RTN recap part 1 and gauged supergravities

I'm back in Santiago after a week at the RTN winter school in CERN. I have a full list of topics that I want to go over from the school, and this time I REALLY will do it (I may have said that before, but this time I really mean it). In particular Henning Samtleben gave an excellent course on gauged supergravity and flux compactifications which was extremely clearly presented for what is not an easy topic to get your head around. There's a reading list which was supposed to be read before the lectures but I feel that I will get more out of them now. I'll put up a link as soon as the videos are up online.

I'll see if I can give a basic overview of gauged supergravities, if only to get it clear in my own head.

Gauged supergravities can be understood from two points of view.

The first route is to take an ungauged supergravity which comes from a compactification of 10 dimensional supergravity on a torus of some number of dimensions. This will give you an effective theory in D=10-d dimensions where d is the dimension of the torus. The second route is to compactify on a less trivial space and to turn on n-form fluxes in your theory.

For the first route, the D-dimensional theory which you obtain from compactifying on a torus will have both gauged and global symmetries. The gauged symmetries will be abelian and there will be a gauge field for each of these U(1) symmetries. Then there will be a global symmetry, which typically will take the form of an exceptional group (one subset of the semi-simple Lie groups).

The theory you now have is an ungauged supergravity (although it does have gauge fields, the name comes from the fact that there are no non-abelian gauge symmetries). A gauged supergravity is formed by gauging some subset of the global symmetries.

There may be many ways to do this (to choose the subset of global symmetries to gauge) which will put the abelian gauge fields that you already have into the adjoint representation of the now non-abelian gauge group. The way in which you do the gauging (choose the subgroup of the global symmetries to gauge) is all defined in terms of a single tensor. This tensor has a series of constraints which must be imposed in order to form a closed algebra when you perform the gauging and also to keep supersymmetry (though you may wish to break some number of the supersymmetries in the gauging - giving you a maximally or non-maximally supersymmetric gauged supergravity).

As I said, there are two routes to gauged supergravity. The second route is to go back to the 10 dimensional supergravity that you started with, and rather than simply compactifying on a torus, you compactify on a more complex manifold - a d-sphere, a torus with torsion, or you may turn on some fluxes which wrap closed cycles in the compactified directions. This process results in the same gauged supergravity theories that you would have obtained by gauging a subgroup of the global symmetries in your 'trivially' compactified theory.

The point about these theories is that you have gone from a 10 dimensional theory to a lower dimensional theory with a non-abelian gauge group. This is clearly an interesting area to explore and has many applications, from the AdS/CFT correspondence to model building and more...

Anyway, that's the very basics (the flow chart as I have picked it up over the last week) of gauged supergravities.

As far as I can tell, the main subtleties come from 1) applying the constraints on the tensor which encodes the gauging of your global symmetries and 2) finding a realisation where the gauge transformations act in a manifestly covariant manner on your fields.

I may try and do the same thing for the other lectures if I get some time. For now I'll upload some photos from the trip.