nLab geometry of physics -- de Rham coefficients

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De Rham coefficients

Model Layer

Lie-algebra valued differential 1-forms

Definition

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

$$\Omega^1(U,\mathfrak{g}) = \Omega^1(U) \otimes_{\mathbb{R}} \mathfrak{g} \,.$$

flat forms:

$$\Omega^1_{flat}(U, \mathfrak{g}) = \left\{ \omega \in \Omega^1(U,\mathfrak{g}) | F_\omega = \mathbf{d} \omega + [\omega, \omega] = 0 \right\} \,.$$

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This is a smooth space

$$\Omega^1_{flat}(-,\mathfrak{g}) \in Smooth 0 Type$$

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

$$\Omega^1(-,Lie(\mathbb{R})) = \Omega^1$$

and we write

$$\Omega^1_{flat}(-,Lie(\mathbb{R})) = \Omega^1_{cl}$$

Below we see

$$\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$$

The de Rham complex

• de Rham complex

• de Rham cohomology

• de Rham theorem

Below we see that

$$\flat_{dR}\mathbf{B}^n \mathbb{R} \simeq \flat_{dR}\mathbf{B}^n U(1) \simeq DK[ \Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to}\cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \,.$$

Semantic Layer

De Rham coefficient objects

Definition

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

$$\flat_{dR} \mathbf{B}G \coloneqq \flat \mathbf{B}G \times_{\mathbf{B}G} *$$

in

$$\array{ \flat_{dR} \mathbf{B}G &\stackrel{UnderlyingConnection}{\to}& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow^{\mathrlap{UnderlyingBundle}} \\ * &\to& \mathbf{B}G } \,.$$

Remark

This pullback diagram expresses that elements of $\flat_{dR}\mathbf{B}G$ are flat $G$-connections $\nabla \colon X \to \flat \mathbf{B}G$, def. equipped with a trivialization of their underlying $G$-principal bundle, def. .

Recovering smooth differential forms from cohesive de Rham coefficients

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

Proposition

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

$$\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g}) \,.$$

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

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

Proposition

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

$$DK \colon : Sh(CartSp, Ch_\bullet) \to Sh(CartSp, sSet) \to L_{lwhe} Sh(CartSp, sSet) \simeq \mathbf{H}$$

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

$$\begin{aligned} \flat_{dR}\mathbf{B}^n U(1) &\simeq \flat_{dR} \mathbf{B}^n \mathbb{R} \\ & \simeq DK[\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)] \\ & \simeq DK[\Omega^1_{cl}(-) \to 0 \to \cdots \to 0] \end{aligned} \,.$$

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

Syntactic Layer

$$\begin{aligned} \flat_{dR}(\mathbf{B}G \colon Type)\; \colon & Type \\ \coloneqq & \;\; \sum_{\nabla \colon \flat \mathbf{B}G} ( UnderlyingBundle(\nabla) = * ) \end{aligned}$$

Maurer-Cartan forms

Model Layer

Maurer-Cartan form on a Lie group

• Maurer-Cartan form

$$\theta_G \colon G \to \Omega^1_{flat}(-,\mathfrak{g})$$

Consider

$$\flat_{dR} \mathbf{B}\mathbb{R} = \Omega^1_{cl}$$

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

$$\theta_{\mathbb{R}} = \mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl} \hookrightarrow \Omega^1 \,.$$

Semantic Layer

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

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

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

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

$$\array{ G &\to& * \\ \downarrow^{\mathrlap{\theta}} & pb & \downarrow \\ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ \downarrow &pb& \downarrow \\ * &\to& \mathbf{B}G }$$

Definition

This is the Maurer-Cartan form on $G$

$$\theta \;\colon\; G \to \flat_{dR} \mathbf{B}G \,.$$

Definition

For $S \;\colon\; X \to G$ a morphism, write

$$S^{-1} \mathbf{d} S \coloneqq S^* \theta_G \;\colon\; X \stackrel{S}{\to} G \stackrel{\theta_G}{\to} \flat_{dR}\mathbf{B}G$$

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

Maurer-Cartan forms on smooth $\infty$-groups

Proposition

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

$$\theta_G \colon G \to \flat_{dR}\mathbf{B}G$$

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

$$\theta_G \in \Omega^1_{flat}(G, \mathfrak{g}) \,.$$

Cohesive differentiation

The Maurer-Cartan form on the line object

$$\theta_{\mathbb{R}} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})$$

is the de Rham differential,

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

Universal curvature characteristic forms

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

$$curv \colon \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1}U(1)$$

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

Syntactic Layer

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