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universal Chern-Simons circle 7-bundle with connection

Context

\infty-Chern-Weil theory

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 16p 2:String 8\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

16p 2:BStringB 7U(1) \frac{1}{6} \mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)

in H:=\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 μ 7:𝔰𝔱𝔯𝔦𝔫𝔤b 6\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

16p^:BString connB 7U(1) conn \frac{1}{6}\hat \mathbf{p} : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn}

hence on cocycle ∞-groupoids

16p^:H conn(X,BString)H 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,)(P,\nabla) a String-principal 2-bundle, we call the image 16p^()H diff(X,B zU(1))\frac{1}{6}\hat\mathbf{p}(\nabla) \in \mathbf{H}_{diff}(X,\mathbf{B}^z U(1)) its Chern-Simons circle 7-bundle with connection.

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

By construction, the curvature 8-form of c^()\hat \mathbf{c}(\nabla) is the curvature characteristic form F F F F \langle F_\nabla \wedge F_\nabla \wedge F_\nabla \wedge F_\nabla\rangle of \nabla and accordingly the 7-form connection on c^()\hat \mathbf{c}(\nabla) is locally a Chern-Simons form CS()CS(\nabla) of \nabla.

Therefore the higher parallel transport induced by 16p^ 2()\frac{1}{6}\hat \mathbf{p}_2(\nabla) over 7-dimensional volumes ϕ:ΣX\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 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 conn(X,BString)H conn(X,B 7U(1)) \mathbf{H}_{conn}(X, \mathbf{B}String) \to \mathbf{H}_{conn}(X, \mathbf{B}^7 U(1))

by postcomposition with the ∞-anafunctor

exp(𝔰𝔱𝔯𝔦𝔫𝔤) conn exp(μ 7) conn exp(b 6) conn cosk 7exp(𝔰𝔱𝔯𝔦𝔫𝔤) conn B 7U(1) conn BString conn \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 μ 7:𝔰𝔱𝔯𝔦𝔫𝔤b 6\mu_7 : \mathfrak{string} \to b^6 \mathbb{R} is the 7-cocycle that classifies the fivebrane Lie 6-algebra.

For

C(U) g BString conn X \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

exp(𝔰𝔱𝔯𝔦𝔫𝔤) conn exp(μ 7) conn exp(b 6) conn C(V) g^ cosk 7exp(𝔰𝔱𝔯𝔦𝔫𝔤) conn B 7U(1) conn C(U) g BString conn X \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 gg to cosk 7exp(𝔰𝔱𝔯𝔦𝔫𝔤)\mathbf{cosk}_7 \exp(\mathfrak{string}). The string-cocycle is itself in first degree a collection of paths in GG, in second a collection of surfaces with labels in U(1)U(1). That lift corresponds to further resolving this to families

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

up to k=7k = 7. That this is indeed always possible is the statement about Lie integration that cosk 7exp(𝔰𝔱𝔯𝔦𝔫𝔤)BString\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)G = SO(n) after π 3\pi_3 is π 7\pi_7.

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

(CS 7(σ i *A), Δ 1g ij *CS 7(A), Δ 2g ijk *CS 7(A), Δ 3g^ ijkl *CS 7(A), Δ 5g^ ijklm *CS 7(A), Δ 6g^ ijklmn *CS 7(A), Δ 7g^ ijklmno *μ(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 AA is a connection form on the total space of the Spin(n)Spin(n)-principal bundle that the string bundle itself is lifted from and CS 7CS_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

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

Revised on August 22, 2013 04:06:44 by Urs Schreiber (151.201.35.138)