7d Chern-Simons theory


\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




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The case of higher dimensional Chern-Simons theory in dimension 7.


We discuss

  1. Abelian 7d CS theory of an abelian 3-form connection;

  2. 7d p2 theory on String 2-connections

  3. 2-species cup-product theory on a G2 manifold

Abelian theory

A basic 7d higher dimensional Chern-Simons theory is the abelian theory, whose extended Lagrangian L\mathbf{L} is the diagonal of the cup product in ordinary differential cohomology

L DDDD:B 3U(1) connΔB 3U(1) conn×B 3U(1) conn^B 7U(1) conn. \mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}} \colon \mathbf{B}^3 U(1)_{conn} \stackrel{\Delta}{\to} \mathbf{B}^3 U(1)_{conn} \times \mathbf{B}^3 U(1)_{conn} \stackrel{\widehat {\cup}}{\to} \mathbf{B}^7 U(1)_{conn} \,.

The transgression of this to codimension 0 hence for Σ 7\Sigma_7 a closed manifold of dimension 7 is the action functional

exp(2πi Σ 7[Σ 7,L DDDD]):[Σ 7,B 3U(1) conn]U(1). \exp\left( 2 \pi i \int_{\Sigma_7} [\Sigma_7, \mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}}] \right) \;\colon\; [\Sigma_7, \mathbf{B}^3 U(1)_{conn}] \to U(1) \,.

A gauge field configuration

ϕ:Σ 7B 3U(1) conn \phi \;\colon\; \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}

here is a circle 3-bundle with connection. In the special case that the underlying circle 3-group principal 3-bundle is trivializable and trivialized, this is equivalently a differential 3-form CΩ 3(Σ 7)C \in \Omega^3(\Sigma_7) and the above action functional takes this to the simple expression

Cexp(2πi Σ 7CdC)U(1), C \mapsto \exp\left( 2 \pi i \int_{\Sigma_7} C \wedge d C \right) \in U(1) \,,

where in the exponent we have the integration of differential forms over the wedge product of CC with its de Rham differential. On general field configurations the action functional is the suitable globalization of this expression.

In (Witten 97), (Witten 98) a slight refinement of this construction (a quadratic refinement induced by an integral Wu structure) was argued to be the holographic dual to the self-dual higher gauge theory of the abelian self-dual 2-form gauge field in the 6d (2,0)-superconformal QFT on the worldvolume of the M5-brane. The issue of the quadratic refinement was discussed in more detail in (HopkinsSinger). A refinement to extended Lagrangians as above is discussed in (FSSII).

By the argument in (Witten98) the above relation holds when we interpret the fields ϕ::Σ 7B 3U(1) conn\phi \colon : \Sigma_7 \to \mathbf{B}^3 U(1)_{conn} as the supergravity C-field after compactification on a 4-sphere in the AdS7-CFT6 setup. By the discussion at 11-dimensional supergravity this field is in general not simply a 3-connection as above but receives corrections by a Green-Schwarz mechanism and “flux quantization” which give it non-abelian components. This, and the resulting non-abelian generalization of the above extended Lagrangian is discussed in (FSSI, FSSII).

The nonabelian 7d action functional this obtained contains the following two examples as summands.

Nonabelian p 2p_2 theory on String 2-connections

The second fractional Pontryagin class

[16p 2]H 8(BString,) [\tfrac{1}{6}p_2] \in H^8(B String, \mathbb{Z})

has a smooth and differential refinement (see at twisted differential fivebrane structure) to an extended Lagrangian

12p^ 2:BString connB 7U(1) conn. \tfrac{1}{2}\hat \mathbf{p}_2 \;\colon\; \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} \,.

where the domain is the smooth moduli 2-stack of String 2-group principal 2-connections (see at differential string structure for more). This modulates the Chern-Simons circle 7-bundle with connection on BString conn\mathbf{B}String_{conn}.

The transgression of this to codimension 0 yields an action functional

exp(2πi Σ 7[Σ 7,16p^ 2]):[Σ 7,BString conn]U(1) \exp\left( 2 \pi i \int_{\Sigma_7} [\Sigma_7, \tfrac{1}{6}\hat \mathbf{p}_2] \right) \;\colon\; [\Sigma_7, \mathbf{B}String_{conn}] \to U(1)

on string 2-connection fields. This is part of the quantum-corrected and flux-quantized extended action functional of the supergravity C-field in 11-dimensional supergravity by the analysis in (FSSII).

Two-species cup-product theory on a G 2G_2 manifold

For XX a G2-manifold with characteristic differential forms

ω 3Ω 3(X) \omega_3 \in \Omega^3(X)


ω 4=ω 3Ω 4(X) \omega_4 = \star \omega_3\in \Omega^4(X)

and for GG a simply connected compact semisimple Lie group with invariant polynomial ,\langle -,-\rangle, consider the action functional on the space of 𝔤\mathfrak{g}-Lie algebra valued 1-forms AA given by the integration of differential forms

Aexp(2πi Xω 4CS(A)), A \mapsto \exp\left( 2 \pi i\int_{X} \omega_4 \wedge CS\left(A\right) \right) \,,

where CS(A)Ω 3(X)CS(A) \in \Omega^3(X) is the Chern-Simons form of AA. This, or some suitable globalization of this, has been considered as an action functional for 7-dimensional Chern-Simons-type theory in (Donaldson-Thomas) and (Baulieu-Losev-Nekrasov). This appears as an action functional in topological M-theory (deBoer et al).

To refine this to an extended Lagrangian and then fully globalize the action functional we can ask for a higher geometric prequantization of ω 4\omega_4, regarded as a 3-plectic structure, by a prequantum 3-bundle G^ 2\hat \mathbf{G}_2

B 3U(1) conn G^ 2 F () X ω 4 Ω cl 4, \array{ && \mathbf{B}^3 U(1)_{\mathrm{conn}} \\ & {}^{\mathllap{\hat \mathbf{G}_2}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega_4}{\to}& \Omega^4_{cl} } \,,

where B 3U(1) conn\mathbf{B}^3 U(1)_{conn} \in Smooth∞Grpd is the smooth moduli ∞-stack of circle 3-bundles with connection.

If moreover we write

c^:BG connB 3U(1) conn \hat \mathbf{c} \;:\; \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}

for the universal differential characteristic map which is the Lie integration of ,\langle-,-\rangle (as discussed at differential string structure), hence the extended Lagrangian for ordinary 3d GG-Chern-Simons theory, then an extended Lagrangian for the above action functional is given by the cup product in ordinary differential cohomology

exp(2πi Σ 7[Σ 7,G^ 4^S^]):X×BG conn(G^ 2,c^)B 3U(1) conn×B 3U(1) conn^B 7U(1) conn. \exp\left( 2 \pi i \int_{\Sigma_7} [\Sigma_7, \hat {\mathbf{G}}_4 \hat \cup \hat \mathbf{S}] \right) \;\colon\; X \times \mathbf{B}G_{conn} \stackrel{(\hat \mathbf{G}_2, \hat \mathbf{c})}{\to} \mathbf{B}^3 U(1)_{conn} \times \mathbf{B}^3 U(1)_{conn} \stackrel{\hat \cup}{\to} \mathbf{B}^7 U(1)_{conn} \,.

(This is an cup product extended Lagrangian of the kind considered in (FSSIII).)

Notice that the prequantization lift to differential cohomology is entirely demanded by the interpretation of ω 4\omega_4 as the field strength of the supergravity C-field in interpretations of this setup in M-theory on G2-manifolds.

Moreover, the above considerations do not really need XX to be a G2-manifold to go through, a manifold with weak G2 holonomy is just as well, hence equipped with ϕΩ 3(X)\phi \in \Omega^3(X) such that

ω 4=λϕ \omega_4 = \lambda \star \phi


dϕ=ω 4. d \phi = \omega_4 \,.

This arises from Freund-Rubin compactifications with cosmological constant λ\lambda (Bilal-Derendinger-Sfetsos).


Moduli of fields (abelian case)

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
n=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian


Abelian theory

The abelian 7d higher dimensional Chern-Simons theory of a circle 3-bundle with connection was considered in

and argued to be the holographic dual to the self-dual higher gauge theory of an abelian 2-form connection on a single M5-brane in its 6d (2,0)-supersymmetric QFT on the worldvolume.

The precise formulation of this functional in terms of differential cohomology and integral Wu structure was given in

Nonabelian theories


the 7d Chern-Simons action obtained by compactifying 11-dimensional supergravity including the quantum corrections of the supergravity C-field on a 4-sphere (the AdS7/CFT6 setup) is considered and refined to an extended Lagrangian. It contains the Donaldson-Thomas-functional XCS(A)G 4\int_X CS(A) \wedge G_4 as one summand and the Witten 97-functional as another.

Further discussion of extended Lagrangians for 7d CS theories is in

On G 2G_2-manifolds

The Chern-Simons type action functionals A XCS(A)ω 4A \mapsto \int_X CS(A) \wedge \omega_4 on a 7d G2-manifold (X,ω 3)(X, \omega_3) was first considered in

and around (3.23) of


this is put into the context of topological M-theory (see around equation (2) in the introduction).

Discussion for weak G2-holonomy is in

  • A. Bilal, J.-P. Derendinger, K. Sfetsos, (Weak) G 2G_2 Holonomy from Self-duality, Flux and Supersymmetry, Nucl.Phys. B628 (2002) 112-132 (arXiv:hep-th/0111274)

Formulation in extended TQFT

Formulation in extended TQFT is discussed in

Revised on September 22, 2014 10:33:22 by Urs Schreiber (