aa r X i v : . [ h e p - t h ] J a n Ninebrane Structures
Hisham SatiJanuary 20, 2015
Abstract
String structures in degree four are associated with cancellation of anomalies of string theoryin ten dimensions. Fivebrane structures in degree eight have recently been shown to be associ-ated with cancellation of anomalies associated to fivebranes in string theory and M-theory. Weintroduce and describe
Ninebrane structures in degree twelve and demonstrate how they cap-ture some anomaly cancellation phenomena in M-theory. Along the way we also define certainvariants, considered as intermediate cases in degree nine and ten, which we call and , respectively. As in the lower degree cases, we also discuss the naturaltwists of these structures and characterize the corresponding topological groups associated toeach of the structures, which likewise admit refinements to differential cohomology.
Contents BO h i and BO h i structures 33 Ninebrane structures 84 The set of lifts 105 Twisted structures 126 Structures not directly defined via the Whitehead tower 157 The (twisted) groups 168 Differential refinement 16 Introduction
The study of higher connected covers of Lie groups in the context of string theory and M-theory, asadvocated in [31, 32, 33], leads to interesting mathematical structures as well as means for cancelinganomalies in string theory and M-theory. Beyond String structures in degree four, obtained bykilling the third homotopy group of the orthogonal group, we have Fivebrane structures in degreeeight obtained by killing the next homotopy group which is in degree seven.We will consider killing – more precisely, co-killing – further homotopy groups. From thehomotopy theoretic point of view one can continue the process of killing indefinitely in a systematicway. However, no systematic understanding of the relevance of all cases exists. What we dois advocate is a natural setting, a description of the higher geometry, as well as provide severalexamples from M-theory and string theory for which performing such killings in the next few degreesis natural. We highlight the structures we consider here in the following table. k π k (O( n )) Z Z Z Z n ) h k i String( n ) Fivebrane( n ) O h i ( n ) O h i ( n ) Ninebrane( n ) ✡ kill π ✡ kill π ☎ kill π = = ☛ kill π The point of view we take here is that the group O h i ( n ) is a ‘shift by 8’ analog of the specialorthogonal group SO( n ). The group O h i ( n ) is a ‘shift by 8’ analog of the Spin group Spin( n ). Themod 8 periodicity of the homotopy groups of the orthogonal group motivates the following for thecorresponding G -structures: The classifying spaces BO h i = B ( O h i ) and BO h i = B ( O h i )correspond to a ‘shift by 8’ analog of orientation and of Spin structure, respectively. Thus to identifythese structures in the second period in the mod 8 periodicity we indicate these as and . We encapsulate the theme in the following diagram of lifts, extending theones in [32] [33], i.e. the higher part of the Whitehead tower of the orthogonal group: B Ninebrane = BO h i (cid:15) (cid:15) BO h i x / / (cid:15) (cid:15) K ( Z , BO h i x / / (cid:15) (cid:15) K ( Z , B Fivebrane x / / (cid:15) (cid:15) K ( Z , X / / ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ < < @ @ B String p / / K ( Z , .
2e identify the obstructions x i , i = 9 ,
10 in Sec. 2 and i = 12 in Sec. 3, and characterize the setof lifts in Sec. 4. The first two will be, in a sense, exotic classes. Then in Sec. 5 we twist thesestructures, and in Sec. 6 we consider variant structures in which the higher obstructions vanishwithout lower classes having to be zero. We characterize the corresponding groups in Sec. 7, andfinally in Sec. 8 we construct differential refinements, and provide a natural M-theoretic setting inway of motivation and examples throughout. BO h i and BO h i structures The topology and geometry of a manifold can be studied via the structures related to its tangentbundle. Starting with a Riemannian manifold X n , its tangent bundle with structure group O ( n ),can be lifted to further structures which in turn imposes topological conditions on X n . The firststep is to lift the structure group to SO ( n ) by equipping X n with an orientation, which is allowedprovided the first Stiefel-Whitney class w of T X n is zero. Further structures can be conditionallygiven. The structure group can be further lifted from SO ( n ) to the double cover Spin( n ) whichallows the existence of spinors provided that the second Stiefel-Whitney class w is zero.Note that the process does not stop here, and we can further continue equipping the tangentbundle with higher structures. Due to the homotopy type of the orthogonal group, the next step inthe process is consider the lifting to the seventh connected cover denoted O h i , which occurs whenthe cohomology class p is zero, where p is the first Pontrjagin class of the tangent bundle. Thenotation G h n i means that all the homotopy groups of order 0 , · · · , n of the original group G arekilled.In fact, the way to obtain the above structures is by pulling back from the universal classifyingspace to our spacetime X n . Since SO ( n ) is obtained from O ( n ) by killing the first homotopygroup, then BSO ≃ BO h i , which is also denoted BO h w i to highlight the condition imposed bysuch a structure. Note that in going from G to BG there is a shift in homotopy, i.e. π i ( G ) ∼ = π i +1 ( BG ). Similarly, B Spin ≃ BO h i ≃ BO h w , w i , this time emphasizing the Spin condition w = 0 = w . Finally, BO h i , sometimes also denoted B String, can be written in the samenotation as BO h w , w , p i . Notice that in this last case, the additional condition is no longer amod 2 condition but rather is one on integral cohomology. The requirement p = 0 is not quite thesame as setting p = 0, because the latter is a rational condition which misses the torsion classesin the former integral condition.Note that p is related to the Stiefel-Whitney classes, namely its mod 2 reduction is given by w . We will see that, in a sense, not all Stiefel-Whitney classes are relevant, but a special role isplayed by the ones of the form w j . For instance, starting from w = w = w = 0 leads to w i = 0for i = 1 , · · ·
7. Thus the only new condition after w = 0 is w = 0. In terms of classifying spaces,what this implies is that there is a lift from BO h r i to BO h w j i for j < r .We now consider the Stiefel-Whitney classes in relatively higher degrees, w i for 2 ≤ i < .For applications in even higher degrees, see [21]. We start with the following observation for thehigher Stiefel-Whitney classes as they arise in the context of M-theory. Lemma 2.1
Let Y be an orientable eleven-manifold. Then we have w ( Y ) = w ( Y ) = w ( Y ) = 0 . ν i ∈ H i ( Y ; Z ) be the Wu class, i.e. the unique cohomology class such that Sq i ( x ) = x · ν i for any x ∈ H − i ( Y ; Z ). From the properties of the Wu classes, we have ν = 1, ν i = 0 for i >
5. Wu’s formula relates the Stiefel-Whitney classes to the Wu classes via w k = X i Sq k − i ν i . (2.1)Using the Adem relation Sq i = Sq Sq i − for i odd, and that Sq : H ( Y ; Z ) → H ( Y ; Z )is trivial for orientable Y , gives that Sq i : H − i ( Y ; Z ) → H ( Y ; Z ) is zero for i odd. Thisimplies, from the definition of the Wu classes, that ν i = 0 for i odd. Hence, for orientable Y , ν i iszero unless i is even and 0 ≤ i ≤
4. From this and expression (2.1) it follows that w i = 0 for i > w ( Y ) = w ( Y ) = w ( Y ) = 0. ✷ Remarks. 1.
Note that the M-theory fivebrane anomaly cancelation requires an
M O h i orien-tation, i.e. a Fivebrane structure [33]. Note that in general for orientable Y with no extra structure, the class w will not be zero. Example.
Consider Y = C P × C P × ( S ) or Y = P (1 , × ( S ) , where P (1 ,
4) is theDold manifold defined as follows. P ( r, s ) is the quotient ( S r × C P s ) / ∼ , where ( x, y ) ∼ ( x ′ , y ′ ) ifand only if x ′ = − x and y ′ = − y . The Dold manifold is the total space of C P s bundle of complexprojective spaces over real projective space R P r whose total Stiefel-Whitney class is given by w ( P ( r, s )) = (1 + e ) r (1 + e + e ) s +1 , (2.2)where e and e are the generators in the cohomology groups of the corresponding projectivespaces H ( R P r ; Z ) and H ( C P s ; Z ), respectively. In particular, P (1 ,
4) is orientable and hasnon-vanishing w .We have seen above that for orientable Y , the class w is not necessarily zero. However,integrality of the one-loop polynomial I = [ p − ( p ) ] appearing in anomaly cancellation inM-theory requires that w be in fact zero. This is because it is the mod 2 reduction of the secondSpin characteristic class. Recall that the integral cohomology of the classifying space of the Spingroup is [38] H ∗ ( B Spin; Z ) = Z [ Q , Q , · · · ] ⊕ γ , (2.3)with γ a 2-torsion factor, i.e. 2 γ = 0. The two relevant degrees are H ( B Spin; Z ) ∼ = Z with generator Q H ( B Spin; Z ) ∼ = Z ⊕ Z with generators Q , Q , where the Spin classes Q and Q are determined by their relation to the Pontrjagin classes p = 2 Q , p = Q + 2 Q . (2.4)Obviously, when inverting is possible, the Spin generators are given by Q = p / Q = p − ( p / . The mod 2 reductions of Q and Q are w and w , respectively. It was explainedin [19] that it is useful to write the one-loop term in terms of the Spin characteristic classes I = Q . (2.5)4ow I is integral when w = 0 [42]. The latter condition allows to define a “Membrane structure” i.e. a structure defined by the condition w = 0 [25]. Then Q is certainly divisible by 2, andhence we have Lemma 2.2
For a manifold Y with a Membrane structure, we have w ( Y ) = 0 . Next, putting together the above discussion, we have the following observation, still motivatedwithin the context of M-theory.
Proposition 2.3
Let Y be a manifold which admits a Fivebrane structure. Then all classes in Y pulled back from universal classes in H n ( BO ; R ) , for n ≤ and R = Z or Z , are trivial. Proof. This follows from statements that can be directly verified. Orientation requires that w ( Y ) = 0. A Fivebrane structure amounts to p ( Y ) = 0 and requires a String structure, i.e. p ( Y ) = 0, which in turn requires a Spin condition, i.e. w ( Y ) = 0, as well as a Membrane con-dition w ( Y ) = 0. The Fivebrane condition further implies w ( Y ) = 0 via mod 2 reduction fromthe Fivebrane condition. Finally, all odd Stiefel-Whitney classes up to that degree are zero. Thisfollows from the Wu formula for the action of the Steenrod algebra which takes the general form Sq i w j = P it =0 (cid:0) j − i − tt (cid:1) w i − t w j + t for i < j (see [36]). First, the Wu formula Sq w = w w + w gives that if w = 0 then w = 0. Second, the formulae Sq w = w w + w , Sq w = w w + w and Sq w = w w + w w + w w + w imply that if w = 0 then the three classes w , w and w arezero. In the next degree, starting with w , the formulae Sq w = w w + w , Sq w = w w + w and Sq w = w w + w w + w w + w imply that if w = 0 then the classes w , w and w arezero. These last three degrees can also be deduced by appealing to the dimension of the manifold Y , i.e. by using Lemma 2.1. ✷ As we saw above, these obstructions mostly vanish for dimension reasons in our range of di-mensions. However, we will consider bundles other than the tangent bundle; for example bundleswith structure group SO(32) rather than SO(10) or SO(11).
Example: Orthogonal gauge structure groups.
Consider the orthogonal group G = SO(32)as a structure group of a gauge bundle over our manifold. This is relevant in type I and heteroticstring theory in ten dimensions. The topological role of this group in relation to global anomaliesis highlighted in [41], which we recast in our language (cf. [31] [32]). The degree seven generator π (SO(32) = Z is used to show invariance of theory on S × S and to derive a quantizationcondition on the H-field, which can be thought of as a curvature of a gerbe in degree three. The nexthomotopy group π (SO(32)) = Z correspond to a Yang-Mills instanton on S via the embeddingSO(10) ֒ → SO(32) by viewing the Spin connection, arising from a spinor representation of thenatural structure group SO(10) of the tangent bundle, as a gauge field, by viewing this in thevector representation of SO(32).What about π (SO(32)) = Z ? We interpret this in essentially the same way as for the case of π (SO(32). However, our setting will be M-theory in eleven dimensions rather than string theory in This name is introduced in Ref. [25] and justified there by the fact that it arises in connection with anomaliesassociated with the membrane in M-theory. Note that the group is more precisely Spin(32) / Z , where Z is the complement of the component of the centerwhich leads to SO(32). However, since we are considering connected covers, such differences at the level of thefundamental group (which is killed) will not matter for us. S with structure group SO(11) andembed this in the group SO(32) and, as above, view the Spin connection of the former as a gaugefield of the latter. One justification for enlarging of the structure group is to form a generalizedconnection, taking into account the C-field terms. For instance, in [9] SO(32) is described as ageneralized holonomy group, while in [15] the group SL(32 , R ) played that role; homotopically, thisis simply the same as SO(32).We now define the first two of the new structures and afterwards we will explain the connectionto the above classes. Definition 2.4 (2-Orientation structure.) A 2-Orientation structure is defined by the lift from BO h i = BFivebrane to BO h i in the following diagram BO h i (cid:15) (cid:15) X f / / ˆ f BFivebrane / / K ( π ( BO ) , . (2.6) Remarks (i)
The existence of the above fibration, as well as all the fbrations that we introducebelow, follows from the work of Stong [36] [37]. (ii)
Corresponding to this diagram is a class x ∈ H (BFivebrane , π ( BO )) = H (BFivebrane , Z ).The map f : X → BFivebrane lifts to ˆ f : X → BO h i if and only if we have the vanishing of theobstruction class f ∗ x = 0 ∈ H ( X ; π ( BO )) = H ( X ; Z ) . (2.7) (iii) One might think that we can identify x = w as the cohomology ring H ∗ ( BO ; Z ) is generatedby the Stiefel-Whitney classes. However, as we will see shortly, this is not the case. (iv) BO h i can also be described as a bundle pulled back from the path fibration in the followingdiagram (as in [33], which builds on [36]) K ( π ( B )) , / / BO h i (cid:15) (cid:15) P K ( π ( BO ) , (cid:15) (cid:15) Ω K ( π ( BO ) , o o X f / / ˆ f BFivebrane / / K ( π ( BO ) , . (2.8)In the next degree we have: Definition 2.5 (2-Spin structure.) A 2-Spin structure is defined by the lift from BO h i to BO h i in the following diagram BO h i (cid:15) (cid:15) X f / / ˆ f BO h i / / K ( π ( BO ) , . (2.9)6 emarks (i) Corresponding to this diagram is a class x ∈ H ( BO h i , π ( BO )) = H ( BO h i , Z ).The map f : X → BO h i lifts to ˆ f : X → BO h i if and only if we have the vanishing of theobstruction class f ∗ x = 0 ∈ H ( X ; π ( BO )) = H ( X ; Z ) . (2.10) (ii) One might think that we can identify x = w as the cohomology ring H ∗ ( BO ; Z ) is generatedby the Stiefel-Whitney classes. However, again, as we will see this is not the case. (iii) BO h i can also be described as a bundle pulled back from the path fibration in the followingdiagram K ( π ( B )) , / / BO h i (cid:15) (cid:15) P K ( π ( BO ) , (cid:15) (cid:15) Ω K ( π ( BO ) , o o X f / / ˆ f BO h i / / K ( π ( BO ) , . (2.11) Identifying the generators x and x . As mentioned above, one might be very tempted toidentify the obstruction classes x and x with the generators of H i ( BO ; Z ) for i = 9 ,
10, i.e. withthe Stiefel-Whitney classes w and w , respectively. However, this would be too simplistic and, aswe will see, is not true. The main subtlety here is that beyond Fivebrane in the Whitehead tower of BO , maps from BO h m i to BO are no longer surjective (see [36] [37]). Thus upon close inspectionone realizes that these will be exotic characteristic classes not arising from the cohomology of BO but rather of BO h i and BO h i . For instance, x arises from pulling back along the map K ( Z , → BFivebrane. This is analogous to the map K ( Z , → BO h i relating gerbes to String structuresand provides interesting geometry. It would be very interesting to identify these obstructions viaexamples, which we expect to be related to the ones on orthogonal structure groups above.One can, in fact, study the generators x and x a little more precisely, by specializing thegeneral disucssion in [36] to our context to relate to more standard classes, namely the fundamentalclasses of Eilenberg-MacLane spaces. We start with 2-Orientations. The spectral sequence of thefibration BO h i → BO h i → K ( Z ,
8) gives the long exact sequence of cohomology groups · · · / / H (BO h i ; Z ) τ / / H ( K ( Z , Z ) / / H (BO h i ; Z ) / / · · · . (2.12)Here the transgression τ is given by x τ Sq ι , where ι is the fundamental cohomology classof the Eilenberg-MacLane space K ( Z , BO h i → BO h i → K ( Z ,
9) gives the long exact sequence of cohomology groups · · · / / H (BO h i ; Z ) τ / / H ( K ( Z , Z ) / / H (BO h i ; Z ) / / · · · . (2.13)The transgression τ is given by x τ Sq ι , with ι the fundamental cohomology class of K ( Z , A be the mod 2 Steenrod algebra and f : BO h i = B Fivebrane → K ( π (BO) ,
9) = K ( Z , f : BO h i = B2-Orientation → K ( π (BO) ,
10) = K ( Z , f : BO h i =BO h i = B2-Spin → K ( π (BO) ,
12) = K ( Z , f ∗ : H i ( K ( Z , Z ) x −→ H i ( B Fivebrane; Z ), f ∗ : H i ( K ( Z , Z ) x −→ i (B2-Orientation; Z ), and f ∗ : H i ( K ( Z , Z ) x −→ H i (B2-Spin; Z ), respectively. Conditionson the subjectivity of these maps can be deduced from [37], and which we will record momentar-ily. We can also deduce from the general results of Stong [37] that the cohomology rings with Z coefficients for our spaces are H i (BO h i ; Z ) ∼ = (cid:0) A / A Sq (cid:1) f ∗ ( ι ) ,H i (BO h i ; Z ) ∼ = (cid:0) A / A Sq (cid:1) f ∗ ( ι ) ,H i (BO h i ; Z ) ∼ = (cid:0) A / A Sq + A Sq (cid:1) f ∗ ( ι ) , (2.14)where ι j is the fundamental class of the appropriate Eilenberg-MacLane space in degree j .We summarize the above discussion with the following Proposition 2.6 (i)
The generators x and x are related to the fundamental classes ι and ι of K ( Z , and K ( Z , via τ ( x ) = Sq ι and τ ( x ) = Sq ι , with τ the transgression in (2.12) and (2.13) , respectively. (ii) The maps f ∗ , f ∗ and f ∗ are surjective for i < , i < and i < , respectively. Note that the above inequalities are certainly within the range of dimensions of interest in stringtheory and M-theory.We now go back to the original question on whether x and x have to do with Stiefel-Whitneyclasses. We will deduce from the results of Bahri-Mahowald [3] and Stong [36] that there is nosimple direct relation. We consider the covering map p : BO h φ ( r ) i → BO, where φ ( r ) is somespecific function of r , the three relevant values of which are given as φ (3) = 8, φ (4) = 9 and φ (5) = 10. A result of [36] asserts that the covering map p maps w i ∈ H ∗ (BO; Z ) to the generatorsin H ∗ (BO h φ ( r ) i ; Z ) if i − r ones in its dyadic (binary) expansion, and the remainingclasses are mapped to decomposables. For our three relevant cases, i = 8 , r = 4 ,
5, and 6, respectively.Furthermore, a result of [3] states that the class p ∗ w n is nonzero in H ∗ (BO h φ ( r ) i ; Z ) if andonly if a certain Poincar´e series has a nonzero entry in dimension n . For r = 3 this series is givenby 1 + t + t + · · · , which explicitly contains a t -term. This is the Fivebrane case. Next, for the2-Orientation case, we have for r = 4 the series − t (1 + t )(1 + t )(1 + t )(1 + t )(1 + t ),which does not have a t term. Similarly for the case of 2-spun structures, the Poincar´e series for r = 5 is likewise sparse and is explicitly seen to no have a t term. Therefore, from both results of[3] and [36] we have Proposition 2.7 (i)
The covering maps p : BO h i → BO and p : BO h i → BO send the Steifel-Whitney classes w ∈ H (BO; Z ) and w ∈ H (BO; Z ) to decomposables. (ii) The classes p ∗ w and p ∗ w are zero in H (BO h i ; Z ) and H (BO h i ; Z ) , respectively. In this section we shift from Stiefel-Whitney classes to Pontrjagin classes to define our third mainstructure. It is easy to see that p is divisible by 2 when we have a String structure. In thiscase, since w is the mod 2 reduction of the first Spin characteristic class Q = p , we have that8 = 0 in the presence of a String structure. From the congruence p = w mod 2 and the factthat w = Sq w we get that p is even under these conditions. In the four-dimensional case, thiswas enough to determine the obstruction. However, in this twelve-dimensional case, we will see anextra division by 2 . Also, as in the case of the Fivebrane structure, there is an extra division by 3and, additionally, by the next prime 5 for for a Ninebrane structure. Another distinction to makeis that, while w = 0 does not imply w = 0, having w = 0 does imply w = 0. The follows fromthe Wu formula Sq w = w w + w , and what distinguishes it from the former is the relativelylow power of the Steenrod square. Definition 3.1
A Ninebrane structure on a 2-Spin manifold M is a lift ˆ f of the classifying map f in the following diagram BO h i = B Ninebrane π (cid:15) (cid:15) M ˆ f f / / BO h i x / / K ( Z , . The obstruction class for lifting the classifying map f : X → BO h i to ˆ f : X → BO h i =BNinebrane is obtained by pulling back the universal class x ∈ H (BO h i ; Z ) ∼ = H ( BO h i ; Z ).Thus a manifold X admits a Ninebrane structure if and only if f ∗ x ∈ H ( X ; Z ) vanishes. Thisis a fraction of the third Pontrjagin class p and can be characterized as follows. For vector bundlesover the sphere S the best possible result on the divisibility of the Pontrjagin class p ( ξ ), for ξ a vector bundle of rank 12 over S , is that p ( ξ ) can be any multiple of (see [18] p. 244, [5] [16])(2 · − , u , that is 240 u , where u is the standard generator of H ( S ). It followsthat the relevant fraction is given by 1 /
240 so that, therefore, we straightforwardly have
Proposition 3.2
The obstruction to a Ninebrane structure is given by p . Remarks. (i)
Note that having simultaneously a String structure, a Fivebrane structure, andpositive scalar curvature leads to a Ninebrane structure. This follows from the Lichnerowicz theoremand the index theorem: the obstruction to positive scalar curvature is given by the b A -genus, whichin dimension 12 is a combination of the String obstruction p , the Fivebrane obstruction p andthe Ninebrane obstruction p . See [26] for extensive discussions in the Spin case. (ii) Note that there is a path fibration K ( π ( B )) , → P K ( π ( BO ) , → K ( π ( BO ) , K ( Z , → P K ( Z , → K ( Z , K ( Z , −→ BNinebrane −→ B2-Spin . (3.1) Example.
Global considerations in M-theory require extension to a 12-dimensional boundingSpin manifold Z . Supersymmetry implies the existence of a Rarita-Schwinger field, i.e. a spinor-valued one-form, which can be viewed as a section of the Spin bundle tensored with the virtualbundle T Z C − O , where the subtraction of 4 O accounts for two Faddeev-Popov ghosts as wellas the two extra directions in relating to the Spin bundle in ten dimensions [8]. The action canbe written in terms of indices of twisted Dirac operators, one of which being the Rarita-Scwingeroperator [42]. The Chern character ch( T Z C − O ) is given by8 + p + ( p − p ) + ( p − p p + 3 p ) . (3.2)9quipping our manifold with a String and a Fivebrane structure, i.e. requiring the vanishing of p and p , we get in dimension twelve the term p . This is twice the Ninebrane obstruction p . The situation here is, in some sense, analogous to the case of the first Pontrjagin class p vs. the first Spin characteristic class Q = p . Note that, concentrating on the prime 2, theNinebrane obstruction is p , whose mod 2 reduction is the Stiefel-Whitney class w = 0. Thesimilar statement in degree eight is that the 2-adic part of the Fivebrane obstruction which is p admits w as its mod 2 reduction.Note, however, that the above example does not amount to a full anomaly cancellation require-ment, but merely that the Ninebrane obstruction appears in the expressions of part of the anomalyor effective action. This is then slightly weaker that the statements in the String and Fivebranecases, which amounted e.g. to the Green-Schwarz anomaly cancellation condition and its dual [31][32]. Invariances of the structures under homotopy equivalences.
We consider whether havingone of the three structures defined above is a property that is invariant under automorphisms.The topological invariance of String structures is considered in [19]. It is interesting to note thatthe obstructions p and p for String and Fivebrane structures, and the class p mod 120 arehomotopy invariant. This follows from the results in [35]. Furthermore, we note that the intersectionform on a closed Spin 12-dimensional manifold is always even, so we have a further division by 2for p . Therefore, we have Proposition 3.3
Having a Fivebrane, Ninebrane, 2-Orient, or 2-Spin structure is a homotopyinvariant property. So if f : X → Y is a homotopy equivalence then there is such a structure on X if and only if there is the same one on Y . Remark.
In [32] [33], the identification of anomalies in M-theory and string theory with Fivebranestructures required some modification to account for further congruences. For example, the class p appeared instead of p . That further division by 8 was accounted for by defining a variantstructure, denoted F h i . Here in the case of Ninebrane structures, the same kind of argumentapplies and we can similarly account for further divisions of p as needed for applications. The set of lifts of familiar structures, such as orientations, Spin structures and String structures isgiven generally by a torsor over a cohomology group of one dimension less than the dimension of theobstruction. This was also shown for Fivebrane structures in [32]. Such a characterization continuesto hold in our case of 2-Orientation = BO h i , 2-Spin = BO h i , and Ninebrane = BO h i structures. We will describe these lifts in a uniform fashion. Let A denote Z for the first and secondstructures and Z for the third structure and let n = 9 ,
10, and 12, respectively, and m = n + 1. Inthe case of the 2-Spin structure we have an automatic further lift one more level to BO h i . We10ncode all this succinctly in the diagram K ( A, n − (cid:15) (cid:15) BO h m i (cid:15) (cid:15) X f / / ˆ f BO h n i , (4.1)in which the fibrations are induced from the path fibrations, including the ones in (2.8) and (2.11).The set of structure is given in the three cases by a torsor for the cohomology group H n − ( X ; A ).Therefore, with an equivalence relation on each of the sets given by homotopy of sections, we have Proposition 4.1 (i) The set of 2-Orientation structures on a given Fivebrane structure is givenby a torsor for the group H ( X ; Z ) .(ii) The set of 2-Spin structures on a given 2-Orientation structure is given by a torsor for thegroup H ( X ; Z ) .(iii) The set of Ninebrane structures on a given 2-Spin structure is given by a torsor for the group H ( X ; Z ) . Remarks.
We note the following: On manifolds Y of dimension eleven, i.e. as in M-theory, the Ninebrane obstruction vanishesidentically by dimension reasons. However, it is still interesting to consider Ninebrane structureson Y as those are parametrized by the group H ( Y ; Z ). This is analogous to the case of Stringstructures on 3-dimensional manifolds M , where these structures are parametrized by H ( M ; Z ),corresponding to a gerbe on the worldvolume and is captured by the volume; see [23] for a char-acterization and application to the M2-brane. The second case is having a Fivebrane structure onthe worldvolume M of the M5-brane. Again this is automatic, but the structures are interestinglyenumerated by the 5-gerbe on the worldvolume. See [33] [10] [30] for detailed accounts. In lower degrees, the lift to a BO h n + 1 i -structure does not depend on the choice of a BO h n i -structure. This is the case for n = 2, where the existence of a Spin structure does not dependon choice of orientation from which to lift, because of homotopy invariance of the second Steifel-Whitney class w . The case n = 4 is similar, where a lift to a String structure does not dependon a choice of underlying Spin structure from which to lift, which can be shown via obstructiontheory (see [6]). However, in higher degrees this changes – see The Manifold Atlas Project [17],which we follow in the ensuing discussion. Starting with Fivebrane structures and going up, onehas dependence of the higher structure on the choice of lower structures. That is, among the setof String structures there might exist one which does not lift to a Fivebrane structure. Let usillustrate the statement in this degree, and the next degrees which we consider in this article willfollow analogously. Consider two String structures given by two classifying maps f, g : X → BStringfor which the composition ˆ f , ˆ g : X → BFivebrane π → BString are homotopic. Then the two maps11 , g differ by a map h : X → K ( Z ,
7) to the homotopy fiber of π . We have the diagram K ( Z , (cid:15) (cid:15) BO h i × K ( Z , id × i / / BO h i × BO h i m / / BO h i π (cid:15) (cid:15) X ˆ f ' ' ˆ g,h ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ f,g / / BString , (4.2)where m is the H -space multiplication. The induced maps on the corresponding cohomology groupsare captured by the sequence H ∗ ( X ; Z ) → H ∗ (BO h i ; Z ) × K ( Z , → H ∗ (BO h i ; Z ) ⊗ H ∗ (BO h i ; Z ) ⊆ H ∗ (BO h i × BO h i ; Z ) −→ H ∗ (BO h i ; Z )ˆ f ∗ x = ˆ g ∗ x + h ∗ i ∗ x o o ✤ x ⊗ ⊗ i ∗ x o o ✤ x ⊗ ⊗ x o o ✤ x := k . Now we take X = K ( Z ,
7) and h the identity. Then in this case, we investigate whether thepullback of the characteristic class k = x ∈ H (BFivebrane; Z ) under the two maps ˆ f and ˆ g agree. Since ˆ f ∗ x = ˆ g ∗ x + h ∗ i ∗ x , this question reduces to whether or not the pullback of x under i : K ( Z , → BFivebrane is zero. Using [36], we have that i ∗ ( x ) = Sq ι , where Sq is theSteenrod square of degree two and ι is the fundamental class of K ( Z , f ∗ x and ˆ g ∗ x do not agree. In particular, if one of them is zerothen the other is not zero. Consequently, it can be arranged that one Fivebrane structure can liftto a BO h i -structure while the other cannot.The discussion for lifting further from BO h i to BO h i one encounters the pullback of adegree ten class x via the map i : K ( Z , → BO h i . Using [36], such a map is given by i ∗ ( x ) = Sq ι , where ι is the fundamental class of K ( Z , X other than anEilenberg-MacLane space. Using the discussion in Chap. XI of [37] the corresponding classes x , x and x belong to (cosets of) Sq ρ H ( X ; Z ), Sq H ( X ; Z ), and Sq H ( X ; Z ), respectively. In this section we define twisted versions of the structures defined above, using the approach in [40][33] [25] [28]. We do not claim that we know the structure of the cohomology ring H ∗ (BO h i ; Z ), but it is enough for us toknow the first generator and that there is an H-space structure. For rational coefficients, this is studied in [32]. efinition 5.1 (Twisted 2-orientation) A twisted 2-orientation on a submanifold (a brane) M embedded in spacetime X is a homotopy in the following diagram, where f is the classifying mapfor Fivebrane bundles and α is a cocycle of degree 9 M f / / i (cid:15) (cid:15) BO h i x (cid:15) (cid:15) X α / / K ( Z , . η t | qqqqqqqqqqqqqqqqqqqqqqqq Remark.
The obstruction to having a twisted 2-orientation on M is given by f ∗ x + i ∗ α = 0 . (5.1) Definition 5.2 (Twisted 2-Spin structure) A twisted 2-Spin structure on a submanifold (a brane) M embedded in spacetime X is a homotopy in the following diagram, where f is the classifying mapof 2-oriented bundles and α is a cocycle of degree 10 M f / / i (cid:15) (cid:15) BO h i x (cid:15) (cid:15) X α / / K ( Z , . η t | qqqqqqqqqqqqqqqqqqqqqqqq Remark.
The obstruction to having a twisted 2-Spin structure on M is given by f ∗ x + i ∗ α = 0 . (5.2)It would be interesting to provide examples of twisted 2-Spin structures and twisted 2-Orientationstructures, along the lines of [25] [28]. Definition 5.3 (Twisted Ninebrane structure) A twisted Ninebrane structure is defined by the fol-lowing diagram M f / / i (cid:15) (cid:15) BO h i p (cid:15) (cid:15) X α / / K ( Z , . η t | qqqqqqqqqqqqqqqqqqqqqqqq Remark.
The obstruction to having a twisted Ninebrane structure is given by f ∗ ( p ) + i ∗ α = 0 . (5.3) Example: Twisted Ninebrane structures via embeddings.
We consider a brane embeddedin spacetime via i : M ֒ → Z . Then the b A -genera of M and Z can be related via a Riemann-Roch formula. In the simplest case where this embedding is a homotopy equivalence, one has that b A ( Z ) / b A ( M ) is an element in the real Chern character chO( Z ), that is a Pontrjagin class of someorthogonal bundle [2]. Considering degree four components gives p ( M ) = p ( Z ) + 12 p ( E ),13here E is an orthogonal bundle on Z . Then a String structure on M leads to a twisted Stringstructure on Z as 12 p ( E ) ∈ Z . This can also be reversed; by writing p ( M ) − p ( E ) = p ( Z ),a String structure on Z amounts to a twisted String structure on M . In the presence of a Spinstructure, the statement can be improved to a further divisibility by two due to the congruence p = w mod 2. By the above general formula of Atiyah-Hirzebruch [2] we can deduce similar statementsin higher degree cases (and following the approach of [24] with sufficiently high dimensions): Degree eight:
In the presence of a String structure on both M and Z , the degree eight componentsgive p ( M ) = p ( Z ) + 240 p ( E ) . (5.4)We view this as an example of a twisted Fivebrane structure on Z determined by a Fivebranestructure on M and vice versa. Degree twelve:
Assuming String structures p ( M ) = 0 = p ( Z ) and Fivebrane structures p ( M ) = 0 = p ( Z ) on both the brane and spacetime, we have in degree twelve p ( M ) = p ( Z ) + 252 p ( E ) . (5.5)Therefore, upon setting each side to zero, this gives an equivalence between a Ninebrane structureon M and a twisted Ninebrane structure on Z , and vice versa. Example: The E index in M-theory. The index of the Dirac operator coupled to an E bundle in M-theory on a Spin manifold Y lifted to twelve dimensions is given by I E = G ∪ G ∪ G − ( p − ( p ) ) ∪ G − p + p p − p . (5.6)Assuming a String structure, the C-field quantization condition [42] G + p = a ∈ H ( Y ; Z )reduces to G = a , the characteristic class of the E bundle. If we further assume a Fivebranestructure, i.e. p = 0, then expression (5.6) reduces to I E = α · p − a ∪ a ∪ a , (5.7)where α = . For any value of α , the second term can viewed as a rational twist for a rationalNinebrane structure. However, when 1 /α times the last term is integral, this latter term serves asan integral twist for the first term, which is an obstruction to the would-be Ninebrane structure.Therefore, the triviality of the Dirac index for E bundles with classes a = 186 n for n ∈ Z , inM-theory on a twelve-dimensional Fivebrane manifold Z is equivalent to a twisted Ninebranestructure on Z , with the twist given by the cubic term in the E characteristic class. Note thatthis kind of twist is composite and is a cubic analog in degree twelve of the composite quadratictwists giving rise to a String c structure in degree four [24] and a Fivebrane K ( Z , structure in degreeeight [25].Again this example highlights the fact that due to the relative high dimension of the Ninebraneobstruction relative to the dimensions of the applications considered, the situation is not as optimalas one had in the cases of lower obstructions, namely of the String and the Fivebrane, in [32] [33].Furthermore, the anomalies encountered should necessarily not be of the usual Green-Schwarz type,since these are always of a factorized form: a product of a degree four piece and a degree eightpiece. However, we will see in Sec. 8 that there is a natural explanation of a new phenomenon,namely the existence of a Chern-Simons term and a top form in M-theory that lends itself to anatural description in terms of Ninebrane structures.14 Structures not directly defined via the Whitehead tower
We know that one can define structures arising from vanishing of (multiples of) higher obstructions,without the lower obstructions necessarily vanishing. Examples of such are abundant: A Pinstructure requires the vanishing of the would-be Spin obstruction w without necessarily havingthe orientation obstruction w vanish. Also, we can have the first Pontrjagin class p vanishingwithout the lower obstruction, the Spin obstruction w , being zero. Such a structure is called a p -structure and is important in Chern-Simons theory and low-dimensional topology. See [25] [28][30] for many examples of structures defined via this general phenomenon.Let X = BO h p i i be the homotopy fiber of the map p i : BO → K ( Z , i ) corresponding to thefirst Pontrjagin class of the universal stable bundle γ over the classifying space BO . Let γ X be thepullback of γ over X . Definition 6.1 A p i -structure on a submanifold (a brane) M is a fiber map from the stable tangentbundle T M of M to γ X . That is, there is the following lifting diagram X = BO h p i i (cid:15) (cid:15) M / / BO p i / / K ( Z , i ) . The Spin/String version of this construction is explained in our context in [23]. So a p -structureis defined when p = 0 but p = 0, and a p -structure is defined when p = 0 while p i = 0, i = 1 , Remark.
We can similarly consider structures defined by Stiefel-Whitney classes and Wu classes,as in [28]. For instance, we define a via x = 0 but x = 0.We now consider twists of the above structures, generalizing the definition in [30] from degreefour to other (higher) degrees. Definition 6.2 An α -twisted p i -structure on a submanifold (a brane) ι : M → Y with a Rieman-nian structure classifying map f : M → BO , is a i -cocycle α : Y → K ( Z , i ) and a homotopy η in the diagram M f / / ι (cid:15) (cid:15) B O( n ) p i (cid:15) (cid:15) Y α i / / K ( Z , i ) . η t | qqqqqqqqqqqqqqqqqqqqqqqq Remarks. (i)
The obstruction is then p i ( M ) + [ α i ] = 0 ∈ H i ( M ; Z ). As in the case oftwisted String, Fivebrane, or Ninebrane structures, the set of such structures will be a torsor for H i − ( M ; Z ). (ii) Similarly we can define a twisted 2-Pin structure and other variants as the case of those givenby Stiefel-Whitney classes and Wu classes. 15
The (twisted) groups
We have defined the structures directly via classifying spaces in previous sections. A naturalquestion then is whether and how to describe the corresponding groups (in the homotopy sense).Here we build the groups as the deloopings of the classifying spaces as in previous cases [33]. Thegeneral machinery there allows similarly that we define new groups here as follows.
Definition 7.1
The homotopy fibers of the structures B -Orient, B -Spin, and B Ninbrane definethe groups 2-Oient, 2-Spin, and Ninebrane, respectively.
Remarks. 1.
Working not necessarily in the stable range, we have the groups 2-Oient( n ), 2-Spin( n ), and Ninebrane( n ). In the notation for connected covers with conventions as in [33], thesegroups are O h i ( n ), O h i ( n ) and O h i ( n ). The group 2-Orient( n ) is the Z double cover (in the homotopy sense, and with a mod 8shift from the classical notion) of the group Fivebrane( n ). We have π (Fivebrane( n )) ∼ = Z while π (2-Orient( n )) = 0. The group 2-Spin( n ) is the Z double cover (also in the above sense) of the group 2-Orient( n ).We have π (2-Orient( n )) ∼ = Z while π (2-Spin( n )) = 0. For any of the above groups G we have π ( G ) = 0. This is a mod 8 shift of the classical factthat π ( G ) = 0 for any Lie group (and hence also for its connected covers).Similarly, we can define groups (again in the homotopy sense) as the homotopy fibers of thecorresponding twisted structures, again as in [33] (see also [24]). Definition 7.2
The twisted groups G c = O h i c ( n ) , O h i c ( n ) and Ninebrane c ( n ) are the homotopyfibers of the corresponding twisted structures BG c . Remark.
As in the cases of String( n ) and Fivebrane( n ), the Whitehead tower construction allowsus to describe the group Ninebrane( n ) via a fibration with an Eilenberg-MacLane space as a fiber K ( Z , −→ Ninebrane( n ) −→ n ) , (7.1)obtained by looping the fibration (3.1).It would be interesting to find explicit geometric/categorical models for the above groups. It is desirable for physics to have differential versions of the topological structures that arise. As inthe case of String and Fivebrane structures [14] [33], one can consider differential refinements of thehigher BO h m i -structures to higher stacks. Via the formulation in [14] [34] we refine the classifyingspaces BG as topological spaces to B G as stacks. This also requires refining Eilenberg-MacLanespaces K ( Z , n ) = B n − U (1) to stacks B n − U (1). Consequently, we have16 roposition 8.1 The above structures refine to moduli stacks described in the diagram B Ninebrane
NinebraneStruc (cid:15) (cid:15) B O h i p / / - SpinStr (cid:15) (cid:15) B U (1) B O h i x / / - OrientStruc (cid:15) (cid:15) B Z B Fivebrane x / / FivebraneStruc (cid:15) (cid:15) B Z B String p / / B U (1) . (8.1) Remarks. 1.
Strictly speaking, the construction used in [14] to Lie integrate the first twoinvariant polynomials of so ( n ) to the smooth p and the smooth p would yield for the thirdinvariant polynomial some multiple of p whose homotopy fiber is the result of killing π = Z inFivebrane, but leaving the π = π = Z alone. The smooth p as displayed above exists, butthis does not follow from the construction in [14]. That construction only kills cocycles at the levelof L ∞ algebras and then integrates that up to smooth higher stacks, but so it cannot kill torsiongroups. We thank Urs Schreiber for illuminating discussions on these matters (see also [34]). We can also provide further refinement to B n U (1) conn by including connections, giving a diagramas above but with the moduli stacks of n -bundles with connections, using the machinery developedin [14] [34]. The corresponding diagram will be one replacing the above, with B n A conn replacing B n A and the refined classes ˆ c replacing the classes c via [14] [33] [34]. Trivialization of the Ninebrane obstruction class.
Recall that in the case of String andFivebrane structures we had trivializations of the corresponding forms given by a degree three class H and a degree seven class H , respectively, in essentially the following form dH = p ( A ) , dH = p ( A ) . (8.2)Furthermore, such expressions arise in physical settings, e.g. essentially in the Green-Schwarzanomaly cancellation and its dual, as explained in [32] [33]. It makes sense to consider for thecohomology class obstructing the Ninebrane structure a trivialization at the level of differentialform representatives given by dH = p ( A ) , (8.3)for some 11-form H . We investigate whether the form H has some physical interpretation. Notethat because of the relatively low dimensions in M-theory and string theory in comparison to ourincreasing levels in the Whitehead tower, such an interpretation becomes harder to get. However,we propose a conjectural relation. Recently, existence of top forms was discovered in string theory(see [4] and references therein); these are “potentials” rather than field strengths, i.e. are higher17onnections rather than higher curvatures. So such a top form H in M-theory can be taken tosatisfy dH = − G ∧ G + · · · . (8.4)Not much is known about the dynamics or the geometry associated with this form. We proposethat H in (8.4) is the trivialization of the Ninebrane form (8.3), i.e. the two expressions arecompatible in the sense that the second admits a correction term by the first. This is analogous indegree twelve to the correction of the equations of motion of the C-field by the one-loop polynomial I in degree eight. We hope that more investigations are made on such forms so that the aboveproposal can be verified. We do, however, provide another possible interpretation. Chern-Simonsterms CS of degree 11 appear in the M-theory action when formulated via the signature (whichis equivalent to formulation via Dirac operators) in [27]. We have the relation to the Ninebranestructures and to p -structures as p ( A ) ∼ dCS ( A ). Secondary 9-brane structures.
Note that while the 12-class ∼ p p is a differential cohomologyclass in degree 12, it is a secondary class/invariant which need not vanish even if its underlying(topological) 12-class vanishes. This is of course just the statement that there may be a non-trivialconnection 11-form, even if its curvature vanishes. So while in 11 dimensions any bare Fivebranestructure always has a lift to a Ninebrane structure, if one considers differential Fivebrane structures(i.e. maps to B Fivebrane conn ) then there is a actual condition to lift to B Ninebrane conn , namelythat not just the curvature 12-class but also the connection 11 form itself vanishes. As indicatedabove, that 11-form is just the Lagrangian for the 11-dimensional Chern-Simons term.
Relation to the M-algebra and the M9-brane.
In the discussion of the algebra correspondingto eleven-dimensional supergravity, and its associated cohomology, it was found in [13] in the supergeometric setting that there exists a spacetime-filling brane in M-theory. In the case of the M9-branethe relation to generalized Wess-Zumino-Witten (WZW) models and generalized Chern-Simons(CS) theories is the last column in the following table, which completes the first two cases studiedextensively in [7] [1] [39] [23] and [23] [10] [11] [12] [13], respectively.String Fivebrane NinebraneWorldvolume Σ Σ Σ WZW WZW WZW WZW Handlebody M M M Chern-Simons CS CS CS Structure String or p Fivebrane or p Ninebrane or p The WZW theory was studied in [22] [20] [29] [13]. Both of the theories (WZW and CS )associated to the ninebrane can be refined to the level of moduli stacks of higher bundles withhigher connections, as lower degrees [14] [10] [11] [12] [13] [34]. Analogously to the degree four anddegree eight cases, i.e. for String and Fivebrane structures, respectively, from the above works, thegeometric and topological ingredients associated to the ninebrane are described by the following18iagram (cf. [34]) B U (1) / / (cid:15) (cid:15) Ninebrane / / WZW10 − bundle (cid:15) (cid:15) ˆ P B U (1)10 − bundle (cid:15) (cid:15) Ninebrane10 − bundle (cid:20) (cid:20) ∗ / / / / (cid:15) (cid:15) WZWLagrangian P / / − Spin9 − bundle (cid:15) (cid:15) B U (1) conn (cid:15) (cid:15) ∗ y / / Y − Spin9 − connection / / ❴❴❴❴❴❴ Chern − SimonsLagrangian , , B Ninebrane conn / / (cid:15) (cid:15) B (2-Spin conn ) p (cid:15) (cid:15) ∗ / / B U (1) conn . Note that this diagram is such that all squares (and hence all composite rectangles) are homotopycartesian (i.e. are homotopy pullback squares) in the ∞ -topos of smooth ∞ -groupoids (i.e. ∞ -stacks over smooth manifolds). This is a much stronger statement than just that the diagramexists, as each item in the top-left of a square is in fact uniquely characterized (up to equivalence)as completing that square to a homotopy pullback square. Similarly, for the case of p -structures B U (1) / / (cid:15) (cid:15) Ninebrane ′ / / WZW10 − bundle (cid:15) (cid:15) ˆ P B U (1)10 − bundle (cid:15) (cid:15) Ninebrane10 − bundle (cid:20) (cid:20) ∗ / / Fivebrane / / (cid:15) (cid:15) WZWLagrangian P / / Fivebrane9 − bundle (cid:15) (cid:15) B U (1) conn (cid:15) (cid:15) ∗ y / / Y Fivebrane9 − connection / / ❴❴❴❴❴❴ Chern − SimonsLagrangian - - B Ninebrane ′ conn / / (cid:15) (cid:15) B (Fivebrane conn ) ˆ p (cid:15) (cid:15) ∗ / / B U (1) conn , with Ninebrane ′ referring to a p -structure. The main difference between the two diagrams aboveis that the second does not involve killing the two Z ’s in degrees 9 and 10. Note that the latter19iagram, unlike the former, is a corollary of the theorem in [14] – see the Remarks at the beginningof this section. Acknowledgements
The author thanks Domenico Fiorenza, Urs Schreiber, and Jim Stasheff for enjoyable collaborationson projects on which this one builds and for useful discussions. The author is grateful to MartinOlbermann for pointing out and emphasizing subtleties in identifying the generators and to UrsSchreiber for valuable comments on the manuscript. The author also thanks the anonymous refereefor useful suggestions to improve the paper. This research is supported by NSF Grant PHY-1102218.
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