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11-dimensional Chern-Simons theory

Contents

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

String theory

Contents

Idea

The most basic version of higher dimensional Chern-Simons theory over a (compact) smooth manifold XX of dimension 11 has as fields cocycles D^:XB 5U(1) conn\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:

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

This is the direct generalization of U(1)U(1)-3d Chern-Simons theory of the abelian 7d 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 7d Chern-Simons theory is richer, namely is, by AdS7/CFT6, the KK-compactification of 11-dimensional supergravity on S 4S^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 11d Chern-Simons theory. However, as a configuration of the RR-field is a cocycle in twisted differential K-theory, so there should be an 11d Chern-Simons theory given (Belov-Moore 06) by the fiber integration in differential cohomology of the cup product in differential cohomology in K-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

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

with more refined discussion in twisted differential KR-theory in

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

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)

Last revised on November 13, 2015 at 06:48:16. See the history of this page for a list of all contributions to it.