Perturbations of the metric in Seiberg-Witten equations

Luca Scala[1]

  • [1] University of Chicago Department of Mathematics 5734 S. University Avenue 60637 Chicago IL (USA)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 3, page 1259-1297
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Scala, Luca. "Perturbations of the metric in Seiberg-Witten equations." Annales de l’institut Fourier 61.3 (2011): 1259-1297. <http://eudml.org/doc/219754>.

@article{Scala2011,
abstract = {Let $M$ a compact connected oriented 4-manifold. We study the space $\Xi $ of $\rm \{Spin\}^c$-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 $\rm \{Spin\}^c$-structures $\Xi $. 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.},
affiliation = {University of Chicago Department of Mathematics 5734 S. University Avenue 60637 Chicago IL (USA)},
author = {Scala, Luca},
journal = {Annales de l’institut Fourier},
keywords = {Seiberg-Witten theory; perturbations of the metric; Kähler surfaces; transversality},
language = {eng},
number = {3},
pages = {1259-1297},
publisher = {Association des Annales de l’institut Fourier},
title = {Perturbations of the metric in Seiberg-Witten equations},
url = {http://eudml.org/doc/219754},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Scala, Luca
TI - Perturbations of the metric in Seiberg-Witten equations
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 3
SP - 1259
EP - 1297
AB - Let $M$ a compact connected oriented 4-manifold. We study the space $\Xi $ of $\rm {Spin}^c$-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 $\rm {Spin}^c$-structures $\Xi $. 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.
LA - eng
KW - Seiberg-Witten theory; perturbations of the metric; Kähler surfaces; transversality
UR - http://eudml.org/doc/219754
ER -

References

top
  1. Daniel Bennequin, Monopôles de Seiberg-Witten et conjecture de Thom (d’après Kronheimer, Mrowka et Witten), Astérisque (1997), Exp. No. 807, 3, 59-96 Zbl0881.57035MR1472535
  2. Arthur L. Besse, Einstein manifolds, 10 (1987), Springer-Verlag, Berlin Zbl1147.53001MR867684
  3. Jean-Pierre Bourguignon, Paul Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992), 581-599 Zbl0755.53009MR1158762
  4. S. K. Donaldson, P. B. Kronheimer, The geometry of four-manifolds, (1990), The Clarendon Press Oxford University Press, New York Zbl0820.57002MR1079726
  5. Jürgen Eichhorn, Thomas Friedrich, Seiberg-Witten theory, Symplectic singularities and geometry of gauge fields (Warsaw, 1995) 39 (1997), 231-267, Polish Acad. Sci., Warsaw Zbl0881.57032MR1458664
  6. Daniel S. Freed, David Groisser, The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group, Michigan Math. J. 36 (1989), 323-344 Zbl0694.58008MR1027070
  7. Daniel S. Freed, Karen K. Uhlenbeck, Instantons and four-manifolds, 1 (1991), Springer-Verlag, New York Zbl0559.57001MR1081321
  8. Thomas Friedrich, Dirac operators in Riemannian geometry, 25 (2000), American Mathematical Society, Providence, RI Zbl0949.58032MR1777332
  9. Olga Gil-Medrano, Peter W. Michor, The Riemannian manifold of all Riemannian metrics, Quart. J. Math. Oxford Ser. (2) 42 (1991), 183-202 Zbl0739.58010MR1107281
  10. Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry. Vol. I, (1996), John Wiley & Sons Inc., New York Zbl0091.34802MR1393940
  11. Andreas Kriegl, Peter W. Michor, The convenient setting of global analysis, 53 (1997), American Mathematical Society, Providence, RI Zbl0889.58001MR1471480
  12. H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, 38 (1989), Princeton University Press, Princeton, NJ Zbl0688.57001MR1031992
  13. Stephan Maier, Generic metrics and connections on Spin- and Spin c -manifolds, Comm. Math. Phys. 188 (1997), 407-437 Zbl0899.53036MR1471821
  14. John W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, 44 (1996), Princeton University Press, Princeton, NJ Zbl0846.57001MR1367507
  15. Christian Okonek, Andrei Teleman, Seiberg-Witten invariants for manifolds with b + = 1 , and the universal wall crossing formula, Internat. J. Math. 7 (1996), 811-832 Zbl0959.57029MR1417787
  16. Alexandru Scorpan, The wild world of 4-manifolds, (2005), American Mathematical Society, Providence, RI Zbl1075.57001MR2136212
  17. N. Seiberg, E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nuclear Phys. B 426 (1994), 19-52 Zbl0996.81510MR1293681
  18. N. Seiberg, E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nuclear Phys. B 431 (1994), 484-550 Zbl1020.81911MR1306869
  19. S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861-866 Zbl0143.35301MR185604
  20. Andrei Teleman, Introduction à la théorie de Jauge, (2005) 
  21. E. Witten, Monopoles and four manifolds, Math. Res. Lett. 3 (1994), 654-675 Zbl0867.57029MR1306021

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.