Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
This will be an introductory course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry. The basic reference will be an article by M. Gualtieri, as listed under textbooks but we will also draw from more recent developments in the physics literature. A basic familiarity with complex and symplectic manifolds will be assumed.
Topics will Include
- Gerbes, B-fields, and Exact Courant Algebroids
- Relation to Sigma Models in Physics and Baby String Theory
- Linear Algebra of a Split-Signature Real Bilinear Form; Pure Spinors
- Generalized Riemannian Structures and The Generalized Hodge Star
- Integrability, Dirac structures, Lie Algebroids and Bialgebroids
- Generalized Complex Structures; Examples of Such
- Generalized Holomorphic Bundles; the Picard Group
- Kodaira-Spencer-Kuranishi Deformation Theory For Generalized Complex Structures
- Local Structure Theory For Generalized Complex Structures
- Generalized Kahler Geometry
- Hodge Decomposition Theorem For Generalized Kahler Structures
- Hermitian Geometry; The Gray-Hervella Classification
- Equivalence Theorem Generalized Kahler=Bihermitian
- Generalized Calabi-Yau Structures and The Hitchin Functional
- Ramond-Ramond Versus Neveu-Schwarz Fluxes; D-Branes
Although the class is geared towards advanced graduate students, there are no specific prerequisites.
This course has no textbook.
Students are instead referred to the following articles:
Courant, T. “Dirac manifolds.” Trans Amer Math Soc, no. 319 (1990): 631-661.
Grana, M., R. Minasian, M. Petrini, and A. Tomasiello. Generalized structures and N=1 vacua.
Gualtieri, M. “Generalized complex geometry.” Oxford D Phil thesis, math.
Hitchin, N. “Generalized Calabi-Yau manifolds.” Q J Math 54 (2003): 281-308.
Kapustin, A., and Y. Li. Topological sigma-models with H-flux and twisted generalized complex manifolds.
Lindstrom, U., M. Rocek, R. von Unge, and M. Zabzine. Generalized Kahler manifolds and off-shell supersymmetry.
Severa, P., and A. Weinstein. “Poisson geometry with a 3-form background.” Prog Theo Phys Suppl 144 (2001): 145-154.
Everyone should be complete the problem sets in order to truly learn the material in the course. As the course progresses, open problems may appear, to pique your interest.
Class Participation 100%