Generalized Complex Geometry (University of Zurich)

Abstract

Generalized complex geometry is a modern approach to unify complex, symplectic, Poisson and more structures. All these can be formulated as a generalized complex structure, an endomorphism of the sum of the tangent and cotangent bundles T+T^* that squares to -1.

Objective

The goal is to develop an understanding of the foundations of generalized complex geometry, compare it with familiar geometric structures and study some applications.

Content

Generalized complex geometry is a modern approach to unify complex, symplectic, Poisson and more structures. All these can be formulated as a generalized complex structure, an endomorphism of the sum of the tangent and cotangent bundles T+T^* that squares to -1. Alternatively one can encode a generalized complex structure by its +i eigenbundle that forms a Dirac structure or as a pure spinor for the Clifford algebra of T+T^*. We will explore T+T^* with its natural split signature metric, Courant bracket and its symmetries which are an extension of smooth diffeomorphisms. A reduction procedure for Courant algebroids and generalized complex structures generalizes both symplectic reduction and holomorphic reduction of complex manifolds. Subobjects in this category are generalized complex branes. We will see how they mediate between Lagrangian submanifolds with a flat bundle and complex submanifolds with a holomorphic bundle. The deformation theory of generalized complex structures extends the deformation theory of complex and symplectic structures. In this context a Kähler structure can be generalized to recover bihermitian geometry discovered by Gates, Hull and Roček. Interesting results and applications include among other topics mirror symmetry and (2,2) supersymmetric sigma models. We will see examples of genralized Kähler structures on CP^2 and on instanton moduli spaces.

Resources