Geodesic
Nontrivial Solutions of Seiberg–Witten Equations on the Noncommutative 4-Dimensional Euclidean Space
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 127-138 
M. Wolf; A. D. Popov; A. G. Sergeev. Nontrivial Solutions of Seiberg–Witten Equations on the Noncommutative 4-Dimensional Euclidean Space. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 127-138. http://geodesic.mathdoc.fr/item/TM_2005_251_a4/
@article{TM_2005_251_a4,
     author = {M. Wolf and A. D. Popov and A. G. Sergeev},
     title = {Nontrivial {Solutions} of {Seiberg{\textendash}Witten} {Equations} on the {Noncommutative} {4-Dimensional} {Euclidean} {Space}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {127--138},
     year = {2005},
     volume = {251},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2005_251_a4/}
}
TY  - JOUR
AU  - M. Wolf
AU  - A. D. Popov
AU  - A. G. Sergeev
TI  - Nontrivial Solutions of Seiberg–Witten Equations on the Noncommutative 4-Dimensional Euclidean Space
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2005
SP  - 127
EP  - 138
VL  - 251
UR  - http://geodesic.mathdoc.fr/item/TM_2005_251_a4/
LA  - ru
ID  - TM_2005_251_a4
ER  - 
%0 Journal Article
%A M. Wolf
%A A. D. Popov
%A A. G. Sergeev
%T Nontrivial Solutions of Seiberg–Witten Equations on the Noncommutative 4-Dimensional Euclidean Space
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2005
%P 127-138
%V 251
%U http://geodesic.mathdoc.fr/item/TM_2005_251_a4/
%G ru
%F TM_2005_251_a4

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Noncommutative Seiberg–Witten equations on the noncommutative Euclidean space $\mathbb R^4_\theta$ are studied that are obtained from the standard Seiberg–Witten equations on $\mathbb R^4$ by replacing the usual product with the deformed Moyal $\star$-product. Nontrivial solutions of these noncommutative Seiberg–Witten equations are constructed that do not reduce to solutions of the standard Seiberg–Witten equations on $\mathbb R^4$ for $\theta \to 0$. Such solutions of the noncommutative equations on $\mathbb R^4_\theta$ exist even when the corresponding commutative Seiberg–Witten equations on $\mathbb R^4$ do not have any nontrivial solutions.

[1] Bak D., “Exact multi-vortex solutions in noncommutative Abelian–Higgs theory”, Phys. Lett. B., 495 (2000), 251–255 ; arXiv: hep-th/0008204 | DOI | MR | Zbl

[2] Berezin F. A., Shubin M. A., Uravnenie Shrëdingera, MGU, M., 1983 | MR

[3] Connes A., Noncommutative geometry, Acad. Press, London, San Diego, 1994 | MR | Zbl

[4] Douglas M. R., Nekrasov N. A., “Noncommutative field theory”, Rev. Mod. Phys., 73 (2001), 977–1029 ; arXiv: hep-th/0106048 | DOI | MR

[5] Gracia-Bondia J. M., Varilly J. C., Figueroa H., Elements of noncommutative geometry, Birkhäuser, Boston, Basel, Berlin, 2001 | MR | Zbl

[6] Jatkar D. P., Mandal G., Wadia S. R., “Nielsen–Olesen vortices in noncommutative abelian Higgs model”, J. High Energy Phys., 2000, no. 09, Pap. 018 ; arXiv: hep-th/0007078 | MR

[7] Nekrasov N. A., Schwarz A. S., “Instantons on noncommutative $\mathbb{R}^4$ and $(2,0)$ superconformal six-dimensional theory”, Commun. Math. Phys., 198 (1998), 689–703 ; arXiv: hep-th/9802068 | DOI | MR | Zbl

[8] Popov A. D., Sergeev A. G., Wolf M., “Seiberg–Witten monopole equations on noncommutative $R^4$”, J. Math. Phys., 44 (2003), 4527–4554 ; arXiv: hep-th/0304263 | DOI | MR | Zbl

[9] Salamon D., Spin geometry and Seiberg–Witten invariants, Preprint, Warwick Univ., Warwick, 1996 | MR

[10] Sergeev A. G., Vortices and Seiberg–Witten equations, Nagoya Math. Lect., 5, Nagoya Univ., Nagoya, 2002