Noncommutative gauge theory for Poisson manifolds. (English) Zbl 0984.81167

Jurčo, Branislav; Schupp, Peter; Wess, Julius

Nucl. Phys., B 584, No. 3, 784-794 (2000).

Summary: A noncommutative gauge theory is associated to every Abelian gauge theory on a Poisson manifold. The semi-classical and full quantum version of the map from the ordinary gauge theory to the noncommutative gauge theory (Seiberg-Witten map) is given explicitly to all orders for any Poisson manifold in the Abelian case. In the quantum case the construction is based on Kontsevich’s formality theorem.

MSC:

81T75	Noncommutative geometry methods in quantum field theory
81T13	Yang-Mills and other gauge theories in quantum field theory
53D17	Poisson manifolds; Poisson groupoids and algebroids

References:

[1]	Callan, C. G.; Lovelace, C.; Nappi, C. R.; Yost, S. A., String loop corrections to beta functions, Nucl. Phys. B, Vol. 288, 525 (1987)
[2]	Abouelsaood, A.; Callan, C. G.; Nappi, C. R.; Yost, S. A., Open strings in background gauge fields, Nucl. Phys. B, Vol. 280, 599 (1987)
[3]	Chu, C.-S.; Ho, P.-M., Noncommutative open string and D-brane, Nucl. Phys. B, Vol. 550, 151 (1999) · Zbl 0947.81136
[4]	Schomerus, V., D-branes and deformation quantization, JHEP, Vol. 9906, 030 (1999) · Zbl 0961.81066
[5]	Connes, A.; Douglas, M. R.; Schwarz, A., Noncommutative geometry and matrix theory: compactification on tori, JHEP, Vol. 9802, 003 (1998) · Zbl 1018.81052
[6]	Douglas, M. R.; Hull, C., D-branes and the noncommutative torus, JHEP, Vol. 9802, 008 (1998) · Zbl 0957.81017
[7]	Morariu, B.; Zumino, B., (Relativity, Particle Physics and Cosmology (1998), World Scientific: World Scientific Singapore)
[8]	Taylor, W., D-Brane field theory on compact spaces, Phys. Lett. B, Vol. 394, 283 (1997)
[9]	Seiberg, N.; Witten, E., String theory and noncommutative geometry, JHEP, Vol. 9909, 032 (1999) · Zbl 0957.81085
[10]	Andreev, O.; Dorn, H., On open string sigma model and noncommutative gauge fields · Zbl 1050.81694
[11]	Jurco, B.; Schupp, P., Noncommutative Yang-Mills from equivalence of star products, Eur. Phys. J. C, Vol. 14, 367 (2000)
[12]	Asakawa, T.; Kishimoto, I., Noncommutative gauge theories from deformation quantization · Zbl 1042.81579
[13]	Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Phys., Vol. 111, 61 (1978) · Zbl 0377.53024
[14]	Kontsevitch, M., Deformation quantization of Poisson manifolds, I
[15]	Sternheimer, D., Deformation quantization: twenty years after · Zbl 0977.53082
[16]	Cornalba, L., Corrections to the Abelian Born-Infeld action arising from noncommutative geometry · Zbl 0989.81618
[17]	Madore, J.; Schraml, S.; Schupp, P.; Wess, J., Gauge theory on noncommutative spaces · Zbl 1099.81524
[18]	Moser, J., On the volume elements on a manifold, Trans. Am. Math. Soc., Vol. 120, 286 (1965) · Zbl 0141.19407
[19]	Ishibashi, N., A relation between commutative and noncommutative descriptions ofD-branes · Zbl 1021.81052
[20]	Okuyama, K., A path integral representation of the map between commutative and noncommutative gauge fields · Zbl 0959.81111
[21]	Cornalba, L., D-brane physics and noncommutative Yang-Mills theory · Zbl 0997.81108
[22]	Arnal, D.; Manchon, D.; Masmoudi, M., Choix des signes pour la formalite de M. Kontsevich · Zbl 1055.53066
[23]	Manchon, D., Poisson bracket, deformed bracket and gauge group actions in Kontsevich deformation quantization · Zbl 0981.53091
[24]	Cattaneo, A. S.; Felder, G., A path integral approach to the Kontsevich quantization formula · Zbl 1038.53088