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A flat connection \nabla on XX is a rule for sending paths (xγy)ΠX(x \stackrel{\gamma}{\to} y) \in \Pi X to group elements, respecting composition.transport():x,y:ΠX(xy)*,*:BG(**)transport(\nabla) \colon \underset{x,y \colon \Pi X}{\sum} \left( x \rightsquigarrow y \right) \to \underset{*,*' \colon \mathbf{B}G}{\sum} (* \rightsquigarrow *') Π(X)transport()BGXBG\frac{\Pi(X) \stackrel{transport(\nabla)}{\to} \mathbf{B}G}{X \stackrel{\nabla}{\to} \flat \mathbf{B}G}.The higher parallel transport trans()trans(\nabla) of a flat connection \nabla: a (higher) gauge field with vanishing field strength.Flat connections
A closed differential form ω\omega is a flat connection \nabla and a trivialization of the underlying bundle. dRBG :BG(UnderlyingBundle()*)\begin{aligned} & \flat_{dR} \mathbf{B} G \coloneqq \\ & \sum_{\nabla \colon \flat \mathbf{B}G} (UnderlyingBundle(\nabla) \simeq *) \end{aligned} dRBG UnderlyingConnection BG UnderlyingBundle * BG\begin{matrix} \flat_{dR}\mathbf{B}G & \stackrel{UnderlyingConnection}{\begin{svg} <svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}}& \flat \mathbf{B}G \\ \begin{svg}<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0v-39" fill="none"/><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \mathclap{\array{\arrayopts{\align{bottom}}\;\begin{svg}<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10.40001pt" height="10.40001pt" viewBox="-0.2 -0.2 10.40001 10.40001 "><g transform="translate(0,10.20001 ) scale(1,-1) translate(0,0.2 )"><g><g stroke="rgb(0.0%,0.0%,0.0%)"><g fill="rgb(0.0%,0.0%,0.0%)"><g stroke-width="0.4pt"><g><path d=" M 0.0 0.0 L 10.00002 0.0 L 10.00002 10.00002 " style="fill:none"/></g></g></g></g></g></g></svg>\end{svg} & \space{10}{0}{30} \\ \space{10}{30}{1} & \swArrow}} & \begin{svg}<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0v-39" fill="none"/><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}{}^{\mathrlap{Underlying \atop Bundle}} \\ * &\stackrel{}{\begin{svg}<svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}}& \mathbf{B}G \end{matrix}The coefficients for de Rham hypercohomology – flat ∞-Lie algebra valued differential forms.de Rham coefficients
A general connection \nabla is the equivalence between the curvature curv(c)curv(\mathbf{c}) of a bundle c\mathbf{c} and a closed differential form ω\omega.:c:B n𝔾ω:Ω cl n+1(curv(c)=ω)\nabla \colon \underset{{\mathbf{c} \colon \mathbf{B}^n \mathbb{G}} \atop { \omega \colon \Omega^{n+1}_{cl} }}\sum \left( curv\left(\mathbf{c}\right) = \omega\right) B n𝔾 conn F () Ω cl n+1 B n𝔾 curv dRB n+1𝔾 \begin{matrix} \mathbf{B}^n \mathbb{G}_{conn} & \stackrel{F_{(-)}}{\begin{svg} <svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}}& \Omega^{n+1}_{cl} \\ \begin{svg}<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0v-39" fill="none"/><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \mathclap{\array{\arrayopts{\align{bottom}}\;\begin{svg}<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="10.40001pt" height="10.40001pt" viewBox="-0.2 -0.2 10.40001 10.40001 "><g transform="translate(0,10.20001 ) scale(1,-1) translate(0,0.2 )"><g><g stroke="rgb(0.0%,0.0%,0.0%)"><g fill="rgb(0.0%,0.0%,0.0%)"><g stroke-width="0.4pt"><g><path d=" M 0.0 0.0 L 10.00002 0.0 L 10.00002 10.00002 " style="fill:none"/></g></g></g></g></g></g></svg>\end{svg} & \space{10}{0}{30} \\ \space{10}{30}{1} & \swArrow}} & \begin{svg}<svg viewBox="-3.99994 -42.00003 7.99988 44.0 " width="8pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="44pt"><g transform="translate(0 2) scale(1 -1) translate(0 42)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0v-39" fill="none"/><g transform="matrix(0 -1 1 0 0 -39)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg} \\ \mathbf{B}^n \mathbb{G} &\stackrel{curv}{\begin{svg}<svg viewBox="-1.99997 -3.99994 44.0 7.99988 " width="44pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="8pt"><g transform="translate(0 4) scale(1 -1) translate(0 4)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><path d="m0 0h39" fill="none"/><g transform="matrix(1 0 0 1 39 0)"><g stroke-width=".4pt"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linecap="round"><g stroke-linejoin="round"><path d="m-2.4 3.2c.2-1.2 2.4-3 3-3.2-.6-.2-2.8-2-3-3.2" fill="none"/></g></g></g></g></g></g></g></g></g></svg>\end{svg}}& \flat_{dR} \mathbf{B}^{n+1}\mathbb{G} \end{matrix} The coefficients for smooth differential cohomology: abelian (higher) gauge fields.Circle principal n-connections
There is a cohesive function from GG-gauge fields to higher 𝔾\mathbb{G}-gauge fields.exp(iS):BG connB n𝔾 conn\vdash \; \exp(i S) \colon \mathbf{B}G_{conn} \to \mathbf{B}^n \mathbb{G}_{conn}A differential universal characteristic class.An extended action functional/prequantum n-bundle for extended higher Chern-Simons-type gauge theory.
Revised on November 20, 2014 16:48:00 by Rod Mc Guire (108.39.129.32)