# nLab geometry of physics -- principal connections

This is a sub-entry of geometry of physics.

# Contents

## Principal connections

### Model Layer

#### Circle-principal connections

Dirac charge quantization says that the electromagnetic field is only locally in general a map

$\array{ && \Omega^1(-) \\ & {}^{\mathllap{A}}\nearrow & \downarrow^{\mathrlap{\mathbf{d}}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl} }$

globally it is instead a map

$\array{ && \mathbf{B}U(1)_{conn} \\ & {}^{\nabla}\nearrow & \downarrow^{F_{(-)}} \\ X &\stackrel{\omega}{\to}& \Omega^2_{cl} }$

where

$\array{ \mathbf{B}U(1)_{conn} &\stackrel{F_{(-)}}{\to}& \Omega^2_{cl} \\ \downarrow &pb& \downarrow \\ \mathbf{B}U(1)_{diff} &\to& \Omega^{1 \leq \bullet \leq 2}_{cl} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}U(1) }$

circle bundle with connection

the smooth groupoid is

$\mathbf{B}U(1)_{conn} = \Omega^1(-) \sslash U(1)$

quotient of $\Omega^1(-)$ by $U(1)$-gauge transformations

for

$A,A' : X \to \Omega^1(-)$

a gauge transformation $A \to A'$ is $\lambda : X \to U(1)$ with

$A' = A + \mathbf{d} log \lambda$

#### Covariant derivatives

$\array{ (V\sslash G)_{conn} \\ \downarrow \\ \mathbf{B}G_{conn} }$
$\array{ \tilde X &&\stackrel{(\sigma, \nabla \sigma)}{\to}&& (V \sslash G)_{conn} \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G_{conn} }$

### Semantic Layer

#### Differential cohomology

Let $G \in Grp(\mathbf{H})$ be a braided ∞-group. Equivalently, let its delooping $\mathbf{B}G \in \mathbf{H}$ be itself equipped with the structure of an ∞-group. Write

$\mathbf{B}^2 G \in \mathbf{H}$

for the corresponding double delooping.

###### Definition

Write

$curv_{G} \coloneqq \theta_{\mathbf{B}\mathbb{G}} \colon \mathbf{B}G \to \flat_{dR} \mathbf{B}^2 G$

for the Maurer-Cartan form on the ∞-group $\mathbf{B}G$, def. \ref{GeneralAbstractMaurerCartanForm}. We call this the universal curvature characteristic of $G$.

###### Definition

The differential cohomology with coefficients in $\mathbf{B}G$ is cohomology in the slice (∞,1)-topos $\mathbf{H}_{/\flat_{dR} \mathbf{B}^2 G}$ with coefficients in $curv_G$

$\mathbf{H}_{/\flat_{dR}\mathbf{B}^2 G}(-, curv_G) \,.$

#### Differential-form curvatures

$\array{ \mathbf{B}^n \mathbb{G} &\to& \Omega^{n+1}_{cl}(-) \\ \downarrow &pb& \downarrow^{\mathrlap{}} \\ \mathbf{B}\mathbb{G} &\stackrel{curv}{\to}& \flat_{dR} \mathbf{B}^2 \mathbb{G} }$

presented by ordinary differential cohomology

#### Higher holonomy

$\exp(2 \pi i \int_{\Sigma}(-)) \colon [\Sigma,\mathbf{B}^n U(1)_{conn}] \stackrel{conc \circ \tau_0}{\to} U(1)$

### Syntactic Layer

#### The dependent curvature type

The universal curvature characteristic, def. 1, has the syntax

$\vdash curv_{G} \colon \mathbf{B}G \to \flat_{dR} \mathbf{B}^2 G \,.$

Regarded as a dependent type in the de Rham coefficient context this is

$\omega \colon \flat_{dR}\mathbf{B}^2 G \; \vdash \; \underset{\mathbf{c} \colon \mathbf{B}G}{\sum} \left( curv_G\left(\mathbf{c}\right) \simeq \omega \right) \colon Type$

Therefore the syntax for a domain object $F \colon X \to \flat_{dR} \mathbf{B}^2 G$ in this context is

$\omega \colon \flat_{dR} \mathbf{B}^2 G \;\vdash\; \underset{x \colon X}{\sum} \left( F_x \simeq \omega \right) \colon Type$

and the syntax for a cocycle

$\array{ X &&\stackrel{\bar P}{\to}&& \mathbf{B}G \\ & {}_{\mathllap{F}}\searrow &\swArrow_{\nabla}& \swarrow_{\mathrlap{curv_G}} \\ && \flat_{dR} \mathbf{B}^2 G }$

in differential cohomology, def. 2, on $(X,F)$ is hence

$\vdash \; (\bar P,\nabla) \colon \underset{\omega \colon \flat_{dR} \mathbf{B}^2 G}{\prod} \left( \left( \underset{x \colon X}{\sum} \left( F_x \simeq \omega \right) \right) \to \left( \underset{\mathbf{c} \colon \mathbf{B}G}{\sum} \left( curv_G(\mathbf{c}) \simeq \omega \right) \right) \right)$

#### Fixed curvature twists

\begin{aligned} (\mathbf{B}^n \mathbb{G} \colon Type)_{conn} \colon & Type \\ \coloneqq & \sum_{\mathbf{c} \colon \mathbf{B}\mathbb{G}} \sum_{\omega \colon \Omega^{n+1}_{cl}} \left( curv(\mathbf{c}) = \omega \right) \end{aligned}
Created on November 2, 2012 16:34:39 by Urs Schreiber (80.187.201.44)