Perturbations of the metric in Seiberg-Witten equations
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 1259-1297.

Let M a compact connected oriented 4-manifold. We study the space Ξ of Spin c -structures of fixed fundamental class, as an infinite dimensional principal bundle on the manifold of riemannian metrics on M. In order to study perturbations of the metric in Seiberg-Witten equations, we study the transversality of universal equations, parametrized with all Spin c -structures Ξ. We prove that, on a complex Kähler surface, for an hermitian metric h sufficiently close to the original Kähler metric, the moduli space of Seiberg-Witten monopoles relative to the metric h is smooth of the expected dimension.

Soit M une variété riemannienne compacte connexe orientée de dimension 4. On étudie l’espace Ξ des structures Spin c de classe fondamentale fixée, comme fibré principal de dimension infinie sur la variété des métriques riemanniennes de M. Afin d’étudier les perturbations de la métrique dans les équations de Seiberg-Witten, on étudie la transversalité des équations universelles, paramétrées par l’espace Ξ de toutes les structures Spin c . On montre que, sur une surface de Kähler, pour une métrique hermitienne h suffisamment proche à la métrique de Kähler de départ, l’espace de modules de monopôles de Seiberg-Witten relatif à la métrique h est lisse de la dimension attendue.

DOI: 10.5802/aif.2640
Classification: 57R57,  58G03,  58D27,  14J80
Keywords: Seiberg-Witten theory, perturbations of the metric, Kähler surfaces, transversality
Scala, Luca 1

1 University of Chicago Department of Mathematics 5734 S. University Avenue 60637 Chicago IL (USA)
     author = {Scala, Luca},
     title = {Perturbations of the metric in {Seiberg-Witten} equations},
     journal = {Annales de l'Institut Fourier},
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Scala, Luca. Perturbations of the metric in Seiberg-Witten equations. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 1259-1297. doi : 10.5802/aif.2640.

[1] Bennequin, Daniel Monopôles de Seiberg-Witten et conjecture de Thom (d’après Kronheimer, Mrowka et Witten), Astérisque (1997) no. 241, pp. Exp. No. 807, 3, 59-96 (Séminaire Bourbaki, Vol. 1995/96) | EuDML | Numdam | MR | Zbl

[2] Besse, Arthur L. Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10, Springer-Verlag, Berlin, 1987 | MR | Zbl

[3] Bourguignon, Jean-Pierre; Gauduchon, Paul Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys., Volume 144 (1992) no. 3, pp. 581-599 | DOI | MR | Zbl

[4] Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1990 (Oxford Science Publications) | MR | Zbl

[5] Eichhorn, Jürgen; Friedrich, Thomas Seiberg-Witten theory, Symplectic singularities and geometry of gauge fields (Warsaw, 1995) (Banach Center Publ.), Volume 39, Polish Acad. Sci., Warsaw, 1997, pp. 231-267 | EuDML | MR | Zbl

[6] Freed, Daniel S.; Groisser, David The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group, Michigan Math. J., Volume 36 (1989) no. 3, pp. 323-344 | DOI | MR | Zbl

[7] Freed, Daniel S.; Uhlenbeck, Karen K. Instantons and four-manifolds, Mathematical Sciences Research Institute Publications, 1, Springer-Verlag, New York, 1991 | MR | Zbl

[8] Friedrich, Thomas Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, 25, American Mathematical Society, Providence, RI, 2000 (Translated from the 1997 German original by Andreas Nestke) | MR | Zbl

[9] Gil-Medrano, Olga; Michor, Peter W. The Riemannian manifold of all Riemannian metrics, Quart. J. Math. Oxford Ser. (2), Volume 42 (1991) no. 166, pp. 183-202 | DOI | MR | Zbl

[10] Kobayashi, Shoshichi; Nomizu, Katsumi Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1996 (Reprint of the 1963 original, A Wiley-Interscience Publication) | MR

[11] Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997 | MR | Zbl

[12] Lawson, H. Blaine Jr.; Michelsohn, Marie-Louise Spin geometry, Princeton Mathematical Series, 38, Princeton University Press, Princeton, NJ, 1989 | MR | Zbl

[13] Maier, Stephan Generic metrics and connections on Spin- and Spin c -manifolds, Comm. Math. Phys., Volume 188 (1997) no. 2, pp. 407-437 | DOI | MR | Zbl

[14] Morgan, John W. The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, 44, Princeton University Press, Princeton, NJ, 1996 | MR | Zbl

[15] Okonek, Christian; Teleman, Andrei Seiberg-Witten invariants for manifolds with b + =1, and the universal wall crossing formula, Internat. J. Math., Volume 7 (1996) no. 6, pp. 811-832 | DOI | MR | Zbl

[16] Scorpan, Alexandru The wild world of 4-manifolds, American Mathematical Society, Providence, RI, 2005 | MR | Zbl

[17] Seiberg, N.; Witten, E. Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nuclear Phys. B, Volume 426 (1994) no. 1, pp. 19-52 | DOI | MR | Zbl

[18] Seiberg, N.; Witten, E. Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nuclear Phys. B, Volume 431 (1994) no. 3, pp. 484-550 | DOI | MR | Zbl

[19] Smale, S. An infinite dimensional version of Sard’s theorem, Amer. J. Math., Volume 87 (1965), pp. 861-866 | DOI | MR | Zbl

[20] Teleman, Andrei Introduction à la théorie de Jauge, 2005 (Cours de D.E.A.

[21] Witten, E. Monopoles and four manifolds, Math. Res. Lett., Volume 3 (1994) no. 7, pp. 654-675 | MR | Zbl

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