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

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
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[1] Alekseev, A., Strobl, T.: Current algebra and differential geometry. JHEP 0503, 035 (2005)
[2] Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591 (2000) · Zbl 1038.53088
[3] Cattaneo, A.S., Felder, G.: Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. Lett. Math. Phys. 69, 157 (2004) · Zbl 1065.53063
[4] Gualtieri, M.: Generalized complex geometry. Oxford University DPhil thesis, http://arxiv.org/list/math.DG/0401221, 2004 · Zbl 1079.53106
[5] Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54, no. 3, 281–308, (2003) · Zbl 1076.32019
[6] Hull, C.M.: Actions for (2,1) sigma models and strings. Nucl. Phys. B 509, 252 (1998) · Zbl 0933.81028
[7] Kapustin, A., Orlov, D.: Vertex algebras, mirror symmetry, and D-branes: The case of complex tori. Commun. Math. Phys. 233, 79 (2003) · Zbl 1051.17017
[8] Kapustin, A.: Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1, 49 (2004) · Zbl 1065.81108
[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, U., Zabzine, M.: N = 2 boundary conditions for non-linear sigma models and Landau-Ginzburg models. JHEP 0302, 006 (2003) · Zbl 1032.81028
[11] Lindström, U.: Generalized N = (2,2) supersymmetric non-linear sigma models. Phys. Lett. B 587, 216 (2004) · Zbl 1246.81375
[12] Lindström, U., Minasian, R., Tomasiello, A., Zabzine, M.: Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257, 235 (2005) · Zbl 1118.53048
[13] Zabzine, M.: Geometry of D-branes for general N = (2,2) sigma models. Lett. Math. Phys. 70, 211 (2004) · Zbl 1070.81523
[14] Zucchini, R.: A sigma model field theoretic realization of Hitchin’s generalized complex geometry. JHEP 0411, 045 (2004)
[15] Zucchini, R.: Generalized complex geometry, generalized branes and the Hitchin sigma model. JHEP 0503, 022 (2005)
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