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For discrete group targets
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For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
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The case of higher dimensional Chern-Simons theory in dimension 7.
We discuss
Abelian 7d CS theory of an abelian 3-form connection;
A basic 7d higher dimensional Chern-Simons theory is the abelian theory, whose extended Lagrangian $\mathbf{L}$ is the diagonal of the cup product in ordinary differential cohomology
The transgression of this to codimension 0 hence for $\Sigma_7$ a closed manifold of dimension 7 is the action functional
A gauge field configuration
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 $C \in \Omega^3(\Sigma_7)$ and the above action functional takes this to the simple expression
where in the exponent we have the integration of differential forms over the wedge product of $C$ 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 $\phi \colon : \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}$ 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.
The second fractional Pontryagin class
has a smooth and differential refinement (see at twisted differential fivebrane structure) to an extended Lagrangian
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 $\mathbf{B}String_{conn}$.
The transgression of this to codimension 0 yields an action functional
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).
For $X$ a G2-manifold with characteristic differential forms
and
and for $G$ a simply connected compact semisimple Lie group with invariant polynomial $\langle -,-\rangle$, consider the action functional on the space of $\mathfrak{g}$-Lie algebra valued 1-forms $A$ given by the integration of differential forms
where $CS(A) \in \Omega^3(X)$ is the Chern-Simons form of $A$. 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 $\omega_4$, regarded as a 3-plectic structure, by a prequantum 3-bundle $\hat \mathbf{G}_2$
where $\mathbf{B}^3 U(1)_{conn} \in$ Smooth∞Grpd is the smooth moduli ∞-stack of circle 3-bundles with connection.
If moreover we write
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 $G$-Chern-Simons theory, then an extended Lagrangian for the above action functional is given by the cup product in ordinary differential cohomology
(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 $\omega_4$ 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 $X$ to be a G2-manifold to go through, a manifold with weak G2 holonomy is just as well, hence equipped with $\phi \in \Omega^3(X)$ such that
and
This arises from Freund-Rubin compactifications with cosmological constant $\lambda$ (Bilal-Derendinger-Sfetsos).
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
The abelian 7d higher dimensional Chern-Simons theory of a circle 3-bundle with connection was considered in
Edward Witten, Five-Brane Effective Action In M-Theory J. Geom. Phys.22:103-133,1997 (arXiv:hep-th/9610234)
Edward Witten, AdS/CFT Correspondence And Topological Field Theory JHEP 9812:012,1998 (arXiv:hep-th/9812012)
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
In
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 $\int_X CS(A) \wedge G_4$ as one summand and the Witten 97-functional as another.
Further discussion of extended Lagrangians for 7d CS theories is in
The Chern-Simons type action functionals $A \mapsto \int_X CS(A) \wedge \omega_4$ on a 7d G2-manifold $(X, \omega_3)$ was first considered in
and around (3.23) of
In
this is put into the context of topological M-theory (see around equation (2) in the introduction).
Discussion for weak G2-holonomy is in
Formulation in extended TQFT is discussed in