nLab
geometry of physics -- principal connections

This is a sub-entry of geometry of physics.

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Principal connections

Model Layer

Circle-principal connections

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

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

globally it is instead a map

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

where

$$\begin{array}{ccc}BU(1{)}_{\mathrm{conn}}& \stackrel{F_{(-)}}{\to }& {\Omega }_{\mathrm{cl}}^2\\ \downarrow & \mathrm{pb}& \downarrow \\ BU(1{)}_{\mathrm{diff}}& \to & {\Omega }_{\mathrm{cl}}^{1\le •\le 2}\\ {\downarrow }^{\simeq }\\ BU(1)\end{array}$$\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

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

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

for

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

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

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

Dirac charge quantization and the electromagnetic field

• Dirac charge quantization

• electromagnetic field

Principal 1-connection

• principal connection

Covariant derivatives

• covariant derivative

$$\begin{array}{c}(V⫽G{)}_{\mathrm{conn}}\\ \downarrow \\ B{G}_{\mathrm{conn}}\end{array}$$\array{ (V\sslash G)_{conn} \\ \downarrow \\ \mathbf{B}G_{conn} }

$$\begin{array}{ccccc}\tilde{X}& & \stackrel{(\sigma ,\nabla \sigma )}{\to }& & (V⫽G{)}_{\mathrm{conn}}\\ & \searrow & ⇙& \swarrow \\ & & B{G}_{\mathrm{conn}}\end{array}$$\array{ \tilde X &&\stackrel{(\sigma, \nabla \sigma)}{\to}&& (V \sslash G)_{conn} \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}G_{conn} }

Deligne cohomology and Cheeger-Simons differential characters

• Deligne cohomology

• Cheeger-Simons differential character

• Beilinson-Deligne cup product

• fiber integration in ordinary differential cohomology

Circle-principal 2-connection

• circle 2-group principal 2-bundle

• circle 2-bundle with connection

Magnetic charge and the B-field

• magnetic charge, B-field

Circle-principal 3-connection

• circle 3-group principal 3-bundle

• circle 3-bundle with connection

The 3d Chern-Simons action functional and the C-field

• supergravity C-field

Circle-principal $n$-connection

• ordinary differential cohomology

• circle n-bundle with connection

Semantic Layer

Differential cohomology

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

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

for the corresponding double delooping.

Definition

Write

$${\mathrm{curv}}_G≔{\theta }_{B𝔾}:BG\to {♭}_{\mathrm{dR}}{B}^{2}G$$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 $BG$, def. \ref{GeneralAbstractMaurerCartanForm}. We call this the universal curvature characteristic of $G$.

Definition

The differential cohomology with coefficients in $BG$ is cohomology in the slice (∞,1)-topos $H_{/{♭}_{\mathrm{dR}}{B}^{2}G}$ with coefficients in ${\mathrm{curv}}_G$

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

Differential-form curvatures

$$\begin{array}{ccc}{B}^n𝔾& \to & {\Omega }_{\mathrm{cl}}^{n+1}(-)\\ \downarrow & \mathrm{pb}& {\downarrow }^{}\\ B𝔾& \stackrel{\mathrm{curv}}{\to }& {♭}_{\mathrm{dR}}{B}^2𝔾\end{array}$$\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

• higher holonomy

$$\exp (2\pi i{\int }_{\Sigma }(-)):[\Sigma ,B^{n}U(1{)}_{\mathrm{conn}}]\stackrel{\mathrm{conc}\circ {\tau }_0}{\to }U(1)$$\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

$$⊢{\mathrm{curv}}_G:BG\to {♭}_{\mathrm{dR}}{B}^{2}G.$$\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

$$\omega :{♭}_{\mathrm{dR}}{B}^{2}G⊢\sum _{c:BG}({\mathrm{curv}}_G\left(c\right)\simeq \omega ):\mathrm{Type}$$\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 $F:X\to {♭}_{\mathrm{dR}}{B}^{2}G$ in this context is

$$\omega :{♭}_{\mathrm{dR}}{B}^{2}G⊢\sum _{x:X}(F_x\simeq \omega ):\mathrm{Type}$$\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

$$\begin{array}{ccccc}X& & \stackrel{\overline{P}}{\to }& & BG\\ & {}_F\searrow & {⇙}_{\nabla }& {\swarrow }_{{\mathrm{curv}}_G}\\ & & {♭}_{\mathrm{dR}}{B}^{2}G\end{array}$$\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

$$⊢(\overline{P},\nabla ):\prod _{\omega :{♭}_{\mathrm{dR}}{B}^{2}G}(\left(\sum _{x:X}(F_x\simeq \omega )\right)\to \left(\sum _{c:BG}({\mathrm{curv}}_G(c)\simeq \omega )\right))$$\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

$$\begin{array}{rl}(B^n𝔾:\mathrm{Type}{)}_{\mathrm{conn}}:& \mathrm{Type}\\ ≔& \sum _{c:B𝔾}\sum _{\omega :{\Omega }_{\mathrm{cl}}^{n+1}}(\mathrm{curv}(c)=\omega )\end{array}$$\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}