Lectures on Generalized Complex Geometry for Physicists

TL;DR

This work provides a concise yet comprehensive introduction to Generalized Complex Geometry (GCG) and its applications to string theory. It develops the mathematical underpinnings—G-structures, generalized tangent bundles, pure spinors, and SU(3)×SU(3) structures—and translates supersymmetry conditions of type II supergravity into differential constraints on pure spinors, including AdS4 backgrounds. It then applies this framework to D-brane physics via generalized calibrations and generalized submanifolds, linking worldvolume flux, RR couplings, and brane stability to the ambient geometric data. The resulting picture unifies complex and symplectic geometry in flux backgrounds and provides a powerful toolkit for constructing and analyzing supersymmetric flux compactifications and brane embeddings, with AdS/CFT contexts exemplified by AdS4/CFT3 dualities.

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.

Figures (4)

• Figure 1: A set of non-degenerate tensors describes a $G$-structure. On the left: in the special case of the figure we assume that the structure group is already reduced to O($d$) (see example \ref{['metricex']}). On the right: an everywhere non-vanishing vector field $v$ is introduced. Because of the existence of this vector field it is possible to construct a reduced frame bundle, where on the overlap between the patches only the rotations that leave the vector invariant are allowed as transition functions, i.e. (proper and improper) rotations in a plane orthogonal to the $v$-axis, making up O($d-1$). The figure is inspired by a similar one from a talk by Davide Cassani.
• Figure 2: When both $T$ and $N$ are integrable one can foliate the manifold in two ways: either with leaves along $T$ or with leaves along $N$.
• Figure 3: Minkowski and AdS$_4$ compactifications indicated in the space of SU(3)$\times$SU(3)-structures.
• Figure 4: $(\Sigma,{\cal F})$ and $(\Sigma',{\cal F}')$ are in the same generalized homology class iff there exists an interpolating D-brane $(\tilde{\Sigma},\tilde{{\cal F}})$.

Theorems & Definitions (77)

• Definition 2.1
• Definition 2.2
• Example 2.1: Metric and orientation
• Example 2.2: Almost complex structure
• Example 2.3: Pre-symplectic structure
• Example 2.4: Hermitian metric
• Example 2.5: Almost product structure
• Definition 2.3
• Definition 2.4
• Definition 2.5