Differential Geometry Seminar on Generalized Complex Geometry

 Differential Geometry Seminar on
Generalized Complex Geometry

A generalized complex structure $\mathcal{J}$ on a smooth manifold M is an assignment to each fiber of the extended tangent bundle $TM\oplus T^*M$ of an orthogonal linear complex structure in a locally compatible manner. The resulting formalism extends both complex and symplectic geometry, and was introduced in 2003 by Nigel Hitchin with an eye to string theory.

The aim of this learning seminar is first to review the foundational material, and then to acquaint ourselves with the state of the art and open questions.

Related constructions include,

• Lie algebroids
• Dirac structures
• Courant algebroids
• moment maps
• generalized Kähler structures
• T-duality

This seminar should appeal to students and researchers with interests in differential geometry and mathematical physics.

Mathematical references:

• Gil Cavalcanti. Introduction to generalized complex geometry. Publicações Matemáticas do IMPA. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2007. 26o Colóquio Brasileiro de Matemática, https://impa.br/wp-content/uploads/2017/04/26CBM_05.pdf
• Marco Gualtieri. Generalized complex geometry. PhD thesis, University of Oxford, November 2003, https://arxiv.org/abs/math/0401221
• Nigel Hitchin. Lectures on generalized geometry. In Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics, volume 16 of Surv. Differ. Geom., pages 79–124. Int. Press, Somerville, MA, 2011, https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2011/0016/0001/SDG-2011-0016-0001-a003.pdf
• Nigel Hitchin. Generalized Calabi–Yau manifolds. Q. J. Math., 54(3):281–308, 2003, https://proxy.library.spbu.ru:2060/10.1093/qmath/hag025

Physical references:

• Paul Koerber. Lectures on generalized complex geometry for physicists. Fortschr. Phys., 59(3-4):169–242, 2011, https://proxy.library.spbu.ru:2150/doi/abs/10.1002/prop.201000083
• Maxim Zabzine. Lectures on generalized complex geometry and supersymmetry. Arch. Math. (Brno), 42(suppl.):119–146, 2006, https://www.emis.de/journals/AM/06-S/zabzine.pdf