From: Jacques Distler <distler@golem.ph.utexas.edu>
Subject: Re: Background free again
Date: Fri, 08 Sep 2000 23:00:18 -0500
Newsgroups: sci.physics.research
Summary: [missing]

In article <8pc0di$17hf$1@mortar.ucr.edu>, baez@galaxy.ucr.edu (John 
Baez) wrote:

>Is it true that there's no known superspace formulation for 11d
>supergravity?  I got that impression from the book "Quantum Fields
>and Strings: A Course for Mathematicians".

As you know, the hard part isn't constructing a supermanifold of the 
appropriate dimension. The hard part is constructing irreducible 
representations of the supersymmetry algebra as "constrained" 
superfields.

For small numbers of supersymmetries (eg, N=1 in D=4), everything works. 
But with more supersymmetries, things becom progressively more difficult.
Typically, one finds that either a) one cannot find a set of constraints 
that does the job, or that b) the constraints are TOO powerful, and that 
imposing them forces the fields to satisfy the equations of motion.

The latter situation is very common and, physically, is just a 
manifestation of the lack of an "off-shell" formulation of the 
supersymmetry transformations. If the supersymmetry algebra only closes 
on-shell (because there do not exist a suitable set of auxiliary fields 
which allow it to close off-shell), then a superfield formulation -- if 
it exists -- also only exists "on-shell", ie, the constraints imply the 
equations of motion.

Such an "on-shell" formulation exists for N=1 supersymmetry in D=10, and 
I recall some vague speculations that one might extend it to type IIA 
(and hence, probably to D=11).

>What's the best place to learn more about 11d supergravity - especially
>attempts to better understand the differential geometry it relies upon?
>I don't like formulas with dozens of terms bristling with indices, 

Oh c'mon! 11-D supergravity (classically, at least) is the simplest of 
all the supergravity theories. The lower dimensional theories are much 
more complicated.

11-D supergravity has just two bosonic fields: a metric, g, and a 3-form 
"gauge field" C.

The bosonic part of the action contains the usual Einstein-Hilbert term, 
a |dC|^2 kinetic term for the 3-form gauge field, and a "Chern-Simons" 
term,

       \integral  C \wedge dC \wedge dC

The only odd thing about this Chern-Simons term is that it has the 
"wrong" normalization. Normally, the coefficient of such a Chern-Simons 
term should be quantized so that it is invariant under "large" gauge 
transformations. This one is actually NOT gauge invariant (its 
coefficient is 1/6, when it should be an integer). To see that 
everything works out, and indeed that this funny coefficient was 
actually NECESSARY to cancel certain anomalies, is a slightly involved 
tale. 

The fermionic lagrangian is also quite simple. There's just one fermion 
field -- the gravitino, and aside from its usual kinetic term, it has a 
coupling to G=dC.

That's it: just 3 fields (2 bosons and one fermion) and a very simple 
Lagrangian.

Now, I suppose you will want to know what sort of geometry is associated 
to the 3-form gauge field (as principal fiber bundles are associated to 
1-form gauge fields).

I wish I could tell you, but if I did,  . . . I'd have to kill you :-).

JD

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