Geometric and topological structures related to M-branes II: Twisted String and String^c structures
aa r X i v : . [ h e p - t h ] O c t Geometric and topological structures related to M-branes II:Twisted String and String c structures Hisham Sati Department of MathematicsUniversity of MarylandCollege Park, MD 20742
Abstract
The actions, anomalies, and quantization conditions allow the M2-brane and the M5-brane to support, in anatural way, structures beyond Spin on their worldvolumes. The main examples are twisted String structures.This also extends to twisted String c structures, which we introduce and relate to twisted String structures. Therelation of the C-field to Chern-Simons theory suggests the use of the String cobordism category to describe theM2-brane. Dedicated to Alan Carey, on the occasion of his 60 th birthday
1. Introduction
In [16] we described various geometric and topological structures related to the M2-brane(or membrane) and the M5-brane (or fivebrane) in M-theory. Some of these structures havealready been established there. Other structures were merely outlined and hence deserve moredetailed and careful elaboration. In addition, there are other structures not covered in the abovework. This is a first in a series of papers which will establish this: expand on structures eludedto in [16] as well as uncover new structures. Consider the C-field in M-theory with field strength G . In [19], the flux quantizationcondition in M-theory on a Spin eleven-manifold Y [23] G − λ = a ∈ H ( Y ; Z ) (1.1)was recast as defining (essentially) a twisted String structure [22] on Y . Here a is the char-acteristic class of an E bundle on Y and λ is half the first Pontrjagin class p ( T Y ) ofthe tangent bundle T Y of Y . A model for G in terms of twisted differential cohomologywas given in [19]. In [16] the C-field ‘potential’ C was identified as (essentially) the Stringclass corresponding to the String structure. It is known that the C-field couples electrically tothe M2-brane, that is the action of the membrane contains a term R W C , where W is the e-mail: [email protected] . Primary, 81T50; Secondary 53C27, 55R65, 55N20. Keywords . String structure, String c structure, Anomalies, M-theory. As that paper was the starting point of this investigation, we will refer to it as part I. Subsequent works willbe numbered accordingly. Thus, this current note will be number II. W will couple to C , the potential corre-sponding to the Hodge dual ∗ G , with respect to the metric g Y on Y . Thus, it is natural toconsider the questions of existence and consequences of String structures on the worldvolumes W and W , rather than just on the target spacetime Y . It is the purpose of this note to dojust that.The embedding of the worldvolumes of the M-branes in spacetime Y allows the decompo-sition of the tangent bundle of Y into tangent bundle of the M-branes and the correspondingnormal bundle. Almost all the structures we are considering satisfy a two out of three principle,that is if two of the bundles above admit a given structure then so does the third. Then, forexample, if we establish that Y and the normal bundle have such structures then so doesthe M-brane worldvolume. However, the situation is not quite as simple, since we seeking anintrinsic characterization of such structures. That is, we discover them via quantization of fluxand requiring the partition functions to be well-defined, so that we are dealing with quantumrather than classical statements, following Witten’s work [23] [24] [25]. Hence, we characterizenot only that such structures exist but also how they occur. Furthermore, “worldvolumes”should be interpreted in the appropriate sense as they could mean extended worldvolumes, thatis higher dimensional manifolds obtained from the original worldvolumes by extending througha circle, taking a bounding manifold, or considering disk bundles. The corresponding normalbundles are modified accordingly. Theorem 1.1.
Consider the M-branes as Spin manifolds inside a Spin eleven-manifold Y .Then1. The tangent bundle and the normal bundle to the M2-brane each admits a twisted Stringstructure. Furthermore, the M2-brane worldvolume supports a (differential) String cobordisminvariant.2. The (extended) tangent bundle and the normal bundle to the M5-brane each admits a twistedString structure. By the extended tangent bundle of a manifold we mean the tangent bundle of the disk bundleover that manifold.In addition to (twisted) String structures, we find that String c structures, as defined in [6],appear on the worldvolumes. We see that such structures are closely related to twisted Stringstructures. In addition, we find that a twisted version of String c structures is also relevant.Thus we are led to define such structures and study some of their elementary properties. Proposition 1.2.
Consider a String c structure of a bundle E with ℓ the Chern class of theline bundle defining the Spin c structure, and let Q α ( E ; ℓ ) denote a twisted String c class, whichis Q ( E ; ℓ ) − α , where α is a degree four integral cocycle.1. Under change of Spin c structure, a String c structure changes as Q ( E ; ℓ + 2 m ) = Q ( E ; ℓ ) − ℓm − m .2. Twisted String c structures are not quite multiplicative; for a fixed α they satisfy Q α ( E ⊕ E ′ ; ℓ + ℓ ′ ) = Q α ( E ; ℓ ) + Q α ( E ′ ; ℓ ′ ) − ℓℓ ′ . Now we consider the application of twisted String c structures to M-branes. Theorem 1.3.
Consider the M-branes as Spin c manifolds inside a Spin c eleven-manifold Y . Then the tangent bundle and the normal bundle of the M-branes each admits a twistedString c structure. c structures,introducing the basic notions in section 3.1, and providing the descriptions for the M2-braneand the M5-brane in sections 3.2 and 3.3, respectively. We will use the notation λ or Q for thefirst Spin characteristic class, Q α for the corresponding twisted class, and λ c or Q ( ; ℓ ) for thethe class in the Spin c case.
2. Twisted String structuresString structure.
The first Pontrjagin class p for a Spin bundle E is divisible by two since p ( E ) ≡ w ( E ) mod 2, where w ( E ) is the second Stiefel-Whitney class of E . This then allowsthe definition of the first Spin characteristic class Q = λ = p , which universally is a generatorof H (BSpin; Z ). For two vector bundles E and E ′ admitting Spin structures, the first Spincharacteristic class is additive (see [20]) Q ( E ⊕ E ′ ) = Q ( E ) ⊕ Q ( E ′ ) . (2.1)Note that this is an improvement over the corresponding formula for the Pontrjagin classes (see[14]) p ( E ⊕ E ′ ) = p ( E ) + p ( E ′ ) mod 2 − torsion , (2.2)in the sense that the 2-torsion is automatically taken care of. A String structure on a bundle E or space X , originally defined via loop spaces [12] [7], is a lift of the structure group fromSpin to String, the 3-connected cover of Spin. That is, the classifying map f : X → B Spin( n )of the natural Spin bundle on an n -manifold X is lifted to a map f ′ : X → B String( n ) via thefibration K ( Z , → B String( n ) → B Spin( n ). Twisted String structure.
Recall from [22] [19] that a twisted String structure on a brane ι : M → X with Spin structure classifying map f : M → B Spin( n ) is a a four-cocycle α : X → K ( Z ,
4) and a homotopy η between f ∗ λ = λ ( M ) and ι ∗ [ α ], as indicated in the diagram M f / / ι (cid:15) (cid:15) B Spin( n ) λ (cid:15) (cid:15) X α / / K ( Z , η u } ssssssssssssssssssssss . (2.3)Then M has a twisted String structure when p ( M ) + ι ∗ [ α ] = 0 ∈ H ( M ; Z ). In [19] the flux quantization conditionin M-theory (1.1) is interpreted essentially as an obstruction to the existence of a twistedString structure, and the role of the corresponding higher connection is highlighted in [16].3he M5-brane six-dimensional worldvolume W admits a map to the target eleven-dimensionalspacetime Y . The tangent bundle then splits as T Y | W = T W ⊕ N W , where N W is thecorresponding normal bundle. Flux quantization and twisted String structure on the M5-brane.
Now we considertopological part of the M5-brane worldvolume action. Such an action is best described topo-logically via a lift to an eight-dimensional disk bundle over the original worldvolume [24] [10],that is, there is a bundle D → X π D −→ W , described as follows. Let M be a Spin 7-manifold,which is a circle bundle over W , and which has a C-field C . Then we have an E bundle on M . Let X be an oriented 8-manifold with boundary M over which C extends. This alwaysexists because of the vanishing of the cobordism group Ω Spin7 ( K ( Z , K ( Z , ∼ E in this range of dimensions. The action includes the term S = Z X G ∪ G − G ∪ λ . (2.4)In [24] the action functional was derived using the Chern-Simons construction. For x ∈ H ( W ; Z ), the construction of the partition function requires defining a Z -valued functionΩ( x ) = ( − h ( x ) , where h ( x ) is an action functional, desired to be even, i.e. zero when takenmod 2. Form the 8-dimensional manifold X as above, i.e. a disk bundle over W . The element x extends to z = u ∪ x on X , where u ∈ H ( S ; Z ). For z = G / π , the action is well-definedmodulo 2 π and given by I ( C ) = 2 π Z X z ∪ z . (2.5)The Chern-Simons construction requires a division by half, as then the construction will give aline bundle L over the intermediate Jacobian T = H ( W ; R ) /H ( W ; Z ) so that c ( L ) equalsthe symplectic form ω on T , via geometric quantization.The inability to define I ( C ) / H ( X ; Z ) is not always even. If this were the case then the even-ness of z would allow thedivision by 2 and hence give I ( C ) /
2, well-defined modulo 2 π . The mechanism to get aroundthis was proposed in [24] as follows. For any a ∈ H ( X ; Z ), a ≡ a · λ mod 2. This is equivalentto the statement that 18 Z X (cid:18) ( a − λ ) − λ (cid:19) ∈ Z . (2.6)This means that the flux quantization condition holds on the eight-dimensional manifold X ,the disk bundle over the worldvolume of the M5-brane, (cid:2) G π (cid:3) = λ − a . The effect is then tomodify the action (2.5) to [24] e I ( C ) = π Z X (cid:18) z − λ (cid:19) . (2.7)Since a is integral and z = λ − a , π e I ( C ) is well-defined mod 2 π and can be used to definethe line bundle L with c ( L ) = ω .At the level of the six-dimensional worldvolume W , a similar condition seems to arise. Thedimensional reduction of the action (2.4) along the disk, i.e. integration over the fiber of thetwo-disk bundle π D : X → W , gives S = Z W C ∪ h − b ∪ λ , (2.8)4here dC = G + exact terms, h = π D ∗ G . Now the variational principle applied to the small B -field b naively gives G − λ = 0 ∈ H ( W ; Z ) . (2.9)However, we note several ambiguities here. First, the action (2.8) is not complete, as there areinevitably other terms. Second, there is a subtlety related to the self-duality of the theory (see[24]). Third, the process of dimensional reduction on the disk assumes λ ( X ) = λ ( W ). Fourth,there will be contributions to the M5-brane worldvolume from π ∗ (Θ), the integration over thefiber of the S bundle over W of the dual Θ of the C-field [8]. This process of dimensionalreduction of the disk over W is mathematically similar to that of taking Y itself to be a diskbundle, and such a process involves ambiguities of division by two as in (the discussion leadingto) equation (11.11) in [8].Now consider a modification of the action (2.5). For X Spin, the values of the integral h ( x ) = Z X ( z ∪ z + λ ∪ z ) (2.10)is always even. The term λ ∪ z in (2.10) means that instead of quantizing the torus T thatparametrizes flat C-fields on W modulo gauge transformations, the modification implies thatone instead is quantizing another torus T ′ , which parametrizes, up to gauge transformations,C-fields that are no longer flat, but instead have curvature λ [24] [25]. The new torus T ′ isisomorphic (not canoncially) to the original torus T , via the map C C + C ′ , where C ′ isany C-field of curvature λ . The transformation is h ( x ) z z − λ / / R X z ∪ z . (2.11)We see that this is simply the shift corresponding to a twisted String structure, where T corresponds to the cocycle (to be viewed as a twist) and T ′ is shifted (by the class that is beingtwisted by the cocycle).The above is strengthened, from another angle, by Witten’s proposal [25] that G | W = θ ,where θ is a torsion class on W , something that was verified in [8]. In this more general case,which includes torsion explicitly, we would still have a twisted String structure, except that nowthe twist is formed out of the original twist and this new class θ . Twisted String structure on the normal bundle of the M5-brane.
Now we can considerthe normal bundle of the M5-brane. As indicated in [19], when the cocycle α represents thecharacteristic class of some bundle E , a twisted String structure on E can be viewed as theString structure on a difference bundle E − E . Hence, we define the class Q α ( E ) = Q ( E ) − α . (2.12)This class satisfies the following additivity Q α ( E ⊕ E ′ ) = Q α ( E ) + Q α ( E ′ ) . (2.13)This formula can be shown, using (2.1) and additivity of cocycles, as follows( λ + α )( E ⊕ E ′ ) = λ ( E ⊕ E ′ ) + α ( E ⊕ E ′ )= λ ( E ) + λ ( E ′ ) + α ( E ) + α ( E ′ )= ( λ + α )( E ) + ( λ + α )( E ′ ) . Y to be Spin, then the flux quantization condition (1.1) will givespacetime a twisted String structure. The above additivity (2.13), applied to the split tangentbundle via the embedding of the M5-brane, will then imply that the normal bundle to theM5-brane will also admit a twisted String structure. The first Spin characteristic class ismultiplicative, as we saw in (2.1). This means that, in general, if any two of the three bundles E , E ′ , E ⊕ E ′ are String, then so is the third. Applying this to the M2-brane we have that if thenormal bundle admits a Spin structure then so does the target space Y . This is because, fordimension reasons, the M2-brane worldvolume trivially admits a String structure. Nevertheless,there are very interesting consequences of requiring the M2-brane to have a String structure [16].On the other hand, we can consider the same question for the normal bundle of the embeddingof W in Y , which is also already considered the same work. Instead, what we do here isconsider differential refinements, via a discussion which is complementary to that in [16]. Consider the M2-branewith worldvolume W , a three-dimensional connected closed Spin manifold. Then W has acanonical topological String structure. A topological String structure α top is by definition atrivialization of the Spin characteristic class Q = p ( T M ) ∈ H ( M ; Z ). Since MSpin = 0,we can find a Spin zero bordism Z of W . An oriented 3-manifold is always Spin. In addition tosuch a manifold admitting a String structure for dimension reasons, one can also get a canonicalString structure via a trivialization of the tangent bundle. In fact, the main example used in [23]is S , which is parallelizable. The physical significance of a String structure for the M2-braneis highlighted in [16].A geometric refinement α of α top trivializes the class Q at the level of differential forms. Achosen connection ∇ W on the tangent bundle T W gives rise to a connection on the Spin(3)-principal bundle given by the Spin structure. Choose an extension ∇ Z of ∇ W from W to Z .The existence of a connection allows for a geometric String structure [15] [21] [4]; a topologicalString structure α top on W gives rise to a 3-form C α which satisfies dC α = p ( ∇ W ). TheChern-Simons aspect is described in [18]. Change of String structure.
The set of topological String structures on W is a torsor under H ( W ; Z ) ∼ = Z . The action of x ∈ H ( W ; Z ) can be written as ( x, α top ) α top + x . Then Z W C α top + x = Z W C α top + h x, [ W ] i , ∀ x ∈ H ( W ; Z ) , (2.14)where h x, [ W ] i is the pairing of the cohomology class x with the fundamental homology class[ W ] of W . String bordism invariant.
We will make use of a String cobordism invariant defined in [5], d Z ( W , α ) := 12 Z Z p ( ∇ Z ) − Z W C α . (2.15)This expression a’ priori takes values in R , but turns out that [5]1. d Z is an integer.2. Furthermore, it is independent of the choice of connections and geometric data of the Stringstructure.3. The corresponding class d ( W , α top ) := [ d Z ( W , α top ] ∈ MString ∼ = Z (2.16)6s a String bordism invariant, so that the map d : MString → Z which takes (cid:2) W , α top (cid:3) to d ( W , α top ) is an isomorphism.From (2.14), the invariant for a shifted topological String structure then takes the form d Z ( W , α top + x ) = d Z ( W , α top ) − h x, [ W ] i . (2.17) A generator for MString . Let S denote the sphere spectrum. There is a unit map from S to any other spectrum. Thus let ǫ : S → MString be the unit of the ring spectrum MString.This is an isomorphism in degree 3, that is MString ∼ = S ∼ = Z [11]. The sphere S ∈ R ,considered as the boundary of the disk D ∈ R , has a preferred orientation, Spin structure,and String structure α top . Let or S ∈ H ( S ; Z ) be the orientation class of S . A generator g ∈ MString is given in [5] as g := [ S , α top − or S ] ∈ MString . (2.18)Then, from (2.17), d ( g ) = [1] ∈ Z , which has order 24 so that g ∈ MString is a generator. M2-brane and 2-framing.
Consider a membrane with worldvolume W , a compact con-nected oriented 3-manifold. At the beginning of this section we considered M2-branes withparallelizable worldvolumes. Now we consider a variation. The double of the tangent bundle2 T W = T W ⊕ T W has a natural Spin structure arising from uniquely lifting the structuregroup to Spin(6) via the following diagram of Lie groups Spin(6) (cid:15) (cid:15) SO (3) diag / / SO (3) × SO (3) / / SO (6) . (2.19)A 2-framing of a closed oriented 3-manifold W is a Spin-trivialization of the double 2 T W of its tangent bundle [1]. Let Z be an oriented zero-bordism of W . Then the 2-framing α at the boundary ∂Z ∼ = W gives rise to a trivialization of the Spin bundle 2 T Z . Thistrivialization refines the Spin class p (2 T Z ) ∈ H ( Z ; Z ) to a relative cohomology class p (2 T Z , α ) ∈ H ( Z , W ; Z ). Then the quantity σ ( α ) := 3sign( Z ) − (cid:28) p (2 T Z , α ) , [ Z , W ] (cid:29) ∈ Z (2.20)gives an integer parametrized by α and does not depend on Z [1]. The canonical 2-framing α is one for which σ ( α ) = 0.A canonical 2-framing gives rise to a canonical String structure [5]. Let α top be any topo-logical String structure on W . The combination σ ( W , , α top ) := 3sign( Z ) − d Z ( W , α top ) ∈ Z (2.21)is independent of the choice of Z and has a cohomology class σ ( W ) := (cid:2) Z ) − d Z ( W , α top ) (cid:3) = [sign( Z )] ∈ Z (2.22)which is also independent of the choice of String structure α top . Then W has a unique topo-logical String structure α top0 characterized by σ ( W , α top0 ) ∈ Z .7 ta invariant and an expression intrinsic on W . An expression for d which does notdepend on the bordism Z is given in [5], which we apply to our situation. Let S ( W ) be theSpin bundle of W . Let V → W be a real E vector bundle with a metric and connection, thenwe can form the Dirac operator D W ⊗ V which acts on sections of the bundle S ( W ) ⊗ R V . Ataming of D W ⊗ V is a self-adjoint operator T acting on section of S ( W ) ⊗ V and given by asmooth integral kernel such that D ′ = D W ⊗ V + T is invertible. The taming is physically amass term which acts as a regulator in the (Pauli-Villars) regularization. This modified operatoris what is used for the eta-invariant. By the Atiyah-Singer index theorem [2], the index of D ′ given by Ind( Z ) = − Z Z p ( ∇ Z ) + η ( W ) , (2.23)so that the following equality of cohomology classes (cid:20) Z Z p ( ∇ Z ) (cid:21) = [12 η ( W )] (2.24)holds in R / Z . Choose a geometric refinement α of the topological String structure α top based onthe Spin connnection induced by ∇ W . Then a formula for the String bordism invariant whichis intrinsic on W is d ( W , α ) = (cid:20) η ( W ) − Z W C α (cid:21) ∈ Z . (2.25)This is our proposed (part of the) topological action for the M2-brane which detects the Stringstructures. Such a quantity would a priori appear in the partition function after multiplicationby an integer between 0 and 23. However, the coefficient 1 is favored by the M2-brane action,and hence by the partition function. See [16] for more on the partition function, in whichone should sum over String structures. It is possible that the other factors appear when oneconsiders M2-brane multi-instantons. The q -expansions. Consider the series θ W ( x, q ) := exp hP ∞ k =2 2(2 k )! G k x k i ∈ Q [[ q ]][[ x ]], where G k are the Eisenstein series, and Θ := K θ W ∈ Q [[ q ]][[ p , p , · · · ]] is the power series correspond-ing to θ W . In [5], a String bordism invariant, which involves a q -expansion, is defined using e Φ := Θ e G p − p . For k = 1 this is given by b = Z W C α ∧ e Φ [0] , (2.26)where e Φ [0] = G , the first Einsenstein series, which is not a modular form. Since G = − + q + · · · , the result is in Z ⊕ Z [[ q ]]. Indeed, for q = 0 this gives − R W C α (cf. [16]).
3. (Twisted) String c Structures3.1. Definitions and constructions Spin c structures in terms of Spin structures. There is a nice geometric criterion for the existence of a Spin c structure (see [13]). Since U (1) = SO (2), there is a natural map f s : SO ( n ) × U (1) → SO ( n + 2). This extends, viaWhitney sum, to a map of bundles. The group Spin c ( n ) can then be defined as the pullback by f s of the covering map Spin( n + 2) → SO ( n + 2)Spin c ( n ) (cid:15) (cid:15) / / Spin( n + 2) (cid:15) (cid:15) SO ( n ) × U (1) f s / / SO ( n + 2) . (3.1)8his implies that a manifold M is Spin c , i.e. T M has a Spin c structure, iff there is a complexline bundle L over M such that T M ⊕ L has a Spin structure.A Spin c manifold M has a two-dimensional class c ∈ H ( M ; Z ), which reduces mod 2 to w ( M ). On such a manifold p − c is divisible by 2, and there is an integral characteristic class λ such that 2 λ = p − c . More generally, if M is Spin c , let J be a real two-dimensional vectorbundle with Euler class c and let E = T M ⊕ J . Then w ( E ) = 0 and λ ( E ) is the correspondingString class for E with 2 λ ( E ) = p ( E ) = p ( T M ) − c , (3.2)and is the String class in the Spin c case. This is simply the String c structure. Remark on coefficients.
The quantization condition G − λ ∈ H ( Y ; Z ) also holds for λ replaced by any integer multiple of λ , that is for λ replaced by (2 k + 1) λ , for any integer k . In this paper we have chosen k = 0 as required by the index theorem calculations to cancelmembrane anomalies, as in [23]. Note also that in the discussion leading to (3.2), p − c canbe replaced by p − (2 m + 1) c , where m is any integer. Indeed, this is compatible with thediscussion in [6] where m is chosen in a dimension-dependent way so as to get a generalizedWitten genus. Since we will not deal with modular forms in this note, we will not make suchdistinctions. (Twisted) String c structures in terms of String structures. From the above discus-sion, it seems natural to define a String structure corresponding to a Spin c structure via onecorresponding instead to a Spin structure. That is, to characterize whether a bundle E ad-mits a String c structure we form another bundle E = E ⊕ L R over the same space X andapply the String construction to E . The condition λ ( E ) = 0 then translates to the condition λ ( E ) − c = 0. If we take the first bundle E to be the tangent bundle T X and E = E ,then we form the direct sum of bundle via the standard diagonal inclusion. Let T X π T −→ X and L R π L −→ X be the two indicated vector bundles on X . Let ∆ : X → X × X be the diagonal map.The Whitney sum T X ⊕ L R of the two bundles T X and L R is the pullback of the Cartesianproduct of T X and L R via ∆, E = T X ⊕ L R / / (cid:15) (cid:15) T X × L R π (cid:15) (cid:15) X ∆ / / X × X . (3.3)Similarly we can provide a definition for the twisted String c structure. In this case we have ahomotopy between λ ( E ) and α E f / / α ( ( PPPPPPPPPPPPPP B Spin( n ) λ (cid:15) (cid:15) K ( Z , η (cid:127) (cid:7) (cid:7)(cid:7)(cid:7)(cid:7) , (3.4)so that λ ( E ) + α = 0 ∈ H ( X ; Z ), which translates to the condition λ ( E ) + α − c = 0 ∈ H ( X ; Z ) . (3.5)This is the condition for the bundle E to admit a twisted String c structure. String c structures directly. We can also give a definition of a String c structure as a specialcase of a twisted String structure. Note that, in general, the latter has a twist given by a degree9our integral cocycle, while the former has a composite cocycle c , which lives in the wedge of K ( Z ,
2) with itself. There is a map from K ( Z , ∧ K ( Z ,
2) to K ( Z ,
4) given by the cup product.We characterize a String c structure via the diagram X f / / c ( X ) (cid:15) (cid:15) α % % JJJJJJJJJJJJJJJJJJJJJJJJJJ B Spin c ( n ) Q (cid:15) (cid:15) K ( Z , ∧ (cid:15) (cid:15) K ( Z , ∧ K ( Z , ∪ / / K ( Z , η u } rrrrrrrrrrrrrr u } η rrrrrrr rrrrrrr . (3.6)The first homotopy η gives the relation Q + α = 0 ∈ H ( X ; Z ) and the second homotopy η gives α + c = 0 ∈ H ( X ; Z ). Combined, the two homotopies then give Q − c = 0 ∈ H ( X ; Z ) . (3.7)This identifies a String c structure as a special case of a twisted String structure. Note thatdiagram (3.6) should be modified to account for the division of c by 2. This is already done in[19] for the case of twisted String structure, and the current case is analogous. Q for a Spin c vector bundle. Let E be a real vector bundle admitting a Spin c structure. Thismeans that w ( E ) is the reduction mod 2 of an integral class ℓ ∈ H ( X ; Z ), i.e. ρ ( ℓ ) = w ( E ).For L a complex line bundle with c ( L ) = ℓ , define the first Spin c characteristic class Q ( E ; ℓ ) = Q ( E − L ) ∈ H ( X ; Z ) . (3.8)Then 2 Q ( E ; ℓ ) = p ( E ) − ℓ and the mod 2 reduction is ρ ( Q ( E ; ℓ )) = w ( E ). Change in Spin c structure. Now consider the change in the Spin c structure. Recall that wedefined a twisted String structure as untwisted String structure of a difference bundle. We can dothe same for a String c structure because, as we have seen above, a String c structure can be seenessentially as a specialization of a twisted String structure. For example, take the original linebundle L and tensor it with a square of a line bundle L of Chern class c ( L ) = m ∈ H ( X ; Z ),so that c ( L ⊗ L ) = ℓ + 2 m . Then the Spin class changes as Q ( E ; ℓ + 2 m ) = Q ( E − L ⊗ L )= Q ( E ; ℓ ) − ℓm − m .Q for a complex vector bundle. Let E be a complex vector bundle. Then E admits a Spinstructure iff the first Chern class c ( E ) is divisible by 2, that is c ( E ) = 2 n for some integer n .Then the first Spin class is Q ( E ) = 2 n − c ( E ) . (3.9)We now consider the String c case. The twisting by a line bundle can be ‘untwisted’ in thefollowing sense. Noting that p = c + 2 c , if E is a complex vector bundle with c ( E ) = ℓ then Q ( E ; ℓ ) = − c ( E ). Differential refinement of twisted String c structures. As in the case of twisted Stringstructure, a twisted String c structure can be refined. The cocycle, the Chern class of the linebundle, and the representative for the String class admit refinements as in [19] [4]. Therefore,we get similarly a refinement of the twisted String c structure, with expressions similar to thosein the twisted String case. 10 .2. Twisted String c structures and the M2-brane M-theory is mostly studied on Spinmanifolds. However, one can also study the theory on Spin c manifolds. This has been discussedextensively in [17]. In this case there is a global gravitino anomaly in the eleven-dimensionalsupergravity description which can be shown to cancel; examples of this are considered in [9].Furthermore, since G couples to the gravitino, there is a correction to the flux quantizationwhich is given in [3] in the case of torus bundles. In general, when dealing with M-theory oneneeds to go beyond the supergravity approximation. Hence it is possible that the would-begravitini need to be replaced by membranes. However, we leave the explicit check to futurework.On the other hand, we can consider M2-branes with worldvolumes admitting a Spin c struc-ture. Since W is a three-dimensional compact oriented manifold, it is Spin, and hence alsoSpin c . By embedding the M2-brane in spacetime, we get a splitting T Y | W = T W ⊕ N W .Spin c structures satisfy a two out of three principle, so that the normal bundle N W will beSpin c . Then the derivation of the flux quantization will be analogous to the Spin case of [23],as was outlined in [17], and will involve index theory on the normal bundle. The result is G + 12 λ + 14 c ( L ) ∈ H ( Y ; Z ) , (3.10)where L is the complex line bundle associated with the Spin c structure (see [17] for details).In fact, the way the condition is derived is really from the same condition on the normalbundle together with the triviality of the condition on W . In this setting, we can interpret(3.10) as defining a twisted String c structure on the normal bundle N W . Therefore, we findtwisted String c structures both on the M2-brane worldvolume as well as on its normal bundle. c structures and the M5-brane Consider the bounding 8-manifold X as a Spin c manifold with a fixed Spin c structure. X has a two-dimensional class c ∈ H ( X ; Z ) which reduces mod 2 to w ( X ) and which is the Euler class of a 2-dimensionalvector bundle E . Furthermore, p − c is divisible by 2 so that, as above, there is an integralclass λ c (or Q ( ; ℓ )) such that 2 λ c = p − c . Consider the trivial rank 3 bundle E = X × R ,and consider the rank five Whiteny sum bundle E = E ⊕ E over X . Consider the unitsphere bundle S ( E ) over X which is a twelve-dimensional Spin manifold Z , and denote theprojection by π E : Z → X .Let x ∈ H ( X ; Z ) and u ∈ H ( Z ; Z ) such that π ∗ ( u ) = 1 and u ∪ u = 0. This u can beconstructed as the Poincar´e dual of a section of π . Now consider an E bundle E over Z withdegree four characteristic class a = u + π ∗ ( x ). The Spin c characteristic class on Z is taken tobe the pullback of the corresponding class on X , that is λ c ( Z ) = π ∗ ( λ c ( X )). The index ofthe Dirac operator for spinors coupled to the E bundle is then [25] i ( E ) = Z X ( x ∪ x + λ c ( X ) ∪ x ) . (3.11)We are now in a situation similar to that of equation (2.10), except that λ is replaced by λ c .Hence, we get 12 λ c ( X ) + x = 0 ∈ H ( X ; Z ) , (3.12)which is a condition for the existence of a twisted String c structure. More properly (andprecisely), we seek, as in [25], that i ( E ) is zero when taken mod 2. A necessary condition toensure this is to require the twisted String c condition. We can again consider the situation on11 rather than on X . We get the same condition if we go through the analysis leading toequation (2.9). Geometric refinement.
Note that we can get a geometric String structure on the M5–brane.We have done this explicitly for the M2-brane in section 2.2.1, and the extension to the M5-brane is somewhat similar. However, there are effects which deserves careful treatment and willbe discussed fully elsewhere.
M5-brane and MString.
The above discussion at the end of section 2.2.1 on 2-framing forthe M2-brane also makes tantalizing connection to the M5-brane, in a special case. To see this,consider the M5-brane with worldvolume W = W × W . A physically appropriate exampleis to take W = S and W = S × S . Then the trivialization of 2 T W can be viewed asa trivialization of T W , by the isomorphism. On the other hand, a trivialization of the Spinbundle gives rise to a canonical topological, and hence geometric, String structure [4]. Acknowledgements
The author would like to thank Ulrich Bunke, Alan Carey, Bai-Ling Wang, and Weipeng Zhangfor useful comments and discussions. This paper was written at the Max-Planck Institute forMathematics in Bonn, whom the author thanks for support and for an inspiring atmosphere.
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