nLab
higher Klein geometry

Contents

Context

Geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          Higher Klein geometry is the generalization of Klein geometry from traditional (differential) geometry to higher geometry:

          where Klein geometry is about (Lie) groups and their quotients, higher Klein geometry is about (smooth) ∞-groups and their ∞-quotients.

          The way that the generalization proceeds is clear after the following observation.

          Observation

          Let GG be a discrete group and HGH \hookrightarrow G a subgroup. Write BG\mathbf{B}G and BH\mathbf{B}H for the corresponding delooping groupoids with a single object. Then the action groupoid G//HG//H is the homotopy fiber of the inclusion functor

          BHBG \mathbf{B}H \to \mathbf{B}G

          in the (2,1)-category Grpd: we have a fiber sequence

          G//HBHBG G//H \to \mathbf{B}H \to \mathbf{B}G

          that exhibits G//HG//H as the GG-principal bundle over BH\mathbf{B}H which is classified by the cocycle BHBG\mathbf{B}H \to \mathbf{B}G.

          Moreover, the decategorification of the action groupoid (its 0-truncation) is the ordinary quotient

          τ 0(G//H)=G/H. \tau_0 (G//H) = G/H \,.
          Proof

          This should all be explained in detail at action groupoid.

          The fact that a quotient is given by a homotopy fiber is a special case of the general theorem discussed at ∞-colimits with values in ∞Grpd

          Remark

          That fiber sequence continues to the left as

          HGG//HBHBG. H \to G \to G//H \to \mathbf{B}H \to \mathbf{B}G \,.
          Observation

          The above statement remains true verbatim if discrete groups are generalized to Lie groups – or other cohesive groups – if only we pass from the (2,1)-topos Grpd of discrete groupoids to the (2,1)-topos SmoothGrpd of smooth groupoids .

          This follows with the discussion at smooth ∞-groupoid – structures.

          Since the quotient G/HG/H is what is called a Klein geometry and since by the above observations we have analogs of these quotients for higher cohesive groups, there is then an evident definition of higher Klein geometry :

          Definition

          Let H\mathbf{H} be a choice of cohesive structure. For instance choose

          Definition

          An \infty-Klein geometry in H\mathbf{H} is a fiber sequence in H\mathbf{H}

          G//HBHiBG G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G

          for ii any morphism between two connected objects, as indicated, hence Ωi:HG\Omega i : H \to G any morphism of ∞-group objects.

          Remark

          For XX an object equipped with a GG-action and f:YXf : Y \to X any morphism, the higher Klein geometry induced by “the shape YY in XX” is given by taking i:HGi : H \to G be the stabilizer ∞-group Stab(f)GStab(f) \to G of ff in XX. See there at Examples – Stabilizers of shapes / Klein geometry.

          Remarks

          Examples

          Higher super Poincaré Klein geometry

          Let H=\mathbf{H} = SuperSmooth∞Grpd be the context for synthetic higher supergeometry.

          Write 𝔰𝔲𝔤𝔯𝔞 11\mathfrak{sugra}_11 for the super L-∞ algebra called the supergravity Lie 6-algebra. This has a sub-super L L_\infty-algebra of the form

          B(𝔰𝔬(10,1)bb 5)B𝔰𝔲𝔤𝔯𝔞 11, \mathbf{B}(\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}) \hookrightarrow \mathbf{B}\mathfrak{sugra}_11 \,,

          where

          The quotient

          𝔰𝔲𝔤𝔯𝔞 11/((𝔰𝔬(10,1)bb 5)) \mathfrak{sugra}_11 / ((\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}))

          is the super translation Lie algebra in 11-dimensions.

          This higher Klein geometry is the local model for the higher Cartan geometry that describes 11-dimensional supergravity. See D'Auria-Fre formulation of supergravity for more on this.

          geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
          differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
          examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
          Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
          anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
          de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
          linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
          conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
          supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
          examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
          super anti de Sitter groupsuper anti de Sitter spacetime
          higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
          cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
          examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

          References

          That there ought to be a systematic study of higher Klein geometry and higher Cartan geometry has been amplified by David Corfield since 2006.

          Such a formalization is offered in

          For more on this see at higher Cartan geometry and Higher Cartan Geometry.

          Last revised on August 19, 2015 at 07:39:19. See the history of this page for a list of all contributions to it.