nLab 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 $\frac{1}{6}p_2 : \mathcal{B}String \to \mathcal{B}^8 \mathbb{Z}$ in Top on the classifying space of the string group has a smooth lift to the smooth second fractional Pontryagin class

$$\frac{1}{6} \mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)$$

in $\mathbf{H} :=$ ∞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 $\mu_7 : \mathfrak{string} \to b^6 \mathbb{R}$ 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

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

hence on cocycle ∞-groupoids

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

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 $\frac{1}{6}\hat\mathbf{p}(\nabla) \in \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$ 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 $\hat \mathbf{c}(\nabla)$ is the curvature characteristic form $\langle F_\nabla \wedge F_\nabla \wedge F_\nabla \wedge F_\nabla\rangle$ of $\nabla$ and accordingly the 7-form connection on $\hat \mathbf{c}(\nabla)$ is locally a Chern-Simons form $CS(\nabla)$ of $\nabla$.

Therefore the higher parallel transport induced by $\frac{1}{6}\hat \mathbf{p}_2(\nabla)$ over 7-dimensional volumes $\phi : \Sigma \to X$ 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 discussion at ∞-Chern-Weil theory and in direct analogy to the construction of the Chern-Simons circle 3-bundle we can model the (∞,1)-functor

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

by postcomposition with the ∞-anafunctor

$$\array{ \exp(\mathfrak{string})_{conn} &\stackrel{\exp(\mu_7)_{conn}}{\to}& \exp(b^6 \mathbb{R})_{conn} \\ \downarrow && \downarrow \\ \mathbf{cosk}_7 \exp(\mathfrak{string})_{conn} &\to& \mathbf{B}^7 U(1)_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}String_{conn} }$$

where $\mu_7 : \mathfrak{string} \to b^6 \mathbb{R}$ is the 7-cocycle that classifies the fivebrane Lie 6-algebra.

For

$$\array{ C(U) &\stackrel{g}{\to}& \mathbf{B}String_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ X }$$

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

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

produces a lift of the transition functions $g$ to $\mathbf{cosk}_7 \exp(\mathfrak{string})$. 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

$$U_{i_1} \cap \cdots U_{i_k} \times \Delta^k \to G$$

up to $k = 7$. That this is indeed always possible is the statement about Lie integration that $\mathbf{cosk}_7 \exp(\mathfrak{string}) \stackrel{\simeq}{\to} \mathbf{B}String$ is a weak equivalence, which in turn is due to the fact that the next nonvanishing homotopy group of $G = SO(n)$ after $\pi_3$ is $\pi_7$.

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

$$\left( CS_7(\sigma_i^* A) \,,\, \int_{\Delta^1} g_{i j}^*CS_7(A) \,,\, \int_{\Delta^2} g_{i j k}^*CS_7(A) \,,\, \int_{\Delta^3} \hat g_{i j k l}^*CS_7(A) \,,\, \int_{\Delta^5} \hat g_{i j k l m}^*CS_7(A) \,,\, \int_{\Delta^6} \hat g_{i j k l m n}^*CS_7(A) \,,\, \int_{\Delta^7} \hat g_{i j k l m n o}^* \mu(A) \right) \,,$$

where $A$ is a connection form on the total space of the $Spin(n)$-principal bundle that the string bundle itself is lifted from and $CS_7$ 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

and identified as part of the 11-dimensional supergravity Chern-Simons terms after KK-reduction on $S^4$ to 7-dimensional supergravity (for AdS/CFT duality with the M5-brane worldvolume 6d (2,0)-superfonformal ∞-Wess-Zumino-Witten theory) in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, 7d Chern-Simons theory and the 5-brane

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The moduli 3-stack of the C-field