nLab universal Chern-Simons circle 7-bundle with connection

Context

Differential cohomology

differential cohomology

Contents

Idea

A Chern-Simons circle 7-bundle is the circle 7-bundle with connection classified by the cocycle in degree-8 ordinary differential cohomology that is canonically associated to a string group-principal 2-bundle with connection.

The characteristic class called the second fractional Pontryagin class $\frac{1}{6}{p}_{2}:ℬ\mathrm{String}\to {ℬ}^{8}ℤ$ in Top on the classifying space of the string group has a smooth lift to the smooth second fractional Pontryagin class

$\frac{1}{6}{p}_{2}:B\mathrm{String}\to {B}^{7}U\left(1\right)$\frac{1}{6} \mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)

in $H:=$ ∞LieGrpd, mapping from the delooping ∞-Lie groupoid of the string Lie 2-group to that of the circle Lie 7-group. This is the Lie integration of the degree 7 ∞-Lie algebra cocycle ${\mu }_{7}:\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to {b}^{6}ℝ$ on the string Lie 2-algebra which classified the fivebrane Lie 6-algebra.

Therefore, by ∞-Chern-Weil theory, there is a refinement of this morphism to ∞-bundles with connection

$\frac{1}{6}\stackrel{^}{p}:B{\mathrm{String}}_{\mathrm{conn}}\to {B}^{7}U\left(1{\right)}_{\mathrm{conn}}$\frac{1}{6}\hat \mathbf{p} : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn}

hence on cocycle ∞-groupoids

$\frac{1}{6}\stackrel{^}{p}:{H}_{\mathrm{conn}}\left(X,B\mathrm{String}\right)\to {H}_{\mathrm{diff}}^{8}\left(X\right)$\frac{1}{6} \hat \mathbf{p} : \mathbf{H}_{conn}(X,\mathbf{B}String) \to \mathbf{H}_{diff}^8(X)

a map from string Lie 2-group-principal 2-bundles with connection to circle 7-bundles with connection, hence degree 8 ordinary differential cohomology.

For $\left(P,\nabla \right)$ a String-principal 2-bundle, we call the image $\frac{1}{6}\stackrel{^}{p}\left(\nabla \right)\in {H}_{\mathrm{diff}}\left(X,{B}^{z}U\left(1\right)\right)$ its Chern-Simons circle 7-bundle with connection.

This is a differential refinement of the obstruction to lifting $P$ to a fivebrane Lie 6-group-bundle.

By construction, the curvature 8-form of $\stackrel{^}{c}\left(\nabla \right)$ is the curvature characteristic form $⟨{F}_{\nabla }\wedge {F}_{\nabla }\wedge {F}_{\nabla }\wedge {F}_{\nabla }⟩$ of $\nabla$ and accordingly the 7-form connection on $\stackrel{^}{c}\left(\nabla \right)$ is locally a Chern-Simons form $\mathrm{CS}\left(\nabla \right)$ of $\nabla$.

Therefore the higher parallel transport induced by $\frac{1}{6}{\stackrel{^}{p}}_{2}\left(\nabla \right)$ over 7-dimensional volumes $\varphi :\Sigma \to X$ is the action functional of degree-7 ∞-Chern-Simons theory. This is the analog of the way the Chern-Simons circle 3-bundle arises from Spin-principal bundles.

Construction

Using the discusson at ∞-Chern-Weil theory and in direct analogy to the constructin of the Chern-Simons circle 3-bundle we can model the (∞,1)-functor

${H}_{\mathrm{conn}}\left(X,B\mathrm{String}\right)\to {H}_{\mathrm{conn}}\left(X,{B}^{7}U\left(1\right)\right)$\mathbf{H}_{conn}(X, \mathbf{B}String) \to \mathbf{H}_{conn}(X, \mathbf{B}^7 U(1))

by postcomposition with the ∞-anafunctor

$\begin{array}{ccc}\mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{\right)}_{\mathrm{conn}}& \stackrel{\mathrm{exp}\left({\mu }_{7}{\right)}_{\mathrm{conn}}}{\to }& \mathrm{exp}\left({b}^{6}ℝ{\right)}_{\mathrm{conn}}\\ ↓& & ↓\\ {\mathrm{cosk}}_{7}\mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{\right)}_{\mathrm{conn}}& \to & {B}^{7}U\left(1{\right)}_{\mathrm{conn}}\\ {↓}^{\simeq }\\ B{\mathrm{String}}_{\mathrm{conn}}\end{array}$\array{ \exp(\mathfrak{string})_{conn} &\stackrel{\exp(\mu_7)_{conn}}{\to}& \exp(b^6 \mathbb{R})_{conn} \\ \downarrow && \downarrow \\ \mathbf{cosk}_7 \exp(\mathfrak{string})_{conn} &\to& \mathbf{B}^7 U(1)_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}String_{conn} }

where ${\mu }_{7}:\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to {b}^{6}ℝ$ is the 7-cocycle that classifies the fivebrane Lie 6-algebra.

For

$\begin{array}{ccc}C\left(U\right)& \stackrel{g}{\to }& B{\mathrm{String}}_{\mathrm{conn}}\\ {↓}^{\simeq }\\ X\end{array}$\array{ C(U) &\stackrel{g}{\to}& \mathbf{B}String_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ X }

an ∞-anafunctor modelling a cocycle for a string 2-group-principal 2-bundle with connection on a 2-bundle the $\infty$-anafunctor composition

$\begin{array}{ccccc}& & \mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{\right)}_{\mathrm{conn}}& \stackrel{\mathrm{exp}\left({\mu }_{7}{\right)}_{\mathrm{conn}}}{\to }& \mathrm{exp}\left({b}^{6}ℝ{\right)}_{\mathrm{conn}}\\ & & ↓& & ↓\\ C\left(V\right)& \stackrel{\stackrel{^}{g}}{\to }& {\mathrm{cosk}}_{7}\mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{\right)}_{\mathrm{conn}}& \to & {B}^{7}U\left(1{\right)}_{\mathrm{conn}}\\ {↓}^{\simeq }& & {↓}^{\simeq }\\ C\left(U\right)& \stackrel{g}{\to }& B{\mathrm{String}}_{\mathrm{conn}}\\ {↓}^{\simeq }\\ X\end{array}$\array{ && \exp(\mathfrak{string})_{conn} &\stackrel{\exp(\mu_7)_{conn}}{\to}& \exp(b^6 \mathbb{R})_{conn} \\ && \downarrow && \downarrow \\ C(V) &\stackrel{\hat g}{\to}& \mathbf{cosk}_7 \exp(\mathfrak{string})_{conn} &\to& \mathbf{B}^7 U(1)_{conn} \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ C(U) &\stackrel{g}{\to}& \mathbf{B}String_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ X }

produces a lift of the transition functions $g$ to ${\mathrm{cosk}}_{7}\mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\right)$. The string-cocycle is itself in first degree a collection of paths in $G$, in second a collection of surfaces with labels in $U\left(1\right)$. That lift corresponds to further resolving this to families

${U}_{{i}_{1}}\cap \cdots {U}_{{i}_{k}}×{\Delta }^{k}\to G$U_{i_1} \cap \cdots U_{i_k} \times \Delta^k \to G

up to $k=7$. That this is indeed always possible is the statement about Lie integration that ${\mathrm{cosk}}_{7}\mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\right)\stackrel{\simeq }{\to }B\mathrm{String}$ is a weak equivalence, which in turn is due to the fact that the next nonvanishing homotopy group of $G=\mathrm{SO}\left(n\right)$ after ${\pi }_{3}$ is ${\pi }_{7}$.

The above composite ∞-anafunctor is manifestly a degree 8-cocycle in Cech-Deligne cohomology given by

$\left({\mathrm{CS}}_{7}\left({\sigma }_{i}^{*}A\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\Delta }^{1}}{g}_{ij}^{*}{\mathrm{CS}}_{7}\left(A\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\Delta }^{2}}{g}_{ijk}^{*}{\mathrm{CS}}_{7}\left(A\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\Delta }^{3}}{\stackrel{^}{g}}_{ijkl}^{*}{\mathrm{CS}}_{7}\left(A\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\Delta }^{5}}{\stackrel{^}{g}}_{ijklm}^{*}{\mathrm{CS}}_{7}\left(A\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\Delta }^{6}}{\stackrel{^}{g}}_{ijklmn}^{*}{\mathrm{CS}}_{7}\left(A\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\Delta }^{7}}{\stackrel{^}{g}}_{ijklmno}^{*}\mu \left(A\right)\right)\phantom{\rule{thinmathspace}{0ex}},$\left( CS_7(\sigma_i^* A) \,,\, \int_{\Delta^1} g_{i j}^*CS_7(A) \,,\, \int_{\Delta^2} g_{i j k}^*CS_7(A) \,,\, \int_{\Delta^3} \hat g_{i j k l}^*CS_7(A) \,,\, \int_{\Delta^5} \hat g_{i j k l m}^*CS_7(A) \,,\, \int_{\Delta^6} \hat g_{i j k l m n}^*CS_7(A) \,,\, \int_{\Delta^7} \hat g_{i j k l m n o}^* \mu(A) \right) \,,

where $A$ is a connection form on the total space of the $\mathrm{Spin}\left(n\right)$-principal bundle that the string bundle itself is lifted from and ${\mathrm{CS}}_{7}$ is the Chern-Simons element in degree 7 defining the fivebrane Lie 6-algebra.

(…)

Applications

The CS 7-bundle serves as the extended Lagrangian for a 7d Chern-Simons theory. See there for more.

References

The CS 7-bundle as an circle 7-bundle with connection on the smooth moduli infinity-stack of string 2-group-2-connections has been constructed in

Revised on January 15, 2013 04:21:39 by Urs Schreiber (203.116.137.162)