nLab geometry of physics -- G-structure and Cartan geometry

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$G$-Structure and Cartan geometry

Model Layer

$G$-Structure

$$\mathbf{B}G \to \mathbf{B}K$$

given a $K$-principal bundle

$$\array{ \tilde X &\to &\mathbf{B}K \\ \downarrow^{\mathrlap{\simeq}} \\ X }$$

a reduction of the structure group along $G \to K$ is

$$\array{ \tilde X &&\to&& \mathbf{B}G \\ & \searrow &\swArrow_{e}& \swarrow \\ && \mathbf{B}K }$$

Examples

Vielbein, orthogonal structure, Riemannian geometry

• vielbein, orthogonal structure

reduction of the structure group along

$\mathbf{B}O(n) \to \mathbf{B}GL(n)$

$$\array{ \tilde X &&\to&& \mathbf{B}O(n) \\ & {}_{\mathllap{\vdash T \Sigma}}\searrow &\swArrow_{e}& \swarrow \\ && \mathbf{B}GL(n) }$$

$e$ is vielbein: definition of an orthonormal frame? at each point

Electromagnetism in gravitational background

example: the other 2 Maxwell equations: $\mathbf{d} \star F = j_{el}$.

Einstein-Maxwell theory

Almost complex structure

• almost complex structure

Almost Hermitean structure

• almost Hermitean structure

Almost symplectic structure

• almost symplectic structure

Metaplectic structure

• metaplectic structure

Metalinear structure

• metalinear structure

Generalized complex geometry

• generalized complex geometry

Type II geometry

• type II geometry

Generalized Calabi-Yau structure

• generalized Calabi-Yau manifold

Exceptional generalized geometry

• exceptional generalized geometry

Spin structure, String structure, Fivebrane structure

• spin structure

• string structure

• fivebrane structure

Semantics Layer

Definition

Given a homomorphism of groups $G \longrightarrow GL(V)$, a G-structure on a $V$-manifold $X$ is a lift $\mathbf{c}$ of the frame bundle $\tau_X$ of prop. through this map

$$\array{ X && \stackrel{}{\longrightarrow} && G \\ & {}_{\mathllap{\tau_X}}\searrow &\swArrow& \swarrow \\ && \mathbf{B}GL(V) } \,.$$

Remark

As in remark , it is useful to express def. in terms of the slice topos $\mathbf{H}_{/\mathbf{B}GL(V)}$. Write $G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(V)}$ for the given map $\mathbf{B}G\to \mathbf{B}GL(V)$ regarded as an object in the slice. Then a $G$-structure according to def. is equivalently a choice of morphism in $\mathbf{H}_{/\mathbf{B}GL(V)}$ of the form

$$\mathbf{c} \;\colon\; \tau_X \longrightarrow G\mathbf{Struc} \,.$$

In other words, $G\mathbf{Struc} \in \mathbf{H}_{/\mathbf{B}GL(v)}$ is the moduli stack for $G$-structures.

Example

A choice of framing $\phi$, def. , on a $V$-manifold $X$ induces a G-structure for any $G$, given by the pasting diagram in $\mathbf{H}$

$$\array{ X &\longrightarrow& \ast &\longrightarrow& \\ & \searrow & \downarrow & \swarrow \\ && \mathbf{B}GL(V) }$$

or equivalently, via remark and remark , given as the composition

$$\mathbf{c}_{li} \;\colon\; \tau_X \stackrel{\phi}{\longrightarrow} V\mathbf{Frame} \longrightarrow G\mathbf{Struc}\,.$$

We call this the left invariant $G$-structure.

Definition

For $X$ a $V$-manifold, then a G-structure on $X$, def. , is integrable if for any $V$-atlas $V \leftarrow U \rightarrow X$ the pullback of the $G$-structure on $X$ to $V$ is equivalent there to the left-inavariant $G$-structure on $V$ of example , i.e. if we have an correspondence in the double slice topos $(\mathbf{H}_{/\mathbf{B}GL(V)})_{/G\mathbf{Struc}}$ of the form

$$\array{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && \swArrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} } \,.$$

The $G$-structure is infintesimally integrable if this holds true at at after restriction along the relative shape modality $\flat^{rel} U \to U$, def. , to all the infinitesimal disks in $U$:

$$\array{ && \tau_{\flat^{rel}U} \\ & \swarrow && \searrow \\ \tau_V && \swArrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} } \,.$$

Definition

Consider an infinity-action of $GL(V)$ on $V$ which linearizes to the canonical $GL(V)$-action on $\mathbb{D}^V_e$ by def. . Form the semidirect product $GL(V) \rtimes V$. Consider any group homomorphism $G\to GL(V)$.

A $(G\to G\rtimes V)$-Cartan geometry is a $V$-manifold $X$ equipped with a $G$-structure, def. . The Cartan geometry is called (infinitesimally) integrable if the $G$-structure is so, according to def. .

Remark

For $V$ an abelian group, then in traditional contexts the infinitesimal integrability of def. comes down to the torsion of a G-structure vanishing. But for $V$ a nonabelian group, this definition instead enforces that the torsion is on each infinitesimal disk the intrinsic left-invariant torsion of $V$ itself.

Traditionally this is rarely considered, matching the fact that ordinary vector spaces, regarded as translation groups $V$, are abelian groups. But super vector spaces regarded (in suitable dimension) as super translation groups are nonabelian groups (we discuss this in detail below in The super-Klein geometry: super-Minkowski spacetime). Therefore super-vector spaces $V$ may carry intrinsic torsion, and therefore first-order integrable $G$-structures on $V$-manifolds are torsion-ful.

Indeed, this is a phenomenon known as the torsion constraints in supergravity. Curiously, as discussed there, for the case of 11-dimensional supergravity the equations of motion of the gravity theory are equivalent to the super-Cartan geometry satisfying this torsion constraint. This way super-Cartan geometry gives a direct general abstract route right into the heart of M-theory.

Syntax Layer

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