# nLab geometry of physics -- de Rham coefficients

This entry is gouing to contain one chapter of geometry of physics.

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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

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

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

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