nLab generalized tangent bundle

Definition

In the context of generalized complex geometry one says for $X$ a manifold, $T X$ its tangent bundle and $T^* X$ the cotangent bundle that the fiberwise direct sum-bundle $T X \oplus T^* X$ is the generalized tangent bundle.

More generally, a vector bundle $E \to X$ that sits in an exact sequence $T^* X \to E \to T X$ is called a generalized tangent bundle, such as notably those underlying a Courant Lie 2-algebroid over $X$.

Properties

As an associated bundle

The ordinary tangent bundle is the canonical associated bundle to the general linear group-principal bundle classified by the morphism

$$g_{T X} : X \to \mathbf{B} GL(n)$$

to the smooth moduli stack of $GL(n)$.

Similarly there is a canonical morphism

$$(g_{T X}, g^{-T}_{T X}) : X \to \mathbf{B} O(n,n)$$

to the moduli stack which is the delooping of the Narain group $O(n,n)$. This classifies the $O(n,n)$-principal bundle to which $T X \oplus T^* X$ is associated.

Reduction of structure group

Where a reduction of the structure group of the tangent bundle along $\mathbf{B} O(n) \hookrightarrow \mathbf{B} GL(n)$ is equivalently a vielbein/orthogonal structure/Riemannian metric on $X$, so a reduction of the structure group of the generalized tangent bundle along $\mathbf{B} (O(n) \times O(n)) \to \mathbf{B}O(n,n)$ is a generalized vielbein, defining a type II geometry on $X$.

Other reductions yield other geometric notions, for instance:

• reduction along $U(n,n) \to O(2n,2n)$ is a generalized complex structure;

• further reduction along $SU(n,n) \to U(n,n) \to O(2n,2n)$ is a generalized Calabi-Yau manifold structure.

Spin(8)-subgroups and reductions to exceptional geometry

reduction	from spin group	to maximal subgroup
Spin(7)-structure	Spin(8)	Spin(7)
G₂-structure	Spin(7)	G₂
CY3-structure	Spin(6)	SU(3)
SU(2)-structure	Spin(5)	SU(2)
generalized reduction	from Narain group	to direct product group
generalized Spin(7)-structure	$Spin(8,8)$	$Spin(7) \times Spin(7)$
generalized G₂-structure	$Spin(7,7)$	$G_2 \times G_2$
generalized CY3	$Spin(6,6)$	$SU(3) \times SU(3)$

see also: coset space structure on n-spheres

Related concepts

• tangent bundle

• Courant Lie 2-algebroid

• exceptional tangent bundle