632 L. PALLA triplet SU(2) singlet of 0 mass fermions and therefore the symmetry group of the (quarks?) as well as 3 SU(3) singlets, SU(2) gauge fields is SU(Z)/2+l)x G\ and in the triplets (leptons?). physically most interesting case (D=6) we The neglected higher mass excitations may can again have 0 mass fermions because CP3 play an important role if we assume that allows a spin structure. strong interactions couple in a way gravitation does,9 as in this case the Planck length becomes References of the order of the size of elementary particles 1. Th. Kaluza: Sitzungberichte der Preuss. Akad. and these geometrical constructions may give Wiss. (1921) 966; O. Klein: Nature 118 (1926) 516; Y. M. Cho and P. G. O. Freund: Phys. a description of the hadronic world. How­ Rev. D12 (1975) 1711; Y. M. Cho and Pong Soo ever in this case we must face the problems Jang: Phys. Rev. D12 (1975) 3789; J. Rayski: arising from the relatively simple spectrum as Acta Phys. Pol. XXVII (1965) 89. well as from the complexity of the resulting 2. Dual Theory, ed. M. Jacob (North Holland, infinite component field theory. Amsterdam, 1974); J. Scherk: Rev. Mod. Phys. 47 (1975) 123. It is obvious that in the D=2 case essentially 3. E. Cremmer and J. Scherk: Nucl. Phys. B103 we used only a topologically non-trivial U(l) (1976) 399; B108 (1976) 409; B118 (1977) 61. gauge field to compactify the extra two dimen­ 4. Z. Horvath, L. Palla, E. Cremmer and J. Scherk: sions on S2. Based on this observation we Nucl. Phys. B127 (1977) 57. determined the minimal gauge groups which 5. J. F. Luciani: Nucl. Phys. B135 (1978) 111. 6. E. Gildener: Phys. Rev. D14 (1976) 1667. allow topologically non-trivial compactifying 7. J. C. Pati and A. Salam: Phys. Rev. D8 (1973) gauge fields on even D dimensional spheres 1240; H. Georgi and L. Glashow: Phys. Rev. with SO(Z>+l) invariance.10 Just like in the Letters 32 (1974) 433; H. Georgi, H. Quinn and D=2 case the "monopole" these gauge fields S. Weinberg: ibid., 33 (1974) 451; M. Fritzsch, also guarantee the exisistence of 0 mass fer­ M. Gell-Mann and P. Minkowski: Phys. Letters mions, but they are now singlets under the 59B (1976) 156. 8. T. T. Wu and C.N. Yang: Phys. Rev. D12 SO(D+l) internal-space-symmetry group. (1975) 3845; Nucl. Phys. B107 (1976) 365. If, however, we compactify the extra even 9. C.J. Isham, A. Salam and J. Strathdee: Phys. D dimensions on CPD/2, then using recent Rev. D3 (1971) 867; D8 (1973) 2600; A. Salam and results from the theory of generalized mono- J. Strathdee: Phys. Letters B67 (1977) 429; Trieste preprint IC/77/153. poles11 it is possible to show that the minimal 10. Z. Horvath and L. Palla: Trieste preprint IC/ compactifying gauge group is U(l). Embedd­ 78/37, to be published in Nucl. Phys. B. ing this U(l) into K raises an interesting pos­ 11. A. Trautman: Int. J. Theor. Phys. 16 (1977) 561. sibility as CP^/2=SU(i)/2+l)/SU(i)/2)xU(l) PROC. 19th INT. CONF. HIGH ENERGY PHYSICS TOKYO, 1978 C 11 World Topology and Gauged Internal Symmetries P. G. O. FREUND Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago 60637 In the theory of gravitation there are two the gravitational metric: topological invariants that are the counter­ parts of the instanton number of Yang-Mills r= \RR*d*x theory.1 Of these I will concentrate on the index r of the space-time manifold defined by with Unified Theories Including Strong Interactions 633 "sufficiently nontrivial topology" above can be made precise: the second Stiefel-Whitney It is related to the first Pontryagin number characteristic class w2 should not vanish. Pi by It contributes to the Adler- Hawking5 has proposed the Taub-NUT Bell-Jackiw axial anomaly solution as a replacement for P2(C). Unfor­ d%5) = ~(Ay4)i?i?* tunately, r(Taub-NUT)=^0, so that this does not work.6 +usual Yang-Mills FF* terms At this point a more general aspect of this 2 as noted by Delbourgo and Salam, and by whole problem surfaces. In the transition Eguchi and myself.3 Here, and below, N D from classical to quantum gravity a functional stands for the number of complex Dirac fields integration over gravitation fields, i.e., over and N =2N is the number of corresponding M D world manifolds is involved. The world con­ real Majorana fields. Thus the integral tains Fermi-matter fields: quarks and leptons. version of the axial anomaly gives the helicity Yet, not every manifold can support spinors 6 chanee JO in terms of the index (i.e., Fermi fields). To obtain a consistent quantum theory two alternatives can be con­ templated :• We proposed3 the complex projective plane (A) restrict the functional integral to mani­ P2(C) as a gravitational instanton for this folds that have a spin structure: spin mani­ anomaly (in terms of three complex homo­ folds, geneous coordinates zl9 z2, z3, P2(C) is defined or; by 2]|^|2=tf2 and by identifying points that (B) require all matter fields to appear in ia differ by an overall phase: (zl9 z2, z3) = (e zl9 suitable multiplets of a gauged symmetry that ia ia e zl9 e zs)). Fubini found a Kahler metric permits the definition of a generalized spin for P2(C). The corresponding 4-dimensional structure on all 4-dimensional Riemann-mani- real (+ + + +) metric is a solution of Einstein's folds. equations with cosmological term (A=3/2a2). Next, I shall explore these two alternatives, 7 The index of P2(C) is T(P2(C))= 1. This leads closely following recent work of Back, Forger 5 to the paradoxical value dQ =—NM/$. The and myself. reason for this is that while fermion triangle First, let me consider alternative (A). It is (and loop) diagrams are responsible for the automatically realized in all forms of super- anomaly, fermions are Lorentz spinors (because gravity theory, since the OSp(A^/4) and SU(N/ of the spin-statistics connection) and spinors 2,2) supergroups contain not the Lorentz group cannot "live" on P2(C). but its covering spin group which thus acts It may be worthwhile to recall4 how spinors in the fibres of the supergravity bundle. In can be denied accomodation on some Riemann such theories a gravitational instanton is to be manifold M. Consider the bundle of oriented a 4-dimensional compact spin-manifold with orthonormal frames of M and a closed curve positive metric forms and first Pontryagin situated completely in one of its fibres, say, number p±^0. To be of special physical in­ over the point P eM. Moreover, let this terest it should also be an Einstein manifold curve correspond to a vierbein rotation by (i.e., Rft^Ag^ for some A). P2(C), not being 2K SO that spinors at P change sign. Then it a spin-manifold, will not do. It is incumbent cannot be contracted to a point while staying on us to find new instantons. The trick is to within the fiber. But, for a sufficiently non- look for algebraic submanifolds of P3(C) other trivial topology of the manifold M, this curve than P2(C). Let z0, zl9 z2, z3 be the homo­ when deformed through the whole bundle geneous complex coordinates of P3(C) and may be shrunk to a point leading to no spinor Fm(Zi) a homogeneous polynomial in zt of sign change. Thus, two curves obtainable degree m. If grad Fm ^ 0 for z =£ 0 then Fm(z)= from each other by continuous deformation 0 defines a Kahler submanifold of P3(C). For give conflicting instructions as to spinor signs. even m=2n the second Stiefel-Whitney class of Spinors cannot sort their signs out and find the V2n vanishes so that it is a spin manifold. manifold "deadly". Fortunately, what I called Standard topological methods then yield: 634 P. G. O. FREUND (w+1 1 each by a Z2 factor. Either Spin(4) -> Spin(4) x r(K1J=-lfr .y" ^ SU(2) homomorphism induces an SO(4)-> Thus, r(F2)=0 while T(V,)=-16. Notice that Spin(4) homomorphism that can be used to s an even now J£?5 i integer and the P2(C) upgrade the SO(4) bundle into a Spin(4)- "paradox" disappears. According to Yau,8 bundle. With a gauged internal SU(2)-sym- Vt admits a metric that is a solution of metry one can thus define spinors on any Einstein's equations without cosmological term, 4-dimensional Riemann manifold. Physically and it is possible that the V2N manifolds are one may attempt to identify this with the Einstein for all n. Moreover, if M is any 4- "weak isospin" factor of the unified weak- dimensional spin manifold without boundary, electromagnetic gauge group. But, there is then there exists a 5-dimensional spin manifold one more consistency requirement. Any re­ whose boundary is the union of M and of presentation A of Spin(4) on a vector space r=—r(M)j\6 copies of F4:Mis spin-cobordant V can be pulled back to a representation of 9 to rV±. In this sense F4's are the fundamental Spin(4) x SU(2)=SU(2) x SU(2) x SU(2). But, gravitational instantons. we must require the element (—1 — 1 — 1) of Supergravity, which implements alternative SU(2)xSU(2)xSU(2) to map to the identity (A) is certainly not experimentally established.