### Chern-Simons Actions for (Super)-Gravities

#### Posted by Urs Schreiber

Just as

electromagnetism is a theory of line 1-bundles with connection coupled to electric 1-particles and magnetic 1-particles,

we have that

supergravity # in eleven dimensions is a theory of line 3- and line 6-bundles with connection coupled to electric 3-particles and magnetic 6-particles.

(There is a beautiful discussion of essentially this statement by D. Freed, which I talked about here, and here. Freed doesn’t say “$n$-bundle with connection”, but instead says “differential cocycle”. But it’s the same kind of thing.)

Wonders never cease, and hence there are indications that there is more to 11-dimensional supergravity than meets the eye. The question is: what? What is 11-dimensional supergravity *really* about?

One idea is: *it is really about 1-particles on the “$E_{10}$-group manifold”*. This we talked about before.

Another idea is: *it is really about the higher Chern-Simons theory # of an invariant degree 6-polynomial on a super Lie algebra not unlike super-$so(n,m)$ #*.

This speculation was put forward in

Petr Hořava
*M-Theory as a Holographic # Field Theory*

(arXiv)

The jargon in the title is such as to make certain physicists excited. A completely different, but possibly just as exciting jargon would be: it is speculated here that, very fundamentally, physics is about those representations of extended cobordism categories which are naturally induced from Chern-Simons $n$-bundles with connection.

I was reminded of that by the appearance of the very nicely written basic review

Jorge Zanelli

Lecture notes on Chern-Simons (super-)gravities

(arXiv)

which was updated a few days ago. (Thanks to It’s equal but It’s different for noticing.)

This reviews the action functionals for theories of gravity one obtains by picking a $d = 2k +1$-dimensional manifold $X$, a structure group $G$ like $SO(d-1,1) \hookrightarrow \left\lbrace \array{ SO(d,1) \\ (ISO(d-1,1)) \\ SO(d-1,2) &\hookrightarrow& OSP(m|N) } \right.$ together with a degree $(d+1)/2$ invariant polynomial $\langle \cdots \rangle$ on its Lie algebra; and takes the action functional to be the corresponding Chern-Simons integral which sends $g$-valued 1-forms $A$ on $X$ to $A \mapsto \int_X \mathrm{CS}(A) \,,$ where the Chern-Simons $d$-form # $CS(A)$ satisfies $d CS(A) = \langle F_A \wedge F_A \wedge \cdots \wedge F_A \rangle$.

For $d=3$ this yields, famously, the ordinary (super) Einstein-Hilbert action in that dimension. For higher (odd) $d$, this yields the (super) Einstein-Hilbert action with higher curvature contributions.

Hořava gave arguments suggesting that and how for $d=11$ the Chern-Simons gravity action reduces to that of ordinary supergravity in the appropriate limit.

Posted at March 12, 2008 6:55 PM UTC
## Re: Chern-Simons Actions for (Super)-Gravities

Whenever I start to read this post I hear the ‘like’ in Americanese – “Like electromagnetism is

soa theory of line 1-bundles…”Staying with the trivial, “wonders never cease” is more common than “wonders never end”.

Now for something more serious. Recasting M-theory as you have, do we get any clearer idea of what’s so special about it? Is there some happenstantial fluke occurring in this dimension? Or is it part of that greater body of theory studied by ‘exceptionology’?

Here’s John from a while back

Any clearer idea now?