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. within A. Label A by three spins (ju j2, 73). Therefore, it is in order to look for alternative This requirement then means e^(2ii+2i2+2:7V = l J resolutions of the "P2(C)-paradox'\ P2(C) so that ji +j2 rjs=integer. Thus, the fer­ itself while not a spin-manifold, still is a com­ mions, for which 71+72=half odd integer be­ plex manifold and as such can be given a cause of the spin-statistics connection, must generalized Spin0 structure.10 This is done11 have a half odd integer weak isospin. Similar­ by having the matter fields carry a conserved ly, bosons must have integer weak isospin. charge, the corresponding current serving as This is a weak isospin-statistics connection. the source of an abelian gauge field. The Unfortunately, as it stands it would rule out extra gauge phase freedom then allows the both the right-handed leptons and Higgs fields consistent definition of spinors on P2(C\ of the Weinberg-Salam model. But, instead provided one has a "charge-statistics" con­ of the minimal SU(2) we can choose an nection: fermions (i.e., spinors) carry odd internal symmetry like SU(2)xSU(2)x(/' and (integer) values while bosons (i.e., tensors) reasonable "internal-spin"-statistics connec­ carry even values of the charge. The gauge tions can be produced. field then also contributes to the anomaly and Notice a remarkable feature of all this. together with the gravitational contribution The natural realization for alternative (A) is 5 gives JQ =integer. There may exist a larger in the framework of supergravity theories. (non-abelian) gauge group G such that coupl­ These theoreies determine both the gauged ing all matter fields to gauge fields of G, all internal symmetry (0(N) or U(N), N<8) and 4-dimensional Riemann manifolds (not only the multiplet spectrum. The seemingly unrelat­ those that are also complex manifolds) can ed alternative (B) again provides informa­ be given a generalized spin structure, again tion on the gauged internal symmetry and on provided a certain "internal spin"-statisties the multiplet spectrum this time in the form connection is enforced. In 4-dimensions such of an "internal spin"-statistics connection. a group G exists and must be at least SU(2) Before concluding I wish to briefly mention or more realistically SU(2)xSU(2) or SU(2)x that one way to experimentally test super- SU(2)xU(l). gravity—and the corresponding solution (A) To see this, note that Spin(4)=SU(2) X SU(2) of the spinor puzzle—is to test its prediction 12 can be homomorphically mapped onto Spin(4) of SU(3)color sextet quarks. In this vein, it XSU(2) by the identity map to Spin(4) and has been remarked recently,13 that the Y'—Y projection onto one of the two SU(2) factors mass splitting and the Y leptonic decay rate as the map to SU(2). Define Spin(4)—Spin(4) could be much more readily accounted for

XSU(2)/Z2 by identifying (g, h)~(-g, -h) if the Y's constituent 6-quark were a color for g e Spin(4), /zeSU(2). Now, SO(4) and sextet.14 Admittedly, the absence of stable Spni(4) differ from Spin(4) and Spin(4) x SU(2) or very long-lived hadrons in the 5-6 GeV/c2 Unified Theories Including Strong Interactions 635 mass range15 makes this interpretation less References probable, although a final verdict should come 1. For a general discussion of anomalies and instan­ from the measurement of the hadronic decay tons in Yang-Mills theories see the beautiful rate of the /"-meson. Of course there could review paper by S. Coleman, Harvard preprint HUTP/A004. exist color sextet quarks of higher mass. If 2. R. Delbourgo and A. Salam: Phys. Letters 16 very long-lived, these would allow for most 40B (1972) 381. exciting practical applications. 3. T. Eguchi and P. G. O. Freund: Phys. Rev. To conclude, let me venture a speculation. Letters 37 (1976) 1251. The new gravitational instantons we found 4. R. Geroch: J. Math. Phys. 9 (1968) 1739. 5. S. Hawking: Phys. Letters 60A (1977) 84. are algebraic manifolds. Remarkably also the 6. T. Eguchi, P. B. Gilkey and A. J. Hanson: Phys. instantons of the SU(2) Yang-Mills theory are Rev. D17 (1978) 423; H. Romer and B. Schroer: algebraic as noted by Atiyah and Ward.17 Phys. Letters 71B (1977) 182; see also, M.J. Maybe in the "euclidean" sector all (finite Duff: Queen Mary College preprints (1976), action) solutions of a gauge theory—whether (1977). 7. A. Back, P. G. O. Freund and M. Forger: Phys. Yang-Mills or gravity—are algebraic just like Letters 77B (1978) 181. the self-dual or anti-self-dual instantons. What 8. S. T. Yau: U.CL.A. preprint. would this mean? Think of a point particle 9. J. Milnor: L'Enseignement Math. 2-Ser. 9 in a central potential. There exist potentials (1963) 198. for which all classical trajectories are algebraic 10. M. Atiyah, R. Bott and A. Shapiro: Topology curves: the Newton-Coulomb and the harmonic 3 (Suppl. 1) (1964) 3. 11. S. Hawking and C. N. Pope: Phys. Letters 73B oscillator potentials. But, for these potentials (1978) 42; G. W. Gibbons and C.N. Pope: both the classical and the quantum-mechanical D.A.M.T.P. preprint, to be published. problems are exactly analytically soluble. 12. A. Salam and J. Strathdee: Phys. Rev. Dll Maybe "euclidean" Yang-Mills and gravity (1975) 1521; M. Gell-Mann and Y. Ne'eman: unpublished. (and supergravity?) are exactly soluble both 13. Y. J. Ng and S.-H. H. Tye: Phys. Rev. Letters classically and at the quantum level. Exactly 41 (1978) 6; P. G. O. Freund and C. T. Hill: soluble 2-dimensional field theories have been Chicago preprint EFI 78/21; H. Fritzsch: Wup- known for a long time. But, here I am sug­ pertal preprint WU-B78-19. gesting the prospect of exactly soluble 4- 14. Color sextet quarks have also been considered dimensional gauge theories. The crucial in­ from different points of view by G. Karl: Phys. Rev. D14 (1976) 2374; by F. Wilczek and A. gredient, the algebraic nature of the classical Zee: ibid., D16 (1977) 860. solutions, would provide a peculiar vindica­ 15. D. Cutts et al.\ Phys. Rev. Letters 41 (1978) tion of Kepler's approach, that emphasized 363; L. Lederman: report at this Conference. precisely the algebraic aspect for planetary 16. P. G. O. Freund and C. T. Hill: Chicago pre­ motion. For gauge theories a global Keplerian print EFI 78/23; R. Cahn: Phys. Rev. Letters 40 (1978) 80. approach may yet be as profitable as the 17. M. Atiyah and R. S. Ward: Comm. Math. prevalent local dynamical Newtonian ap­ Phys. 55 (1977) 117. proach. 18. In a different context A. M. Polyakov has also speculated on the solubility of gauge theories. PI a: Hadron-Hadron Reactions, Low Multiplicity

Chairman: E. L. GOLDWASSER Speaker: V. A. TSAREV Scientific Secretaries: F. TAKASAKI K. OGAWA

Plb: Hadron-Hadron Reactions, High Multiplicity

Chairman: L. VAN HOVE Speaker: R. E. DIEBOLD Scientific Secretaries: T. HIROSE H. KICHIMI

(Monday, August 28, 1978; 9: 00-11: 10)