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

Contents

This entry contains one chapter of geometry of physics. See there for background and context

previous chapter geometry of physics – manifolds and orbifolds

Contents

GG-Structure and Cartan geometry

Model Layer

GG-Structure

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

given a KK-principal bundle

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

a reduction of the structure group along GKG \to K is

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

Examples

Vielbein, orthogonal structure, Riemannian geometry

reduction of the structure group along

BO(n)BGL(n)\mathbf{B}O(n) \to \mathbf{B}GL(n)

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

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

Electromagnetism in gravitational background

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

Einstein-Maxwell theory

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

Definition

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

X G τ X BGL(V). \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 H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)}. Write GStrucH /BGL(V)G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(V)} for the given map BGBGL(V)\mathbf{B}G\to \mathbf{B}GL(V) regarded as an object in the slice. Then a GG-structure according to def. is equivalently a choice of morphism in H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)} of the form

c:τ XGStruc. \mathbf{c} \;\colon\; \tau_X \longrightarrow G\mathbf{Struc} \,.

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

Example

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

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

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

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

We call this the left invariant GG-structure.

Definition

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

τ U τ V τ X c li c GStruc. \array{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && \swArrow && \tau_X \\ & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G \mathbf{Struc} } \,.

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

τ relU τ V τ X c li c GStruc. \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)GL(V) on VV which linearizes to the canonical GL(V)GL(V)-action on 𝔻 e V\mathbb{D}^V_e by def. . Form the semidirect product GL(V)VGL(V) \rtimes V. Consider any group homomorphism GGL(V)G\to GL(V).

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

Remark

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

Traditionally this is rarely considered, matching the fact that ordinary vector spaces, regarded as translation groups VV, 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 VV may carry intrinsic torsion, and therefore first-order integrable GG-structures on VV-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

(…)

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