nLab D=11 Chern-Simons theory

Idea

The most basic version of higher dimensional Chern-Simons theory over a (compact) smooth manifold $X$ of dimension 11 has as fields cocycles $\hat D \colon X \to \mathbf{B}^5 U(1)_{conn}$ in degree-5 ordinary cohomology and whose action functional is given by the fiber integration in ordinary differential cohomology of the cup product in ordinary differential cohomology of such a field with itself:

$$\hat D \mapsto \exp\left(\tfrac{i}{\hbar}\int_X \hat D \cup \hat D\right) \,.$$

This is the direct generalization of $U(1)$-D=3 Chern-Simons theory of the abelian D=7 Chern-Simons theory, and all three are related by holography to the self-dual higher gauge field in dimension 2,6, and 10, respectively.

However, for applications in string theory more refined versions of these theories matter. For instance in 7d the full 6d (2,0)-superconformal QFT contains not just a single abelian higher self-dual gauge field and accordingly the corresponding D=7 Chern-Simons theory is richer, namely is, by AdS7/CFT6, the KK-compactification of 11-dimensional supergravity on $S^4$. Similarly, in 10-dimensions the RR-field of type II superstring theory is a higher self-dual gauge field whose quantization law is of the form that makes it qualify (Moore-Witten 99) as the holographic boundary theory of an D=11 Chern-Simons theory. However, as a configuration of the RR-field is a cocycle in twisted differential K-theory, so there should be an D=11 Chern-Simons theory given (Belov-Moore 06) by the fiber integration in differential cohomology of the cup product in differential cohomology in K-theory.

Related concepts

• D=11 supergravity

• 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

References

The self-dual higher gauge field nature (see there for more) in terms of a quadratic form on differential K-theory is discussed originally around

• Gregory Moore, Edward Witten, Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 0005:032 (2000) [arXiv:hep-th/9912279, doi:10.1088/1126-6708/2000/05/032]

and (Freed 00) for type I superstring theory, and for type II superstring theory in

• Edward Witten, Duality Relations Among Topological Effects In String Theory, JHEP 0005:031,2000 (arXiv:hep-th/9912086)

• Daniel Freed, Michael Hopkins, On Ramond-Ramond fields and K-theory, JHEP 0005 (2000) 044 (arXiv:hep-th/0002027)

• D. Diaconescu, Gregory Moore, Edward Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv.Theor.Math.Phys.6:1031-1134,2003 (arXiv:hep-th/0005090), summarised in A Derivation of K-Theory from M-Theory (arXiv:hep-th/0005091)

with more refined discussion in twisted differential KR-theory in

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)

See at orientifold for more on this. The relation to 11d Chern-Simons theory is made manifest in

• Dmitriy Belov, Greg Moore, Type II Actions from 11-Dimensional Chern-Simons Theories (arXiv:hep-th/0611020)

Review is in

• Richard Szabo, section 3.6 and 4.6 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology (arXiv:1209.2530)