# nLab generalized complex geometry

Contents

complex geometry

### Examples

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Generalized complex geometry is the study of the geometry of symplectic Lie 2-algebroid called standard Courant algebroids $\mathfrak{c}(X)$ (over a smooth manifold $X$).

This geometry of symplectic Lie 2-algebroids turns out to unify, among other things, complex geometry with symplectic geometry. This unification notably captures central aspects of T-duality.

## Definition

### On a vector space

Let $V$ be a finite dimensional vector space over the real numbers.

Recall that a complex structure on $V$ is a linear map

$J : V \to V$

such that $J \circ J = - id_V$. And a symplectic structure on $V$ is equivalently a linear isomorphism

$\omega : V \to V^*$

such that

$\omega^* = - \omega \,,$

where $V^*$ denotes the dual vector space and $\omega^*$ the dual linear map.

The following definition may be thought of as combining these two concepts.

###### Definition

A generalized complex structure on $V$ is a linear map

$\mathcal{J} : V \oplus V^* \to V \oplus V^*$

(an endomorphism of the direct sum of $V$ with its dual vector space)

such that it is both

1. a complex structure on $V \oplus V^*$ in that $\mathcal{J}^2 = - id$;

2. a symplectic structure on $V \oplus V^*$ in that $\mathcal{J}^* = - \mathcal{J}$.

The following shows that this is indeed a joint generalization of complex and symplectic structures.

###### Examples

Let $J : V \to V$ be an ordinary complex structure on $V$. Then the linear endomorphism of $V \oplus V^*$ defined by matrix calculus as

$\mathcal{J}_j := \left( \array{ -J & 0 \\ 0 & J^* } \right)$

is a generalized complex structure on $V$.

Similarly, let $\omega : V \to V^*$ be an ordinary symplectic structure on $V$. then the endomorphism

$\mathcal{J}_\omega := \left( \array{ 0 & - \omega^{-1} \\ \omega & 0 } \right)$

is a generalized complex structure on $V$.

### On a manifold

A generalized complex structure on a manifold is a generalized complex structure on the fibers of the generalized tangent bundle.

(…)

### In terms of reduction of the structure group

A generalized complex structure on $V \oplus V^*$ is equivalently a reduction of the structure group along the inclusion

$U(n,n) \hookrightarrow O(2n,2n) \,,$

where the left hand is identified as $U(n,n) = O(2n,2n) \cap GL(2n, \mathbb{C})$ (Gualtieri, prop. 4.6).

The analog of this with the unitary group replaced by the orthogonal group yields type II geometry.

## Properties

One finds (as described at standard Courant algebroid) that

In components these are structures found on the vector bundle

$T X \oplus T^* X \,,$

the direct sum of the tangent bundle with the cotangent bundle of $X$.

Generalized complex geometry thus generalizes and unifies

It was in particular motivated by the observation that this provides a natural formalism for describing T-duality.

## Examples

#### General

Generalized complex geometry was proposed by Nigel Hitchin as a formalism in differential geometry that would be suited to capture the phenomena that physicists encountered in the study of T-duality. It was later and is still developed by his students, notably Gualtieri and Cavalcanti.

A standard reference is the PhD thesis

A survey set of slides with an eye towards the description of the Kalb-Ramond field and bundle gerbes is

#### As targets for $\sigma$-models

Generalized complex structures may serve as target spaces for sigma-models. Relations to the Poisson sigma-model and the Courant sigma-model are discussed in

#### Mirror symmetry

• Oren Ben-Bassat, Mirror symmetry and generalized complex manifolds. I. The transform on vector bundles, spinors, and branes, J. Geom. Phys. 56 (2006), no. 4, 533–558 math.AG/0405303 MR2006k:53067 doi

#### Geometry of supergravity

Generalized complex geometry and variants of exceptional generalized complex geometry are natural for describing supergravity background compactifications in string theory with their T-duality and U-duality symmetries (non-geometric vacua).