nLab generalized tangent bundle

Contents

Context

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

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

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:

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto 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)$