Topics in geometry: dirac geometry

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.

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.

Zucchini, R. Generalized complex geometry, generalized branes and the Hitchin sigma model.

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%

Don't show me this again

This is one of over 2,200 courses on OCW. Find materials for this course in the pages linked along the left.

MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.

No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.

Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.

Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse (just remember to cite OCW as the source.)

Learn more at Get Started with MIT OpenCourseWare