characterize some ofthe terms in the would-be action up in 27 dimensions. In the absence of a clear handle, we take the mosteconomical approach and concentrate on the topological terms, which in any case are the terms we can trust.We also make some remarks on other terms as well.The simplest topological term coming from O P at the rational level would be some differential form ofdegree sixteen. This could also be decomposable, i.e. a wedge product of differential forms of lower degreessuch that the total degree is 16. We should seek forms that naturally occur on O P . Looking at the questionfrom a 27-dimensional perspective, a Kaluza-Klein mechanism comes to mind. We do not attempt to discussthis problem fully here but merely provide some possibilities that are compatible with the structures thatwe have. In dimensional reduction from ten and eleven dimensions to lower dimensions, holonomy plays animportant role as it gives some handle on the differential forms involved, as well as on supersymmetry.From the cohomology of O P , the possible topological terms generated from this internal space comefrom X i ∈ H i ( O P ) for i = 8 ,
16, so that their linear combination generates a candidate degree sixteen term ρ := aX + bX , (2.42)12here X and X are eight- and sixteen-forms, respectively, and a and b are some parameters. Remarks1.
Since the degree 16 generator is built out of the degree 8 generator, namely the first is proportional to u and the second is u , then equation (2.42) is redundant as X is really built out of X . Thus equation(2.42) should be replaced by ρ = bX . In terms of the generator u of H ( O P ; Z ), the expression at the integral level should be ρ = αu , (2.43)with α ∈ Q . The term ρ would be thought of as a degree 16 analog of the one loop term I in M-theory and typeIIA string theory from [15]. It would appear as a topological term in the action, rationally as S top(27) = Z M L top(27) = Z M ρ ∧ L (11) top , (2.44)where L top(11) is the topological Lagrangian in eleven dimensions given by L top(11) = 16 G ∧ G ∧ C − I ∧ C . (2.45)Then we have S top(27) = Z Y L top(11) Z O P ρ = α Z Y L top(11) = α S top(11) . At the rational level we can thus use ω to build a Spin(9)-invariant degree sixteen expression ρ R = ω ∧ ω that we integrate and insert as part of the action as R O P ρ R .The integration of ρ over O P in the second step of equation (2.46) requires the existence of a fun-damental class [ O P ] for the Cayley plane. The Cayley 8-form J allows for such an evaluation at therational and integral level. The next question is about torsion. The existence of such a fundamental classat that level is neither automatic nor obvious. In order to state the following result we recall some nota-tion. Let β : H i ( Y ; Z ) → H i +1 ( Y ; Z ) be the Bockstein homomorphism corresponding to the reductionmodulo 3, r : Z → Z , i.e. associated to the short exact sequence 0 → Z → Z → Z → P : H j ( Y ; Z ) → H j +4 ( Y ; Z ) be the Steenrod reduced power operation at p = 3. Then we have Theorem 2.15.
A fundamental class exists provided that βP x = 0 , where x is the mod 3 class on Y pulled back from BF via the classifying map.Proof. Consider the fiber bundle E → Y with fiber O P and structure group F . There is a universalbundle of this type. O P bundles over Y are pullbacks of the universal bundle O P = F / Spin(9) −→ B Spin(9) −→ BF (2.46)by the classifying map f : Y → BF . Since BF is path-connected and O P is connected then we canapply the Serre spectral sequence to the fibration (2.46). We consider two cases for the coefficients of thecohomology: Z p (or any field in general), p a prime, and Z coefficients. Coefficients in Z p : The important primes are p = 2 , F . For p = 2 theinclusion map i : Spin(9) ֒ → F induces a map on the classifying spaces so that H ∗ ( B Spin(9); Z p ) is a free H ∗ ( BF ; Z p )-module on generators 1 , x, x with x ∈ H ( B Spin(9); Z p ) the universal Leray-Hirsch generator13hat maps to x ∈ H ( O P ; Z p ). Here we use the fact [36] that the Serre spectral sequence for a fiber bundle F → E → B collapses if and only if the corresponding Poincar´e series P ( − ) := P n ≥ t n dim Z p H n ( − ; Z p )satisfies P ( E ) = P ( F ) P ( B ). In our case the Serre spectral sequence of (2.46) collapses [26]. This followsfrom the equality of the corresponding Poincar´e polynomials P ( B Spin(9)) P ( BF ) = (1 − t ) − (1 − t ) − (1 − t ) − (1 − t ) − (1 − t ) − (1 − t ) − (1 − t ) − (1 − t ) − (1 − t ) − (1 − t ) − = 1 − t − t = 1 + t + t , (2.47)which is just the Poincar´e polynomial P ( O P ) of the Cayley plane. This implies that the Leray-Hirschtheorem holds, i.e. that the map H ∗ ( O P ) ⊗ H ∗ ( BF ) → H ∗ ( B Spin(9)) is an isomorphism of H ∗ ( BF )-modules. This implies in particular that H ∗ ( B Spin(9)) is a free BF -module on 1 , x, x , where x is either w or w + w . The Wu formula with w = w = 0 for both cases gives that Sq x = Sq x = Sq x = Sq x = 0so that Sq x = x + Sq x + Sq x + Sq x + x . (2.48)The elements x , Sq x , Sq x ∈ H ∗ BF are mapped to the elements w , w = Sq w , w = Sq w ∈ H ∗ B Spin(9). The Leray-Hirsch theorem holds for the universal bundle, and consequently for all O P bundles [26].For p = 3 the argument is similar except that now the generators in degrees 4 and 8 are related as p = p and p = p + p , respectively [61]. Here p i are the Pontrjagin classes (see the appendix). Coefficients in Z : We would like to find the differentials for H ∗ ( B Spin(9); Z ) ⇐ = H ∗ (cid:0) BF , H ∗ ( O P ; Z ) (cid:1) . (2.49)The class u maps under the differential to a Z class of degree 9 which we will call α . The lowest degreeclass on the fiber is x , so the differentials begin with d . The differential is d on x so that the class is βP x , where x is the mod 3 class on Y coming from BF Y −→ BF −→ K ( Z , . (2.50)We thus have a 3-torsion class of O P bundles. The obstruction in H ( Y ; Z ) coming from H ( BF ; Z )is zero if and only if there exists a degree 16 class, say ρ , that restricts on each fiber to the fundamentalclass.Thus the vanishing of d provides us with a fundamental class which we use to integrate over O P . Remark.
The Pontrjagin classes p and p of O P are divisible by three. There is always a class in M that restricts on the fiber to three times the generator of the cohomology of O P . In this section we consider the question of extension of the theories in eleven and twenty-seven dimensionsto bounding theories in twelve and twenty-eight dimensions, respectively, assuming the spaces to be
String and taking into account the F bundles. As mentioned in the introduction, our discussion will make contact14ith a version of elliptic cohomology constructed by Kreck and Stolz [29]. In that paper the emphasis wason the Spin case corresponding geometrically to quaternionic projective plane H P bundles, but the authorsassert the existence of a BO h i version corresponding to octonionic projective plane O P bundles. Let usdenote this theory by E h i or, equivalently, by E O .We consider the String condition from an eleven-dimensional point of view. One point that we utilizeis that Ω spin11 (pt), the Spin cobordism group in eleven dimensions, is zero. This means that any eleven-dimensional Spin manifold bounds a twelve-dimensional one. It is also the case that the BO h i cobordismgroup Ω h i (pt) is zero [17], so that the extension from an eleven-dimensional String manifold to the corre-sponding boundary is unobstructed. Thus, if the space Y in which M-theory is defined admits a String structure then this always bounds a twelve-dimensional
String manifold Z .Generalized cohomology theories can, in fact, be obtained as quotients of cobordism (see [30] for someexposition on this for physicists) by classic results [10]. For instance, Spin cobordism Ω spin ∗ = Ω h i∗ is closelyrelated to real K-theory KO , a fact we used in section 2.1. For a space X , KO ∗ (pt) can be made into anΩ spin ∗ -module and there is an isomorphism of KO ∗ ( X ) with Ω spin ∗ ( X ) ⊗ Ω spin ∗ KO ∗ (pt). As we have seen, thisis related to the mod 2 index of the Dirac operator with values in real bundles in ten dimensions whichappears in the mod 2 part of the partition function [13]. There is an analogous construction for ellipticcohomology, where there the starting point is Ω h i∗ . This fact is related to the elliptic refinement of the mod2 index which then has values in a real version of elliptic cohomology [30]. BO h i -manifolds with fiber O P Now we go back to our main discussion of relating the cobordisms of the eleven- and twenty-seven-dimensionaltheories together with the F - O P bundles. Thus we are led to the study of the cobordism groups Ω h i i ( BF )for i = 11 and 27. We will also be interested in relating these two groups.We have an 11-dimensional base manifold Y , assumed to admit a String(11) structure, with an O P bundle such that the total space is M and the structure group is F . Let I ∈ Ω h i be the ideal generatedby elements of the form [ M ] − [ O P ][ Y ] where, as before, M → Y is a fibration with fiber O P andstructure group F . We have Proposition 3.1.
Let Y be a compact manifold with a String structure on which M-theory is taken, andlet M be the String manifold on which the 27-dimensional theory is taken, realizing the Euler tripletsgeometrically. Then such 27-manifolds M are in the ideal I of Ω h i generated by O P bundles. Our setting is given in the following diagram O P / / M f ′ ! ! BBBBBBBBBBBBBBBBB π (cid:15) (cid:15) Y f / / N . (3.1)First we ignore the structure group and consider N to be a point. As in Section 3.1, let Ω h i∗ be the cobordismring of manifolds with w = w = p = 0. This ring has only 2-torsion and 3-torsion, with the 3-torsionbeing a Z summand in dimensions 3, 10, and 13 (this is known only up to roughly dimension 16).15ote that cobordim groups Ω h n i∗ arise as homotopy groups of the Thom spectra M O h n i , in the sensethat the former groups are the homotopy groups of the spectra (this is general for any type of cobordism).Hence the Thom spectrum for the String cobordism ring is
M O h i , and Ω h i∗ = π ∗ ( M O h i ). We can actuallygain information about Ω h i∗ by looking at topological modular forms. This is due to the following fact. Let M O h i → tmf be any multiplicative map whose underlying genus is the Witten genus. Then the inducedmap on the homotopy groups π ∗ M O h i → π ∗ tmf is surjective [23]. The low-dimensional homotopy groupsof tmf are [23] k π k tmf Z Z / Z / Z /
24 0 0 Z / Z ⊕ Z / Z / Z / Z Z / Z / Z / (2) Ω h i∗ of Ω h i∗ are given by [17] (see also [26] [60]) k (2) Ω h i k Z Z / Z / Z / Z / Z ⊕ Z / Z / Z / Z Z / Z / Z ) By comparing the two tables, we can indeed see the ‘missing’ Z / h i = 0 since the 2-primary part is zero andthere is no torsion in that dimension. There does not seem to be a computation for dimensions as high as27. This implies that the map ̺ : Ω h i (pt) −→ Ω h i (pt) (3.2)is a map whose domain is 0, and is thus not interesting.We next allow the structure group F so that there is a map from Y to its classifying space BF . Thuswe are considering N = BF and the classifying map to be f in (3.1). In this case, instead of the map ̺ wewill consider the map ̺ ′ : Ω h i ( BF ) −→ Ω h i ( BF ) (3.3)[ Y , f ] [ M , f ′ ] , (3.4)which maps bordism classes of 11-manifolds, together with a map f to BF , to bordism classes of 27-manifolds together with a map f ′ to BF . Now both the domain and the range are in general non-emptyunless certain condition are applied. Remarks1.
The classifying space BF has at least interesting degree four cohomology. However, we have seen thatfor the String condition to be multiplicative on O P bundles then we must kill x coming from BF . Thiswould then mean that we should in this case consider BF h x i instead of BF . Here we prefer to use the notation for cyclic groups used in homotopy theory, e.g. Z / Z . We hope this will beclear. . Killing x as above would lead to the rational homotopy type BF h x i ∼ S × higher spheres , (3.5)so that the first homotopy is in dimension 12. This then would mean that should consider Ω h i ( BF h x i ),which is zero, by dimension. If we use BF h x i instead of BF , then this might cause some problems for the description of the fields ofM-theory in terms of O P bundles, since there we used the Lie group F on the nose. In other words, unlikethe case for compact E in eleven dimensions, F appears not merely topologically, but via representationtheory. However, compare to the arguments in [51] for the E model of the C -field in M-theory. It shouldbe checked that the representations coming from the Lie 2-group F h x i respect the discussion in section 2.We can actually say more about the extensions of the F bundle. We have Proposition 3.2.
The F bundle on a String manifold Y can be extended to Z where ∂Z = Y .Proof. We look for cobordism obstructions. Extending the bundle would be obstructed by Ω h n i ( BF ). Sincethe homotopy type of F is (3 , , ,
23) then that of BF is (4 , , ,
24) so that up to dimension 11 theclassifying space BF has the homotopy type of K ( Z, E does (and in fact all exceptionalLie groups except E ) in that range. Now we reduce the problem to checking whether Ω h n i h ( K ( Z , i ) iszero. This is indeed so by calculations of Stong [59], for n = 4, and Hill ([20], motivated by this question),for n = 8.Let T h i ( BF ) be the subgroup of Ω h i ( BF ) consisting of bordism classes [ M , f ◦ π ], i.e. the classesthat factor through the base Y . It could happen that some of the classes [ Y , f ] of the bordism groupof the base are zero. Let e T h i ( BF ) be the subgroup whose elements satisfy the additional assumption that[ Y , f ] = 0 in Ω h i ( BF ). Corresponding to the diagram (2.20) there is a classifying map ψ : Ω h i ( BF ) −→ Ω h i (pt) (3.6)which takes the class [ Y , f ] to the class [ M = f ∗ E ]. The image T h i = im ψ of this map is the set oftotal spaces of O P bundles in Ω h i . If we forget the classifying map f then instead of (3.6) we can map λ : Ω h i ( BF ) −→ Ω h i (pt) , (3.7)where now the class [ Y , f ] lands in the class [ Y ] by simply forgetting f . Obviously, the kernel of λ makesup the classes [ Y , f ] which map to [ Y ] that are zero in Ω h i . Such classes [ Y , f ] map under ψ to totalspaces of O P bundles with zero-bordant bases in Ω h i . It is clear that ψ (ker λ ) is the subgroup e T h i . Thatis, we have T h i := im ψ = n total spaces of O P bundles in Ω h i (pt) o (3.8) e T h i := ψ (ker λ )= n total spaces of O P bundles with zero bordant base in Ω h i (pt) o . (3.9)Note that, as mentioned above, the 2-primary part of Ω h i n for n ≤
16 is calculated in [17]. For n = 11this is zero. This implies that the kernel of λ is all of Ω h i ( BF ), i.e. all cobordism classes of total spaceshave zero bordant bases. Then we have 17 roposition 3.3. T h i and e T h i coincide for base String manifolds of dimension eleven. There are two cases to consider in order to determine whether or not the above spaces are trivial: If Ω h i turns out to be zero, then the map ψ will be trivial in that degree. If it turns out that Ω h i = 0, then the map ψ is not trivial. It would then mean that T h i = e T h i = ∅ .However, looking carefully at the map ψ we notice that its domain is zero. This is because the homotopytype of F is K ( Z ,
3) up to dimension ten, so that the homotopy type of BF is K ( Z ,
4) up to dimensioneleven. This means that Ω h i ( BF ) = Ω h i (( K Z , ψ is trivial. Inmodding out by the corresponding equivalence to form E O = E h i = Ω h i /T h i , (3.10)we simply get Proposition 3.4.
The homology theory is just the bordism ring E O = Ω h i . Remarks1.
Proposition 3.4 implies that in dimension 27 we do not get anything smaller or simpler than bordism. The two spaces (3.9) and (3.9) have been characterized in the quaternionic case, i.e. when the fiber is H P with structure group P Sp (3), as T h i = ker( α ) (3.11) e T h i = ker(Φ och ) , (3.12)i.e. as the kernels of the Atiyah invariant in [57] and the Ochanine genus in [29], respectively. We see thatin our case, α ( M ) = 0, but Φ och ( M ) is not necessarily zero. This provides another justification for thecalculations leading to theorem 2.14. In fact, we can use the nontriviality of the Ochanine genus to checkwhether or not the homology theory is empty. Since, using Theorem 2.14, we can find a 27-dimensionalmanifold M with Φ och ( M ) = 0, the Spin cobordism group is nonzero Ω h i = 0. Consequently, we havethe following result for the corresponding String cobordism group.
Theorem 3.5. Ω h i = 0 . Remark.
Alternatively, the theorem can proved using information about tmf . Since the orientation mapfrom M String =
M O h i to tmf is surjective [4] then it is enough to know that the homotopy group of tmf in dimension 27 is nonzero. Indeed, at least π ( tmf ) ⊃ Z /
3, so that Ω = π ( M String) = 0.In [60], the Witten genus was proposed as a candidate for the replacement of α in the octonionic case,so that T h i (pt) = ker( α O ) := ker(Φ W ) . (3.13)Indeed, we have shown in Proposition 2.12 that the Witten genus is zero for our 27-dimensional manifolds,which are O P bundles. The extension of the the ‘new Atiyah invariant’ α O would be to a ‘new Ochaninegenus’ Φ O och : Ω h i∗ −→ Q [ E , E ][[ q ]] , (3.14)i.e. to the power series ring over rationalized coefficients of level 1 elliptic cohomology, such that the constantterm is the Witten genus. We have seen in theorem 2.4 that the Witten genus of O P is zero, so that in I thank Mike Hill for pointing out the Z / K X → Ω h i∗ ( X ) / I , where I is the ideal introduced in the beginning of this section. Thequestion is whether this is a generalized (co)homology theory. The desired homology theory E O n is formedby dividing Ω h i∗ by e T n and inverting the primes 2 and 3 [60]. However, there is one extra condition required,which is the invertibility of the element v = O P . By taking the limit in E O n ( X )[ O P ] − = lim j E O n +16 j ( X ) (3.15)over the sequence of homomorphisms given by multiplying by O P the resulting theory is eℓℓ O ∗ ( X ) = E O ∗ ( X )[ O P ] − = M k ≥ Ω ∗ +16 k ( X ) / ∼ , (3.16)where the equivalence relation ∼ is generated by identifying [ Y, f ] ∈ Ω h i∗ ( X ) with [ M, f ◦ π ] ∈ Ω h i∗ +16 k ( X )for an O P bundle π : M → Y , with structure group Isom O P = F , i.e. the total space of an O P bundleis identified with its base. A full construction of this theory is not yet achieved by homotopy theorists butit is believed that this should be possible in principle. We mentioned towards the end of Section 3.1 that KO ∗ (pt) can be made into an Ω spin ∗ -module and the existence of an isomorphism relating KO ∗ ( X ) and KO ∗ (pt). The octonionic version of Kreck-Stolz theory is arrived at by replacing KO ∗ (pt) by eℓℓ O n (pt), i.e.Ω h i∗ ( X ) ⊗ Ω h i∗ eℓℓ O ∗ (pt) −→ e ℓℓ O ∗ (X) (3.17)is an isomorphism away from the primes 2 and 3 [60]. Remarks.1.
The model for elliptic homology in fact involves indefinitely higher cobordism groups in increments of 16, eℓℓ O ( Y ) = M k ≥ Ω k / ∼ , (3.18)where ∼ is an equivalence that provides a correlation between topology in M-theory and topology in dimen-sions 27 , , · · · ,
11 + 16 k, · · · , ∞ . We have two points to make: • The first bundle with total space an O P bundle over Y is related to Ramond’s Euler multiplet. • As the pattern continues in higher and higher dimensions, one is tempted to seek physical interpreta-tions for such theories as well. While this direction is tantalizing, we do not pursue it in this paper. There is another homology theory that one can form, namely by identifying the image of ψ with thetrivial bundle as in [29]. The construction is analogous. The advantage here is that we do not kill Ø P , asdividing by T has the effect of killing the fiber.We have seen connections between eleven-dimensional M-theory and the putative theory in twenty-sevendimensions. If the latter theory in twenty-seven dimensions is fundamental, then it should ultimately bestudied also without restricting to the relation to M-theory. This is analogous to the case of M-theory itselfin relation to ten-dimensional type IIA string theory. Since M-theory is, as far as we know, a fundamentaltheory, then it should be (and it is being) studied without necessarily assuming a circle bundle for theeleven-dimensional manifold. In other words, what about 27-dimensional manifolds that are not the totalspace of O P bundles over eleven-manifolds? Hence 19 roposal. To study the bosonic theory as a fundamental theory in twenty-seven dimensions we shouldalso consider modding out by the equivalence relation (the ideal).For example, extension problems can be studied in this way.
It is desirable to consider the O P bundle as a family problem of objects on the fiber of M parametrizedby points in the base Y . The family of these 16-dimensional String manifolds will define an element of thecobordism group
M O h i − ( Y ) . (3.19) Remarks 1.
We have seen in section 2.4.1 that the total space of an O P bundle is not necessarily String even if Y is String . However, we do get a family of
String manifolds provided we kill the degree four classpulled back from BF (see Prop. 2.11). Unfortunately, genera are multiplicative on fiber bundles so that the vanishing of Φ W ( O P ) will forcethe Witten genus of M to be zero as well. Also taking higher and higher bundles – so as to get fibers ofdimensions higher than 16– as in (3.18) will not help in making the Witten genus nonzero. tmf is the homeof the parametrized version of the Witten genus, but we do not see modular forms in this picture. This isto be contrasted with the H P case where the Witten genus is E / Nevertheless, the elliptic genus Φ ell of O P is not zero, so the total space will not automatically have azero elliptic genus. However, elliptic genera are defined for Spin manifolds of dimension divisible by 4. Ourbase space Y is eleven-dimensional and so will automatically have zero elliptic genus. This also applies forthe Witten genus. One way out of this is instead to consider the bounding twelve-dimensional theory, i.e.the extension of the topological terms from Y = ∂Z to Z as in [64]. If we also take a 28-dimensionalcoboundary for M , i.e. ∂W = M , we would then have O P / / M (cid:31) (cid:127) / / π (cid:15) (cid:15) W π (cid:15) (cid:15) " " DDDDDDDDDDDDDDDDDD Y (cid:31) (cid:127) / / Z f / / BF . (3.20)Such an extension would involve cobordism obstructions. The manifolds extend nicely, as Ω h n i = 0 for both n = 4 (Spin) and n = 8 ( String ). The bundles also extend as shown in Proposition 3.2. It is tempting topropose that the theories should be defined on the (12 + 16 m )-dimensional spaces, and then restriction tothe boundaries would be a special instance.We have provided evidence for some relations between M-theory and an octonionic version of Kreck-Stolz elliptic homology. Strictly speaking, both theories are conjectural, and we hope that this contributionmotivates more active research both on completing the mathematical construction of this elliptic homologytheory (part of which is outlined in [60]) as well as making more use of the connection to M-theory. In doingso, we even hope that M-theory itself would in turn give more insights into the homotopy theory.In closing we hope that further investigation will help shed more light on the mysterious appearance ofthe exceptional groups E and F and to give a better understanding of their role in M-theory.20 cknowledgements The author thanks the American Institute of Mathematics for hospitality and the “ Algebraic Topologyand Physics” SQuaRE program participants Matthew Ando, John Francis, Nora Ganter, Mike Hill, MikhailKapranov, Jack Morava, Nitu Kitchloo, and Corbett Redden for very useful discussions. The author wouldlike to thank the Hausdorff Institute for Mathematics in Bonn for hospitality and the organizers of the“Geometry and Physics” Trimester Program at HIM for the inspiring atmosphere during the writing of thispaper. Special thanks are due to Matthew Ando, Mark Hovey, Nitu Kitchloo, Matthias Kreck, and PierreRamond for helpful remarks and encouragement, to Stephan Stolz for useful comments on the draft, and toArthur Greenspoon for many useful editorial suggestions.
References [1] J. F. Adams, Lectures on exceptional Lie groups, University of Chicago Press, Chicago, IL, 1996.[2] A. Adams and J. Evslin,
The loop group of E and K-theory from 11d , J. High Energy Phys. (2003) 029, [ arXiv:hep-th/0203218 ].[3] D. W. Anderson, E. H. Brown, Jr., and F. P. Peterson, SU -cobordism, KO -characteristic numbers, andthe Kervaire invariant , Ann. of Math. (2) 83 (1966) 54–67.[4] M. Ando, M. J. Hopkins, and N. P. Strickland, Elliptic spectra, the Witten genus and the theorem ofthe cube , Invent. Math. (2001), no. 3, 595–687.[5] M. F. Atiyah, R. Bott, and A. Shapiro,
Clifford modules , Topology (1964) suppl. 1, 3–38.[6] J. Baez, The octonions , Bull. Amer. Math. Soc. (2002) 145-205. Erratum ibid. (2005) 213,[ arXiv:math/0105155 ] [math.RA].[7] D. Belov and G. M. Moore, Holographic action for the self-dual field , [ arXiv:hep-th/0605038 ].[8] D. Belov and G. M. Moore,
Type II actions from 11-dimensional Chern-Simons theories ,[ arXiv:hep-th/0611020 ].[9] A. Borel, Sur l’homologie et la cohomologie des groupes de Lie compacts connexes , Amer. J. Math. (1954) 273–342.[10] P. Conner and E. Floyd, The relation of cobordism to K -theories , Lecture Notes in Mathematics, no. Springer-Verlag, Berlin-New York 1966.[11] E. Cremmer, B. Julia, and J. Scherk,
Supergravity theory in eleven-dimensions , Phys. Lett.
B76 (1978)409-412.[12] E. Diaconescu, D. S. Freed, and G. Moore,
The M-theory 3-form and E gauge theory , Elliptic coho-mology, 44–88, London Math. Soc. Lecture Note Ser., 342, Cambridge Univ. Press, Cambridge, 2007,[ arXiv:hep-th/0312069 ].[13] E. Diaconescu, G. Moore, and E. Witten, E gauge theory, and a derivation of K-Theory from M-Theory , Adv. Theor. Math. Phys. (2003) 1031, [ arXiv:hep-th/0005090 ].[14] M. J. Duff, M-theory (the theory formerly known as strings) , Int. J. Mod. Phys.
A11 (1996) 5623-5642,[ arXiv:hep-th/9608117v3 ]. 2115] M. J. Duff, J. T. Liu, and R. Minasian,
Eleven dimensional origin of string/string duality: A one looptest , Nucl. Phys.
B452 (1995) 261, [ arXiv:hep-th/9506126 ].[16] T. Friedrich,
Weak
Spin (9) -structures on 16-dimensional Riemannian manifolds , Asian J. Math. (2001) 129–160, [ arXiv:math/9912112 ] [math.DG].[17] V. Giambalvo, On h i -cobordism , Illinois J. Math. (1971) 533–541; erratum ibid (1972) 704.[18] M. B. Green, J. H. Schwarz, and E. Witten, Superstring theory Vol. 2, second edition, CambridgeUniversity Press, Cambridge, 1988.[19] F. R. Harvey, Spinors and Calibrations, Academic Press, Boston, 1990.[20] M. Hill, The String bordism of BE and BE × BE through dimension 14 , [ arXiv:0807.2095 ][math.AT].[21] F. Hirzebruch and P. Slodowy, Elliptic genera, involutions, and homogeneous spin manifolds , Geom.Dedicata (1990), no. 1-3, 309–343.[22] N. Hitchin, Harmonic spinors , Advances in Math. (1974) 1–55.[23] M. J. Hopkins, Algebraic topology and modular forms , Proceedings of the ICM, Beijing 2002, vol. ,283–309, [ arXiv:math/0212397v1 ] [math.AT].[24] G. Horowitz and L. Susskind, Bosonic M-theory , J. Math. Phys. (2001) 3152,[ arXiv:hep-th/0012037 ].[25] M. Hovey, Spin bordism and elliptic homology , Math. Z. (1995), no. 2, 163–170.[26] S. Klaus,
Brown-Kervaire invariants , PhD thesis, University of Mainz, Shaker Verlag, Aachen, 1995.[27] S. Klaus,
The Ochanine k -invariant is a Brown-Kervaire invariant , Topology (1997), no. 1, 257–270.[28] A. Kono, H. Shiga, and M. Tezuka, A note on the cohomology of a fiber space whose fiber is a homo-geneous space , Quart. J. Math. Oxford Ser. (2) (1989), no. 159, 291–299.[29] M. Kreck and S. Stolz, H P -bundles and elliptic homology , Acta Math. (1993) 231–261.[30] I. Kriz and H. Sati, M Theory, type IIA superstrings, and elliptic cohomology , Adv. Theor. Math. Phys. (2004) 345, [ arXiv:hep-th/0404013 ].[31] I. Kriz and H. Sati, Type IIB string theory, S-duality and generalized cohomology , Nucl. Phys.
B715 (2005) 639, [ arXiv:hep-th/0410293 ].[32] I. Kriz and H. Sati,
Type II string theory and modularity , J. High Energy Phys. (2005) 038,[ arXiv:hep-th/0501060 ].[33] A. Lichnerowicz, Spineurs harmoniques , C. R. Acad. Sci. Paris (1963) 7–9.[34] M. Mahowald and M. Hopkins,
The structure of 24-dimensional manifolds having normal bundleswhich lift to B O[8], in Recent progress in homotopy theory (Baltimore, MD, 2000), 89–110, Contemp.Math., 293, Amer. Math. Soc., Providence, RI, 2002.2235] V. Mathai and H. Sati,
Some relations between twisted K-theory and E gauge theory , J. High EnergyPhys. (2004) 016, [ arXiv:hep-th/0312033 ].[36] J. McCleary, User’s Guide to Spectral Sequences, Publish or Perish, Wilmington, Delaware, 1984.[37] J. Milnor and J. Stasheff, Characteristic classes, Princeton University Press, Princeton, NJ, 1974.[38] S. Ochanine, Signature modulo , invariants de Kervaire g´en´eralis´es et nombres caract´eristiques dansla K -th´eorie r´eelle , Mm. Soc. Math. France (N.S.) Genres elliptiques ´equivariants , in Elliptic curves and modular forms in algebraic topology(Princeton, NJ, 1986), 107–122, Lecture Notes in Math., 1326, Springer, Berlin, 1988.[40] S. Ochanine,
Elliptic genera, modular forms over K O ∗ and the Brown-Kervaire invariant , Math. Z. (1991), no. 2, 277–291.[41] T. Pengpan and P. Ramond, M(ysterious) Patterns in
SO(9), Phys. Rept. (1999) 137-152,[ arXiv:hep-th/9808190 ].[42] D. Quillen,
The mod 2 cohomology rings of extra-special -groups and the spinor groups , Math. Ann. (1971) 197–212.[43] P. Ramond, Boson-fermion confusion: The string path to supersymmetry , Nucl. Phys. Proc. Suppl. (2001) 45-53, [ arXiv:hep-th/0102012 ].[44] P. Ramond,
Algebraic dreams , Meeting on Strings and Gravity: Tying the Forces Together, Brussels,Belgium, 19-21 Oct 2001, [ arXiv:hep-th/0112261 ].[45] S.-J.Rey,
Heterotic M(atrix) strings and their interactions , Nucl. Phys.
B502 (1997) 170,[ arXiv:hep-th/9704158 ].[46] H. Salzmann, D. Betten, T. Grundh¨ofer, H. H¨ahl, R. L¨owen, and M. Stroppel, Compact ProjectivePlanes, Walter de Gruyter & Co., Berlin, 1995.[47] H. Sati,
M-theory and Characteristic Classes , J. High Energy Phys. (2005) 020,[ arXiv:hep-th/0501245 ].[48] H. Sati,
Flux quantization and the M-theoretic characters , Nucl. Phys.
B727 (2005) 461,[ arXiv:hep-th/0507106 ].[49] H. Sati,
Duality symmetry and the form-fields in M-theory , J. High Energy Phys. (2006) 062,[ arXiv:hep-th/0509046 ].[50] H. Sati,
The Elliptic curves in string theory, gauge theory, and cohomology , J. High Energy Phys. (2006) 096, [ arXiv:hep-th/0511087 ][51] H. Sati, E gauge theory and gerbes in string theory , [ arXiv:hep-th/0608190 ].[52] H. Sati, On higher twist in string theory , [ arXiv:hep-th/0701232 ].[53] H. Sati,
Loop group of E and targets for spacetime , [ arXiv:hep-th/070123 ].[54] H. Sati, An approach to anomalies in M-theory via K Spin, J. Geom. Phys. (2008) 387,[ arXiv:0705.3484 ] [hep-th]. 2355] H. Sati, U. Schreiber and J. Stasheff, L ∞ -connections and applications to String- and Chern-Simons n -transport , in Recent Developments in QFT , eds. B. Fauser et al., Birkh¨auser, Basel (2008),[ arXiv:0801.3480 ] [math.DG].[56] H. Sati, U. Schreiber, and J. Stasheff,
Fivebrane structures , to appear in Rev. Math. Phys.,[ arXiv:0805.0564 ] [math.AT].[57] S. Stolz,
Simply connected manifolds of positive scalar curvature , Ann. of Math. (2) 136 (1992), no.3, 511–540.[58] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968.[59] R. Stong,
Calculation of Ω spin ( K ( Z , Elliptic Cohomology , Kluwer Academic/Plenum Publishers, New York, 1999.[61] H. Toda,
Cohomology mod 3 of the classifying space BF of the exceptional group F , J. Math. KyotoUniv. (1973) 97–115.[62] P. K. Townsend, Four lectures on M-theory , Summer School in High Energy Physics and CosmologyProceedings, E. Gava et. al (eds.), Singapore, World Scientific, 1997, [ arXiv:hep-th/9612121v3 ].[63] E. Witten,
String theory dynamics in various dimensions , Nucl. Phys.
B443 (1995) 85-126,[ arXiv:hep-th/9503124v2 ].[64] E. Witten,
On Flux quantization in M-theory and the effective action , J. Geom. Phys. (1997) 1,[ arXiv:hep-th/9609122 ].[65] E. Witten, Five-brane effective action in M-theory , J. Geom. Phys. (1997) 103-133,[ arXiv:hep-th/9610234 ].[66] E. Witten, Duality relations among topological effects in string theory , J. High Energy Phys. (2000) 031, [ arXiv:hep-th/9912086 ].[67] D. Zagier,