geometry of physics -- G-structure and Cartan geometry

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geometry of physics. See there for background and contextprevious chapter

geometry of physics – manifolds and orbifolds

**$G$-Structure and Cartan geometry**- Model Layer
- $G$-Structure
- Examples
- Vielbein, orthogonal structure, Riemannian geometry
- Almost complex structure
- Almost Hermitean structure
- Almost symplectic structure
- Metaplectic structure
- Metalinear structure
- Generalized complex geometry
- Type II geometry
- Generalized Calabi-Yau structure
- Exceptional generalized geometry
- Spin structure, String structure, Fivebrane structure
- Semantics Layer
- Syntax Layer

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

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

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

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

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

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

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

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.

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*.

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 $U$:

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

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. .

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.

(…)

Created on March 19, 2015 at 22:06:34. See the history of this page for a list of all contributions to it.