Perturbations of the metric in Seiberg-Witten equations
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 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 : https://doi.org/10.5802/aif.2640
Classification:  57R57,  58G03,  58D27,  14J80
Keywords: Seiberg-Witten theory, perturbations of the metric, Kähler surfaces, transversality
@article{AIF_2011__61_3_1259_0,
     author = {Scala, Luca},
     title = {Perturbations of the metric in Seiberg-Witten equations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {1259-1297},
     doi = {10.5802/aif.2640},
     mrnumber = {2918729},
     zbl = {1238.57029},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_3_1259_0}
}
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. http://www.numdam.org/item/AIF_2011__61_3_1259_0/

[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) | Numdam | MR 1472535 | Zbl 0881.57035

[2] Besse, Arthur L. Einstein manifolds, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 10 (1987) | MR 867684 | Zbl 0613.53001 | Zbl 1147.53001

[3] Bourguignon, Jean-Pierre; Gauduchon, Paul Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys., Tome 144 (1992) no. 3, pp. 581-599 | Article | MR 1158762 | Zbl 0755.53009

[4] Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1990) (Oxford Science Publications) | MR 1079726 | Zbl 0820.57002

[5] Eichhorn, Jürgen; Friedrich, Thomas Seiberg-Witten theory, Symplectic singularities and geometry of gauge fields (Warsaw, 1995), Polish Acad. Sci., Warsaw (Banach Center Publ.) Tome 39 (1997), pp. 231-267 | MR 1458664 | Zbl 0881.57032

[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., Tome 36 (1989) no. 3, pp. 323-344 | Article | MR 1027070 | Zbl 0694.58008

[7] Freed, Daniel S.; Uhlenbeck, Karen K. Instantons and four-manifolds, Springer-Verlag, New York, Mathematical Sciences Research Institute Publications, Tome 1 (1991) | MR 1081321 | Zbl 0559.57001

[8] Friedrich, Thomas Dirac operators in Riemannian geometry, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 25 (2000) (Translated from the 1997 German original by Andreas Nestke) | MR 1777332 | Zbl 0949.58032

[9] Gil-Medrano, Olga; Michor, Peter W. The Riemannian manifold of all Riemannian metrics, Quart. J. Math. Oxford Ser. (2), Tome 42 (1991) no. 166, pp. 183-202 | Article | MR 1107281 | Zbl 0739.58010

[10] Kobayashi, Shoshichi; Nomizu, Katsumi Foundations of differential geometry. Vol. I, John Wiley & Sons Inc., New York, Wiley Classics Library (1996) (Reprint of the 1963 original, A Wiley-Interscience Publication) | MR 1393940

[11] Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 53 (1997) | MR 1471480 | Zbl 0889.58001

[12] Lawson, H. Blaine Jr.; Michelsohn, Marie-Louise Spin geometry, Princeton University Press, Princeton, NJ, Princeton Mathematical Series, Tome 38 (1989) | MR 1031992 | Zbl 0688.57001

[13] Maier, Stephan Generic metrics and connections on Spin- and Spin c -manifolds, Comm. Math. Phys., Tome 188 (1997) no. 2, pp. 407-437 | Article | MR 1471821 | Zbl 0899.53036

[14] Morgan, John W. The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 44 (1996) | MR 1367507 | Zbl 0846.57001

[15] Okonek, Christian; Teleman, Andrei Seiberg-Witten invariants for manifolds with b + =1, and the universal wall crossing formula, Internat. J. Math., Tome 7 (1996) no. 6, pp. 811-832 | Article | MR 1417787 | Zbl 0959.57029

[16] Scorpan, Alexandru The wild world of 4-manifolds, American Mathematical Society, Providence, RI (2005) | MR 2136212 | Zbl 1075.57001

[17] Seiberg, N.; Witten, E. Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nuclear Phys. B, Tome 426 (1994) no. 1, pp. 19-52 | Article | MR 1293681 | Zbl 0996.81510

[18] Seiberg, N.; Witten, E. Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nuclear Phys. B, Tome 431 (1994) no. 3, pp. 484-550 | Article | MR 1306869 | Zbl 1020.81911

[19] Smale, S. An infinite dimensional version of Sard’s theorem, Amer. J. Math., Tome 87 (1965), pp. 861-866 | Article | MR 185604 | Zbl 0143.35301

[20] Teleman, Andrei Introduction à la théorie de Jauge (2005) (Cours de D.E.A. http://www.cmi.univ-mrs.fr/~teleman/documents/cours-sw.pdf)

[21] Witten, E. Monopoles and four manifolds, Math. Res. Lett., Tome 3 (1994) no. 7, pp. 654-675 | MR 1306021 | Zbl 0867.57029