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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{7d Chern-Simons theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory} [[!include infinity-Chern-Simons theory - contents]] \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{AbelianTheory}{Abelian theory}\dotfill \pageref*{AbelianTheory} \linebreak \noindent\hyperlink{OnString2Connections}{Nonabelian $p_2$ theory on String 2-connections}\dotfill \pageref*{OnString2Connections} \linebreak \noindent\hyperlink{TwoSpeciesCupProductTheoryOnG2Manifold}{Two-species cup-product theory on a $G_2$ manifold}\dotfill \pageref*{TwoSpeciesCupProductTheoryOnG2Manifold} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{moduli_of_fields_abelian_case}{Moduli of fields (abelian case)}\dotfill \pageref*{moduli_of_fields_abelian_case} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{abelian_theory_2}{Abelian theory}\dotfill \pageref*{abelian_theory_2} \linebreak \noindent\hyperlink{ReferencesNonabelianTheories}{Nonabelian theories}\dotfill \pageref*{ReferencesNonabelianTheories} \linebreak \noindent\hyperlink{on_manifolds}{On $G_2$-manifolds}\dotfill \pageref*{on_manifolds} \linebreak \noindent\hyperlink{formulation_in_extended_tqft}{Formulation in extended TQFT}\dotfill \pageref*{formulation_in_extended_tqft} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The case of [[higher dimensional Chern-Simons theory]] in [[dimension]] 7. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} We discuss \begin{enumerate}% \item \hyperlink{AbelianTheory}{Abelian 7d CS theory} of an abelian 3-form connection; \item \hyperlink{OnString2Connections}{7d p2 theory on String 2-connections} \item \hyperlink{TwoSpeciesCupProductTheoryOnG2Manifold}{2-species cup-product theory on a G2 manifold} \end{enumerate} \hypertarget{AbelianTheory}{}\subsubsection*{{Abelian theory}}\label{AbelianTheory} 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]] \begin{displaymath} \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} \,. \end{displaymath} The [[transgression]] of this to [[codimension]] 0 hence for $\Sigma_7$ a [[closed manifold]] of [[dimension]] 7 is the [[action functional]] \begin{displaymath} \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) \,. \end{displaymath} A [[gauge field]] configuration \begin{displaymath} \phi \;\colon\; \Sigma_7 \to \mathbf{B}^3 U(1)_{conn} \end{displaymath} here is a [[circle n-bundle with connection|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 \begin{displaymath} C \mapsto \exp\left( 2 \pi i \int_{\Sigma_7} C \wedge d C \right) \in U(1) \,, \end{displaymath} 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 (\hyperlink{Witten97}{Witten 97}), (\hyperlink{Witten98}{Witten 98}) a slight refinement of this construction (a [[quadratic refinement]] induced by an [[integral Wu structure]]) was argued to be the [[holographic principle|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 (\hyperlink{HopkinsSinger}{HopkinsSinger}). A refinement to [[extended Lagrangians]] as above is discussed in (\hyperlink{FSSII}{FSSII}). By the argument in (\hyperlink{Witten98}{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 [[Kaluza-Klein mechanism|compactification]] on a 4-[[sphere]] in the [[AdS-CFT|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 (\hyperlink{FSSI}{FSSI}, \hyperlink{FSSII}{FSSII}). The nonabelian 7d action functional this obtained contains the following two examples as summands. \hypertarget{OnString2Connections}{}\subsubsection*{{Nonabelian $p_2$ theory on String 2-connections}}\label{OnString2Connections} The [[second fractional Pontryagin class]] \begin{displaymath} [\tfrac{1}{6}p_2] \in H^8(B String, \mathbb{Z}) \end{displaymath} has a smooth and differential refinement (see at \emph{[[twisted differential fivebrane structure]]}) to an [[extended Lagrangian]] \begin{displaymath} \tfrac{1}{2}\hat \mathbf{p}_2 \;\colon\; \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} \,. \end{displaymath} where the domain is the [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|moduli 2-stack]] of [[String 2-group]] [[connection on a 2-bundle|principal 2-connections]] (see at \emph{[[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]] \begin{displaymath} \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) \end{displaymath} 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 (\hyperlink{FSSII}{FSSII}). \hypertarget{TwoSpeciesCupProductTheoryOnG2Manifold}{}\subsubsection*{{Two-species cup-product theory on a $G_2$ manifold}}\label{TwoSpeciesCupProductTheoryOnG2Manifold} For $X$ a [[G2-manifold]] with characteristic [[differential forms]] \begin{displaymath} \omega_3 \in \Omega^3(X) \end{displaymath} and \begin{displaymath} \omega_4 = \star \omega_3\in \Omega^4(X) \end{displaymath} 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]] \begin{displaymath} A \mapsto \exp\left( 2 \pi i\int_{X} \omega_4 \wedge CS\left(A\right) \right) \,, \end{displaymath} 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 (\hyperlink{DonaldsonThomas}{Donaldson-Thomas}) and (\hyperlink{BaulieuLosevNekrasov}{Baulieu-Losev-Nekrasov}). This appears as an action functional in [[topological M-theory]] (\hyperlink{deBoerEtAl}{deBoer et al}). To refine this to an [[extended Lagrangian]] and then fully globalize the action functional we can ask for a [[higher geometric quantization|higher geometric prequantization]] of $\omega_4$, regarded as a [[n-plectic structure|3-plectic structure]], by a [[prequantum n-bundle|prequantum 3-bundle]] $\hat \mathbf{G}_2$ \begin{displaymath} \itexarray{ && \mathbf{B}^3 U(1)_{\mathrm{conn}} \\ & {}^{\mathllap{\hat \mathbf{G}_2}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega_4}{\to}& \Omega^4_{cl} } \,, \end{displaymath} where $\mathbf{B}^3 U(1)_{conn} \in$ [[Smooth∞Grpd]] is the smooth [[moduli ∞-stack]] of [[circle n-bundle with connection|circle 3-bundles with connection]]. If moreover we write \begin{displaymath} \hat \mathbf{c} \;:\; \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} \end{displaymath} for the universal differential characteristic map which is the [[Lie integration]] of $\langle-,-\rangle$ (as discussed at \emph{[[differential string structure]]}), hence the [[extended Lagrangian]] for ordinary [[3d Chern-Simons theory|3d]] $G$-[[Chern-Simons theory]], then an [[extended Lagrangian]] for the above [[action functional]] is given by the [[cup product in ordinary differential cohomology]] \begin{displaymath} \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} \,. \end{displaymath} (This is an cup product extended Lagrangian of the kind considered in (\hyperlink{FSSIII}{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 \begin{displaymath} \omega_4 = \lambda \star \phi \end{displaymath} and \begin{displaymath} d \phi = \omega_4 \,. \end{displaymath} This arises from [[Freund-Rubin compactifications]] with [[cosmological constant]] $\lambda$ (\hyperlink{BilalDerendingerSfetsos}{Bilal-Derendinger-Sfetsos}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{moduli_of_fields_abelian_case}{}\subsubsection*{{Moduli of fields (abelian case)}}\label{moduli_of_fields_abelian_case} [[!include moduli of higher lines -- table]] \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[schreiber:∞-Chern-Simons theory]] \item [[higher dimensional Chern-Simons theory]] \begin{itemize}% \item [[1d Chern-Simons theory]] \item [[2d Chern-Simons theory]] \item [[3d Chern-Simons theory]] \item [[4d Chern-Simons theory]] \item [[5d Chern-Simons theory]] \item [[6d Chern-Simons theory]] \item \textbf{7d Chern-Simons theory} \item [[11d Chern-Simons theory]] \item [[AKSZ sigma-models]] \item [[string field theory]] \item [[infinite-dimensional Chern-Simons theory]] \end{itemize} \item [[M-theory on G2-manifolds]], [[topological M-theory]] \item [[Hitchin functional]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{abelian_theory_2}{}\subsubsection*{{Abelian theory}}\label{abelian_theory_2} The abelian 7d [[higher dimensional Chern-Simons theory]] of a [[circle n-bundle with connection|circle 3-bundle with connection]] was considered in \begin{itemize}% \item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory} J. Geom. Phys.22:103-133,1997 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234}) \item [[Edward Witten]], \emph{AdS/CFT Correspondence And Topological Field Theory} JHEP 9812:012,1998 (\href{http://arxiv.org/abs/hep-th/9812012}{arXiv:hep-th/9812012}) \end{itemize} and argued to be the [[holographic principle|holographic dual]] to the [[self-dual higher gauge theory]] of an abelian 2-form connection on a \emph{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 \begin{itemize}% \item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology, and M-Theory]]} \end{itemize} \hypertarget{ReferencesNonabelianTheories}{}\subsubsection*{{Nonabelian theories}}\label{ReferencesNonabelianTheories} In \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:7d Chern-Simons theory and the 5-brane]]} \end{itemize} \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The moduli 3-stack of the C-field]]} \end{itemize} the 7d Chern-Simons action obtained by [[Kaluza-Klein reduction|compactifying]] [[11-dimensional supergravity]] including the quantum corrections of the [[supergravity C-field]] on a 4-sphere (the [[AdS-CFT|AdS7/CFT6]] setup) is considered and refined to an [[extended Lagrangian]]. It contains the \hyperlink{DonaldsonThomas}{Donaldson-Thomas}-functional $\int_X CS(A) \wedge G_4$ as one summand and the \hyperlink{Witten97}{Witten 97}-functional as another. Further discussion of [[extended Lagrangians]] for 7d CS theories is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Extended higher cup-product Chern-Simons theories]]} \end{itemize} \hypertarget{on_manifolds}{}\subsubsection*{{On $G_2$-manifolds}}\label{on_manifolds} 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 \begin{itemize}% \item [[S. Donaldson]], R. Thomas, \emph{Gauge theory in higher dimensions} (\href{http://www2.imperial.ac.uk/~rpwt/skd.pdf}{pdf}) \end{itemize} and around (3.23) of \begin{itemize}% \item L. Baulieu, A. Losev, [[Nikita Nekrasov]], \emph{Chern-Simons and Twisted Supersymmetry in Higher Dimensions}, Nucl.Phys. B522 (1998) 82-104 (\href{http://arxiv.org/abs/hep-th/9707174}{arXiv:hep-th/9707174}) \end{itemize} In \begin{itemize}% \item [[Jan de Boer]], Paul de Medeiros, Sheer El-Showk, Annamaria Sinkovics, \emph{Open $G_2$ Strings} (\href{http://arxiv.org/abs/hep-th/0611080}{arXiv:hep-th/0611080}) \end{itemize} this is put into the context of [[topological M-theory]] (see around equation (2) in the introduction). Discussion for [[weak G2-holonomy]] is in \begin{itemize}% \item A. Bilal, J.-P. Derendinger, K. Sfetsos, \emph{(Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry}, Nucl.Phys. B628 (2002) 112-132 (\href{http://arxiv.org/abs/hep-th/0111274}{arXiv:hep-th/0111274}) \end{itemize} \hypertarget{formulation_in_extended_tqft}{}\subsubsection*{{Formulation in extended TQFT}}\label{formulation_in_extended_tqft} Formulation in [[extended TQFT]] is discussed in \begin{itemize}% \item [[Dan Freed]], \emph{[[4-3-2 8-7-6]]}, talk at \emph{\href{https://people.maths.ox.ac.uk/tillmann/ASPECTS.html}{ASPECTS of Topology}} Dec 2012 \end{itemize} [[!redirects 7-dimensional Chern-Simons theory]] [[!redirects 7d Chern-Simons theories]] [[!redirects 7-dimensional Chern-Simons theories]] \end{document}