Lectures on Generalized Complex Geometry for Physicists

TL;DR

This paper reviews Generalized Complex Geometry and its applications in string theory, focusing on supersymmetric flux compactifications and D-brane embeddings, with detailed examples like AdS4 compactifications relevant to AdS/CFT correspondence.

Contribution

It provides a comprehensive overview of Generalized Complex Geometry and introduces new differential conditions for supersymmetry and generalized calibrations in string theory contexts.

Findings

Derived supersymmetry conditions using pure spinors

Explored generalized calibrations for D-branes

Analyzed AdS4 compactifications in detail

Abstract

In these lectures we review Generalized Complex Geometry and discuss two main applications to string theory: the description of supersymmetric flux compactifications and the supersymmetric embedding of D-branes. We start by reviewing G-structures, and in particular SU(3)-structure and its torsion classes, before extending to Generalized Complex Geometry. We then discuss the supersymmetry conditions of type II supergravity in terms of differential conditions on pure spinors, and finally introduce generalized calibrations to describe D-branes. As examples we discuss in some detail AdS4 compactifications, which play a role as the geometric duals in the AdS4/CFT3-correspondence.