Chiral Fermions and Spinc structures on Matrix approximations to manifolds

arXiv:hep-th/0207007 v2 24 Sep 2002 Preprint typeset in JHEP style - HYPER VERSION DIAS-STP-02=08 Chiral Fermions and SpincStructures on Matrix Approximations to Manifolds Brian P. Dolan and C. Nash Dept. of Mathematical Physics, NUI, Maynooth, Ireland and School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Rd., Dublin 8, Ireland [email protected], [email protected] Abstract: The Atiyah-Singer index theorem is investigated on various compact manifolds which admit finite matrix approximations (“fuzzy spaces”) with a view to applications in a modified Kaluza-Klein type approach in which the internal space consists of a finite number of points. Motivated by the chiral nature of the standard model spectrum we investigate manifolds that do not admit spinors but do admit Spincstructures. It is shown that, by twisting with appropriate bundles, one generation of the electroweak sector of the standard model, including a right-handed neutrino, can be obtained in this way from the complex projective space CP2. The unitary Grassmannian U(5)/(U(3) ×U(2)) yields a spectrum that contains the correct charges for the Fermions of the standard model, with varying multiplicities for the different particle states. Keywords: Non-Commutative Geometry, Field Theories in Higher Dimensions, Differential and Algebraic Geometry. Contents 1. Introduction 1 2. Chiral Fermions on CP25 3. Chiral Fermions on Sp(2)/U(2) 7 4. Unitary Grassmannians 8 5. Conclusions 10 A. Spin and spincstructures on a manifold 12 B. Cohomology and spincon CP216 C. Cohomology and spincin six dimensions 17 D. Cohomology and generalised spinors for a 12 dimensional Grassmannian. 18 1. Introduction Non-commutative geometry has recently come to the fore as contender for a possible modification of physics with applications in attempts to unify gravity and gauge theories (for reviews see [1]). Long before the current surge of interest via superstrings it was suggested by Connes and Lott that the standard model of particle physics could be derived from non-commutative geometry, [2] [3]. A related concept is that of “matrix manifolds”—these are a version of non-commutative geometry in which continuous spaces with an infinite number of degrees of freedom are replaced with finite dimensional non-commutative matrix algebras approximating the continuum space. As the size of the matrices is taken to infinity the algebra becomes commutative and the continuum space is recovered. These algebras are often called “fuzzy spaces” but we shall refer to them as “matrix manifolds” in order to avoid the negative connotations of the word “fuzzy”. The matrix manifold approach has much in –1– common with generalised coherent states in quantum mechanics, [4] [5] and examples of manifolds which admit a finite matrix approximation are S2[6] [7], and more generally CPn, [9] as well as unitary Grassmannians U(n) U(k)×U(n−k)(1.1) [10] (star products on continuous complex projective spaces and unitary Grassmannians were constructed in [11] [12]). One of the attractive features of matrix manifolds is that they have the same symmetries as the continuum space, so a matrix version (G/H)Mof a coset space G/H has all the same symmetries of its continuous parent, despite being a finite approximation. Matrix manifolds are also closely related to harmonic expansions of functions on coset spaces, indeed the matrix algebras are nothing more than cunning rearrangements of the expansion coefficients into a matrix, and it is natural to ask if matrix manifolds might have a rˆole to play in Kaluza–Klein theory. This question was investigated in [13] and is one of the motivations for the present work. Another motivation is the calculation of the spectrum of the Dirac operator on CP2 Min [14] [15], the calculation in the latter reference being built on a construction which bears a remarkable resemblance to the electroweak sector of the standard model. This naturally leads one to ask if there might be a larger matrix manifold which could incorporate the whole standard model in its spectrum. One of the problems with the Kaluza–Klein programme was the realisation that it was unlikely to generate a chiral gauge theory in 4-dimensions without some modification, [16]. To obtain a chiral gauge theory it seems necessary to introduce fundamental gauge fields and then one is faced with the difficulty of anomaly cancellation, which is more difficult in higher dimensions because there are more potentially anomalous graphs to worry about. The introduction of fundamental gauge fields also negates the whole Kaluza–Klein philosophy whose aim is to derive the gauge fields purely from a metric. If the internal space is a matrix manifold however fundamental gauge fields are more natural, as a matrix manifold has no simple definition of a metric, but it does have symmetries. To call a theory with a matrix manifold as an internal space a Kaluza–Klein theory is really a misnomer as all it has in common with the usual Kaluza–Klein approach is an ‘internal’ space with a symmetry—if a metric is not defined there are no induced gauge fields so they must be added by hand. Nevertheless there is a symmetry, the symmetry of the isometries and holonomy of the coset space are there even at the finite level— like the grin of the Cheshire cat the symmetries remain even though the metric has gone.1For this reason we shall continue to refer to matrix Kaluza–Klein theory because the concept has much in common with continuum Kaluza–Klein theory, though there are also strong differences. 1For brevity we shall refer to Gand Has the isometry and the holonomy group, even when no metric or connection are defined. –2– Fundamental gauge fields are therefore natural in matrix Kaluza–Klein theories, but we must still worry about anomalies. To our knowledge the question of gauge anomalies on matrix manifolds has not yet been investigated, though chiral anomalies have been [7] [8]. If one takes a model consisting of 4-dimensional Minkowski spacetime with a matrix internal space one can hope that it may be sufficient for the 4-dimensional gauge anomalies to cancel, without worrying about graphs with more external legs that would be important if the internal space were continuous. To a large extent this is a question of dynamics on the internal space, if the dynamics reproduces that of the continuum in the continuum limit (though it doesn’t have to if we don’t want to take that limit) then these other graphs would have to be important in the limit. But as long as the internal space consists of a (small) number of finite degrees of freedom it seems not unreasonable to assume that only the usual 4-dimensional graphs contribute to a potential anomaly. For example the matrix manifold representing the 2-sphere S2 Mhas an approximation consisting of only 2 points. A matrix Kaluza–Klein theory based on S2 Mwould look like two copies of Minkowski space with an SU(2) action on the 2 points (much like the Higgs sector in Connes’ version of the standard model). There seems no compelling reason to believe that such a model would exhibit a six-dimensional gauge anomaly. For the reasons outlined above it seems worthwhile investigating the possibility of obtaining chiral gauge theories in 4-dimensions from a matrix Kaluza–Klein theory with internal space (G/H)Mand fundamental gauge fields. The tool that we use will be standard differential geometry and the Atiyah–Singer index theorem for the Dirac operator on continuous manifolds. Though the aim is to apply the concepts to finite matrix geometries it is not unreasonable to expect that the usual index theorem applies since it makes statements about topology by counting finite data. Indeed a Dirac operator can be defined on matrix manifolds, even though they are finite dimensional. The spectrum of the Dirac operator on some specific matrix manifolds has already been investigated, notably S2 M[7] [8] and CP2 M, [14] [15]. The construction of the spectrum on CP2in [15] is built on a 4-dimensional reducible representation of SU(2) ×U(1) which is that of electroweak sector of the standard model of particle physics, including a state with the quantum numbers of a right-handed neutrino which is a chiral zero-mode. As is well known CP2∼ =SU(3)/U(2) is not a spin manifold, it has an obstruction to the global definition of spinors, but coupling spinors to an appropriate background U(1) gauge field allows spinors to be defined—a construction which is called a spincstructure in the mathematical literature—and this gives rise to the right-handed neutrino in [15]. Since CP2 Mis a finite matrix algebra approximation to continuum CP2, which captures all the topological features of the continuum manifold, the topology is reflected at the matrix level and the emergence of chiral spinors on CP2 Mis a direct consequence of the Atiyah-Singer index theorem for spinors on CP2. –3– Since the holonomy group of CP2is U(2) the spectrum of the Dirac operator can be decomposed into representations of SU(2) ×U(1) and the representations in [15] are built on that of the electroweak sector of the standard model with the addition of a right-handed neutrino 10=VR1−2=eR2−1=VL eL.(1.2) There is a zero-mode state in the construction of [15], the 10, and its existence requires a background Abelian ‘monopole’ field on CP2. In fact, as we shall see, coupling Fermions to monopole fields of higher charge and non-Abelian background fields as well allows every state in (1.2) to be realised as a zero-mode of the Dirac operator on CP2. The fact that a right-handed neutrino appears naturally in the construction is particularly appealing in view of the recent evidence for solar neutrino oscillations [17] [18] whose simplest interpretation requires a right-handed neutrino. Another manifold which has holonomy group U(2) and does not admit spinors is Sp(2) U(2) ∼ =SO(5) SO(3) ×SO(2).(1.3) This space has a finite matrix approximation and has been proposed as a matrix version of the cotangent bundle to S3[19]. One might wonder if the spectrum in (1.2) is generic for spincstructures on manifolds with holonomy U(2) and this space is a counter-example. We shall see that a spectrum emerges which contains the correct charges of the electroweak sector but the Dirac operator for electron-neutrino doublet has zero index. Nevertheless it is useful to include this as an example of a space of dimension 2 mod 4 which is not spin (the significance for chiral spinors of a distinction between spaces of dimension 0 mod 4 and dimension 2 mod 4 was emphasised in [16]). The last space which we shall examine is the matrix version of the unitary Grassmannian U(5) U(3) ×U(2).(1.4) This space has a finite matrix approximation and an explicit local formula for a star-product, in terms of a finite sum of derivatives, was derived in [10]. It is not a spin manifold but admits a spincstructure, and so seems a good candidate for chiral spinors. Furthermore the holonomy group is exactly right for the standard model, since U(5) U(3) ×U(2) ∼ =SU(5) S[U(3) ×U(2)] (1.5) and the particle spectrum of the standard model is really such that the Fermions fall into a representation whose true group is precisely S[U(3) ×U(2)], [20]. For this space the spectrum contains one generation of the full standard model, including a right-handed neutrino, though the multiplicities are different for different states, some of them having zero index. –4– Section 2 contains an index theorem analysis of spinors on CP2and reproduces the zero-mode spectrum (1.2). Section 3 contains a discussion of the 6-dimensional manifold Sp(2)/U(2). Section 4 analyses spincstructures, and their non-Abelian generalisations, on the unitary Grassmannian U(5)/(U(3) ×U(2)) and its relation to the standard model spectrum. Our results are summarised in section 5. The analysis relies on the index theorem for the Dirac operator for various bundles over these three spaces. The derivation of the relevant index for the cases under study is given in four appendices, where a general discussion of spincstructures is also given as an aid to those who may not be familiar with the construction. 2. Chiral Fermions on CP2 The complex projective space CP2∼ =SU(3)/U(2) was actively investigated in the 1980s as an interesting candidate for an Euclidean gravitational instanton [21]. The Euler characteristic of CP2is χ= 3 and the signature is τ= 1. It is not a spin manifold, there is a global obstruction to putting spinors on this space, but one can put spinors on it provided fundamental gauge fields are introduced and an appropriate topologically non-trivial background gauge field is introduced. This fact was used in [21] to construct a “generalised spin structure”, where spinors with an Abelian charge move in the field of the K¨ahler 2-form on CP2, which is somewhat analogous to a monopole field on CP1∼ =S2. The holonomy group of CP2is U(2) ⊂SO(4) ∼ =SU(2) ×SU(2) Z2 .(2.1) If spinors could be defined this would be lifted to SU(2)×U(1) ⊂Spin(4) ∼ =SU(2)× SU(2), and the two different chiralities of Weyl spinors would transform under the different factors of SU(2) ×U(1) as, for example, ψ+=11+1−1and ψ−=20,(2.2) where the subscript denotes the U(1) charge. But since spinors cannot be defined globally (cf. appendix B), the spinor bundle does not exist. This can be cured by introducing a U(1) gauge field with non-trivial topology and correlating the charge with that of the U(1) subgroup of Spin(4). Mathematically, on a complex manifold X, we take the square root of the canonical line bundle K, as described in appendix A, and tensor it with the the spin bundle S(X). Neither of these bundles exists separately but S(X)⊗K−1/2does. In fact, if Lis a generating line bundle (cf. the appendix) with RS2c1(L) = −1, where S2is a non-trivial two sphere embedded in X,2then S(X)⊗Lpis a well defined bundle for any half-integral p. For CP2it is 2This is ambiguous if H2(X;Z) has dimension greater than one, but in all the examples we shall consider in this paper H2(X;Z) is one dimensional and this integral is uniquely defined. –5– shown in appendix B that S(X)⊗Lp=∧0,∗TX ⊗K1/2⊗Lp=∧0,∗T X ⊗L−q, X =CP2(2.3) where q=−p−3 2, since the canonical line bundle for CP2is given by K=L3. The net number of zero modes depends on qand for CP2is given in (B.5) of appendix B as ν=1 2(q+ 1)(q+ 2).(2.4) In fact qcan be interpreted as the effective U(1) charge. The charge is not pbecause there is a contribution from the angular momentum associated with the spinor bundle S(X). To evaluate the charge we use a general argument concerning spinor bundles over complex manifolds. We define the U(1) charge, which will be identified later with the hypercharge Y, using the Chern character of the generating line bundle raised to the appropriate power, in this case L−q, by taking a non-trivial S2embedded in the manifold Xand defining q=ZS2 ch(L−q) = −qZS2 ch(L) since ZS2 c1(L) = −1.(2.5) For comparison with the usual charge assignments of the standard model below, we rescale this by 2/3 to Y= 2q/3. For q= 0 for example ν= 1 and, identifying positive chirality with right-handed spinors, this would appear as a neutral righthanded particle: a right-handed neutrino VR. A spinor with q=−3 also has ν= 1, so would be right-handed, with Y=−2: the right-handed electron, eR. If a fundamental SU(2) gauge field is added with the spinors taken to be SU(2) doublets then spinors can be obtained from the bundle ∧0,∗TCP2⊗F⊗L−q, where Fis the rank 2 vector bundle defined by F⊕L=I3(I3denoting a trivial rank 3 bundle). The structure group of Fis U(2) corresponding to a SU(2) ×U(1)-gauge field. In fact Fis associated to the principal U(2) bundle induced by the coset construction U(2) −→ SU(3) ↓ CP2. (2.6) The Dirac index for ∧0,∗TCP2⊗F⊗L−qis derived in appendix B and is given by (B.11) ν= (q+ 1)(q+ 3).(2.7) Zero modes would give rise to chiral SU(2) doublets. The U(1) charge is now calculated as the Chern character ch(F⊗L−q) evaluated on a topologically non-trivial S2embedded in CP2, the result is 2q+1. As the Chern character involves tracing over a 2 ×2 matrix the U(1) generator is (2q+1) 21, where 1 is the 2 ×2 identity matrix, so the individual charges are q+1 2. Re-scaling by 2/3, –6– as above, gives Y=2q+1 3. In particular q=−2 yields Y=−1 with ν=−1 and, identifying positive chirality with right-handed particles, we get a single generation of a left-handed doublet with charge −1, the electron-neutrino doublet. So we can obtain a single generation of the electroweak sector of the standard model from CP2by taking SU(2) singlets with q= 0 and q=−3 (ν= +1) and a single SU(2) doublet with q=−2 (ν=−1), that is 10=VR1−2=eR2−1=VL eL, ν > 0 right-handed.(2.8) 3. Chiral Fermions on Sp(2)/U(2) As an example of a six-dimensional space which does not admit a spin structure, but does admit a Spincstructure, consider Sp(2)/U(2). This space has Euler characteristic χ= 4. In fact Sp(2) U(2) ∼ =SO(5) SO(3) ×SO(2) (3.1) and this space admits a matrix approximation. The spinor bundle does not exist but a Spincstructure can be defined using S(X)⊗Lp, with Lthe generating line bundle and phalf-integral. The canonical line bundle is related to the generating line bundle by K=L3(see appendix C) so that S(X)⊗Lp=∧0,∗TX ⊗L−q, X =Sp(2) U(2) (3.2) where q=−p−3 2. The Dirac index of this bundle is derived in appendix C and is given in (C.18): ν=1 6(2q+ 3)(q+ 1)(q+ 2).(3.3) The zero-modes will give rise to particles in 4-dimensions whose U(1) charge is q, which we re-scale by 2/3 to bring it line with the usual standard model conventions below. so, for example, q=−3 gives a single generation of negative chirality particles with charge −2 while q= 0 would give a single generation of positive chirality neutral particles. As before we can also couple the Fermions to a fundamental SU(2) gauge field by introducing a rank 2 vector bundle Fassociated to the principal bundle U(2) −→ Sp(2) ↓ Sp(2)/U(2) (3.4) with structure group U(2). It is shown in appendix C that the index of ∧0,∗TX ⊗ F⊗L−qis now ν=2 3q(q+ 1)(q+ 2).(3.5) –7– The Chern character ch(F⊗L−q) evaluates to 2q+ 1 on a non-trivial S2. Again this is the trace of a 2 ×2 matrix and the individual states have charge q+1 2which is rescaled by 2/3 to give the U(1) charge as Y=2q+1 3. For example q= 1 gives Y= 1 and ν= 4 and thus four copies of positive chirality doublets while q=−2 gives Y=−1 and ν= 0. We can try to get the electroweak charges from this construction. For example interpreting positive chirality as left-handed the singlets would be the right-handed electron eRand a left-handed anti-neutrino (V)L. But the doublets with Y= 1 would have to have negative chirality to fit with the standard model (the righthanded positron and anti-neutrino) and νis positive. If we interpret positive chirality as right-handed, the doublet could the positron–anti-neutrino doublet (V)R (e)R, but then the singlet with Y=−2 has the wrong chirality to be the right-handed electron. On the other hand choosing a doublet with q=−2 giving Y=−1, in addition to the singlets above, gives ν= 0 for the doublet: in general the Dirac operator will have no zero modes for this doublet though it may have for specific choices of the U(2) connection, but even then the zero modes will occur in pairs of opposite chirality. The spectrum contains one generation of the electroweak sector of the standard model, but there is an additional unwanted doublet of the wrong chirality. 4. Unitary Grassmannians The final source of examples that we wish to discuss is the unitary Grassmannians U(n) U(k)×U(n−k)∼ =SU(n) S(U(n−k)×U(k)) (4.1) of which the complex projective spaces, k= 1, are special cases. The first Chern class of the tangent bundle for these space evaluates to n, [23], and the second Stiefel-Whitney class is nmod 2—so these spaces admit a spin structure if and only if nis even. We shall focus on the particular case of n= 5 and k= 2, this is an interesting case because the holonomy group of SU(5)/S(U(3) ×U(2)) is precisely that of the standard model, [20]. This condition dictates that the Fermions actually sit in representations of SU(3)×SU(2)×U(1) in which the generators are traceless— whence S(U(3) ×U(2)). As a matrix manifold SU(5)/S(U(3) ×U(2)) was studied in [10], where a star product was explicitly constructed in terms of derivatives. The Grassmannian SU(5)/S(U(3) ×U(2)) has Euler characteristic χ= 10 and signature τ= 2. It is not a spin manifold but a Spincstructure exists. Taking the bundle ∧0,∗T X ⊗L−q, with Xthe Grassmannian and Lthe generating line bundle, the Dirac index is calculated in appendix D as (D.27), ν{q,1,1}=1 144(q+ 1)(q+ 2)2(q+ 3)2(q+ 4),(4.2) –8– We shall call Lthe generating line bundle and, in each case, Kwill be some power of L—this power will be odd if Xis not a spin manifold—so that K=Lm, m ∈Z.(A.22) Hence a general spincstructure will have the spincbundle ∧0,∗TX ⊗L−q=Sc(X)⊗L−q, q ∈Z(A.23) (the minus sign in the exponent is for later convenience). When q= 0 we have the canonical spincstructure; there is also a dependence of the spincstructure on an element of H1(X;Z2) but our examples have H1(X;Z) = 0 so we do not need to consider this. If we use the fact that K=Lmthen the corresponding Dirac operator then becomes /∂L−(q+m/2) which we shall neaten up slightly by writing it as /∂Lpwhere p=−q−m/2.(A.24) There is also an index formula for the zero modes of /∂Lpwhich involves the usual ˆ Agenus of Xand the ‘Chern class’ of the line bundle Lp. Let /∂Lpdenote the Dirac operator coupled to Lpthen its index is given by 3 index (/∂Lp) = ch (Lp)ˆ A(X)[X] (A.25) = exp [pc1(L)] ˆ A(X)[X].(A.26) We will also need the case where the Dirac operator is further coupled to a second vector bundle Eof rank possibly greater than one; in this case the requisite index formula is index (/∂Lp⊗E) = ch (Lp⊗E)ˆ A(X)[X] (A.27) = ch (E) exp [pc1(L)] ˆ A(X)[X],(p=−q−m/2).(A.28) In the next section we treat an actual spincexample in four dimensions. 3We could equally have used instead the formula for index (¯ ∂Lp) which would have involved ch (L) and the Todd class td(X). In fact this realisation of the Dirac operator as ¯ ∂Lpenables one to easily understand why index (/∂L−q−m/2) is equal to unity for q= 0: it is because, when q= 0, index (¯ ∂L−m/2) gives the arithmetic genus P(−1)sh0,s of the complex manifold Xwhere the Hodge number hr,s denotes the dimension of the space of holomorphic forms of type (r, s). Now for the manifolds Xwe consider in this paper the only holomorphic forms are of type (s, s) a fact which reduces the arithmetic genus to h0,0which is trivially unity. – 15 – B. Cohomology and spincon CP2 On CP2the Chern class is [25] c(CP2) = 1 −3c1(L) + 3c2 1(L) (B.1) where the generating line bundle Lhas c(L) = 1 + c1(L) with −c1(L) generated by the K¨ahler 2-form. The Euler characteristic is 3 so c2 1(L)[CP2] = 1 and in this case c1(CP2) = −3c1(L) so m= 3. Since the coefficient of c1(L) is odd w26= 0 and CP2 does not admit a spin structure. The index of the Dirac operator coupled to Lpis index (/∂Lp) = ch (Lp)ˆ A(X)[X] = exp [pc1(L)] ˆ A(X)[X] (B.2) =1 + pc1(L) + 1 2p2c2 1(L)1−p1(X) 24 [X], X =CP2(B.3) =1 8(4p2−1),(B.4) where p1is the Pontrjagin class and p1(X) = c2 1(X)−2c2(X) = 3c2 1(L) on CP2. This index is integral for half-integral pand, setting p=−q−3/2, we obtain index (/∂Lp) = 1 2(q+ 2)(q+ 1).(B.5) We can define a non-trivial rank 2 bundle Fover CP2with structure group U(2) by F⊕L∼ =I3where I3is the trivial rank 3 bundle. Then c(F)c(L) = 1 so c1(F) = −c1(L) and c2(F) = c2 1(L); tensoring this with pcopies of the generating line bundle Lthen gives, for the Chern character, ch(Lp⊗F) = ch(Lp)ch(F) (B.6) =1 + pc1(L) + 1 2p2c2 1(L) + ···2−c1(L)−1 2c2 1(L) + ···(B.7) = 2 + (2p−1)c1(L) + p2−p−1 2c2 1(L) + ···,(B.8) leading to index (/∂Lp⊗F) = ch (F⊗Lp)ˆ A(X)[X] = exp [pc1(L)] ch (F)ˆ A(X)[X] (B.9) =2 + (2p−1)c1(L) + p2−p−1 2c2 1(L)1−p1 24[X] =1 4(2p−3)(2p+ 1), X =CP2(B.10) = (q+ 1)(q+ 3),again using p=−q−3/2.(B.11) – 16 – C. Cohomology and spincin six dimensions In this section Xis the complex manifold given by X=Sp(2) U(2) (C.1) whose real dimension is 6. The cohomology ring of Xis generated by the even dimensional classes σ1∈H2(X;Z) and σ2∈H4(X;Z) subject to the single relation σ2 1= 2σ2.(C.2) Now Xis not a spincmanifold because we can compute that c(X) = 1 + c1(X) + c2(X) + c3(X) (C.3) = 1 + 3σ1+ 8σ2+ 4σ1σ2(C.4) ⇒c1(X) = 3σ1=−3c1(L).(C.5) We note that σ1generates H2(X;Z) and so deduce that c1(X) is odd and so w2(X)6= 0 ⇒Xis not spin.(C.6) We also see that K=L3(C.7) so that the integer mof appendix A is equal to 3. The index of the Dirac operator /∂Lpcan now be computed from the expansions of ch (Lp) and ˆ A(X) giving us the formula index (/∂Lp) = 1 + pc1(L) + 1 2p2c2 1(L) + ···1−p1(X) 24 +···[X] (C.8) =−pc1(L)p1(X) 24 +1 3!p3c3 1(L)[X].(C.9) But we can calculate that p1(X) = c2 1(X)−2c2(X) (C.10) = 9σ2 1−16σ2,(C.11) with σ1=−c1(L). Hence we find that index (/∂Lp) = p 24σ1(9σ2 1−16σ2)−1 3!p3σ3 1[X] (C.12) =−(4p3−p)σ3 1 24[X] (C.13) =−1 12(4p3−p) = −1 12p(2p−1)(2p+ 1),(C.14) – 17 – where we have used the Gauss–Bonnet theorem which says that c3(X)[X] = χ(X) (C.15) = 4 = 2σ3 1[X] (C.16) to deduce that σ3 1[X] = 2. Before finishing we should check that the index is integral. Recall that p=−q−m/2, q ∈Z, m = 3 (C.17) This fact immediately gives us the formula index (/∂Lp) = 1 6(2q+ 3)(q+ 1)(q+ 2), q ∈Z(C.18) and this is easily checked to give an integer index for integral qas it should. If we tensor product with a further rank 2 bundle F, with c1(F) = σ1then we find that index (/∂Lp⊗F) = ch(Lp⊗F)ˆ A(X)[X] (C.19) =ch(Lp)ch(F)ˆ A(X)[X] (C.20) =−1 12(2p+ 3)(2p−1)(2p+ 1) (C.21) =2 3q(q+ 1)(q+ 2), p =−q−3/2, q ∈Z(C.22) and again this gives an integral index. D. Cohomology and generalised spinors for a 12 dimensional Grassmannian. In this section Xis the 12 dimensional Grassmannian given by X=U(5) U(3) ×U(2).(D.1) Xis a perfectly standard complex manifold (of complex dimension 6) and its cohomology ring H∗(X;Z) has 3 generators σi∈H2i(X;Z), i = 1,2,3 (D.2) which obey the single relation σ3= 2σ1σ2−σ3 1.(D.3) Its Chern class is given by c(X) = (1 + c1(X) + c2(X) + c3(X) + c4(X) + c5(X) + c6(X)) (D.4) = (1 −5σ1+ 12σ2 1−15σ3 1+ 8σ4 1+ 2σ2 1σ2+ 7σ2 2+ 4σ5 1−25σ1σ2 2(D.5) −29σ6 1+ 7σ2 1σ2 2+ 56σ4 1σ2−27σ3 2) (D.6) – 18 – from which we see that c1(X) = −5σ1(D.7) and hence we deduce, as we did in the previous section, that w2(X)6= 0 (D.8) and so Xis not spin. We now pass to the spincbundle Sc(X) and to the calculation of the index of its Dirac operator /∂Lpwhere Lis the generating line bundle as it was in the previous section. But this time we need the fact that σ1is actually a negative generator of H2(X, Z) with our orientation conventions and so we have c1(X) = −5σ1, σ1negative, σ1=c1(L) (D.9) ⇒K=L5(m= 5) (D.10) ˆ A(X) = 1−p1(X) 24 +1 5760(7p2 1(X)−4p2(X)) (D.11) −1 210 ·945(16p3(X)−44p1(X)p2(X) + 31p3 1(X)) + ···(D.12) as well as p1(X) = c2 1(X)−2c2(X) = σ2 1+ 2σ2(D.13) p2(X) = −2c1(X)c3(X) + c2 2(X) + 2c4(X) = 10σ4 1−20σ2 1σ2+ 15σ2 2(D.14) p3(X) = 2c1(X)c5(X)−2c2(X)c4(X) + c2 3(X)−2c6(X) (D.15) = 51σ6 1+ 72σ2 1σ2 2−144σ4 1σ2+ 68σ3 2.(D.16) This information allows to compute that index (/∂Lp) = exp [pc1(L)] ˆ A(X)[X] (D.17) =−1 60480σ3 2[X]−41 15120 +1 360p2σ2 1σ2 2[X] (D.18) +353 161280 +3 320p2−1 288p4σ4 1σ2[X] (D.19) +−407 967680 −11 3840p2−1 576p4+1 720p6σ6 1[X].(D.20) Now use the cohomology generators and the fact that Xclearly has Euler characteristic 10 we discover that σ3 2[X] = 1 (D.21) σ2 1σ2 2[X] = 2 (D.22) σ4 1σ2[X] = 3 (D.23) σ6 1[X] = 5.(D.24) – 19 – This all gives the formulae index (/∂Lp) = −1 1024 +19 2304p2−11 576p4+1 144p6(D.25) =1 9.210 (4p2−9)(4p2−1)2(D.26) =1 144(q+ 1)(q+ 2)2(q+ 3)2(q+ 4),using p=−q−5/2,(D.27) and this index is an integer for integral qas required. We shall finish by calculating the index when we couple the Dirac operator to some higher rank bundles. We shall give the results for two bundles Eand Fwhich are naturally associated to Xand also for the tensor product E⊗F. Let Ebe the rank 3 vector bundle over X=U(5) U(3) ×U(2) (D.28) whose fibre over a point x∈Xis the 3-plane xitself. This describes the bundle E. Now consider the product rank 5 bundle X×C5then Fis the rank 2 bundle created by forming the quotient X×C5 E.(D.29) The bundles Eand Fsatisfy E⊕F∼ =I5(D.30) where I5is a trivial rank 5 bundle and it is not difficult to work out that c(E)c(F) = 1 (D.31) i.e. (1 + c1(E) + c2(E) + c3(E))(1 + c1(F) + c2(F)) = 1 (D.32) ch(E) + ch(F) = 5.(D.33) In fact equation (D.33) can be used to derive the relation (D.3) since the classes σi are just the classes ci(E) and so this allows all of c(E) and c(F) to be expressed in terms of the σi. The Chern characters of Eand Fare given by ch(E) = 3 + c1(E) + 1 2c2 1(E)−2c2(E)+1 3! c3 1(E)−3c1(E)c2(E) + 3c3(E)+1 4! c4 1(E) −4c2 1(E)c2(E) + 4c1(E)c3(E) + 2c2 2(E)+1 5! c5 1(E)−5c3 1(E)c2(E) + 5c2 1(E)c3(E) +5c1(E)c2 2(E)−5c2(E)c3(E)+1 6! c6 1(E)−6c4 1(E)c2(E) + 6c3 1(E)c3(E) +9c2 1(E)c2 2(E)−12c1(E)c2(E)c3(E)−2c3 2(E) + 3c2 3(E) ch(F) = 5 −ch(E). – 20 – Now we can calculate the index of the appropriate Dirac operators: Forming the product Lp⊗Ewe have index (/∂Lp⊗E) = ch (Lp⊗E)ˆ A(X)[X] (D.34) = ch (E) exp [pc1(L)] ˆ A(X)[X],(D.35) and we find that index (/∂Lp⊗E) = −15 1024 +3 128p+59 768p2−5 48p3−5 64p4+1 24p5+1 48p6 =1 3.210 (2p+ 5)(2p−1)(4p2−9)(4p2−1) (D.36) =1 48q(q+ 1)(q+ 2)(q+ 3)2(q+ 4),(p=−q−5/2) (D.37) and for the product Lp⊗F index (/∂Lp⊗F) = 5 512 −3 128p−41 1152p2+5 48p3−5 288p4−1 24p5+1 72p6 =1 9.29(2p−5)(2p−1)(4p2−9)(4p2−1) (D.38) =1 72(q+ 1)(q+ 2)(q+ 3)2(q+ 4)(q+ 5),(p=−q−5 2).(D.39) Finally for the bundle Lp⊗E⊗Fwe have index (/∂Lp⊗E⊗F) = −25 512 −25 384p+103 384p2+13 48p3−29 96p4−1 24p5+1 24p6 =1 3.29(4p2−25)(2p−3)(4p2−1)(2p+ 1) (D.40) =1 24q(q+ 2)2(q+ 3)(q+ 4)(q+ 5),(p=−q−5/2) (D.41) and in each case one can verify that the index is an integer. 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