The Spectrum of the Dirac Operator on Coset Spaces with Homogeneous Gauge Fields

JHEP05(2003)018 Published by Institute of Physics Publishing for SISSA/ISAS Received: April 10, 2003 Accepted: May 9, 2003 The spectrum of the Dirac operator on coset spaces with homogeneous gauge fields Brian P. Dolan Dept. of Mathematical Physics, NUI, Maynooth, Ireland and School of Theoretical Physics Dublin Institute for Advanced Studies 10 Burlington Rd., Dublin 8, Ireland E-mail: [email protected]e Abstract: The spectrum and degeneracies of the Dirac operator are analysed on compact coset spaces when there is a non-zero homogeneous background gauge field which is compatible with the symmetries of the space, in particular when the gauge field is derived from the spin-connection. It is shown how the degeneracy of the lowest Landau level in the recently proposed higher dimensional quantum Hall effect is related to the Atiyah-Singer index theorem for the Dirac operator on a compact coset space. Keywords: Field Theories in Higher Dimensions, Differential and Algebraic Geometry. c °SISSA/ISAS 2003 http://jhep.sissa.it/archive/papers/jhep052003018 /jhep052003018 .pdf JHEP05(2003)018 Contents 1. Introduction 1 2. Symmetric spaces 3 3. Non-symmetric spaces 11 4. Conclusions 13 A. Metric and connection on G/H 14 B. Spectrum of the Dirac operator on CP 216 C. Index theorem on SU(3)/U(1) ×U(1) 18 1. Introduction There has long been a fruitful interplay between condensed matter physics and field theory in particle physics, many concepts that were first developed in the former later being applied to the latter and vice versa. The quantum Hall effect [1] has attracted the interest of many high energy theorists, not least because the fractional QHE exhibits collective excitations which mimic a fractional electric charge, but also because there are deeper connections between the Hall effect and string theory [2]. Recently Zhang and Hu proposed a higher dimensional analogue of the quantum Hall effect, on S4[3], based on Haldane’s description of the Hall effect on S2with a magnetic monopole at the centre, [4]. Zhang and Hu’s idea was developed further in [5] and extended to complex projective spaces in [6]. The connection between the higher dimensional quantum Hall effect and string theory was analysed in [7]. The higher dimensional quantum Hall effect involves a generalisation of the Landau problem to particles moving on a compact coset space G/H, in the presence of a background gauge field: such as a U(1) monopole on CP nor a homogeneous SU(2) instanton field on S4. A common ingredient of the these analyses is the calculation of the degeneracy of the ground state for particles moving in a homogeneous background, i.e. a background field which has the symmetry of the isometry group G. In previous works, and in this paper also, the gauge group will be restricted to be the holonomy group H(or a factor group of same if His a product of smaller groups). In [3, 6] the degeneracy of the ground state was calculated using group theory: the non-relativistic hamiltonian for a spinless particle moving in a homogeneous background field involves the quadratic Casimirs of the groups Gand Hand the allowed states involve – 1 – JHEP05(2003)018 irreducible representations of Gthat contain pre-ordained representations of H. The dimension of the representation of Gcorresponding to the ground state is identified with the degeneracy of the ground state. It is common in discussions of the quantum Hall effect to ignore the electron’s spin. Zhang and Hu treated scalar particles satisfying the exclusion principle, as did Karabali and Nair: this is perfectly justified when the Zeeman splitting is large enough that transitions between spin states can be ignored and the ground state is effectively isolated from the next highest spin state. Nevertheless one is tempted to ask what is the rˆole of electron spin in the higher dimensional quantum Hall effect, and it will be argued here that there is an important quantitative relic of the Fermionic nature of the particles in the degeneracy of the ground state, over and above the trivial consequences of the exclusion principle. It is shown in section 2 that the degeneracies calculated in [3, 4, 6], for S4,S2and CP n respectively, are related to the index of the Dirac operator for Fermions moving in the appropriate background field: in fact the degeneracy is the number of zero-modes of the Dirac operator and, generically, this is the modulus of the index. Furthermore the ground state wave-functions, the higher dimensional analogues of the Laughlin wave-functions, are precisely the zero-modes of the Dirac operator. We do not have to look far to discover the reason for this — the square of the Dirac operator is nothing other than the hamiltonian for a non-relativistic Fermion moving in a static background field, (i\ D)2=−DαDα+R 41−i 2Fαβγαβ ,(1.1) where Ris the Ricci scalar and the last term represents the Zeeman splitting. (There is an extra term on the right hand side of (1.1) if the spin connection involves torsion, this equation must therefore be modified for non-symmetric coset spaces with torsion as considered in section 3.) Thus a non-relativistic particle moving in a static background magnetic field is an example of supersymmetric quantum mechanics with a self-dual pre-potential [8]. Since the Dirac operator is hermitean, the eigenvalues of the hamiltonian (1.1) are positive semidefinite and a zero-mode requires exact cancellation of all three terms on the right hand side. It is shown in sections 2 and 3 that, for specific homogeneous background fields (analogues of monopole and instanton fields on S2and S4) all three terms on the right hand side of (1.1) are mutually commuting and so can be simultaneously diagonalised so that all spin components decouple from each other. The Dirac laplacian ∆ = −DαDαis itself a positive operator on a compact space with positive curvature, so a zero-mode of the Dirac operator, if one exists, requires a cancellation of the lowest eigenvalue of the laplacian with the lowest eigenvalue of the sum of curvature and Zeeman terms on the right hand side of (1.1). In fact, for the homogeneous background fields that are considered here, the eigenvalues of (1.1) can be determined purely in terms of certain quadratic Casimirs of the isometry group G and the holonomy group Hand are given by equation (2.15) or, more generally, (3.9). Even without calculating the full spectrum it is possible to find the representation of G with the lowest eigenvalue for any Fermion in a given representation of the gauge group. If the chosen representation of the gauge group allows for a zero-mode of the Dirac operator – 2 – JHEP05(2003)018 then the dimension of the representation corresponding to the lowest eigenvalue generically gives the number of zero-modes. Since this calculation involves only the lowest eigenvalue of the Zeeman and the curvature terms, which are fixed in advance, the Fermionic nature of the particles can be ignored and the problem reduces to choosing the correct representations of Gto scan in minimising the laplacian. This is precisely what was done in [3] and [6]. The net result is that the number of zero-modes of the Dirac operator, for Fermions in a given representation of the gauge group, can be calculated simply from a knowledge of the quadratic Casimirs of Gand the decompositions of its representations under H7→ G. This generalises the results of [3] and [6] to the quantum hall effect on any coset space G/H with compact Lie groups Gand H. Of course an analysis of the spectrum of the Dirac operator is of intrinsic interest, even without reference to the quantum Hall effect. In particular it is of obvious importance in Kaluza-Klein theories and and string theory. The layout of the paper is as follows. In section 2 the case of symmetric spaces G/H is treated in detail and it is shown how the eigenvalues of the Dirac operator in the presence of a homogeneous background gauge field can be expressed in terms of quadratic Casimirs C2(G) and C2(H). The examples of S2,S4and CP 2are worked out and compared to known results. Section 3 extends the analysis to non-symmetric spaces with torsion and the example of SU(3)/U(1)×U(1) is treated in detail — this space is of interest in string theory where it is arises in the context of seven dimensional spaces with G2holonomy and their conical singularities [9]. Section 4 gives a summary of the results. Some technical details are relegated to three appendices: appendix A reviews aspects of the geometry of homogeneous spaces used in the text. Appendix B presents the spectrum of the Dirac operator on CP 2, in the presence of a homogeneous background SU(2) ×U(1) gauge field. Appendix C presents a standard analysis of the Atiyah-Singer index theorem on SU(3)/U(1) ×U(1) for comparison with the results of section 3. 2. Symmetric spaces Consider the Dirac operator for a Fermion moving on a d-dimensional compact space, without boundary, in the presence of a background gauge field: i\ D=iγαDα=iemαγαµ∂m+1 4ωm,αβγαβ +iAi mti¶,(2.1) where ωαβ =ωαβ,mdxmis the spin connection, Ai=Ai mdxmthe gauge connection and tiare generators of the gauge group (α, β = 1,2,···, d are orthonormal indices and m= 1,2,···, d is a co-ordinate index). The γ-matrices satisfy the usual Clifford algebra, nγα, γβo= 2δαβ ,with γαβ := 1 2hγα, γβi,(2.2) and emαare d-beins for the metric. The curvature and field strength follow from [Dα, Dβ] = iF i αβti+1 4Rαβγδγγδ.(2.3) – 3 – JHEP05(2003)018 For a torsion free connection squaring the Dirac operator gives (i\ D)2= ∆ + R 41−i 2Fαβγαβ ,(2.4) with ∆ = −DαDαthe Dirac laplacian. We shall refer to ∆ + R 41as the kinetic energy and −i 2Fαβγαβ =−i 2Fi αβtiγαβ as the Zeeman energy. The laplacian here is the laplacian acting on spinors, including the spin and the gauge connection, so its spectrum depends on both the metric and the background field. On a coset space G/H, with Gand Hcompact groups, it is natural to use the Ginvariant metric, for which the generators of Gare Killing vectors and the holonomy group is H⊆SO(d). Furthermore we shall consider background gauge fields which are compatible with the isometries, in the sense that Lie transport of the field strength Fby a Killing vector Kgenerates a gauge transformation, LKF=g−1Fg , (2.5) where g∈ G, the group of gauge transformations. In particular this will be the case if we identify the gauge group with the holonomy group and the gauge connection with the spin connection — the details of this identification are given in appendix A. (A variation on this is if the holonomy group factorises into simple groups and U(1) factors. When this is the case the gauge group can be taken to be one of the factors. For example this is the situation for the homogeneous SU(2) instanton on S4, where S4= SO(5)/SO(4) and, at the level of the algebras, H= SU(2) ×SU(2) so we can take the gauge group to be just SU(2).) Let tAbe the generators of the isometry group G, with [tA, tB] = ifABCtC, and tithe generators of the holonomy group H. Then the curvature 2-forms of a G-invariant metric for a symmetric space can be taken to be (see appendix A), Rαβ=1 2Rαβγδeγ∧eδ=1 2fαβifiγδeγ∧eδ.(2.6) Identifying the gauge connection with the spin connection gives rise to the field strength, Fi=1 2Fi αβeα∧eβ=1 2fiαβeα∧eβ.(2.7) For a symmetric space the Riemann tensor is co-variantly constant and this means that the above field strength is co-variantly constant, DαFi βγ = 0 .(2.8) In particular the laplacian commutes with the Zeeman term in the hamiltonian. With this choice of background field the commutator (2.3) simplifies, [Dα, Dβ] = ifiαβ ½(1⊗ti)−i 4fiγδ ³γγδ ⊗1´¾.(2.9) Now Ti:= −i 4fiγδγγδ (2.10) – 4 – JHEP05(2003)018 are a representation of the gauge group (which may be reducible, in general), [Ti, Tj] = ifijkTk,(2.11) so [Dα, Dβ] = fiαβ Di(2.12) with Di:= i{(1⊗ti) + (Ti⊗1)}(2.13) being the generators of Hin the tensor product representation of tiwith the spinor representation Ti. This allows the laplacian to be expressed as the difference of quadratic Casimirs, ∆ = −DαDα=−DADA+DiDi=C2(G, ·)−C2(H, Di).(2.14) For spinors in a given representation tiof the gauge group C2(H, Di) in this expression is always calculated in the fixed representation (2.13), which in general involves reducible representations of G, while the representations used in C2(G, ·) range over all irreducible representations of Gthan contain (2.13). In particular the cross-term −2ti⊗Tifrom (Di)2 in (2.14) exactly cancels the Zeeman energy in (2.4) and, as described in appendix A, the second order Casimir for the representation Tiis related to the Ricci scalar by C2(H, Ti) = R/8. The eigenvalues of the square of the Dirac operator (2.4) can then be expressed purely in terms of quadratic Casimirs: E=C2(G, ·)−C2(H, ti) + R 81.(2.15) This construction will now be illustrated with some examples. (i) S2∼ =SO(3)/SO(2).This was the geometry originally studied by Haldane in the context of the quantum Hall effect [4]. The isometry group is generated by the algebra of SU(2) [tA, tB] = i²ABCtC(2.16) and we are free to choose t3to generate the U(1) holonomy. Formula (A.9) of appendix A gives Rαβ =1 2²αβ3²3γδeγ∧eδ(2.17) so R12 = e1∧e2(2.18) are the curvature 2-forms for a sphere of unit radius. Also F3= e1∧e2(2.19) is the field strength if a magnetic monopole at the centre of the sphere. Actually this corresponds to a monopole of charge 2, since 1 2πZS2 e1∧e2= 2 (2.20) – 5 – JHEP05(2003)018 is the Chern class of the tangent bundle (which is equal to the Euler characteristic). In general we can put a monopole of any integral charge at the centre of the sphere F3=M 2e1∧e2.(2.21) (Alternatively we can work with a monopole of charge 2 and consider Fermions of any half-integral charge in this background.) Choosing γ1=σ1and γ2=σ2, with σ1and σ2Pauli matrices, we have i 2F3 αβγαβ =iF 3 12(iσ3) = −M 2µ1 0 0−1¶.(2.22) The Ricci scalar for a sphere of unit radius is 2, so equation (2.4) gives (i\ D)2= ∆ + 1 21+M 2µ1 0 0−1¶.(2.23) For positive Mthis indicates that there are spin down zero-modes of the Dirac operator if ∆ + 1 2=M 2(2.24) while for negative Mthere are spin up zero-modes if ∆ + 1 2=−M 2.(2.25) There are of course no zero-modes for M= 0 as required by Lichnerowicz theorem. In this example tiof (2.13) is just a number, M/2, and Tiis σ3/2 so D3=iµM+1 20 0M−1 2¶⇒D3D3=−µ¡M+1 2¢20 0¡M−1 2¢2¶.(2.26) The eigenvalues of the laplacian (2.14) are therefore ∆j=j(j+ 1) −µM±1 2¶2 ,(2.27) as discussed in [3], so eigenvalues of (2.23) are Ej=(2j+ 1)2−M2 4,(2.28) which can also be obtained directly from (2.15). For M= 0 this reproduces the well-known result that the spectrum of the Dirac operator is linear in angular momentum (see e.g. [10]). For M6= 0 the representations jof SU(2) that appear in a harmonic expansion of ∆ are restricted to those that contain the U(1) representation of charge M±1 2, i.e. j=|M|−1 2+k with ka non-negative integer, Ej=k(k+|M|).(2.29) – 6 – JHEP05(2003)018 There are zero-modes for k= 0, and jmin =M−1 2for positive Mor −(M+1 2) for negative M. In either case the degeneracy of the ground state is d(jmin) = 2jmin + 1 = |M|,(2.30) which is the number of zero-modes of the Dirac operator. Note the shift of jmin away from |M|by 1/2, due to the intrinsic spin of the Fermion. The degeneracy (2.30) relates to the Atiyah-Singer index theorem which states that the the index of the Dirac operator is minus the first Chern class [11], ν=ν+−ν−=−1 2πZS2 F3=−M , (2.31) where ν+is the number of positive chirality zero-modes and ν−the number of negative chirality zero-modes. Indeed the ground state wave-functions in [4] for the integer quantum Hall effect, spherical analogues of the Laughlin wave-functions, are precisely these zeromodes. The case |M|= 1 corresponds to jmin = 0, in this case the gauge connection exactly cancels the spin connection for the relevant chirality and single zero-mode of the Dirac operator is a constant spinor. The above calculation can be represented graphically using Young tableaux, which will be useful in more complicated situations to follow. The fundamental of SU(2) decomposes as SU(2) →U(1) 2→11+1−1.(2.32) Denoting 11by ×and 1−1by •the (p+1)-dimensional irreducible representation of SU(2) contains × ·· × | {z } s • ·· • | {z } r ⊂·· | {z } p (2.33) with p=r+s. Fixing the U(1) charge to be Qconstrains s−r=Qso p= 2r+Q. The ground state energy for a Fermion in this background can now be found by minimising ∆, since all the other terms in the energy are constants for fixed Q, that is by minimising p 2³p 2+ 1´=r(r+ 1) + Q 2(2r+ 1) + Q2 4.(2.34) If Q>0 this is minimised by r= 0, so p=Qand the degeneracy of the ground state is Q+1. If Q<0 it is minimised by r=pand then p=−Q, so the degeneracy is −Q+1. In either case the ground state has p=|Q|and the degeneracy is |Q|+ 1. Clearly |Q|= 2jmin and the U(1) charge Qis not just the monopole charge M, but includes a shift to account for the intrinsic spin of the Fermion, |Q|=|M|−1. This method, using representation theory to describe the kinetic energy and calculate the degeneracy of the ground state, was used in [6]: though in that reference the particles were treated as scalars so there was no intrinsic spin — there was therefore no Zeeman energy to make the total ground state energy vanish and no shift in the charge to account for the intrinsic spin of the particles. The technique is however applicable to both the laplacian for scalars and the square of the Dirac operator because, for a given gauge background and representation, they only differ by constants. It has the advantage of avoiding an explicit calculation of the full eigenvalue spectrum of the Dirac operator. – 7 – JHEP05(2003)018 (ii) S4∼ =SO(5)/SO(4).The next example, S4, was the case studied in the first paper on the higher dimensional quantum Hall effect, [3]. In this case the algebra of the holonomy group is SU(2) ×SU(2) and we can take the gauge group to be just one SU(2) factor. The Riemann tensor can be split into self-dual and anti-self-dual parts and these correspond to the curvatures arising from the two SU(2) factors of the holonomy group. Choosing, for example, the self-dual SU(2) factor the resulting SU(2) background gauge field is the homogeneous instanton of charge one, which has SO(5) symmetry on S4[12] (this paper was published a little after the BPST instanton [13], but the techniques are very enlightening and highlight the analogy with the Wu-Yang monopole — Yang calls this homogeneous instanton configuration a non-abelian monopole). Representations of SO(5) can be labelled by two integers pand qwith p≥q. The second order Casimir and dimension are given by C2(p, q) = p2+q2 2+ 2p+q(2.35) and d(p, q) = 1 6(p+q+ 3)(p−q+ 1)(p+ 2)(q+ 1) (2.36) respectively. Now suppose we have a particle on S4in the representation Iof SU(2) in the background of a homogeneous instanton. Demanding that an SO(5) irreducible representation contains the Iof SU(2) implies [12] p−q= 2I , (2.37) and so C2(q+ 2I, q) = q2+q(2I+ 3) + 2I2+ 4I . (2.38) The Ricci scalar for the unit four-sphere is R= 12 so the eigenvalues (2.15) of (i\ D)2for a Fermion in the representation Jof the gauge group are thus E=q2+q(2I+ 3) + 2I2+ 4I−2J(J+ 1) + 3 2.(2.39) (The factor of two in front of the gauge Casimir J(J+ 1) here is due to the fact that the Dirac operator is non-chiral, the holonomy group is SU(2) ×SU(2), and both chiralities couple to the gauge group in the same way.) The total isospin Iis a combination of the gauge isospin Jand the intrinsic spin of the Fermion, I=J±1/2, so the energy levels are labelled by the integer qand E+(q) = q2+q(2I+ 3) + 2(2I+ 1) E−(q) = q2+q(2I+ 3) (2.40) both with degeneracies d(2I+q,q) = 1 6(2q+ 2I+ 3)(2I+ 1)(q+ 2I+ 2)(q+ 1) .(2.41) – 8 – JHEP05(2003)018 Then the subset eαare orthonormal 1-forms for a G-invariant metric on G/H and the remaining 1-forms eican be expanded on G/H as ei= Πiαeα. The torsion free H-valued2connection ωαβis then defined by deα+ωαβ∧eβ= 0 (A.5) and evaluates to ωαβ=µ1 2fαβγ +fαβiΠiγ¶eγ.(A.6) The curvature 2-forms can then be calculated from Rαβ=dωαβ+ωαγ∧ωγβ(A.7) resulting in Rαβ=1 4¡2fαβifiγδ +fαβ²f²γδ −fαγ²f²βδ¢eγ∧eδ.(A.8) On a symmetric space these reduce to the simpler form Rαβ=1 2¡fαβifiγδ¢eγ∧eδ,(A.9) so the Riemann tensor has components Rαβγδ =fαβifiγδ .(A.10) On a non-symmetric space there is a second, very useful, connection that comes from introducing a torsion tensor which is identified with the non-symmetric structure constants: Tαβγ =fαβγ (A.11) giving torsion 2-forms Tα=1 2fαβγeβ∧eγ.(A.12) Then the connection with torsion is defined via deα+ωαβ∧eβ=Tα(A.13) which leads to ωαβ=1 2fαβiΠiγeγ.(A.14) The resulting curvature 2-forms are Rαβ=1 2fαβifiγδeγ∧eδ,(A.15) giving curvature tensor Rαβγδ =fαβifiγδ .(A.16) 2For notational simplicity we do not distinguish between the group and the algebra here. – 15 – JHEP05(2003)018 The Ricci scalar for the connection with torsion is then easily evaluated as R=Rαβαβ =fαβifiαβ =fABifiAB −fjkifijk ,(A.17) which can be determined using the appropriate quadratic Casimirs of H. A particular instance of this is when His trivial so G/H ∼ =G. Then ωαβ= 0 and Rαβ= 0, all co-variant derivatives are trivial and Tαis called the parallelising torsion for G. On a symmetric space, of course, (A.8) and (A.15) are identical because fαβγ = 0. In fact it is not difficult to show, using (A.14), (A.15) and the Jacobi identity, that Rαβγδ in (A.16) is co-variantly constant, ∇²Rαβγδ = 0 .(A.18) On a generic d-dimensional manifold the curvature 2-forms (A.7) are SO(d) Lie algebra valued 2-forms, but on G/H both (A.8) and (A.15) are H valued 2-forms, where H⊆SO(d). This means that we can take linear combinations of (A.15) that lie in H without losing any information. For example, if His semi-simple, taking the combination fiαβRαβ =1 2³C2(G, adj)−C2(H, adj)´fiγδeγ∧eδ(A.19) suggests defining Fi:= 1 2fiγδeγ∧eδ(A.20) and then Fiare H-valued 2-forms which are equivalent to (A.15) (this formula is easily adapted to the case where Hcontains U(1) factors). B. Spectrum of the Dirac operator on CP 2 The calculation of the full spectrum of the Dirac operator on CP 2proceeds as follows (the spectrum on CP n, with nodd and no background gauge field, has been considered in [17]). For SU(3) the second order Casimir and dimension are C2(p, ¯p) = 1 3³p(p+ 3) + ¯p(¯p+ 3) + p¯p´(B.1) and d(p, ¯p) = 1 2(p+ ¯p+ 2)(p+ 1)(¯p+ 1) (B.2) respectively. With p=r+sand ¯p= ¯r+ ¯s, as in the text, the constraints read (s−¯s)−2(r−¯r) = Yand s+ ¯s 2=I , (B.3) where Yis even (odd) for Iintegral (half-integral). Now the spectrum depends on whether |Y| ≥ 2Ior |Y| ≤ 2I: – 16 – JHEP05(2003)018 •If Y≥2Ithen ¯r≥r: in this case let n= ¯s, so n= 0,...,2I,r=kand ¯r= k+n−I+Y 2, for ka non-negative integer. If Y≤ −2Ithen r≥¯r: in this case let n=s, so n= 0,...,2I,r=k+n−I−Y 2and ¯r=k, for ka non-negative integer. In either case C2(p, q) = kµk+n+ 2 + I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y| 2+Y2 12 +I(I+ 1) . (B.4) For C2(H, ti) in (2.15) take the U(1) background to have fixed charge Mand the SU(2) background to have isospin J, so C2(H, ti) = M2 12 +J(J+ 1) ,(B.5) (the 1 12 here is because the U(1) gauge field is a multiple of 1 2√3to conform with the normalisation of t8in appendix C). Finally the Ricci scalar for CP 2can be evaluated from (A.17) and the structure constants in appendix C to be R= 6 so, putting all this together, the eigenvalues of (2.15) are E(k, n) = kµk+n+ 2 + I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+ +(|Y|+ 3)2 12 −M2 12 +I(I+ 1) −J(J+ 1) ,(B.6) while the degeneracies are d(k, n) = 1 2µ2k+n+I+ 2 + |Y| 2¶µk+ 2n−I+ 1 + |Y| 2¶(k−n+ 2I+ 1) , (B.7) with n= 0,...,2Iand k≥0 an integer. It is important to understand how the gauge charges Mand Jare related to the total charges Yand I(which include the spin connection). There are four cases to consider: 1. M=Y±3 and I=J, these are states that couple to the U(1) part of the spin connection and not the SU(2) part (the ±3 relates to the fact that the first Chern class of the tangent bundle for CP 2is 3). The spectrum is E(k, n) = kµk+n+ 2 + I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y|∓Y 2; (B.8) For SU(2) singlets I=J= 0, so n= 0, this spectrum agrees with the results of [19] (in the notation of that reference M= 2m+ 3, so Y/2 = mor m+ 3). 2. M=Yand I=J±1 2, these are states that couple to the SU(2) part of the spin connection and not the U(1) part. The spectrum is E(k, n) = kµk+n+2+I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y| 2+I+ 1 ; (B.9) E(k, n) = kµk+n+2+I+|Y| 2¶+nµn+ 1 −I+|Y| 2¶+|Y| 2−I . (B.10) – 17 – JHEP05(2003)018 For |Y|>2Ionly case 1 above allows for zero-modes (when k=n= 0). For |Y|= 2I there are zero-modes in both cases. •If 0 ≤Y≤2I, let n=¯s−s+Y 2, so n=−I+Y/2,...,I +Y/2. Then: either r=k+|n|and ¯r=k; or r=kand ¯r=k+|n|. If −2I≤Y≤0, let n=s−¯s−Y 2, so n=−I−Y/2,...,I −Y/2. Then: either r=k+|n|and ¯r=k; or r=kand ¯r=k+|n|. In either case: C2(p, q) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|+Y2 12 +I2+ 2I . (B.11) The eigenvalues of (2.15) are therefore E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|+ +Y2−M2 12 +I2+ 2I−J(J+ 1) + 3 4,(B.12) with degeneracies d(k, n)= ½(k+I+ 1)2+|n|(k+I+ 1) −(4n−|Y|)(2n−|Y|) 4¾µk+I+ 1 + |n| 2¶, (B.13) where −I+|Y| 2≤n≤I+|Y| 2and k≥0. (The degeneracy is always an integer because of the restriction that Yis odd when Iis half-integral and even if Iis integral.) Again there are four possibilities: 1. M=Y±3 and I=J, E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n|Y| 2+I∓Y 2; (B.14) 2. M=Yand I=J±1 2, E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|+ 2I+ 1 ; (B.15) E(k, n) = k(k+ 2I+ 2) + n2+|n|(k+I+ 1) −n 2|Y|.(B.16) These eigenvalues are bounded below by zero and, for |Y|<2I, only (B.16) allows for zero-modes (when k=n= 0). When |Y|= 2Ithe spectrum agrees with equations (B.8)-(B.10). C. Index theorem on SU(3)/U(1) ×U(1) In this section we give an explicit evaluation of the index of the Dirac operator on SU(3)/U(1) ×U(1), using standard differential-geometric techniques. Let λA;A= 1,...,8 be the Gell-Mann matrices for SU(3), so [tA, tB] = ifABCtCwith tA=λA 2(C.1) – 18 – JHEP05(2003)018 and f123 = 1 , f453 =−f673 =f471 =−f561 =f462 =f572 =1 2, f458 =f678 =√3 2. (C.2) In the notation of appendix A, i= 3,8 and α= 1,2,4,5,6,7, when t3=1 2  100 0−1 0 000  and t8=1 2√3  1 0 0 0 1 0 0 0 −2  (C.3) are chosen as the U(1) ×U(1) generators. This space is not symmetric, because some of the fαβγ 6= 0. The curvature 2-forms, for the spin connection with torsion described in appendix A, are R12 =1 2¡2 e1∧e2+ e4∧e5−e6∧e7¢ R45 =1 2¡e1∧e2+ 2 e4∧e5+ e6∧e7¢ R67 =1 2¡−e1∧e2+ e4∧e5+ 2 e6∧e7¢. These are not independent, since R45 =R12 +R67, they are associated with two U(1) field strengths: F3t3=1 2fαβ3³eα∧eβ´t3=1 4¡2 e1∧e2+ e4∧e5−e6∧e7¢  1 0 0 0−1 0 0 0 0   F8t8=1 2fαβ8³eα∧eβ´t8=1 4¡e4∧e5+ e6∧e7¢  1 0 0 0 1 0 0 0 −2  . We extract U(1) singlets by projecting out the top left-hand components F(3) =1 4¡2 e1∧e2+ e4∧e5−e6∧e7¢ F(8) =1 4¡e4∧e5+ e6∧e7¢. A monopole field with charges (M, N) is a linear combination of these F=MF(3) +NF(8) (C.4) from which F∧F∧F=3 16M¡N2−M2¢e124567 (C.5) (we use the shorthand e124567 = e1∧e2∧e4∧e5∧e6∧e7). The index of the Dirac operator is [20] ν=1 (2π)3Z½1 6F∧F∧F+1 48F∧tr(R∧R)¾.(C.6) – 19 – JHEP05(2003)018 Explicit calculation reveals that F∧tr(R∧R) = 0, so ν=1 256π3M¡N2−M2¢V, where V=Re124567 is the volume of SU(3)/U(1) ×U(1). The normalisation can be fixed by using the fact SU(3)/U(1) ×U(1) has Euler characteristic χ= 6, so χ=1 3! 1 (4π)3Z²α1···α6Rα1α2∧Rα3α4∧Rα5α6= 6 .(C.7) this fixes V= 32π2so ν=1 8M¡N2−M2¢.(C.8) Note that Mand Nmust be either both even or both odd for νto be an integer. References [1] S.M. Girvin, The quantum hall effect: novel excitations and broken symmetries, cond-mat/9907002. [2] B.A. Bernevig, J.H. Brodie, L. Susskind and N. Toumbas, How bob laughlin tamed the giant graviton from taub-nut space,J. High Energy Phys. 02 (2001) 003 [hep-th/0010105]. [3] S.-C. Zhang and J.-p. Hu, A four dimensional generalization of the quantum hall effect, Science 294 (2001) 823 [cond-mat/0110572]; J.-p. Hu and S.-C. Zhang, Collective excitations at the boundary of a 4D quantum hall droplet,cond-mat/0112432. [4] F.D.M. Haldane, Fractional quantization of the hall effect: a hierarchy of incompressible quantum fluid states,Phys. Rev. Lett. 51 (1983) 605. [5] Y.-X. Chen, B.-Y. Hou and B.-Y. 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