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Configurations de Particules et Espaces de Modules

Published online by Cambridge University Press:  20 November 2018

J. C. Hurtubise
J. C. Hurtubise*
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 rue Sherbrooke O. Montréal, Québec H3A 2K6 e-mail:hurtubis@gauss.math.mcgill.ca
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Résumé

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Cet article de survol est le résumé de la conférence Coxeter-James de l'auteur, prononcée à la réunion d'hiver 1993 de la Société Mathématique du Canada.

La théorie de Morse décrit les liens entre la topologie d'une variété et la topologie des points critiques d'une fonction sur cette variété. La fonctionnelle d'énergie pour les applications d'une surface dans une variété, dont les points critiques seront des applications harmoniques et parfois holomorphes, et la fonctionnelle de Yang-Mills pour des connections sur une variété de dimension quatre sont deux cas en dimension infinie pour lesquels la théorie de Morse ne tient pas. Néanmois, dans les deux cas, on peut récupérer une quantité étonnante d'information, pourvu qu'on stabilise par rapport à un degré ou une charge qui sont des données du problème. Les preuves recyclent des résultats de la théorie de l'homotopie des années '70, et les combinent à des idées de géométrie complexe pour donner de jolis modèles des espaces en cause en termes de "particules". Nous espérons donner un survol général et accessible des idées utilisées.

Abstract

Abstract

This survey is the written summary of the author's Coxeter-James lecture, delivered at the 1993 Winter Meeting of the Canadian Mathematical Society.

Morse theory relates the topology of the critical set of a function on a manifold to the topology of the whole manifold. The energy functional for maps of surfaces into a manifold, whose critical points are harmonic and occasionally holomorphic maps, and the Yang-Mills functional for connections on a four-manifold are two infinite dimensional cases where Morse theory fails. Nevertheles, in both cases a surprising amount can be said, providing one stabilises with respect to a natural charge or degree. The proofs borrow from the homotopy theory of the 1970's and combine it with some input from complex geometry to give some nice "particle" models of the spaces involved. This paper gives a fairly general and, it is hoped, accessible survey of the ideas involved.

Keywords

58D2758E1553C07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 1 , 01 March 1995 , pp. 66 - 79
Copyright
Copyright © Canadian Mathematical Society 1995

References

[A] Atiyah, M. F., Instantons in two and four dimensions, Comm. Math. Phys. 93(1984), 437–451.Google Scholar
[AJ] Atiyah, M. F. and Jones, J. D., Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61(1978), 97–118.Google Scholar
[B] Bott, R., Lectures on Morse Theory, Old and New, Bull. Amer. Math. Soc. (N.S.) 7(1982), 331–358 Google Scholar
[BE] R. Baston, J. and Eastwood, M. G., The Penrose transform, Oxford University Press, 1989.Google Scholar
[BHMM] Boyer, C. P., Hurtubise, J. C., Mann, B. M. and Milgram, R. J., The topology of the space of rational maps into generalised flag manifolds, A paraître aux Acta Math., p. 35. Google Scholar
[BHMM2] Boyer, C. P., The topology of instanton moduli spaces. I: The Atiyah-J ones Conjecture, Ann. of Math. 137(1993), 561–609.Google Scholar
[BHMM3] Holomorphic maps of Riemann surfaces into almost Lie groups, en preparation.Google Scholar
[BPV] Barth, W., Peters, C. and Van de Ven, A., Compact Complex Surfaces, Springer Verlag, New York, 1984.Google Scholar
[CCMM] Cohen, F. R., Cohen, R. L., Mann, B. M. and Milgram, R. J., The topology of rational functons and divisors of surfaces, Acta Math. 166(1991), 163–221.Google Scholar
[Dl] Donaldson, S. K., Instantons and geometric invariant theory, Comm. Math. Phys. 93(1984), 453–461.Google Scholar
[D2] Donaldson, S. K., Anti-self-dual connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50(1985), 1–26.Google Scholar
[DK] Donaldson, S. K. and Kronheimer, P. B., The geometry of four-manifolds, Oxford University Press, Oxford, 1990.Google Scholar
[EL] Eells, J. andLemaire, L., Another Report on Harmonic Maps, Bull. London Math. Soc. 20(1988), 385–524.Google Scholar
[EW] Eells, J. and Wood, J., Harmonic maps from surfaces to complex projective spaces, Adv. Math. 49( 1983), 217–263.Google Scholar
[G] Grothendieck, A., Sur la classification des fibres holomorphes sur la sphère de Riemann Amer. J. Math. 79(1957), 121–138 Google Scholar
[Gui] Guest, M. A., Topology of the space of absolute minima of the energy functional, Amer. J. Math. 106 (1984), 21–42.Google Scholar
[Gu2] Guest, M. A., The topology of the space of rational curves on a toric variety, prétirage, Rochester University, 1993.Google Scholar
[Gr] Gravesen, J., On the topology of spaces of holomorphic maps, Acta Math. 162(1989), 247–286.Google Scholar
[H] Hurtubise, J. C., Holomorphic maps of a Riemann surface into a flag manifold, prétirage CRM, 1994, J. Differential Geom., à paraître.Google Scholar
[HM] Hurtubise, J. C. and Milgram, R. J., The Atiyah-Jones conjecture for ruled surfaces, prétirage, 1993.Google Scholar
[L] Lawson, B., Algebraic cycles and homotopy theory, Ann. of Math. 129(1989), 253–291.Google Scholar
[Kil] Kirwan, F. C., On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Mat. (2)24(1986), 221–275.Google Scholar
[Ki2] Kirwan, F. C., Geometric invariant theory and the Atiyah-J ones conjecture, prétirage d'Oxford University, 1992.Google Scholar
[May] May, J. P., The Geometry of Iterated Loop Spaces, Springer-Verlag, Lecture Notes in Math. 271, 1972.Google Scholar
[Mi] Milgram, R. J., Iterated loop spaces, Ann. of Math. 84(1966) 386–403.Google Scholar
[MM] Mann, B. M. and Milgram, R. J., On the moduli space of S\J(n) monopoles and holomorphic maps to flag manifolds, J. Differential Geom. (1) 38(1993), 39–103.Google Scholar
[S] Segal, G., The topology of rational functions, Acta Math., 143 (1979), 39–72.Google Scholar
[Tl] Taubes, C. H., Path-connected Yang-Mills moduli spaces, J. Differential Geom. 19(1984), 337–392.Google Scholar
[T2] Taubes, C. H., The stable topology of self-dual moduli spaces, J. Differential Geom. 29(1989), 163–230.Google Scholar
[Ti] Tian, Y., The based su(nyinstanton moduli spaces, Math. Ann. 298(1994), 117–140.Google Scholar
[U] Uhlenbeck, K., Variational Problems for gauge fields. Dans Seminar on Differential Geometry, (ed. S.-T. Yau), Ann. of Math. Stud. 102, Princeton University Press, 1982.Google Scholar