On Cartan Geometries and the Formulation of a Gravitational Gauge Principle

On Cartan Geometries and the Formulation of a Gravitational Gauge Principle.

Gabriel Catren SPHERE (UMR 7219) - Université Paris Diderot/CNRS

On Cartan Geometries and the Formulation of a Gravitational Gauge Principle. - Gabriel Catren - Workshop Reﬂections on Space, Time and their Quantum Nature, AEI, Golm 26-28 November, 2012 - p. 1/44 Introduction

Introduction ● Introduction Cartan geometries ! gravitation as a gauge theory? ● General Relativity vs. Yang-Mills Theory ● Bridging the gap between GR and Y-M Gauge theories of gravity: Utiyama (1956), Sciama, Kibble, Trautman, Hehl, Ne’eman, ● Historical landmarks • Isham, Macdowell & Mansouri, Stelle & West, etc. Gauging Gravity

Cartan’s Program The theory of Cartan connections seems to provide the adequate geometric framework Symmetry breaking • for accomplishing this task (c.f. Wise, Randono). Cartan connection

Metric Main bibliography: Cartan Curvature •

Development .Mathematical: E. Cartan, R.W. Sharpe, S. Kobayashi, P.W. Michor, and S. Sternberg. Conclusion

.Physical: K.S. Stelle & P.C. West, Spontaneously broken de Sitter symmetry and the gravitational holonomy group, Phys. Rev. D, 21, 6, 1980.

Introduction General relativity: ● Introduction • ● General Relativity vs. Yang-Mills Theory g ● Bridging the gap between GR [g] = Unique metric & torsionless connection (Levi-Civita connection) and Y-M Diff(M) ● Historical landmarks

Gauging Gravity

Cartan’s Program Yang-Mills theory: • Symmetry breaking ωG Cartan connection [ωG] = AutV (PG) Metric where P M is a G-principal bundle with G a Lie group and ω an Ehresmann conn. G → G Cartan Curvature

Development Whereas a gauge ﬁeld is represented by an Ehresmann connection on the Conclusion internal spaces of a G-principal bundle over space-time,...

...the gravitational ﬁeld is represented by a metric on the space-time itself.

Introduction “Metric” Einstein-Hilbert formulation: ● Introduction • ● General Relativity vs. Yang-Mills Theory ● Bridging the gap between GR .metric g. and Y-M ● Historical landmarks “Tetrad-connection” Palatini formulation: Gauging Gravity •

Cartan’s Program .Ehresmann conn. ω for the local Lorentz group (called spin conn.), Symmetry breaking

Cartan connection .tetrads, vierbeine, or moving frames θ √g. Metric ∼

Cartan Curvature Cartan formulation: • Development

Conclusion .ω + θ is a connection (!)... for the local Poincaré group?

... but not an Ehresmann conn., but rather a Cartan connection.

The difference between Y-M theory & GR is reduced to the difference between • Ehresmann and Cartan connections.

Introduction In the article ● Introduction • ● General Relativity vs. Yang-Mills Theory Sur les variétés à connexion afﬁne et la théorie de la relativité généralisée (première partie), ● Bridging the gap between GR and Y-M e ● Historical landmarks (Annales scientiﬁques de l’E.N.S, 3 série, tome 40, pp. 325-412, 1923)

Gauging Gravity ... E. Cartan introduced the Cartan connections and proposed a generalization of GR to Cartan’s Program geom. with non-zero torsion. Symmetry breaking

Cartan connection In the article Metric •

Cartan Curvature Les connexions inﬁnitésimales dans un espace ﬁbré différentiable

Development (Seminaire N. Bourbaki, 1948-1951, exp. n◦24, pp. 153-168, 1950), Conclusion ... 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, namely the notion of Ehresmann conn.

Cartan’s conception was overshadowed by the notion of Ehresmann connection.

Introduction Pragmatic motivations: • Gauging Gravity ● Motivations for “gauging” gravity ● Gauge Principle in Yang-Mills .Since we know how to quantize Y-M theory, the reduction of the gap between Theory ● On Fiber Bundles Y-M and GR might be useful for quantizing gravity (c.f. LQG). ● On locality & interactions ● Different roles played by symmetry groups ● Towards a Gauge Principle for Gravity .It might be helpful for unifying gravity with the other Y-M interactions.

Cartan’s Program

Symmetry breaking Conceptual motivations: Cartan connection •

Metric

Cartan Curvature Y-M theory can be (partially) obtained from an astonishing heuristic ♣ Development argument, namely the gauge principle (GP):

Conclusion

Symmetry ! Locality ! Interactions

Introduction Ehresmann connection ψ(x) eiαψ(x) / ψ(x) eiα(x)ψ(x) Gauging Gravity ∼ ∼ ● Motivations for “gauging” gravity ● Gauge Principle in Yang-Mills Theory ● On Fiber Bundles In order to construct a locally invariant theory it is necessary to introduce ● On locality & interactions ● Different roles played by physical interactions in the form of Ehresmann connections. symmetry groups ● Towards a Gauge Principle for Gravity

Cartan’s Program “Kretschmann” objection: • Symmetry breaking A mere epistemic requirement regarding the permissible coordinate transformations Cartan connection seems to imply non-trivial new physics. Metric

Cartan Curvature Solution: Development •

Conclusion Local gauge invariance is the epistemic consequence of the ontological commitment of the theory regarding the fund. geom. structure that it presupposes: ﬁber bundles.

Introduction Common conception: • Gauging Gravity ● Motivations for “gauging” Fiber bundles = globally twisted generalizations of the Cartesian product of two gravity ● Gauge Principle in Yang-Mills spaces. Theory ● On Fiber Bundles π However, even locally a G-principal bundle P M is not a product space ● On locality & interactions • G −→ ● Different roles played by Ui G, with Ui M... symmetry groups × ⊂ ● Towards a Gauge Principle for −1 Gravity ... since π (x) = G, but is rather a G-torsor or a principal homog. space ,... 6 Cartan’s Program ... i.e. a set on which G acts in a free and transitive manner. Symmetry breaking

Cartan connection π−1(x) is isomorphic to G in a non-canonical way since it does not have a privileged • Metric origin.

Cartan Curvature Each fiber can be identified with G only by fixing a local section σ : Ui PG: Development • → ≃ ψ : U G π−1(U ) (local trivialization of P ) Conclusion i × −→ i G ψ(x, g) σ(x)g. 7→ Internal states in different fibers cannot be intrinsically compared.

Introduction The requirement of local gauge invariance is just the “epistemic” counterpart of the fact... • Gauging Gravity ● Motivations for “gauging” gravity ... that internal states are not endowed with an intrinsic “qualitative suchness”. ● Gauge Principle in Yang-Mills Theory ● On Fiber Bundles ● On locality & interactions ● Different roles played by symmetry groups In Y-M theory, physical interactions in the form of Ehresmann conn. are necessary to ● Towards a Gauge Principle for • Gravity overcome...

Cartan’s Program

Symmetry breaking ... (in a path-dependent or curved way)...

Cartan connection

Metric ... the disconnection introduced by the spatio-temporal localization of matter ﬁelds.

Cartan Curvature

Development A connection reconnects what space-time disconnects. Conclusion

Introduction It is not the same to say... • Gauging Gravity ● Motivations for “gauging” gravity ... that the observables of the theory must be invariant under a symmetry ● Gauge Principle in Yang-Mills group... Theory ● On Fiber Bundles ● On locality & interactions ● Different roles played by ... than saying that the very degrees of freedom of the theory must be symmetry groups ● Towards a Gauge Principle for introduced in order to guarantee the invariance of the theory under a symmetry group. Gravity

Cartan’s Program

Symmetry breaking While the Y-M gauge ﬁelds are introduced in order to guarantee the invariance of the Cartan connection • theory under AutV (PG)... Metric

Cartan Curvature ... it is not clear to what extent the invariance under Diff(M) plays such a Development “constructive” role in RG.

Conclusion

Introduction Is it possible to reformulate GR as a theory that describes a dynamical conn. on a • Gauging Gravity ﬁbration over M? ● Motivations for “gauging” gravity ● Gauge Principle in Yang-Mills Theory ● On Fiber Bundles ● On locality & interactions And, what is the kind of locality guaranteed by such a gravitational connection? ● Different roles played by • symmetry groups ● Towards a Gauge Principle for Gravity

Cartan’s Program First evident answer: • Symmetry breaking

Cartan connection Spacetime is endowed with a natural bundle, namely the tangent bundle T M...

Metric

Cartan Curvature ... and the Levi-Civita conn. is a law for -transporting vectors in T M. k Development

Conclusion

Introduction Levi-Civita viewpoint: • Gauging Gravity Tangent local model of vacuum (LMV) = Minkowski vector space. Cartan’s Program ● Cartan’s criticism ● Generalized local models of Cartan viewpoint: the Levi-Civita notion of connection has two ﬂaws: vacuum • ● Klein Geometries ● Locals models of gravitational Mink. S-T is a homog. space, i.e. there is a group vacuum ♠ ● Limitations of Klein’s Erlangen 4 ↑ Program Π(3, 1) = R ⋊ SO (3, 1) Poincaré group ● Riemann + Klein = Cartan ● Summary (I) ● Summary (II) that acts transitively on M.

Symmetry breaking .However, the Levi-Civita conn. only takes into account the rotational part Cartan connection SO↑(3, 1) of Π(3, 1),...

Metric ... neglecting in this way the symmetry associated to the fact that ﬂat S-T does Cartan Curvature not have a privileged origin.

Development If the topology of M is not that of Mink. S-T, i.e. if Mink. S-T is not the ground Conclusion ♣ state of the theory...

... why should we use it Mink. S-T as a LMV?

Introduction Cartan bypasses these two ﬂaws by using more general LMV. • Gauging Gravity

Cartan’s Program Cartan starts with an affine fiber bundle (rather than a vector bundle)... ● Cartan’s criticism ♠ ● Generalized local models of vacuum ● Klein Geometries ... in which the local structural group is the whole affine group of the LMV... ● Locals models of gravitational vacuum ● Limitations of Klein’s Erlangen Program ... incorporating the fact that the ground state lacks a privileged origin. ● Riemann + Klein = Cartan ● Summary (I) ● Summary (II) Rather than using Mink. S-T, Cartan uses as LMV a homog. space adapted ♣ Symmetry breaking to the topology of S-T,...

Cartan connection

Metric ...models that are given by the so-called Klein geometries.

Cartan Curvature

Development All in all • Conclusion Tangent Minkowski vector space Tangent afﬁne Klein geometry.

Introduction An homog. space (M, G) is a connected space M endowed with a transitive action • Gauging Gravity of a Lie group G.

Cartan’s Program ● Cartan’s criticism Given x0 M, the surjective map: ● Generalized local models of • ∈ vacuum ● π : G M Klein Geometries x0 → ● Locals models of gravitational vacuum g g x0. ● Limitations of Klein’s Erlangen 7→ · Program ● Riemann + Klein = Cartan ● Summary (I) induces a bijection ● Summary (II) G/H M, 0 → Symmetry breaking where H = π−1(x ) G is the isotropy group of x . 0 x0 0 0 Cartan connection ⊂

Metric Whereas H leaves x invariant, G/H generates translations in M. • 0 0 0 Cartan Curvature The pair (G, H) with G/H a connected homog. space is called a Klein geometry. Development •

Conclusion A KG (G, H) induces a canonical H-ﬁbration G G/H, where G is called the • → principal group (“Haugtgruppe”) of the geometry.

Introduction The relevant examples of KG (G, H) in the framework of gravitational theories are • Gauging Gravity given by the vacuum solutions of the Einstein’s equations with a cosmological constant Λ:

Cartan’s Program ● Cartan’s criticism .Minkowski space-time (Λ=0): ● Generalized local models of vacuum ● Klein Geometries ● Locals models of gravitational (R4 ⋊ SO↑(3, 1),SO↑(3, 1)), vacuum ● Limitations of Klein’s Erlangen Program ● Riemann + Klein = Cartan .Anti-de Sitter space-time (Λ < 0): ● Summary (I) ● Summary (II)

Symmetry breaking (SO↑(3, 2),SO↑(3, 1)),

Cartan connection

Metric .de Sitter space-time (Λ < 0):

Cartan Curvature

Development (SO↑(4, 1),SO↑(3, 1)).

Conclusion

Note: in order to obtain a Klein geometry KG (G, H) from a homogeneous space, we • have to choose an origin in the latter.

Introduction Klein’s marriage between geometry and group theory only works for homog. (i.e. • Gauging Gravity symmetric) spaces:

Cartan’s Program ● Cartan’s criticism “At ﬁrst look, the notion of group seems alien to the geometry of Riemannian ● Generalized local models of spaces, as they do not possess the homogeneity of any space with a principal vacuum ● Klein Geometries group.” ● Locals models of gravitational vacuum E. Cartan, La théorie des groupes et les recherches récentes de géométrie ● Limitations of Klein’s Erlangen Program différentielle. ● Riemann + Klein = Cartan ● Summary (I) ● This limitation of Klein’s Erlangen program can be bypassed by using the Klein Summary (II) • symmetric spaces as local tangent models: Symmetry breaking

Cartan connection “In spite of this, even though a Riemannian space has no absolute homogeneity, Metric it does, however, possess a kind of inﬁnitesimal homogeneity; in the immediate

Cartan Curvature neighborhood it can be assimilated to a Kleinian space.”

Development E. Cartan, Ibid.

Conclusion This generalization of Klein’s program requires to go beyond the stance according to • which the inﬁn. models of a curved geom. must be given by Euclidean space.

Introduction Euclidean Geometry / Klein Geometries Gauging Gravity

Cartan’s Program ● Cartan’s criticism ● Generalized local models of / vacuum Riemannian Geometry / Cartan Geometries. ● Klein Geometries ● Locals models of gravitational vacuum Riemannian Geometry is locally modeled on Euclidean Geometry but globally ● Limitations of Klein’s Erlangen • Program deformed by curvature. ● Riemann + Klein = Cartan ● Summary (I) ● Summary (II) Klein Geometries provide more general symmetric spaces that Euclidean Geometry. • Symmetry breaking Cartan’s twofold generalization: Cartan connection •

Metric Cartan Geometries are locally modeled on Klein Geometries but globally Cartan Curvature deformed by curvature

Development

Conclusion In particular, a Riemannian geometry on M is a torsion-free Cartan geometry on M • modeled on Euclidean space.

Introduction We shall start with a Y-M geometry (i.e., with a theory with purely internal afﬁne • Gauging Gravity symmetries):

Cartan’s Program ● Cartan’s criticism PG ● Generalized local models of vacuum ● Klein Geometries ωG ● Locals models of gravitational vacuum ● Limitations of Klein’s Erlangen Program M ● Riemann + Klein = Cartan ● Summary (I) where ωG is an Ehresmann connection and G is the Poincaré, de Sitter or anti-de Sitter ● Summary (II) afﬁne group that acts transitively on the vacuum solution of the theory. Symmetry breaking We shall then “externalize” some of the internal symmetries in order to induce geom. Cartan connection • structures on M itself. Metric

Cartan Curvature We shall consider a KG (G, H) with dim(G/H) = dim(M) where • Development .G = Poincaré, de Sitter or anti-de Sitter afﬁne group. Conclusion .H = Lorentz group.

.G/H = group of translations of the vacuum solution.

Introduction Since G is now an affine group,... • Gauging Gravity ... the fibers do not have a privileged point of attch. to M as it is the case for tg. vector Cartan’s Program ● Cartan’s criticism bundles. ● Generalized local models of vacuum In order to attach the fibers to M, i.e. to solder the internal geometry to the geometry of ● Klein Geometries • ● Locals models of gravitational M,... vacuum ● Limitations of Klein’s Erlangen Program ... we have to “break” the Poincaré symmetry down to the Lorentz group H by selecting a ● Riemann + Klein = Cartan ● Summary (I) point of attch. in each fiber. ● Summary (II) This amounts to reduce the Ehresmann-connected G-bundle P to a Cartan-connected Symmetry breaking • G H-bundle PH : Cartan connection ι PH / PG Metric y yy Cartan Curvature y yy yy Development ∗ ω ι (ωG)=ωH +θ yyG Conclusion y yy yy |y M

Introduction An Ehresmann conn. on P M is a horizontal equivariant distribution H • G → Gauging Gravity deﬁned by means of a G-eq. and g-valued 1-form ωG on PG such that

Cartan’s Program

Symmetry breaking Hp = Ker (ωG)p TpPG. ● Ehresmann connections ⊂ ● Associated bundle in homogeneous spaces The conn. form ωG satisﬁes: ● Reduced H-bundle • ● Reduction in a nutshell ● Attaching the LMV to M ● Symmetry breaking or partial ∗ .RhωG = Ad(h−1)ωG, where Ad is the adj. repr. of G on g. gauge ﬁxing? ● H-reductions as partial trivializations ● Canonical G-extension of an ♯ .ωG(ξ ) = ξ (“vertical parallelism”) where H-bundle

Cartan connection

Metric ♯ : g VpPG Cartan Curvature → ♯ d Development ξ ξ (f(p)) = (f(p exp(λξ))) . 7→ dλ · |λ=0 Conclusion Important: ωG has values in the Lie algebra g of the structural group G.

Introduction It might be possible to reduce (non-canonically) P to an H-bundle P M. • G H → Gauging Gravity

Cartan’s Program

Symmetry breaking To do so, we have to consider the associated G-bundle in homog. spaces ● Ehresmann connections • ● Associated bundle in homogeneous spaces ● Reduced H-bundle ● Reduction in a nutshell PG G G/H M. ● Attaching the LMV to M × → ● Symmetry breaking or partial gauge ﬁxing? ● H-reductions as partial trivializations ● Canonical G-extension of an This bundle is obtained by attaching to each x a LMV G/H. H-bundle • ≃

Cartan connection

Metric It can be shown that Cartan Curvature •

Development

Conclusion P G/H = P /H. G ×G ∼ G

Introduction The reduction of P to P can be deﬁned either by a global section • G H Gauging Gravity σ : M PG G G/H ∼= PG/H Cartan’s Program → ×

Symmetry breaking or, equivalently, by an equivariant function ● Ehresmann connections ● Associated bundle in −1 ϕ : PG G/H, ϕ(pg) = g ϕσ (p). homogeneous spaces → ● Reduced H-bundle ● Reduction in a nutshell The reduced H-bundle PH M is given either by ● Attaching the LMV to M • → ● Symmetry breaking or partial −1 gauge ﬁxing? PH = ϕσ ([e]) ● H-reductions as partial trivializations ● Canonical G-extension of an or by the pullback of PG along the section σ: H-bundle

Cartan connection ∗ / PH = σ PG / PG rrr Metric rr rrr Cartan Curvature rrr π xr Development ox M o P: G/H. Conclusion : σ

Introduction All in all, there is one-to-one correspondence between • Gauging Gravity Reduced H-subbundles PH of PG Cartan’s Program

Symmetry breaking ● Ehresmann connections m ● Associated bundle in Global sections σ : M PG/H homogeneous spaces → ● Reduced H-bundle or ● Reduction in a nutshell ● Attaching the LMV to M ● Symmetry breaking or partial Equivariant functions ϕ : PG G/H gauge ﬁxing? → ● H-reductions as partial trivializations ● Canonical G-extension of an ι ϕσ H-bundle −1 ∗ P = ϕ ([e]) = σ P / PG / / G/H H σ G qq Cartan connection qq qqq Metric qq qqq Cartan Curvature qq qqq Development qqq qq σϕ Conclusion qqq xq * M o PG/H = PG G G/H ∼ ×

Introduction The reduction amounts to select a point of attachment σ(x) in each LMV G/H at • ≃ Gauging Gravity each x.

Cartan’s Program

Symmetry breaking By doing so, we shall identify ● Ehresmann connections • ● Associated bundle in homogeneous spaces ● Reduced H-bundle .each σ(x) with x, ● Reduction in a nutshell ● Attaching the LMV to M ● Symmetry breaking or partial .each TxM to the vertical tangent space to σ(x). gauge ﬁxing? ● H-reductions as partial trivializations ● Canonical G-extension of an H-bundle In this way, the LMV attached to x will be tangent to M at σ(x). • Cartan connection

Metric By selecting a point of att. for each x, we “break” the translational symm. of the LMV. Cartan Curvature •

Development

Conclusion H = SO↑(3, 1) encodes the “unbroken” rotational symmetry. •

Introduction Since the LMV G/H in which σ is valued are analogous to the manifold that • ≃ parameterizes the = degenerated vacua in a theory with symmetry breaking.... Gauging Gravity 6

Cartan’s Program

Symmetry breaking ● Ehresmann connections ● Associated bundle in ... the field σ is sometimes called a Goldstone field... homogeneous spaces ● Reduced H-bundle ● Reduction in a nutshell ● Attaching the LMV to M ● Symmetry breaking or partial gauge fixing? ... and the reduction process is understood as a symmetry breaking (c.f. Stelle & West). ● H-reductions as partial trivializations ● Canonical G-extension of an H-bundle ...Cartan connection However, the reduction P ֒ P defined by σ might also be understood as a • H → G Metric ... gauge fixing of the G/H-translational local invariance... Cartan Curvature

Development

Conclusion ... that is, as a partial gauge ﬁxing of the G-invariance.

Introduction In particular, a reduction of P to a id -principal bundle is given either by a section • G { G} Gauging Gravity s : M PG G (G/ idG ) ∼= PG/ idG = PG Cartan’s Program → × { } { }

Symmetry breaking or by a G-equivariant function ● Ehresmann connections ● Associated bundle in ϕ : PG G/ idG = G, homogeneous spaces → { } ● Reduced H-bundle ● Reduction in a nutshell where the reduced idG -bundle is ● Attaching the LMV to M { } ● Symmetry breaking or partial ∗ −1 gauge ﬁxing? P = s PG = ϕ (idG). idG ● H-reductions as partial { } trivializations ● Canonical G-extension of an Hence, a complete reduction with H = id is a trivialization s : M P of P . H-bundle • { G} → G G Cartan connection Instead of selecting a unique frame for each x as the trivialization s does... Metric •

Cartan Curvature ... a H-reduction can be considered a sort of partial trivialization of PG that Development selects a non-trivial H-set of frames for each x.

Conclusion

Introduction Whereas the reduction of the G-bundle P depends on the existence of a global section • G Gauging Gravity σ : M PG/H ∼= PG G G/H Cartan’s Program → ×

Symmetry breaking ● Ehresmann connections ... a H-bundle PH M can be canonically extended to the associated G-bundle ● Associated bundle in → homogeneous spaces P G M, ● Reduced H-bundle H ×H → ● Reduction in a nutshell ● Attaching the LMV to M where the G-action if given by ● Symmetry breaking or partial gauge ﬁxing? ′ ′ ● H-reductions as partial [(p, g)] g = [(p,gg )] trivializations · ● Canonical G-extension of an H-bundle ... and where the inclusion is given by Cartan connection

Metric ι : P ֒ P G H → H ×H Cartan Curvature p [(p, e)]. 7→ Development

Conclusion While the reduction of PG to PH is not canonical,

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

Introduction Let’s suppose that ω satisﬁes • G Gauging Gravity Ker(ωG) ι∗TPH = 0, Cartan’s Program ∩ ... where ι : P ֒ P Symmetry breaking H → G Cartan connection ● Induced Cartan connection on ... or, equivalently, that ωG has no null vectors when restricted to PH : PH ● Reductive decomposition of . ∗ g Ker(A = ι (ωG)) = 0. ● Coordinate & geometric soldering form ● Soldering the LMV to M The 1-form A : TPH g deﬁnes a Cartan connection if Metric • →

Cartan Curvature .for each p P , A induces a linear iso. T P = g (“absolute parallelism”) ∈ H p H ∼ Development .(R∗ A) = Ad A for all p P and h H. h p (h−1) p ∈ H ∈ Conclusion .A(ξ♯) = ξ for any ξ g where ♯ : g V P . ∈ → p G

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

Introduction Let’s suppose that the KG (G, H) is reductive, i.e. that there exists a Ad(H)-module • Gauging Gravity decomposition g = h m, Ad(H) m m Cartan’s Program ⊕ · ⊂ . Symmetry breaking (in what follows m = g/h). By composing with the projections, this decomp. of g induces a decomp. of A: Cartan connection • ● Induced Cartan connection on P H A = ωH + θ, ● Reductive decomposition of g ● Coordinate & geometric where the so-called spin connection soldering form ● Soldering the LMV to M A πh ωH : TPH g h Metric −→ −−→

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

Development π A g/h θ : TP g g/h Conclusion H −→ −−−−→

is a g/h-valued 1-form on PH that is .horizontal: θ(η)=0 for vertical vectors η VP ) ∈ H ∗ −1 .H-eq.: Rhθ = h θ

Introduction While σ attaches the LMV to x M at σ(x), the g/h-part θ of A = ω + θ identiﬁes • ∈ H Gauging Gravity each TxM to the vertical tg. space to the LMV at σ(x).

Cartan’s Program This is a consequence of Ker(A)=0, since • Symmetry breaking A(vh) = ωH (vh)+ θ(vh) = θ(vh) = 0 g/h, Cartan connection 6 ∈ ● Induced Cartan connection on PH where vh is a horizontal vector. ● Reductive decomposition of g Given the coordinate soldering form ● Coordinate & geometric • soldering form ● Soldering the LMV to M θ : TP g/h, H → Metric the isomorphism Cartan Curvature Ωq (P , g/h)H Ωq (M,P g/h), hor H ≃ H ×H Development induces a geometric soldering form Conclusion θ˜ : T M P g/h → H ×H The bundle P g/h can be identiﬁed with the bundle of vertical tg. vectors to • H ×H P G/H along σ : M P G/H: G ×G → G ×G P g/h V (P G/H). H ×H ≃ σ G ×G

Introduction The geometric soldering form • Gauging Gravity θ˜ : T M PH H g/h Cartan’s Program → × identiﬁes each vector v in T M with a geometric vector θ˜(v) in P g/h,... Symmetry breaking x H ×H Cartan connection ● Induced Cartan connection on ... that is with a vertical vector tangent to PG G G/H at σ(x). PH × ● Reductive decomposition of g ● Coordinate & geometric The tg. space TxM is thus soldered to the tg. space to the homog. ﬁber G/H of soldering form • ≃ ● Soldering the LMV to M P G/H at σ(x). G ×G Metric

Cartan Curvature The LMV are soldered to T M along the section σ.

Development The coordinate soldering form Conclusion • θ : T P g/h p p H → deﬁnes the g/h-valued coordinates of θ˜(v) in the frame p.

Introduction The existence of a soldering form θ on P implies that P is isomorphic to a • H H Gauging Gravity GL(g/h)-structure,...

Cartan’s Program

Symmetry breaking ... that is to a subbundle of the GL(g/h)-principal bundle LM of linear frames on M. Cartan connection

Metric Indeed, θ deﬁnes an application ● Reducing the frame bundle • ● Recovering the metric Soldering vs. canonical form f θ : P ֒ LM ● H → Cartan Curvature p f θ (p) : g/h T M 7→ → π(p) Development

Conclusion that identiﬁes each element p in P with a frame f θ (p) LM over π(p) M. H ∈ ∈

Since f θ : P LM is an H-morphism, f θ(P ) is a H-subbundle of LM. • H → H

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

Introduction The metric gθ on M can be explicitly deﬁned in terms of the Ad(H)-invariant scalar • Gauging Gravity product , of g/h by means of the expression h· ·i Cartan’s Program θ g (v,w) = θ˜p(v), θ˜p(w) , Symmetry breaking D E Cartan connection where θ˜ (x) : T M g/h. p x → Metric ● Reducing the frame bundle θ ● Recovering the metric The H-invariance of , implies that g (v,w) does not depend on the frame p over x. ● Soldering vs. canonical form • h· ·i

Cartan Curvature

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

... by inducing an isomorphism between PH and a SO(3, 1)-subbundle of LM...

... induces a Lorentzian metric gθ on M.

Introduction The soldering form θ Ω1(P , g/h) is the pullback by f θ of the canonical form • ∈ H Gauging Gravity θc : T (LM) g/h Cartan’s Program →

Symmetry breaking on LM given by:

Cartan connection θ : T (LM) g/h c e(x) → Metric −1 ● Reducing the frame bundle v˜ e(x) (π∗(˜v)). ● Recovering the metric 7→ ● Soldering vs. canonical form where Cartan Curvature e(x) : g/h T M → x Development is a frame on TxM. Conclusion Contrary to the canonical form θ on LM, the soldering form θ on P is not canonical... • c H

... 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 The Cartan curvature F Ω2(P , g) of a Cartan geom. (P ,A) is given by • ∈ H H Gauging Gravity

Cartan’s Program 1 F = dA + [A,A] = Fh + Fg/h. Symmetry breaking 2

Cartan connection The curvature R Ω2(P , h) of a Cartan geom. (P ,A) is given by Metric • ∈ H H Cartan Curvature ● Curvature of the Cartan connection . 1 1 ● On Cartan ﬂatness R = dωH + [ωH ,ωH ] = Fh [θ,θ]h. ● (Maurer-)Cartan ﬂat 2 − 2 connection

Development 2 The torsion T Ω (PH , g/h) of a Cartan geom. (PH ,A) is given by Conclusion • ∈

. 1 1 T = dθ + ([ωH ,θ]+[θ,ωH ]) = Fg h [θ,θ]g h. 2 / − 2 /

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

Cartan’s Program 1 R = [θ,θ]h F = 0 − 2 Symmetry breaking 1 ⇔ ( T = [θ,θ]g/h − 2 Cartan connection

Metric The standard for Cartan flatness F = 0 is given by the “curved” LMV (G, H). Cartan Curvature • ≃ ● Curvature of the Cartan connection ● On Cartan flatness The so-called symmetric models satisfy ● (Maurer-)Cartan flat • connection

Development [g/h, g/h] h, Conclusion ⊆ which implies 1 T = Fg h F = (R + [θ,θ]h)+ T. / 2

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

Introduction Let’s consider the canonical H-ﬁbration G G/H of the KG (G, H). • → Gauging Gravity

Cartan’s Program The Maurer-Cartan form A of G is given by • G Symmetry breaking A (g) : T G g Cartan connection G g → Metric ξ (L ) ξ, 7→ g−1 ∗ Cartan Curvature −1 ● Curvature of the Cartan where Lg−1 : G G is the left translation defined by Lg−1 (a) = g a and it satisfies connection → ● On Cartan flatness ● (Maurer-)Cartan flat connection ∗ R AG = Ad(g)AG Development g 1 Conclusion dAG + [AG,AG] = 0 2

Maurer-Cartan form = Flat Cartan connection on G G/H. →

Introduction ω deﬁnes parallel transports in the associated bundle • H Gauging Gravity PH H g/h Vσ (PG G G/H), Cartan’s Program × ≃ ×

Symmetry breaking that is parallel transports of vectors tg. to the LMV along the section

Cartan connection σ : M P G/H. → G ×G Metric Since the geom. soldering form θ˜ deﬁnes an identiﬁcation Cartan Curvature •

Development ≃ T M PH H g/h, ● H-parallel transports −→ × ● Development ● Inﬁnitesimal developments the Ehresmann conn. ωH transports vectors tg. to M.

Conclusion The ω -parallel transports coincide with the levi-Civita parallel transports. • H

Now, A = ω + θ does not only -transport “internal” states (tg. vectors in this case) • H k as in Y-M theory (by means of ωH )...

... but also the spatiotemporal locations themselves (by means of θ).

Introduction Let γ : [0, 1] M be a curve on M and γ˜ : [0, 1] P any lift of γ. • → → H Gauging Gravity Since PH PG, the curve γ˜ is in PG. Cartan’s Program • ⊂ −1 Symmetry breaking If we use ωG for -transporting γ˜(t) to π (x0) along γ for all t [0, 1], we obtain a • −1 k ∈ curve γˆ in π (x0). Cartan connection

Metric By using the projection • Cartan Curvature ̺ PG PG/H PG G G/H, Development −→ ≃ × ● H-parallel transports we can define a curve γ∗ = ̺(ˆγ) in the fiber of P G/H over x called the ● Development G ×G 0 ● Infinitesimal developments development of γ over x0.

Conclusion In this way, any curve γ : [0, 1] M can be “printed” on the LMV over x . • → 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 Since the development is obtained by projecting a -transport deﬁned by ω onto • k G Gauging Gravity G/H,...

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

Symmetry breaking Given an infin. displacement v T M, the geometric soldering form Cartan connection • ∈ x Metric θ˜ : T M P g/h V (P G/H) → H ×H ≃ σ G ×G Cartan Curvature defines an infin. displacement in the LMV on x0 at the point of attachment σ(x). Development ● H-parallel transports ● Development ● Infinitesimal developments This means that the point of attachment at x + v will be developed in the fiber above x • ˜ Conclusion into the point σ(x)+ θ(v).

In other terms, the translational part θ of A deﬁnes the γ-dependent image of any • x M in the LMV at x . ∈ 0

Translational locality: this identification is dynamically defined by the translational component of the Cartan gauge field A.

Introduction The theory of Cartan geometries allows us to put together ω and θ into a unique • H Gauging Gravity Cartan connection

Cartan’s Program ωH Gauges the local Lorentz symmetry Symmetry breaking A  Gauges the local translational symmetry θ  θ Cartan connection  ( Induces a metric g on M Metric  ... that gauges the local affine gauge invariance defined by the affine group G that acts Cartan Curvature transitively on the vacuum solution of the theory. Development

Conclusion This can be done by reducing a Y-M geometry ● Conclusion (I) • ● Conclusions (II) ● Further Research... (PG M,ωG) ● The End → by means of a partial gauge ﬁxing

σ : M P G/H → G ×G that breaks the translational invariance of the LMV G/H. ≃

Introduction In GR (where T = 0), the consideration of a local afﬁne symmetry instead of the • Gauging Gravity smaller local Lorentz symmetry has no effects.

Cartan’s Program

Symmetry breaking

Cartan connection If we relax the condition T = 0, then ωH and θ are indep. geom. structures and the Metric • gravitational ﬁeld must be described by the whole A = ωH + θ. Cartan Curvature

Development

Conclusion ● Conclusion (I) ● Conclusions (II) Since the LMV is not necessarily Minkowski S-T, the afﬁne group G is not necessarily ● • Further Research... the Poincaré group. ● The End

Introduction Clarify the relationship between the local translational invariance gauged by θ and the • Gauging Gravity invariance under diffeomorphisms of M...

Cartan’s Program

Symmetry breaking ... being these symmetries related by the soldering

Cartan connection θ˜ : M V (P G/H) → σ G ×G Metric which identiﬁes the external inﬁnitesimal translations in M with the internal translations in Cartan Curvature the internal LMV. Development

Conclusion ● Conclusion (I) ● Conclusions (II) ● Further Research... Clarify the nature of the reduction process: dynamical symmetry breaking or partial • ● The End gauge ﬁxing?

Analyze the different actions S that can be constructed from the Cartan connection A. •

Introduction

Gauging Gravity

Cartan’s Program

Symmetry breaking

Cartan connection Thanks for your attention !!!

Metric

Cartan Curvature

Development

Conclusion ● Conclusion (I) ● Conclusions (II) ● Further Research... ● The End