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geometry of physics -- de Rham coefficients

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De Rham coefficients

Model Layer

Lie-algebra valued differential 1-forms

Definition

Let GG be a Lie group, and write 𝔀\mathfrak{g} for its Lie algebra. The set of Lie algebra valued differential 1-forms is the tensor product

Ξ© 1(U,𝔀)=Ξ© 1(U)βŠ— ℝ𝔀. \Omega^1(U,\mathfrak{g}) = \Omega^1(U) \otimes_{\mathbb{R}} \mathfrak{g} \,.

flat forms:

Ξ© flat 1(U,𝔀)={Ο‰βˆˆΞ© 1(U,𝔀)|F Ο‰=dΟ‰+[Ο‰,Ο‰]=0}. \Omega^1_{flat}(U, \mathfrak{g}) = \left\{ \omega \in \Omega^1(U,\mathfrak{g}) | F_\omega = \mathbf{d} \omega + [\omega, \omega] = 0 \right\} \,.

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This is a smooth space

Ξ© flat 1(βˆ’,𝔀)∈Smooth0Type \Omega^1_{flat}(-,\mathfrak{g}) \in Smooth 0 Type

For 𝔀=Lie(ℝ)\mathfrak{g} = Lie(\mathbb{R}) we have

Ξ© 1(βˆ’,Lie(ℝ))=Ξ© 1 \Omega^1(-,Lie(\mathbb{R})) = \Omega^1

and we write

Ξ© flat 1(βˆ’,Lie(ℝ))=Ξ© cl 1 \Omega^1_{flat}(-,Lie(\mathbb{R})) = \Omega^1_{cl}

Below we see

β™­ dRBG≃Ω flat 1(βˆ’,𝔀) \flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})

The de Rham complex

Below we see that

β™­ dRB nℝ≃♭ dRB nU(1)≃DK[Ξ© 1(βˆ’)β†’dΞ© 2(βˆ’)β†’dβ‹―β†’dΞ© cl n(βˆ’)]. \flat_{dR}\mathbf{B}^n \mathbb{R} \simeq \flat_{dR}\mathbf{B}^n U(1) \simeq DK[ \Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to}\cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \,.

Semantic Layer

De Rham coefficient objects

Definition

For G∈Gpr(H)G \in Gpr(\mathbf{H}), its de Rham coefficient object is the homotopy pullback

β™­ dRBG≔♭BGΓ— BG* \flat_{dR} \mathbf{B}G \coloneqq \flat \mathbf{B}G \times_{\mathbf{B}G} *

in

β™­ dRBG β†’UnderlyingConnection β™­BG ↓ pb ↓ UnderlyingBundle * β†’ BG. \array{ \flat_{dR} \mathbf{B}G &\stackrel{UnderlyingConnection}{\to}& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow^{\mathrlap{UnderlyingBundle}} \\ * &\to& \mathbf{B}G } \,.
Remark

This pullback diagram expresses that elements of β™­ dRBG\flat_{dR}\mathbf{B}G are flat GG-connections βˆ‡:Xβ†’β™­BG\nabla \colon X \to \flat \mathbf{B}G, def. equipped with a trivialization of their underlying GG-principal bundle, def. .

Recovering smooth differential forms from cohesive de Rham coefficients

Let H=\mathbf{H} = Smooth∞Grpd. All smooth manifolds and sheaves on smooth manifolds etc. in the following are canonically regarded as objects in this H=Sh ∞(CartSp)\mathbf{H} = Sh_\infty(CartSp).

Proposition

For GG a Lie group, the de Rham coefficient object β™­ dRBG\flat_{dR}\mathbf{B}G, def. of its delooping is given by the sheaf of flat Lie algebra valued differential 1-forms Ξ© flat 1(βˆ’,𝔀)\Omega^1_{flat}(-,\mathfrak{g}), def. , for 𝔀\mathfrak{g} the Lie algebra of GG:

β™­ dRBG≃Ω flat 1(βˆ’,𝔀). \flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g}) \,.

This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for BG with G a Lie group.

Write U(1)U(1) for the circle group regared as a Lie group in the standard way.

Proposition

For nβˆˆβ„•n \in \mathbb{N}, the de Rham coefficient object β™­ dRB nU(1)\flat_{dR}\mathbf{B}^n U(1), def. , of the nn-fold delooping of U(1)U(1) is given by the image under the Dold-Kan correspondence

DK::Sh(CartSp,Ch β€’)β†’Sh(CartSp,sSet)β†’L lwheSh(CartSp,sSet)≃H DK \colon : Sh(CartSp, Ch_\bullet) \to Sh(CartSp, sSet) \to L_{lwhe} Sh(CartSp, sSet) \simeq \mathbf{H}

of the truncated de Rham complex of sheaves of differential forms,

β™­ dRB nU(1) ≃♭ dRB nℝ ≃DK[Ξ© 1(βˆ’)β†’dβ‹―β†’dΞ© cl n(βˆ’)] ≃DK[Ξ© cl 1(βˆ’)β†’0β†’β‹―β†’0]. \begin{aligned} \flat_{dR}\mathbf{B}^n U(1) &\simeq \flat_{dR} \mathbf{B}^n \mathbb{R} \\ & \simeq DK[\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \\ & \simeq DK[\Omega^1_{cl}(-) \to 0 \to \cdots \to 0] \end{aligned} \,.

This is discussed at smooth ∞-groupoid - structures - de Rham coefficients for the circle n-groups.

Syntactic Layer

β™­ dR(BG:Type): Type ≔ βˆ‘ βˆ‡:β™­BG(UnderlyingBundle(βˆ‡)=*) \begin{aligned} \flat_{dR}(\mathbf{B}G \colon Type)\; \colon & Type \\ \coloneqq & \;\; \sum_{\nabla \colon \flat \mathbf{B}G} ( UnderlyingBundle(\nabla) = * ) \end{aligned}

Maurer-Cartan forms

Model Layer

Maurer-Cartan form on a Lie group

ΞΈ G:Gβ†’Ξ© flat 1(βˆ’,𝔀) \theta_G \colon G \to \Omega^1_{flat}(-,\mathfrak{g})

Consider

β™­ dRBℝ=Ξ© cl 1 \flat_{dR} \mathbf{B}\mathbb{R} = \Omega^1_{cl}

the Maurer-Cartan form on ℝ\mathbb{R} is the de Rham differential

ΞΈ ℝ=d:ℝ→Ω cl 1β†ͺΞ© 1. \theta_{\mathbb{R}} = \mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl} \hookrightarrow \Omega^1 \,.

Semantic Layer

Maurer-Cartan form on a cohesive ∞\infty-group

Let H\mathbf{H} be a cohesive (infinity,1)-topos (Ξ βŠ£β™­βŠ£β™―)(\mathbf{\Pi} \dashv \flat \dashv \sharp). We discuss a general formulation of Maurer-Cartan forms on cohesive infinity-groups

Let G∈Grp(H)G \in Grp(\mathbf{H}) be a group object.

Use the pasting law together with the fact that β™­\flat is right adjoint and hence preserves limits to get ΞΈ\theta in

G β†’ * ↓ ΞΈ pb ↓ β™­ dRBG β†’ β™­BG ↓ pb ↓ * β†’ BG \array{ G &\to& * \\ \downarrow^{\mathrlap{\theta}} & pb & \downarrow \\ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow \\ * &\to& \mathbf{B}G }
Definition

This is the Maurer-Cartan form on GG

θ:G→♭ dRBG. \theta \;\colon\; G \to \flat_{dR} \mathbf{B}G \,.
Definition

For S:X→GS \;\colon\; X \to G a morphism, write

S βˆ’1dS≔S *ΞΈ G:Xβ†’SGβ†’ΞΈ Gβ™­ dRBG S^{-1} \mathbf{d} S \coloneqq S^* \theta_G \;\colon\; X \stackrel{S}{\to} G \stackrel{\theta_G}{\to} \flat_{dR}\mathbf{B}G

for its composite with the map of def. , hence the pullback of the Maurer-Cartan form along SS. We also call this the de Rham differential of SS.

Maurer-Cartan forms on smooth ∞\infty-groups

Proposition

For GG a Lie group canonically regarded in H=\mathbf{H} = Smooth∞Grpd the general abstract morphism

θ G:G→♭ dRBG \theta_G \colon G \to \flat_{dR}\mathbf{B}G

is identified, via the identification β™­ dRBG≃Ω flat 1(βˆ’,𝔀)\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g}) of prop. and the Yoneda lemma, with the traditional Maurer-Cartan form

ΞΈ G∈Ω flat 1(G,𝔀). \theta_G \in \Omega^1_{flat}(G, \mathfrak{g}) \,.

Cohesive differentiation

The Maurer-Cartan form on the line object

ΞΈ ℝ:ℝ→Ω cl 1(βˆ’,ℝ) \theta_{\mathbb{R}} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})

is the de Rham differential,

d=ΞΈ ℝ. \mathbf{d} = \theta_{\mathbb{R}} \,.

Universal curvature characteristic forms

For G=B nU(1)G = \mathbf{B}^n U(1)

curv:B nU(1)β†’β™­ dRB n+1U(1) curv \colon \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1}U(1)

sends a circle nn-bundle to the curvature of a pseudo-connection on it.

Syntactic Layer

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Created on March 6, 2015 at 08:27:06. See the history of this page for a list of all contributions to it.