nLab generalized complex geometry

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

• choices of sub-Lie algebroids of $\mathfrak{c}(X)$ encode (almost) Dirac structures and – after complexification – generalized complex structures;

• choices of sections of the canonical morphism $\mathfrak{c}(X) \to T X$ 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

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

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.

Examples

• generalized Calabi-Yau manifold

Related concepts

• generalized tangent bundle, generalized vielbein

• type II geometry

• exceptional generalized geometry

• Bn-geometry

• generalized contact 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 Oxford Durham Symposium 2005 (pdf)

• Nigel Hitchin, Lectures on generalized geometry (arxiv/1008.0973)

With emphasis on the role of G-structures:

• Paul Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59: 169-242, 2011 (arXiv:1006.1536)

In relation to Cartan geometry::

• Falk Hassler, Ondrej Hulik, David Osten: Current Algebra and Generalised Cartan Geometry [arXiv:2409.00176]

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, euclid MR2012a:81255 pdf

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).

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

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

• Mariana Graña, Ruben Minasian, Michela Petrini, Daniel Waldram, T-duality, generalized geometry and non-geometric backgrounds, J. High Energy Phys. 2009, no. 4, 075, 39 pp. arXiv:0807.4527 MR2010i:81323 doi

• David Andriot, Ruben Minasian, Michela Petrini, Flux backgrounds from twists, J. High Energy Phys. 2009, no. 12, 028 arXiv:0903.0633 MR2011c:81201 doi

• Anthony Ashmore, Charles Strickland-Constable, David Tennyson, Daniel Waldram. Generalising $G_2$ geometry: involutivity, moment maps and moduli. JHEP 2021, no. 1 (2021): 1-66. (doi158)).