Hamiltonian perspective on generalized complex structure. (English) Zbl 1104.53077

Zabzine, Maxim

Commun. Math. Phys. 263, No. 3, 711-722 (2006).

Author’s summary: In this note we clarify the relation between extended world-sheet super-symmetry and generalized complex structure. The analysis is based on the phase space description of a wide class of sigma models. We point out the natural isomorphism between the group of orthogonal automorphisms of the Courant bracket and the group of local canonical transformations of the cotangent bundle of the loop space. Indeed this fact explains the natural relation between the world-sheet and the geometry of \(T \oplus T^*\). We discuss D-branes in this perspective.

Reviewer: A. D. Osborne (Keele)

MSC:

53C80	Applications of global differential geometry to the sciences
53C26	Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
81T60	Supersymmetric field theories in quantum mechanics

Keywords:

world-sheet; sigma models; loop space; D-branes

References:

[1]	Alekseev, JHEP, 0503, 035 (2005) · doi:10.1088/1126-6708/2005/03/035
[2]	Cattaneo, Commun. Math. Phys., 212, 591 (2000) · Zbl 1038.53088 · doi:10.1007/s002200000229
[3]	Cattaneo, Lett. Math. Phys., 69, 157 (2004) · Zbl 1065.53063 · doi:10.1007/s11005-004-0609-7
[4]	Gualtieri, M.: Generalized complex geometry. Oxford University DPhil thesis, http://arxiv.org/list/math.DG/0401221, 2004 · Zbl 1235.32020
[5]	Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54, no. 3, 281-308, (2003) · Zbl 1076.32019
[6]	Hull, Nucl. Phys. B, 509, 252 (1998) · Zbl 0933.81028 · doi:10.1016/S0550-3213(97)00492-6
[7]	Kapustin, Commun. Math. Phys., 233, 79 (2003) · Zbl 1051.17017 · doi:10.1007/s00220-002-0755-7
[8]	Kapustin, Int. J. Geom. Meth. Mod. Phys., 1, 49 (2004) · Zbl 1065.81108 · doi:10.1142/S0219887804000034
[9]	Kapustin, A., Li, Y.: Topological sigma-models with H-flux and twisted generalized complex. http://arxiv.org/list/hep-th/0407249, 2004 · Zbl 1192.81310
[10]	Lindström, JHEP, 0302, 006 (2003) · doi:10.1088/1126-6708/2003/02/006
[11]	Lindström, Phys. Lett. B, 587, 216 (2004) · Zbl 1246.81375 · doi:10.1016/j.physletb.2004.03.014
[12]	Lindström, Commun. Math. Phys., 257, 235 (2005) · Zbl 1118.53048 · doi:10.1007/s00220-004-1265-6
[13]	Zabzine, Lett. Math. Phys., 70, 211 (2004) · Zbl 1070.81523 · doi:10.1007/s11005-004-4296-1
[14]	Zucchini, JHEP, 0411, 045 (2004) · doi:10.1088/1126-6708/2004/11/045
[15]	Zucchini, JHEP, 0503, 022 (2005) · doi:10.1088/1126-6708/2005/03/022