Geodesic
Noncommutative unitons
Teoretičeskaâ i matematičeskaâ fizika, Tome 154 (2008) no. 2, pp. 220-239 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

By Uhlenbeck's results, every harmonic map from the Riemann sphere $S^2$ to the unitary group $U(n)$ decomposes into a product of so-called unitons: special maps from $S^2$ to the Grassmannians $\mathrm{Gr}_k(\mathbb C^n)\subset U(n)$ satisfying certain systems of first-order differential equations. We construct a noncommutative analogue of this factorization, applicable to those solutions of the noncommutative unitary sigma model that are finite-dimensional perturbations of zero-energy solutions. In particular, we prove that the energy of each such solution is an integer multiple of $8\pi$, give examples of solutions that are not equivalent to Grassmannian solutions, and study the realization of non-Grassmannian zero modes of the Hessian of the energy functional by directions tangent to the moduli space of solutions.
Keywords: noncommutative sigma model, uniton factorization. 
@article{TMF_2008_154_2_a1,
     author = {A. V. Domrin},
     title = {Noncommutative unitons},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {220--239},
     year = {2008},
     volume = {154},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2008_154_2_a1/}
}
TY  - JOUR
AU  - A. V. Domrin
TI  - Noncommutative unitons
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2008
SP  - 220
EP  - 239
VL  - 154
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2008_154_2_a1/
LA  - ru
ID  - TMF_2008_154_2_a1
ER  - 
%0 Journal Article
%A A. V. Domrin
%T Noncommutative unitons
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2008
%P 220-239
%V 154
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2008_154_2_a1/
%G ru
%F TMF_2008_154_2_a1
A. V. Domrin. Noncommutative unitons. Teoretičeskaâ i matematičeskaâ fizika, Tome 154 (2008) no. 2, pp. 220-239. http://geodesic.mathdoc.fr/item/TMF_2008_154_2_a1/

[1] W. J. Zakrzewski, Low Dimensional Sigma Models, Adam Hilger, Bristol, 1989 | MR | Zbl

[2] I. Davidov, A. G. Sergeev, UMN, 48:3 (1993), 3–96 | DOI | MR | Zbl

[3] M. A. Guest, “An update on harmonic maps of finite uniton number, via the zero curvature equation”, Integrable Systems, Topology, and Physics. A Conference on Integrable Systems in Differential Geometry (Tokyo, Japan, 2000), Contemp. Math., 309, eds. M. Guest et al., AMS, Providence, RI, 2002, 85–113 | DOI | MR | Zbl

[4] K. Uhlenbeck, J. Differential Geom., 30:1 (1989), 1–50 | DOI | MR | Zbl

[5] G. Valli, Topology, 27:2 (1988), 129–136 | DOI | MR | Zbl

[6] J. A. Harvey, Komaba lectures on noncommutative solitons and $D$-branes, arXiv: hep-th/0102076

[7] J. M. Gracia-Bondia, J. C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser Adv. Texts Basler Lehrbucher, Birkhäuser, Boston, 2001 | MR | Zbl

[8] L. Khermander, Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, T. III. Psevdodifferentsialnye operatory, Mir, M., 1987 | MR | MR | Zbl

[9] O. Lechtenfeld, A. D. Popov, JHEP, 11 (2001), 040 | DOI | MR

[10] A. V. Domrin, O. Lechtenfeld, S. Petersen, JHEP, 03 (2005), 045 | DOI | MR

[11] A. V. Domrin, “Prostranstva modulei reshenii nekommutativnoi sigma-modeli”, TMF, 2008 (to appear)

[12] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | MR | Zbl | Zbl

[13] R. Rochberg, N. Weaver, Proc. Amer. Math. Soc., 129:9 (2001), 2679–2687 | DOI | MR | Zbl

[14] I. fon Neiman, Matematicheskie osnovy kvantovoi mekhaniki, Nauka, M., 1964 | MR | MR | Zbl | Zbl