nLab D=7 Chern-Simons theory

Idea

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

Examples

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 D=7 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

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

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

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

A gauge field configuration

$$\phi \;\colon\; \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}$$

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 $C \in \Omega^3(\Sigma_7)$ and the above action functional takes this to the simple expression

$$C \mapsto \exp\left( 2 \pi i \int_{\Sigma_7} C \wedge d C \right) \in U(1) \,,$$

where in the exponent we have the integration of differential forms over the wedge product of $C$ 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.

Nonabelian $p_2$ theory on String 2-connections

The second fractional Pontryagin class

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

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

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

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 $\mathbf{B}String_{conn}$.

The transgression of this to codimension 0 yields an action functional

$$\exp\left( 2 \pi i \int_{\Sigma_7} [\Sigma_7, \tfrac{1}{6}\hat \mathbf{p}_2] \right) \;\colon\; [\Sigma_7, \mathbf{B}String_{conn}] \to U(1)$$

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 $G_2$ manifold

For $X$ a G₂-manifold with characteristic differential forms

$$\omega_3 \in \Omega^3(X)$$

and

$$\omega_4 = \star \omega_3\in \Omega^4(X)$$

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

$$A \mapsto \exp\left( 2 \pi i\int_{X} \omega_4 \wedge CS\left(A\right) \right) \,,$$

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 $\hat \mathbf{G}_2$

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

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

If moreover we write

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

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 $G$-Chern-Simons theory, then an extended Lagrangian for the above action functional is given by the cup product in ordinary differential cohomology

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

(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 $\omega_4$ 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

$$\omega_4 = \lambda \star \phi$$

and

$$d \phi = \omega_4 \,.$$

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

Properties

Moduli of fields (abelian case)

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

Related concepts

• ∞-Chern-Simons theory

• higher dimensional Chern-Simons theory

  • D=1 Chern-Simons theory

  • D=2 Chern-Simons theory

  • D=3 Chern-Simons theory

  • D=4 Chern-Simons theory

  • D=5 Chern-Simons theory

  • D=6 Chern-Simons theory

  • D=7 Chern-Simons theory

  • D=11 Chern-Simons theory

  • AKSZ sigma-models

  • string field theory

  • infinite-dimensional Chern-Simons theory

• M-theory on G₂-manifolds, topological M-theory

• Hitchin functional

References

Abelian theory

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

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory

Nonabelian theories

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

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Extended higher cup-product Chern-Simons theories

On $G_2$-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

• S. Donaldson, R. Thomas, Gauge theory in higher dimensions (pdf)

and around (3.23) of

• L. Baulieu, A. Losev, Nikita Nekrasov, Chern-Simons and Twisted Supersymmetry in Higher Dimensions, Nucl.Phys. B522 (1998) 82-104 (arXiv:hep-th/9707174)

In

• Jan de Boer, Paul de Medeiros, Sheer El-Showk, Annamaria Sinkovics, Open $G_2$ Strings (arXiv:hep-th/0611080)

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) $G_2$ 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

• Dan Freed, 4-3-2 8-7-6, talk at ASPECTS of Topology Dec 2012