nLab
generalized complex geometry

Idea

What is called generalized complex geometry is the study of the geometry of symplectic Lie 2-algebroid called standard Courant algebroids $𝔠(X)$ (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

Generalized complex structure 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

$$J:V\to V$$J : V \to V

such that $J\circ J=-{\mathrm{id}}_V$. And a symplectic structure on $V$ is equivalently a linear isomorphism

$$\omega :V\to V^{*}$$\omega : V \to V^*

such that

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

$$𝒥:V\oplus V^{*}\to V\oplus V^{*}$$\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 ${𝒥}^2=-\mathrm{id}$;

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

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

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

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

is a generalized complex structure on $V$.

Generalized complex structure on a manifold

(…)

Properties

One finds (as described at standard Courant algebroid) that

• choices of sub-Lie algebroids of $𝔠(X)$ encode Dirac structure?s and – after complexification? – generalized complex structures;

• choices of sections of the canonical morphism $𝔠(X)\to TX$ to the tangent Lie algebroids encode generalized Riemannian metrics: pairs consisting of a (possibly pseudo-)Riemannian metric and a 2-form.

  In applications in string theory, this encodes the field of gravity and the Kalb–Ramond field, respectively. (There are also proposals for how the dilaton field appears in this context.)

In components these are structures found on the vector bundle

$$TX\oplus T^{*}X,$$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

• complex geometry,
• symplectic geometry, and
• Riemannian geometry.

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

Related concepts

• generalized tangent bundle

• exceptional generalized complex geometry

References

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

• Marco Gualtieri, Generalized complex geometry (arXiv:math/0401221)

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

• Nigel Hitchin, B-Fields, gerbes and generalized geometry (pdf)
• Nigel Hitchin, Lectures on generalized geometry (arxiv/1008.0988)

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

• Alberto Cattaneo, Jian Qiub, Maxim Zabzine, 2D and 3D topological field theories for generalized complex geometry (pdf)

Geometry of supergravity

Generalized complex geometry and variant of exceptional generalized complex geometry are natural for describing supergravity background compactifications in string theory with their T-duality and U-duality symmetries.

See for instance

• Ian Ellwood, NS-NS fluxes in Hitchin’s generalized geometry (arXiv:hep-th/0612100)

• Mariana Graña, Ruben Minasian, Michela Petrini, Daniel Waldram, T-duality, Generalized Geometry and Non-Geometric Backgrounds (arXiv:0807.4527)

• Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362)

• David Andriot, Ruben Minasian, Michela Petrini, Flux backgrounds from Twists (arXiv:0903.0633)