On Cartan Connections and the Geometric Structure of Space...

On Cartan Connections and the Geometric Structure of Space-Time - Gabriel Catren - Approaches to Quantum Gravity, Université Blaise-Pascal, Clermont-Ferrand, France. Jan. 6-10th 2014 - p. 1/53

On Cartan Connections and the GeometricStructure of Space-Time

Gabriel CatrenLaboratoire SPHERE - Sciences, Philosophie, Histoire (UMR 7219) - Université Paris Diderot/CNRS

ERC Starting Grant Philosophy of Canonical Quantum Gravity

Introduction

Outline

Objective

General Relativity vs.

Yang-Mills Theory

Gauging Gravity

Physical Unification of ω and

θ Mathematical Unification of

ω and θ Historical Landmarks

Cartan’s Program

Implementing Cartan’s Program

Cartan Connection

Cartan Curvature

Development


Outline

Euclidean Geometry

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Riemann’s Princ. of Local Flatness

**VVVVVVVVVVVVVVVVVVKlein’s Erlangen Program

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Cartan’s Princ. of Local Affine Symmetry

Einstein-Cartan’s Gravity

Einstein’s Background Independ.

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Einstein’s Geometrization Program

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Weyl’s infinitesimal geom.

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Mach’s Principle

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Introduction

Outline

Objective


Yang-Mills Theory

Gauging Gravity




Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Objective

• We shall introduce the geometric underpinning of some current attempts...

... trying to reformulate General Relativity as a gauge theory,

... namely, the theory of Cartan geometries .

Introduction

Outline

Objective


Yang-Mills Theory

Gauging Gravity




Cartan’s Program


Cartan Connection

Cartan Curvature

Development


General Relativity vs. Yang-Mills Theory

• General relativity describes the metric geometry of space-time M :

[g] =g

Diff(M) Unique metric & torsionless Levi-Civita connection

• Yang-Mills theory describes the connective geometry of “internal” spaces overspace-time:

[ωG] =ωG

AutV (PG)

where PG → M is a G-principal bundle with G a Lie group and ωG an Ehresmannconnection .

• Since we know how to quantize the latter,...

... we could try to use this knowledge to quantize gravity...

... by trying to reduce as much as possible the formal differences between GRand Yang-Mills theories.

Introduction

Outline

Objective


Yang-Mills Theory

Gauging Gravity




Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Gauging Gravity

• In order to “gauge” gravity, we have to pass from the “Metric” Einstein-Hilbertformulation ...

... to a theory describing a bundle over space-time M ...

... endowed with a dynamical connection “gauging” a local symmetry.

• “Tetrad-connection” Palatini formulation :

.Ehresmann connection ω for the local Lorentz group (called spin conn.),

.tetrads , vierbeine, or moving frames θ ∼ √g.

• Now, the tetrads are additional fields with no analog in Y-M...

... and the spin conn., far from being a fundamental structure, can be obtainedfrom the tetrad.

• However, in the 70’ some people started to consider the spin connection ω and thetetrad θ...

... as different components on a single field ω + θ.

Introduction

Outline

Objective


Yang-Mills Theory

Gauging Gravity




Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Physical Unification of ω and θ

• Witten (1988): quantization of 2+1 gravity by formulating the latter as a pureChern-Simons action with an affine structural group.

(2+1 Dimensional Gravity as an Exactly Soluble System, Nuclear Physics B311).

• MacDowell & Mansouri (1977): reformulation of GR in terms of a Yang-Mills action witha gauge symmetry partially broken “by hand”.

(Unified geometric theory of gravity and supergravity, Phys. Rev. Lett., 38).

• Stelle & West (1980): analysis of the MacDowell-Mansouri action in terms of aspontaneous symmetry breaking mechanism.

(Spontaneously broken de Sitter symmetry and the gravitational holonomy group, PhysicalReview D, vol. 21, n 6)

• Freidel-Starodubtsev (2005): reobtention of the MacDowell-Mansouri action byperturbating a topological BF-theory.

(Freidel & Starodubtsev, Quantum gravity in terms of topological observables,hep-th/0501191)

• Now, what is the geometric counterpart of this unification betwee n ω and θ?

Introduction

Outline

Objective


Yang-Mills Theory

Gauging Gravity




Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Mathematical Unification of ω and θ

• In the framework of the theory of Cartan geometries ...

... the unified field ω + θ is understood as a connection (!!!)

• However, far from being an Ehresmann connection, ω + θ is a Cartan connection .

• The difference between GR & Y-M theory,...

... is reduced to the difference between Ehresmann and Cartan connections.

• Ref.: Derek Wise (2010), MacDowell–Mansouri gravity and Cartan geometry, Class.Quantum Grav., 27.

Introduction

Outline

Objective


Yang-Mills Theory

Gauging Gravity




Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Historical Landmarks

• In the article

Sur les variétés à connexion affine et la théorie de la relativ ité généralisée (première partie) ,

(Annales scientifiques de l’E.N.S, 3e série, tome 40, pp. 325-412, 1923),

... E. Cartan introduced the Cartan connections and proposed a generalization of GR togeometries with non-zero torsion.

• In the article

Les connexions infinitésimales dans un espace fibré différen tiable ,

(Seminaire N. Bourbaki, 1948-1951, exp. n24, pp. 153-168, 1950),

... C. Ehresmann formalized the notion of conn. by means of the theory of fiber bundles...

... and showed that Cartan conn. are a particular case of a more grl. notion, i.e. the notionof Ehresmann conn.

Introduction

Cartan’s Program

Generalizations of Euclidean

Geometry

Riemann’s Generalization

Klein’s Generalization

Klein Geometries

The Limitations of Both

Generalizations Cartan’s Sublation of the

Chiasm Curving vs. Symmetry

Breaking Local Flatness vs. Local

Symmetry

Cartan’s Principle of Local

Affine Symmetry

Poincaré Group

Adapted Local Klein Models


Cartan Connection

Cartan Curvature

Development


Generalizations of Euclidean Geometry

• Euclidean geometry was generalized in two independent directions:

Euclidean Geometry

// Klein Geometries

Riemannian Geometry

... namely,

... by allowing curvature (Riemann’s generalization) ,

... by allowing symmetry groups others than the Euclidean group (Klein’sgeneralization) .

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development



• Riemann’s Principle of Local Flatness :

A curved manifold can be infinitesimally approximated by flat tangent models .

• This means that Euclidean geometry lost the status of being the unique possiblegeometry...

... and at the same time acquired a universal scope given...

... by the fact that it can be used to infinitesimally model any curved space.

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development



• Euclidean geometries are homogeneous spaces ,...

... i.e. a maximally symmetric spaces endowed with a transitive action of theEuclidean group G.

• Klein’s generalization of Euclidean geometry :

Why should we privilege Euclidean geometry instead of considering otherpossible homogeneous (or maximally symmetric) spaces associated to other symmetrygroups G?

• Klein’s Erlangen program amounts to study the “Galoisian” correspondence between

Maximally symmetric spaces ! Symmetry Groups.

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development


Klein Geometries

• A Klein geometry (M0, G) is a homogeneous G-space , i.e. a connected space M0

endowed with a transitive action of a Lie group G.

• Given x ∈ M0, the surjective map:

πx : G → M0

g 7→ g · x.

induces a bijection

G/H≃−→ M0,

where H = π−1x (x) ⊂ G is the isotropy group of x.

• Whereas H is the “group of rotations ” leaving invariant x, G/H is the “group oftranslations ” along M0.

• The Klein geometry (G/H, G) induces a canonical H-fibration G → G/H, where G

is called the principal group (“Haugtgruppe”) of the geometry.

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development


The Limitations of Both Generalizations

• Riemann’s and Klein’s generalization of Euclidean geometry are both limited, since...

... Riemann only considers infinitesimal Euclidean models (instead ofhomogeneous models with other symmetry groups)...

... Klein only considers homogeneous geometries (instead of generalnon-homogeneous manifolds).

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development


Cartan’s Sublation of the Chiasm

• Cartan bypasses this antagonism between Riemannian and Kleinian geometry...

... by considering “generalized spaces” (Cartan geometries )...

... which are infinitesimally modeled by Klein geometries (therebygeneralizing Riemannian geometry)...

... but locally deformed by (a new form of) “Cartan curvature” (therebygeneralizing Kleinian geometry).

Euclidean Geometry

// Klein Geometries

Riemannian Geometry // Cartan Geometries .

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development


Curving vs. Symmetry Breaking

• Now, the local Klein models,...

... far from being necessarily flat,...

... are by definition maximally symmetric.

• Hence, the local deformation encoded by the Cartan curvature should not beunderstood as a curving...

... but rather as a symmetry breaking of the symmetries of the local Kleinmodels.

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development


Local Flatness vs. Local Symmetry

• All in all, in Cartan geometries the tangent models will not be given by flat models as inRiemannian geom., but rather by symmetric models .

• Cartan’s principle of local symmetry amounts ...

... to promote to the foreground the notion of local symmetry

... to the detriment of Riemann’s notion of local flatness .

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development


Cartan’s Principle of Local Affine Symmetry

• Now, there are different Klein models M0 = G/H...

... that differ in theirs principal groups G and share the same isotropy group H.

• For instance,

.Minkoswki space,

.de Sitter space,

.anti-de Sitter space

... are Klein geometries with different G and the same H, namely the Lorentzgroup.

• Therefore, Cartan’s twofold generalization would be trivial if we only considered theinfinitesimal linear H-symmetries of a point.

• Cartan’s principle of local symmetry must be understood as a principle of local affinesymmetry ...

... encoding both the rotational and the translational symmetries of the localKlein models.

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development


Poincaré Group

• For instance, Mink. space-time M is a homog. space , which means that there is agroup

Π(3, 1) = R4

⋊ SO(3, 1), (Poincaré group)

... that acts transitively on M .

• Now, the principle of local affine symmetry requires to take into account not only theLorentz subgroup SO(3, 1) of Π(3, 1),...

... but also the translational symmetry of Minkowski space-time.

• In Cartan’s approach,

.ω “gauges” the local Lorentz symmetry,

.θ “gauges” the translational symmetry.

• Hence, ω + θ “gauges” all the symmetries of the local model.

Introduction

Cartan’s Program


Geometry



Klein Geometries





Symmetry


Affine Symmetry

Poincaré Group



Cartan Connection

Cartan Curvature

Development



• Cartan’s ideas open the possibility of choosing tangent Klein models M0 “adapted” tothe base manifold M .

• For instance, we could choose a Klein model M0 that has the same topology than M .

• After all, if the topology of M is not that of Mink. S-T, why should we use Mink. S-T astangent model?

• By choosing a Klein model M0, one fixes the homogeneous standard with respect towhich we shall compare the curvature and the torsion of M .

• The Cartan “curvature” and “torsion” of M are zero when their ordinary curvature andtorsion are that of M0.

• Roughly speaking, we can “absorbe” the curvature of M by using a curvedhomogeneous tangent model.

Introduction

Cartan’s Program


Building a Cartan Geometry

Attaching and Soldering

Ehresmann Connection

Vertical Parallelism

Affine Frames

Associated Bundle in

Homogeneous Spaces

Reduced H-bundle

Reduction in a Nutshell

Attaching Klein Models to M

Partial Gauge Fixing

H-reductions as Partial

Trivializations Canonical G-extension of an

H-bundle

Cartan Connection

Cartan Curvature

Development



• In order to build a Cartan geometry on M , we shall select a local Klein modelM0 ≃ G/H such that dim(M0) = dim(M)

• In the cases that are relevant in gravitational physics:

.the affine group G is...

... the Poincaré group R4

⋊ SO(3, 1) for Λ = 0,

... the de Sitter group SO(4, 1) for Λ > 0

... the anti-de Sitter group SO(3, 2) for Λ < 0.

.H = Lorentz group SO(3, 1).

.G/H = group of translations of the Klein model M0.

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development



• In order to infinitesimally model the geometry of M by means of M0 ≃ G/H, we haveto

1) attach (zero-order identification)

2) solder (first-order identification)

... copies Mx0 of the Klein model M0 to each x ∈ M .

• To do so, we shall need additional geometric data, namely

.a symmetry breaking section (reduction of a G-principal bundle to anH-bundle).

.a soldering form .

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development



• We shall start with a G-principal bundle

PG → M

... endowed with an Ehresmann connection , i.e. horizontal equivariant distribution H

defined by means of a G-eq. and g-valued 1-form ωG on PG such that

Hp = Ker (ωG)p ⊂ TpPG.

• The conn. form ωG satisfies:

.R∗gωG = Ad(g−1)ωG, where Ad is the adj. repr. of G on g (G-equivariance ).

.ωG(ξ♯) = ξ (vertical parallelism ), where

♯ : g → TpPG

ξ 7→ ξ♯(f(p)) =d

dλ(f(p · exp(λξ)))|λ=0.

• Important: ωG has values in the Lie algebra g of the structural group G.

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development



• The fundamental vector fields defined by the map

♯ : g → TpPG

define a vertical parallelism , i.e. a trivialization of the vertical bundleV PG

.= ker(dπ) → PG given by

PG × g → VpPG

(p, ξ) 7→ ξ♯(p).

• The curvature of ωG

F = dωG +1

2[ωG, ωG]

is a G-equivariant and horizontal g-valued 2-form on PG.

• The horizontality of F reflects the fact that

ωG([ξ♯, ζ

♯]) = [ωG(ξ

♯), ωG(ζ

♯)],

i.e. that the vertical parallelism ♯ : g → VpPG implementing the vertical G-action is a Liealgebra homomorphism.

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development


Affine Frames

• Since the structural group G of the fiber bundle PG → M is the affine group of theKlein model M0...

... a fiber in PG is the set of all the affine frames (m, e) in M0, where an affineframe consists of:

1) a point m in M0

2) a basis e of the vector space TmM0.

• For the moment, these “internal” affine frames have no relation whatsoever to thetangent spaces to M .

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development


Associated Bundle in Homogeneous Spaces

• In order to attach a copy Mx0 of M0 ≃ G/H to each x ∈ M we shall first consider the

associated G-bundle in homog. spaces

PG ×G G/H → M.

• It can be shown that

PG ×G G/H ∼= PG/H.

• Each fiber of PG ×G G/H is a copy Mx0 of the local Klein model M0 ≃ G/H.

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development


Reduced H-bundle

• The reduction of PG to PH can be defined either by a global section

σ : M → PG ×G G/H ∼= PG/H

or, equivalently, by a G-equivariant function

ϕ : PG → G/H, ϕ(pg) = g−1ϕσ(p).

• The reduced H-bundle PH → M is given either by

PH = ϕ−1σ ([e])

or by the pullback of PG along the section σ:

PH = σ∗PG

// PG

M

σ

::PG/H.oo

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development



• All in all, there is one-to-one correspondence between

Reduced H-subbundles PH of PG

m

Global sections σ : M → PG/H

or

Equivariant functions ϕ : PG → G/H

PH = ϕ−1σ ([e]) = σ∗PG

ι //

PG

ϕσ // //

xxqqqqqqqqqqqqqqqqqqqqqqqqqq

G/H

M

σϕ

**PG/H ∼= PG ×G G/Hoo

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development



• The reduction amounts to select a point of attachment σ(x) in each local Klein modelMx

0 ≃ G/H.

• By selecting a point of attachment for each x, we “break” the G-symmetry of Mx0 down

to the isotropy group H.

• The group H encodes the “unbroken” symmetry that fixes the point of attachment σ(x)

in each Mx0 .

• The restriction of the Ehresmann connection ωG to the reduced bundle PH defines aso-called Cartan connection on PH .

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development



• The reduction PH → PG defined by σ can be understood as a...

... gauge fixing of the G/H-translational local invariance...

... i.e., as a partial gauge fixing of the G-invariance.

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development


H-reductions as Partial Trivializations

• In particular, a reduction of PG to a idG-principal bundle is given either by a section

s : M → PG ×G (G/ idG) ∼= PG/ idG = PG

or by a G-equivariant function

ϕ : PG → G/ idG = G,

where the reduced idG-bundle is

PidG = s∗PG = ϕ

−1(idG).

• Hence, a complete reduction with H = idG is a trivialization s : M → PG of PG.

• Instead of selecting a unique frame for each x as the trivialization s does...

... a H-reduction can be considered a sort of partial trivialization of PG thatselects a non-trivial H-set of frames for each x.

Introduction

Cartan’s Program






Affine Frames


Homogeneous Spaces

Reduced H-bundle






H-bundle

Cartan Connection

Cartan Curvature

Development


Canonical G-extension of an H-bundle

• Whereas the reduction of the G-bundle PG depends on the existence of a global section

σ : M → PG/H ∼= PG ×G G/H

... a H-bundle PH → M can be canonically extended to the associated G-bundle

PH ×H G → M,

where the G-action if given by

[(p, g)] · g′ = [(p, gg′)]

... and where the inclusion is given by

ι : PH → PH ×H G

p 7→ [(p, e)].

While the reduction of PG to PH is not canonical,

PH can always be extended to a G-bundle PH ×H G.

Introduction

Cartan’s Program


Cartan Connection

Induced Cartan Connection

on PH Cartan Connections from

Scratch Absolute Parallelism

Reductive Decomposition of

g

Coordinate & Geometric

Soldering Form

Soldering the Local Klein

Models to M Reducing the Frame Bundle

Recovering the Metric

Soldering vs. Canonical Form

Cartan Curvature

Development


Induced Cartan Connection on PH

• Let’s suppose that ωG satisfies

Ker(ωG) ∩ ι∗TPH = 0, ι : PH → PG

... or, equivalently, that the restriction

A.= ι∗(ωG) : TPH → g

of ωG to PH has no null vectors :

Ker(A) = 0.

• This condition means that the horizontal lift of v ∈ TxM cannot be a mere rotation in H.

• Then g-valued 1-form A : TPH → g defines a so-called Cartan connection on PH .

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development


Cartan Connections from Scratch

• A Cartan connection can be directly defined on PH without passing by the reductionmechanism.

• To do so, we do not need a Lie group G,...

... but just a model geometry , i.e. a pair (g, H) and a representation Ad of H

on g.

• A Cartan geometry (PH , A) on M is given by a 1-form (Cartan connection ) on PH

A : TPH → g

such that

.(R∗hA)p = Ad(h−1)Ap for all p ∈ PH and h ∈ H (H-equivariance ).

.A induces a linear iso. TpPH∼= g for all p ∈ PH (absolute parallelism )

.A(ξ♯) = ξ for any ξ ∈ h where ♯ : h → VpPG (vertical parallelism ).

• The 1-form A cannot be an Ehresmann conn. on PH since it is not valued in h.

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development


Absolute Parallelism

• A Cartan connection is a g-valued H-equivariant 1-form on PH ...

... extending the vertical parallelism

VpPH≃−→ h

... to an absolute parallelism

TpPH≃−→ g

... that trivializes the tangent bundle TPH → PH :

PH × g → TPH

(p, ξ) 7→ A−1(ξ)(p).

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development


Reductive Decomposition of g

• Let’s suppose that the model geometry (g, H) is reductive , i.e. that there exists aAd(H)-module decomposition

g = h ⊕ m, Ad(H) · m ⊂ m

(in what follows m.= g/h).

• By composing with the projections, this decomp. of g induces a decomp. of A:

A = ωH + θ,

where the so-called spin connection

ωH : TPHA−→ g

πh−−→ h

is an Ehresmann conn. on PH and the so-called soldering form

θ : TPHA−→ g

πg/h−−−−→ g/h

is a g/h-valued 1-form on PH that is

.horizontal: θ(ξ♯) = 0 for vertical vectors ξ♯ ∈ V PH

.H-eq.: R∗hθ = h−1θ

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development


Coordinate & Geometric Soldering Form

• While σ attaches the local Klein models Mx0 to x ∈ M at σ(x),...

... the g/h-part θ of A = ωH + θ identifies each TxM to the vertical tg. spaceto the local Klein model at σ(x).

• Given the coordinate soldering form

θ : TPH → g/h,

the isomorphism

Ωqhor(PH , g/h)

H ≃ Ωq(M, PH ×H g/h),

induces a geometric soldering form

θ : TM → PH ×H g/h

• The bundle PH ×H g/h can be identified with the bundle of vertical tg. vectors to

PG ×G G/H along σ : M → PG ×G G/H:

PH ×H g/h ≃ Vσ(PG ×G G/H).

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development


Soldering the Local Klein Models to M

• The geometric soldering form

θ : TM → PH ×H g/h

identifies each vector v in TxM with a geometric vector θ(v) in PH ×H g/h,...

... that is with a vertical vector tangent to PG ×G G/H at σ(x).

• The tg. space TxM is thus soldered to the tg. space to the homog. fiber ≃ G/H ofPG ×G G/H at σ(x).

The local Klein models are soldered to M along the section σ.

• The coordinate soldering form

θp : TpPH → g/h

defines the g/h-valued coordinates of θ(v) in the frame p.

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development


Reducing the Frame Bundle

• The existence of a soldering form θ on PH implies that PH is isomorphic to aGL(g/h)-structure , i.e. to a subbundle of the GL(g/h)-principal bundle LM of linearframes on M .

• Indeed, θ defines an application

fθ : PH → LM

p 7→ fθ(p) : g/h → Tπ(p)M

given by

fθ(p)(ξ) = θ

−1p (ξ),

where

θp(x) : Tπ(p)Mθ−→ PH ×H g/h

τ(p,·)−−−−→ g/h

with τ : PH ×M (PH ×H g/h) → g/h the canonical map.

• This application identifies each element p in PH with a frame fθ(p) ∈ LM .

• Since fθ : PH → LM is an H-morphism, fθ(PH) is a H-subbundle of LM .

• Such a reduction of LM amounts to define a Lorentzian metric on M .

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development



• The metric gθ on M can be explicitly defined in terms of an Ad(H)-invariant scalarproduct 〈·, ·〉 of g/h by means of the expression

gθ(v, w) =

⟨

θp(v), θp(w)⟩

,

where θp(x) : TxM → g/h.

• The H-invariance of 〈·, ·〉 implies that gθ(v, w) does not depend on the frame p over x.

The translational part θ of the Cartan connection A...

... by inducing an isomorphism between PH and a H-subbundle of LM ...

... induces a Lorentzian metric gθ on M .

Introduction

Cartan’s Program


Cartan Connection





g


Soldering Form





Cartan Curvature

Development



• The soldering form θ ∈ Ω1(PH , g/h) is the pullback by fθ of the canonical form

θc : T (LM) → g/h

on LM given by:

θc : Te(x)(LM) → g/h

v 7→ e(x)−1

(π∗(v)).

where

e(x) : g/h → TxM

is a frame on TxM .

• Contrary to the canonical form θc on LM , the soldering form θ on PH is not canonical...

... since it comes from the restriction of the arbitrary Ehresmann conn. ωG on PG to PH .

• This is consistent with the fact that θ defines a degree of freedom of the theory.

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Curvature of the Cartan

Connection On Cartan Flatness

(Maurer-)Cartan Flat

Connection Flatness vs. Symmetry

Development


Curvature of the Cartan Connection

• The Cartan curvature F ∈ Ω2(PH , g) of a Cartan geom. (PH , A) is given by

F = dA +1

2[A, A] = Fh + Fg/h

• The curvature R ∈ Ω2(PH , h) of a Cartan geom. (PH , A) is given by

R.= dωH +

1

2[ωH , ωH ] = Fh − 1

2[θ, θ]h

• The torsion T ∈ Ω2(PH , g/h) of a Cartan geom. (PH , A) is given by

T.= dθ + [ωH , θ] = Fg/h − 1

2[θ, θ]g/h

where Fh.= πh F and Fg/h

.= πg/h F

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature





Development


On Cartan Flatness

• In general., Cartan flatness does not imply R = 0 and T = 0:

F = 0 ⇔

R0 = − 12 [θ, θ]h

T0 = − 12 [θ, θ]g/h

• The so-called symmetric models satisfy in addition

[g/h, g/h] ⊆ h,

which implies

Fg/h = T F = (R +1

2[θ, θ]h) + T.

• T naturally appears as the “translational” component of F .

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature





Development


(Maurer-)Cartan Flat Connection

• The canonical examples of flat Cartan connections are given by the “curved” Kleingeometries G/H themselves.

• Indeed, the Maurer-Cartan form AG on the canonical H-fibration G → G/H given by

AG(g) : TgG → g

ξ 7→ (Lg−1 )∗ξ,

is a Cartan connection satisfying the Maurer-Cartan structure equation

dAG +1

2[AG, AG] = 0,

... which means that AG is Cartan-flat .

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature





Development


Flatness vs. Symmetry

• The standard for Cartan flatness F = 0 is given by the “curved” Klein model G/H...

... which means that F does not measure the deficiency of Riemannian flatness,

... but rather the deficiency of symmetry.

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development

H-parallel Transports

Development

Rolling the Local Klein Models

This is the End



• The spin (Ehresmann) connection ωH defines parallel transports in the associatedbundle

PH ×H g/h ≃ Vσ(PG ×G G/H),

that is parallel transports of vectors tg. to the local Klein models along the section

σ : M → PG ×G G/H.

• Since the geom. soldering form θ defines an identification

TM≃−→ PH ×H g/h,

the Ehresmann conn. ωH transports vectors tg. to M .

• Now, the Cartan connection A = ωH + θ does not only define ‖-transport of tg.vectors (by means of ωH )...

... but also “transports” the spatiotemporal locations themselve s (bymeans of θ).

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Development


This is the End


Development

• Let γ : [0, 1] → M be a curve on M and γ : [0, 1] → PH any lift of γ.

• Since PH ⊂ PG, the curve γ is in PGπ′−−→ M .

• If we use ωG for ‖-transporting γ(t) to π′−1(x0) along γ for all t ∈ [0, 1], we obtain acurve γ in π′−1(x0).

• By using the projection

PG−→ PG/H ≃ PG ×G G/H,

we can define a curve γ∗ = (γ) in the local Klein model Mx0 over x0 called the

development of γ over x0.

• In this way, any curve γ : [0, 1] → M with γ(0) = x can be developed into the localKlein model Mx

0 .

• It can be shown that:

.γ∗ only depends on γ and is independent from the choice of γ.

.The devel. of a closed curve might fail to close by an amount given by T .

The torsion measures the non-commutativity of the translational parallel transports.

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Development


This is the End



• Since the development is obtained by projecting a ‖-transport defined by ωG ontoG/H,...

... the only relevant part of ωG is the g/h-valued part, namely θ.

• The development process maps curves γ in M starting at γ(0) into curves γ∗ in the

internal local Klein model Mγ(0)0 over γ(0).

• This process can be understood as the result of “rolling” the local Klein model Mγ(1)0

along γ until γ(0).

• This means that the point γ∗(1) in Mγ(0)0 is the position reached by the point of

attachment of the local Klein model Mγ(1)0 when the latter is rolled back to γ(0) along γ.

• In this way, the translational part θ of A defines the γ-dependent image of any x ∈ M

in the local Klein model at x0.

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development


Development


This is the End


This is the End

Thanks for your attention !!!

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development

Symmetries

Diff(M) vs. Gauge

Group Internalizing external

displacements

Internal Gauge

Transformations External Diffeomorphisms

Obstruction to

Diff(M) =

AutV (PG)


Diff(M) vs. Gauge Group

• Whereas the symmetry group of GR is the group Diff(M)...

... the symmetry group of a YM theory is the gauge group G = AutV (PG) of verticalautomorphisms of the G-principal bundle PG.

• Since we have recasted the geometric structures (M, g) of GR in terms of a partiallygauge fixed YM theory (PG, ωG, σ)...

... we could compare the transformations of A = ωH + θ under Diff(M) with thetransformations under G.

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development

Symmetries

Diff(M) vs. Gauge


displacements

Internal Gauge


Obstruction to

Diff(M) =

AutV (PG)


Internalizing external displacements

• Given an external displacement v ∈ TxM , the form

θ : TM → PH ×H g/h ≃ Vσ(PG ×G G/H)

defines an internal displacement θ(v(x)) ∈ Tσ(x)Mx0 in the local Klein model on x0 at

the point of attachment σ(x).

• This means that the point of attachment of the local Klein model Mx+v(x)0 will be

developed into the point σ(x) + θ(v(x)) of the local Klein model Mx0 when M

x+v(x)0 is

rolled back to x along v(x).

• The form θ defines a correspondence between

.external spatiotemporal diffeomorphisms generated by vector fields v onM ,

.internal (or local ) gauge translations generated by θ(v) ∈ Γ(E).

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development

Symmetries

Diff(M) vs. Gauge


displacements

Internal Gauge


Obstruction to

Diff(M) =

AutV (PG)


Internal Gauge Transformations

• The natural representation of a local gauge transformation λ ∈ CG(PG, g) on ωG inPG is given by the usual expression

δλωG = dωG λ.

• The induced gauge transformations of (ωH , θ) is given by

δ(Λ,ξ)θ = [θ, ξ]g/h + [θ, Λ] + dωH ξ,

δ(Λ,ξ)ωH = [θ, ξ]h + dωH Λ

where

Λ.= πh λ ∈ CH(PH , h)

and

ξ.= πg/h λ ∈ CH(PH , g/h).

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development

Symmetries

Diff(M) vs. Gauge


displacements

Internal Gauge


Obstruction to

Diff(M) =

AutV (PG)


External Diffeomorphisms

• Let’s consider now the lifts v = vh + vv of v ∈ TM to every p ∈ PH , where

.vh is the horizontal lift of v (i.e. ωH(vh) = 0),

.vv ∈ V PH is a vertical component given by vv(p) = κ(Λ(p)) whereκ : h → V PH is the canonical map between h and the fundamental vector fields on PH .

• We can thus recover the H-equivariant maps Λ : PH → h and ξ : PH → g/h bymeans of the expressions

Λ(p) = ωH(vv(p)),

ξ(p) = θ(vh(p)).

• The Lie derivatives of (θ, ωH) along v are:

Lvθ = ivT + [θ, Λ] + dωH ξ

LvωH = ivR + dωH Λ

Introduction

Cartan’s Program


Cartan Connection

Cartan Curvature

Development

Symmetries

Diff(M) vs. Gauge


displacements

Internal Gauge


Obstruction to

Diff(M) =

AutV (PG)


Obstruction to Diff(M) = AutV (PG)

• The difference between a local gauge transformation of (ωH , θ) defined by (Λ, ξ)...

... and a dragging of (ωH , θ) along the field v = vh + vv ...

... is given by the components of the Cartan curvature:

Lvθ − δ(Λ,ξ)θ = ivT − [θ, ξ]g/h = ivFg/h

LvωH − δ(Λ,ξ)ωH = ivR − [θ, ξ]h = ivFh.

On Cartan Connections and the Geometric Structure of Space...

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Transcript of On Cartan Connections and the Geometric Structure of Space...