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International Journal of Pure and Applied Mathematics ————————————————————————– Volume 52 No International Journal of Pure and Applied Mathematics ————————————————————————– Volume 52 No. 5 2009, 759-766 ON ATIYAH-SINGER INDEX OF DIRAC OPERATOR ON SIX DIMENSIONAL SPHERE WITH A SO(6)-GAUGE THEORY H¨unkar Kayhan1, Cenap Ozel¨ 2 § 1Department of Physics Kamp¨us¨uBolu Abant Izzet˙ Baysal University G¨olk¨oy, 14280, TURKEY e-mail: hunkar [email protected] 2Department of Mathematics Kamp¨us¨uBolu Abant Izzet˙ Baysal University G¨olk¨oy, 14280, TURKEY e-mail: [email protected] Abstract: We consider the Dirac operator on a six dimensional sphere with gravitational and Yang-Mills gauge fields. Here we calculate the Atiyah-Singer index of Dirac operator on a six dimensional sphere with a SO(6)-gauge theory. Moreover, we take the vector field as to be Hedgehog-connection which gives an appropriate result for the field strength and therefore the topological charge. AMS Subject Classification: 53C07, 53Z05, 57R56, 57R20, 58J20 Key Words: Atiyah-Singer index, Dirac operator, anomaly, topological charge, Yang-Mills fields, gravitational fields, Hedgehog connection 1. Introduction This part of the paper gives a very basic idea for anomaly and topology which heavily borrows from the textbook of R.A. Bertlmann [2]. Dirac operator was considered on different manifolds. Roberto Camporesi and Atsushi Higuchi [3] constructed and solved the Dirac operator on a n-dimensional sphere Sn Received: May 12, 2009 c 2009 Academic Publications §Correspondence author 760 H. Kayhan, C. Ozel¨ and hyperbolic spaces Hn by the work “On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces”. For other manifolds, some other works have been done by others [6], [4], [8], [1], [5], [9]. In these works, the Dirac operator was considered mostly on the spheres to reveal some important properties of the operator and hence the global properties of the considered manifolds. The basic of Quantum Field Theory (OFT)-gauge theory is the principle of gauge symmetry which is formulated in Yang-Mills gauge theory. There an anomaly which is the violation of a classical conserved current signal the break- down of the gauge symmetry and, in consequence, the ruin of the consistency of the axial or the chiral anomaly corresponds to an axial or a chiral fermion current. The singlet anomaly is determined by an index theorem. The reason is that the anomaly can be expressed by a sum of eigenfunctions of the Dirac operator, which are only the zero-modes of a given chirality. The anomalies in gravitation are the analogues of the anomaly in Yang- Mills gauge theory. Particularly, the pure Einstein anomaly which represents the non-conservation of the energy-momentum tensor, is the precise analogue to the gauge anomaly, which expresses the non-conservation of the gauge current. Anomaly has a natural explanation; it occurs as an obstruction in certain nontrivial bundles and it is determined completely by a topological quantity called index. The singlet anomaly is related to the index of a differential oper- ator. This triggered a new era of topological investigations for the anomaly. In the eighties the field of anomalies began to boom. Modern mathematical tech- niques such as topology and differential geometry played an important role and revealed important properties of anomalies. One advantage is that anomalies can be written in terms of differential forms. Another boom of such topological approaches to anomalies began with the works of Atiyah and Singer and Al- varez, Singer and Zumino. They say that the non-Abelian anomaly is related to index theorem. In fact, the index of the operator gives the topological charge which is determined by the only topology of the bundle. Let φn(x) be the eigenfunction of the Dirac operator, Dφn(x)= λnφn(x) 0 0† and φn(x) is the zero-mode eigenfunction of the operator (λn = 0) and φn± (x)= P±φn(x) where the projection operator P±. Consider the following expression dx φ0† (x)φ0 (x) − φ0† (x)φ0 (x)= n − n . n+ n+ n− n− + − n This difference is called the index of the operator, index D+ = n+ − n−, ON ATIYAH-SINGER INDEX OF DIRAC OPERATOR ON... 761 where D+ = DP+. The difference in the chirality zero-modes represents the index of the Weyl operator (D+). 2. Dirac Operator We consider the Dirac operator (D) with gravitational and Yang-Mills gauge fields. The nabla operator ∇ is defined as ∇ = d + A + ω , (1) where d is the exterior derivative, A is the electromagnetic connection and ω is the spin connection i d = ∂idx , (2) i A = Aidx , (3) i jk i ω = ωjkΓ /2dx , (4) ab 1 where Ai = Ai Γab/2 and Γab = 4 [Γa, Γb], Γab = −Γba. The gauge group SO(6) is generated by these gamma matrices Γa (a = 1, 2,... 6). The dimensions of these matrices are 26/2 = 8 with complex coeffi- cients. So, the Lie algebra is embedded in su(8) algebra. These matrices satisfy Clifford algebra relations as {Γa, Γb} = 2δab , where δab = (1, 1, 1, 1, 1, 1) and the commutation relation 1 Γ = {Γ , Γ }. ab 2 a b The products of the elements are 1 Γ = sign(σ)Γ .Γ . Γ . ab...f s! a b f σ We also define the chirality operator γ7 = −iΓ1 . Γ6. TheΓab s satisfy the following commutation and anti-commutation relations; {Γab, Γcd} = 2(δbcδad − δbdδac +Γabcd) (5) and [Γab, Γcd] = 2(δbcΓad − δbdΓac + δadΓbc) . (6) The operator γ7 anticommutes with Γa and commutes with Γab. Our base space is a six dimensional sphere S6. The sphere is covered by the 762 H. Kayhan, C. Ozel¨ two patches that are diffeomorphic to R6. We take into account one of these patches. Take the space coordinates by xi. We take the standard metric on the sphere δ ds2 = ij dxidxj . (7) x2 1+ 2 4R0 We use sechsbein ei to construct the Clifford algebra dxi ei = . x2 1+ 2 4R0 They form an orthonormal frame on which the metric takes the form of ds2 = i j δije e . Then the Dirac operator becomes i D = e (∇i) , (8) where ei, (i, 1,... 6) are the Clifford elements of the algebra. They satisfy the Clifford algebra relations. {ei, ej } = 2δij , (9) where δij is the Krocneker delta. Then, Dirac equation for a massless particle becomes Dψ = 0 , (10) where ψ is the spinor. In the Clifford algebra language the spinor is an element of the spin space S, ψ ∈ S which is the left ideal of Cl6, S = Cl6f and f is the primitive idempotent. In the S space, ψ can be expanded in the basis of this space si as ψ = j j ψjs (j, 1,... 16). There are four primitive idempotent in this algebra 1 1 f = (1 ± e1) (1 ± e12345) . (11) 2 2 There are basis of the S space corresponding to these idempotents and the representations in each idempotent are isomorphic to each other. By choosing the idempotent as 1 1 f = (1 + e1) (1 + e12345) . 2 2 The corresponding basis in S-space are 1 s1 = (e234 − e1234 − e5 + e15) , 4 1 s2 = (e34 + e134 − e25 − e125) , 4 ON ATIYAH-SINGER INDEX OF DIRAC OPERATOR ON... 763 1 s3 = (e24 + e124 + e35 + e135) , 4 1 s4 = (e4 − e14 + e235 − e1235) , 4 1 s5 = (e23 + e123 − e45 − e145) , 4 1 s6 = (e3 − e13 − e245 + e1245) , 4 1 s7 = (e2 − e12 + e345 − e1345) , 4 1 s8 = f = (1 + e1 + e2345 + e12345) , (12) 4 1 s9 = (e2346 + e12346 − e56 − e155) , 4 1 s10 = (e346 − e1346 − e256 + e1256) , 4 1 s11 = (e246 − e1246 + e356 − e1356) , 4 1 s12 = (e46 + e146 + e2356 + e12356) , 4 1 s13 = (e236 − e1236 − e456 + e1456) , 4 1 s14 = (e36 + e136 − e2456 − e12456) , 4 1 s15 = (e26 + e126 + e3456 + e13456) , 4 1 s16 = (e6 − e16 + e23456 − e123456) . 4 3. The Index of the Dirac Operator 3.1. Atiyah-Singer Index Theorem Atiyah and Singer have shown that the index is determined by topology-called topological index. It is a topological invariant which depends only on the fibre bundles considered and on the manifold. Moreover, their topological invariant can be expressed as an integral over certain classes. We use Atiyah-Singer Index Theorem to calculate the index of the Dirac 764 H. Kayhan, C. Ozel¨ operator. indexD+ = ch(F )Aˆ(M) , (13) M2n where ch(F ) is the Chern character of electromagnetic field strength F and Aˆ(M) is the Dirac genus of the 2n-dimensional manifold M: 1 i ch(F )= ( )n TrF n , (14) n! 2π M2n x /2 Aˆ(M)= a , (15) sinh x /2 a a where xas are the eigenvalues of the Ricci tensor Rab 0 x1 .. −x1 ... R . ... ab = , (16) 2π . ... . ... 0 xn . ... −xn . 1 1 Aˆ(M)=1+ TrR2 ... (17) (4π)2 12 2n 2n 2n For the sphere S with radius R0, TrR = 0 and Aˆ(S ) = 1. So the Atiyah-Singer Index Theorem for manifolds with dimensions 2, 4, 6, 8 are given as follows: i dim = 2, indexD+ = TrF , (18) 2π M2 1 1 2 r 2 dim = 4, indexD+ = 2 − TrF + TrR , (19) (2π) M4 2 48 1 i 3 i 2 dim = 6, indexD+ = 3 − TrF + TrF TrR , (20) (2π) M6 6 48 1 1 4 1 2 2 dim = 8, indexD+ = 4 − TrF − TrF TrR (2π) M8 24 96 r r + (TrR2)2 + TrR4 , (21) 4608 5760 where r denotes the dimension of the gauge group representation. So the Atiyah-Singer Index Theorem gives an interesting connection be- tween the local information about the solutions of a differential operator and its global properties on a manifold. Moreover, the integral in the theorem is ON ATIYAH-SINGER INDEX OF DIRAC OPERATOR ON..
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