geometry of physics -- principal connections

This is a sub-entry of geometry of physics.


Principal connections

Model Layer

Circle-principal connections

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

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

globally it is instead a map

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


BU(1) conn F () Ω cl 2 pb BU(1) diff Ω cl 12 BU(1) \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) conn=Ω 1()U(1) \mathbf{B}U(1)_{conn} = \Omega^1(-) \sslash U(1)

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


A,A:XΩ 1() A,A' : X \to \Omega^1(-)

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

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

Dirac charge quantization and the electromagnetic field

Principal 1-connection

Covariant derivatives

(VG) conn BG conn \array{ (V\sslash G)_{conn} \\ \downarrow \\ \mathbf{B}G_{conn} }
X˜ (σ,σ) (VG) conn BG conn \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

Circle-principal 2-connection

Magnetic charge and the B-field

Circle-principal 3-connection

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

Circle-principal nn-connection

Semantic Layer

Differential cohomology

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

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

for the corresponding double delooping.



curv Gθ B𝔾:BG dRB 2G 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\mathbf{B}G, def. \ref{GeneralAbstractMaurerCartanForm}. We call this the universal curvature characteristic of GG.


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

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

Differential-form curvatures

B n𝔾 Ω cl n+1() pb B𝔾 curv dRB 2𝔾 \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(2πi Σ()):[Σ,B nU(1) conn]concτ 0U(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

curv G:BG dRB 2G. \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

ω: dRB 2Gc:BG(curv G(c)ω):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 dRB 2GF \colon X \to \flat_{dR} \mathbf{B}^2 G in this context is

ω: dRB 2Gx:X(F xω):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

X P¯ BG F curv G dRB 2G \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)(X,F) is hence

(P¯,):ω: dRB 2G((x:X(F xω))(c:BG(curv G(c)ω))) \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

(B n𝔾:Type) conn: Type c:B𝔾 ω:Ω cl n+1(curv(c)=ω) \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 (