10D SuGra 2-Connection
Posted by Urs Schreiber
We have seen () that 11-dimensional supergravity is a gauge theory of a 3-connection, taking values in a certain Lie 3-algebra, , which is an extension of the super-Poincaré-1-algebra by a 4-cocycle.
I claimed () that non-fake flat -connections with values in an -algebra are to be interpreted in terms of their curvatures, which are flat -connections with values in
the -algebra of inner derivations of , where flatness encodes the -Bianchi identity.
The task is hence to compute .
That’s straightforward, but pretty hard. I am still hoping to figure out a shortcut, computing directly at the level of FDAs. But I don’t see the pattern yet.
Meanwhile, it might be a good idea to study related but simpler examples. As John has already mentioned () it might be easier to look at 10-dimensional supergravity first.
Here I present a discussion of what should be the bosonic part of the 2-connection governing 10-dimensional supergravity. The main point is to understand, from a categorical point of view, the relation
between the curvature 3-form , the Kalb-Ramond 2-form and the -connection , which governs the Green-Schwarz anomaly cancellation () (Phys.Lett.B149:117-122,1984, ()).
Following Killingback (, ), we expect 10-dimensional supergravity to be governed by the 2-group (, ), where is the the 10-dimensional Lorentz group times an internal factor or .
The Lie-2-algebra of this 2-group has a weak skeletal incarnation which is, as noticed first by André Henriques () nothing but the Baez-Crans Lie 2-algebra (see example 50 of Alissa Crans’ thesis). The Koszul-dual FDA of this guy is particularly simple:
Let be some Lie algebra, and let . On the free graded-commutative algebra
with in degree 1 and in degree 2, we define a differential of grade 1 in terms of a basis of and of by setting
Here are the structure constants of in the chosen basis and , where is the Killing form of .
This is nilpotent due to the Jacobi identity in .
We would like to find the Lie 3-algebra of inner derivations of this Lie 2-algebra.
Instead of trying to strictly derive this, I’ll notice that due to various constraints there is not much of a choice and an obvious ansatz will do the job.
From the study of for an arbitrary strict Lie-2-algebra (example 12 of the FDA Lab ()), we expect the algebra of to be based on a vector space consisting of two copies of , one of which shifted by one in degree. So consider
with the first in degree 1, the expression in brackets in degree 2 and the last in degree 3.
Let be a basis of in degree 1, a basis of in degree 2, a basis of in degree 2 and a basis of in degree 3.
We know (example 12 of the FDA Lab ()) that the 2-algebra of inner derivations of itself is encoded in the differential defined by
All available modifications of the original are hence given by
for some real parameter. For the particular choice we find that is nilpotent precisely if
In summary, we have the FDA on determined by
I dare to call this FDA , though I have only shown that it is one particular choice from a 1-parameter collection of admissable ansätze.
Given this, it is easy to derive the degrees of freedom of a flat (curvature-)-connection with values in this Lie 3-algebra.
First of all, this is a degree-preserving map from to the deRham complex of our base manifold, compatible with the algebra structure. In other words, we have a -valued 1-form
a -valued 2-form
and an ordinary 2-form
and finally a 3-form
In order for this to be a chain map, we need to have , where . This condition is equivalent to requiring that
is the curvature of and that
is the 3-form curvature, where denotes the Chern-Simons 3-form () of .
The last condition - the Bianchi-identity on - is then automatic
This is exactly the bosonic field content of the gauge sector that we expect.
In closing, I note that the above should essentially be the local infinitesimal version of a connection on the String-gerbe which Danny Stevenson describes in section 6 of his notes. Notice how in these notes, too, the String gerbe is governed by a flat CS-2-gerbe.
You can find notes on the computations involved here in the FDA Laboratory.
Re: 10D SuGra 2-Connection
The progress you’re making is really impressive!
Let’s try to figure out the Lie 3-algebra for a Lie 2-algebra g without resorting to any guesses. I’m sure you’re on the right track, but it would be nice to have a systematic approach, especially for when we categorify.
The only way I can imagine is fairly strenuous. You may have already tried it. If is the Lie 2-algebra of a Lie 2-group , should be the Lie 3-algebra of the Lie 3-group - the Lie 3-group of inner automorphisms of . I think we can, at least with sufficient energy, figure out the definition of and then work out its Lie 3-algebra . In fact, should only depend on ; we should be able to see how. Then, we can use this as a definition of for any Lie 2-algebra , regardless of whether it has a corresponding Lie 2-group.
(In fact any Lie n-algebra will come from a Lie n-group in the sense of Henriques . For the present purposes, this simply amounts to saying we can locally integrate the Lie n-algebra to a Lie n-group. In fact Henriques goes further, but we don’t need to worry about that here.)
Before we dive into this, a question: why are you sure we need instead of the potentially larger ?
There’s something funny about , after all. It should be analogous to . When is a group, the 2-group is equivalent to the trivial 2-group, right? I assume you mean to define this 2-group using a crossed module with . So, all objects in this 2-group are uniquely isomorphic. So, it’s equivalent to the trivial 2-group. By analogy, I expect that for any Lie algebra , the Lie 2-algebra should be equivalent to the trivial Lie 2-algebra. And, categorifying this idea rather blindly, I’d guess that for any Lie n-algebra , the Lie (n+1)-algebra is equivalent to the trivial one!
Can we really get something interesting with a -connection when is equivalent to something trivial?
I guess it’s easiest to check this for n = 1, and I assume you already have. If you could explain what’s really going on here, that would be great. You may recall a similar discussion over on David’s blog, where I was attacking the idea of Klein 2-geometry being interesting when the relevant 2-group is equivalent to a trivial one.
(Could there be a difference between 2-groups and Lie 2-groups here??? A smooth category with all objects uniquely isomorphic may not be smoothly equivalent to the trivial category. But in fact, I don’t think that’s a way out - I think is even smoothly equivalent to the trivial 2-group.)
Anyway, if we wanted to compute , I’d first ponder and then differentiate. If is a 2-group, is the 3-group with
Sitting inside here should be , but I’m not quite sure how to define it.