Abstract
We prove the existence of non-self-dual Yang-Mills connections onSU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal±1. We study connections equivariant under anSU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dualSU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.
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[Ad] Adams, R.A.: Sobolev spaces. New York: Academic Press 1975
[ADHM] Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., Manin, Y.I.: Construction of instantons. Phys. Lett. A65, 185 (1978)
[AJ] Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97 (1978)
[ASSS] Avron, J.E., Sadun, L., Segert, J., Simon, B.: Chern numbers, quaternions, and Berry's phases in fermi systems. Commun. Math. Phys.124, 595 (1989)
[Au] Aubin, T.: Nonlinear analysis on manifolds. Monge-Amperè equations. Berlin, Heidelberg, New York: Springer 1982
[Ber] Berger, M.S.: Nonlinearity and functional analysis. New York: Academic Press 1977
[BoMo] Bor, G., Montgomery, R.:SO(3) invariant Yang-Mills fields which are not self-dual. Proceedings of the MSI Workshop on Hamiltonian Systems, Transformation Groups, and Spectral Transform Methods, held in Montreal, Canada, Oct. 1989
[BL] Bourguignon, J.P., Lawson, H.B.: Stability and isolation phenomena for Yang-Mills equations. Commun. Math. Phys.79, 189 (1982)
[BLS] Bourguignon, J.P., Lawson, H.B., Simons, J.: Stability and gap phenomena for Yang-Mills fields. Proc. Natl. Acad. Sci. USA76, 1550 (1979)
[BPST] Belavin, A.A., Polyakov, A.M., Schwartz, A.S., Tyupkin, Yu.: Pseudo-particle solutions of the Yang-Mills equations. Phys. Lett. B59, 85 (1975)
[CDD] Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, manifolds, and physics. Amsterdam: North-Holland 1982
[FHP1] Forgacs, P., Horvath, Z., Palla, L.: An exact fractionally charged self-dual solution. Phys. Rev. Lett.46 392 (1981)
[FHP2] Forgacs P., Horvath, Z., Palla, L.: One can have noninteger topological charge. Z. Phys. C—Particles and Fields12, 359–360 (1982)
[FU] Freed, D.S., Uhlenbeck, K.K.: Instantons and four-manifolds. Berlin, Heidelberg, New York: Springer 1984
[I] Itoh, M.: Invariant connections and Yang-Mills solutions. Trans. Am. Math. Soc.267, 229 (1981)
[JT] Jaffe, A., Taubes, C.: Vortices and monopoles. Boston: Birkhäuser 1980
[LU] Ladyzhenskaya, O., Ural'tseva, N.: Linear and quasilinear elliptic partial differential equations. New York: Academic Press 1968
[Ma1] Manin, Yu.: New exact solutions and cohomology analysis of ordinary and supersymmetric Yang-Mills equations. Proc. Steklov Inst. of Math.165, 107 (1984)
[Ma2] Manin, Yu.: Gauge field theory and complex geometry. Berlin, Heidelberg, New York: Springer 1988
[P1] Parker, T.: Unstable Yang-Mills fields. Preprint 1989
[P2] Parker, T.: Non-minimal Yang-Mills fields and dynamics. Invent. Math. (in press)
[Pal] Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys.69, 19 (1979)
[RS] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. I, II. New York: Academic Press 1980
[SaU] Sacks, J., Uhlenbeck, K.: On the existence of minimal immersions of 2-spheres. Ann. Math.113(2), 1–24 (1982)
[Sed] Sedlacek, S.: A direct method for minimizing the Yang-Mills functional over 4-manifolds. Commun. Math. Phys.86, 515–527 (1982)
[SS1] Sadun, L., Segert, J.: Chern numbers for fermionic quadrupole systems. J. Phys. A22, L111 (1989)
[SS2] Sadun, L., Segert, J.: Non-self-dual Yang-Mills connections with nonzero Chern number. Bull. Am. Math. Soc.24, 163–170 (1991)
[SS3] Sadun, L., Segert, J.: Stationary points of the Yang-Mills action. Commun. Pure Appl. Math. (in press)
[SiSi1] Sibner, L.M., Sibner, R.J.: Singular Sobolev connections with holonomy. Bull. Am. Math. Soc.19, 471–473 (1988)
[SiSi2] Sibner, L.M., Sibner, R.J.: Classification of singular Sobolev connections by their holonomy. Commun. Math. Phys. (to appear)
[SSU] Sibner, L.M., Sibner, R.J., Uhlenbeck, K.: Solutions to Yang-Mills equations which are not self-dual. Proc. Natl. Acad. Sci USA86, 860–863 (1989)
[T1] Taubes, C.H.: Stability in Yang-Mills theories. Commun. Math. Phys.91, 235 (1983)
[T2] Taubes, C.H.: On the equivalence of the first and second Order equations for gauge theories. Commun. Math. Phys.75, 207 (1980)
[Uh1] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11–29 (1982)
[Uh2] Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31–42 (1982)
[Uh3] Uhlenbeck, K.: Variational problems for gauge fields. In Seminar on Differential Geometry. Yau, S.-T. (ed.). Princeton: Princeton University Press 1982
[Ur] Urakawa, H.: Equivariant theory of Yang-Mills connections over Riemannian manifolds of cohomogeneity one. Indiana Univ. Math. J.37, 753 (1988)
[Bo] Bor, G.: Yang-Mills fields which are not self-dual. Commun. Math. Phys.145, 393–410 (1992)
[P3] Parker, T.H.: A Morse Theory for Equivariant Yang-Mills. Duke Math. J. (in press)
[W] Wang, H.-Y.: The existence of non-minimal solutions to the Yang-Mills equation with groupSU(2) onS 2×S 2 andS 1×S 3. J. Diff. Geom.34, 701–767 (1991)
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Communicated by T. Spencer
Research partially supported by NSF Grant DMS-8806731
Most of this research was done while the author was a Bantrell Fellow at the California Institute of Technology, and was partially supported by NSF Grant DMS-8801918
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Sadun, L., Segert, J. Non-self-dual Yang-Mills connections with quadrupole symmetry. Commun.Math. Phys. 145, 363–391 (1992). https://doi.org/10.1007/BF02099143
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DOI: https://doi.org/10.1007/BF02099143