nLab
generalized complex geometry

Context

Complex geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive Unknown characterUnknown character discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \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 𝔠(X)\mathfrak{c}(X) (over a smooth manifold XX).

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 VV be a finite dimensional vector space over the real numbers.

Recall that a complex structure on VV is a linear map

J:VV J : V \to V

such that JJ=id VJ \circ J = - id_V. And a symplectic structure on VV is equivalently a linear isomorphism

ω:VV * \omega : V \to V^*

such that

ω *=ω, \omega^* = - \omega \,,

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

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

Definition

A generalized complex structure on VV is a linear map

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

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

such that it is both

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

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

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

Examples

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

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

is a generalized complex structure on VV.

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

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

is a generalized complex structure on VV.

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 VV *V \oplus V^* is equivalently a reduction of the structure group along the inclusion

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

where the left hand is identified as U(n,n)=O(2n,2n)GL(2n,)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

TXT *X, T X \oplus T^* X \,,

the direct sum of the tangent bundle with the cotangent bundle of XX.

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

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

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

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

Revised on August 19, 2015 11:23:14 by Urs Schreiber (131.227.23.37)