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universal Chern-Simons circle 7-bundle with connection

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 $\frac{1}{6}{p}_2:ℬ\mathrm{String}\to {ℬ}^8\mathbb{Z}$ in Top on the classifying space of the string group has a smooth lift to the smooth second fractional Pontryagin class

$$\frac{1}{6}{p}_2:B\mathrm{String}\to B^{7}U(1)$$\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 ${\mu }_7:\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to b^6ℝ$ 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

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

hence on cocycle ∞-groupoids

$$\frac{1}{6}\hat{p}:H_{\mathrm{conn}}(X,B\mathrm{String})\to H_{\mathrm{diff}}^8(X)$$\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 $(P,\nabla )$ a String-principal 2-bundle, we call the image $\frac{1}{6}\hat{p}(\nabla )\in H_{\mathrm{diff}}(X,B^{z}U(1))$ 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 $\hat{c}(\nabla )$ 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 $\hat{c}(\nabla )$ is locally a Chern-Simons form $\mathrm{CS}(\nabla )$ of $\nabla$.

Therefore the higher parallel transport induced by $\frac{1}{6}{\hat{p}}_2(\nabla )$ 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

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

by postcomposition with the ∞-anafunctor

$$\begin{array}{ccc}\exp (\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{)}_{\mathrm{conn}}& \stackrel{\exp ({\mu }_7{)}_{\mathrm{conn}}}{\to }& \exp (b^6ℝ{)}_{\mathrm{conn}}\\ \downarrow & & \downarrow \\ {\mathrm{cosk}}_7\exp (\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{)}_{\mathrm{conn}}& \to & B^{7}U(1{)}_{\mathrm{conn}}\\ {\downarrow }^{\simeq }\\ B{\mathrm{String}}_{\mathrm{conn}}\end{array}$$\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 ${\mu }_7:\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to b^6ℝ$ is the 7-cocycle that classifies the fivebrane Lie 6-algebra.

For

$$\begin{array}{ccc}C(U)& \stackrel{g}{\to }& B{\mathrm{String}}_{\mathrm{conn}}\\ {\downarrow }^{\simeq }\\ X\end{array}$$\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 $\mathrm{\infty }$-anafunctor composition

$$\begin{array}{ccccc}& & \exp (\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{)}_{\mathrm{conn}}& \stackrel{\exp ({\mu }_7{)}_{\mathrm{conn}}}{\to }& \exp (b^6ℝ{)}_{\mathrm{conn}}\\ & & \downarrow & & \downarrow \\ C(V)& \stackrel{\hat{g}}{\to }& {\mathrm{cosk}}_7\exp (\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}{)}_{\mathrm{conn}}& \to & B^{7}U(1{)}_{\mathrm{conn}}\\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }\\ C(U)& \stackrel{g}{\to }& B{\mathrm{String}}_{\mathrm{conn}}\\ {\downarrow }^{\simeq }\\ X\end{array}$$\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 ${\mathrm{cosk}}_7\exp (\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤})$. The string-cocycle is itself in first degree a collection of paths in $G$, in second a collection of surfaces with labels in $U(1)$. That lift corresponds to further resolving this to families

$$U_{i_1}\cap \cdots U_{i_k}\times {\Delta }^k\to G$$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 ${\mathrm{cosk}}_7\exp (\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤})\stackrel{\simeq }{\to }B\mathrm{String}$ is a weak equivalence, which in turn is due to the fact that the next nonvanishing homotopy group of $G=\mathrm{SO}(n)$ after ${\pi }_3$ is ${\pi }_7$.

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

$$({\mathrm{CS}}_7({\sigma }_i^{*}A),{\int }_{{\Delta }^1}{g}_{ij}^{*}{\mathrm{CS}}_7(A),{\int }_{{\Delta }^2}{g}_{ijk}^{*}{\mathrm{CS}}_7(A),{\int }_{{\Delta }^3}{\hat{g}}_{ijkl}^{*}{\mathrm{CS}}_7(A),{\int }_{{\Delta }^5}{\hat{g}}_{ijklm}^{*}{\mathrm{CS}}_7(A),{\int }_{{\Delta }^6}{\hat{g}}_{ijklmn}^{*}{\mathrm{CS}}_7(A),{\int }_{{\Delta }^7}{\hat{g}}_{ijklmno}^{*}\mu (A)),$$\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 $\mathrm{Spin}(n)$-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

• differential fivebrane structure

• dual heterotic string theory

Related concepts

• universal Chern-Simons n-bundle

  • Chern-Simons circle 3-bundle

  • Chern-Simons circle 7-bundle

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

• Domenico Fiorenza, Urs Schreiber, Hisham Sati, Cech cocycles for differential characteristic classes