Gauged SUGRA

Lets continue with compactifications of maximal supergravities. As it is now clear from the previous post the Kaluza-Klein reduction leads to a theory whose vector multiplets are abelian. In other words the gauge group is $U(1)^{n_V}$ with $n_V$ as number of vector multiplets. And none of the matter fields is charged under the gauge group. At the same time both scalars and tensors transform non-trivially under the global duality group. Scalar are elements of the coset space (see the previous post) and vectors transform in some linear representation $\mathcal{R}_{n_V}$. This is what is called ungauged supergravity. 

Ungauged SUGRA

At the moment we have 5 different string theories that are connected to each other by duality transformations.

As it is shown on this picture low energy limits of IIA and Heterotic $E_8\times E_8$ string theories can be obtained by compactification of the 11D SUGRA on a circle $S^1$ and a line $I$ respectively.

What we know is that the $11$ dimensional supergravity is a low energy limit of the mysterious M-theory. Actually, this is more or less the only thing we know about M-theory. This means that it is worth understanding supergravity.

W-algebras

Some time ago I was reading articles about W-algebras and W-gravity by Chris Hull http://arxiv.org/abs/hep-th/9302110 and Christopher Pope http://arxiv.org/abs/hep-th/9112076 . This appeared to be very fascinating thing.

Every textbook on the string theory more or less starts with Virasoro algebra and Virasoro operators $L_n$. These operators are modes of the energy-momentum tensor of a string that is an object of spin 2. This could be found from it OPE
$$T(z)T(w)\sim \frac{\partial T(w)}{z-w}+\frac{2T(w)}{(z-w)^2}+\frac{c/2}{(z-w)^4}.$$
The Operator Product Expansion simly represents commutator in the radial quantization and can be understood as some fancy way of writing of Virasoro commutation relations. And 2 in the second term of OPE directly related to the fact, that $T(z)$ has spin 2.