Why M2 and M5 branes are called electric and magnetic?

M-theory, that is a theory describing dynamics of 2 and 5 dimensional membranes, is believed to be a certain generalisation of 11 dimensional supergravity. It is often said that 11D SURGA is a low energy limit of M-theory. Basically, this is the only information known about M-theory to the moment. So why the fundamental objects of M-theory are known to be M2 and M5 branes? In one of the previous posts it has been figured out that this can be related to 11-dimensional superalgebra and Fierz identities. However, this does not explain why M5 branes are called magnetic. Here we will approach from the side of the supergravity in 11 dimensions and see that its fundamental gauge fields interact with electric and magnetic currents represented by 3- and 6-forms.

Database of Hep-TH people

Almost a year ago being in Seoul I was thinking about visiting say the Far East Federal University with a seminar. Trying to find any information on their official website about people working on HEP I realized, how a complicated task it is. Almost all webpages of institutions contain very fragmented, insufficient, outdated or even wrong information on the subject, that makes it very hard to understand who is working at which institute and what are his/her research interests. Precisely to overcome this problem I started developing a website, storing a database for all HEP-people.

Why there are only M2 and M5 branes?

A short answer to the question the title asks would be: Because Clifford algebra in odd dimensions is not simple.

An extended and more understandable explanation obviously needs some calculations, that will relate Fierz identities in 11 dimensions to central charges of $\mathcal{N}=1$ superalgebra. And the reducibility argument will allow to include the correct amount of central charges.

Certainly, one should realise that the arguments below do not give an exhaustive answer to the question, one necessarily needs to consider solutions of EOM of 11 dimensional sugra, look at the realisation of superalgebra and susy multiplets. But these simple calculations certainly give a hint to at least intuitive understanding, that is obviously necessary for our work.

Exotic branes

In the theory of open string one imposes boundary conditions on string ends. in general there are two types of these conditions: Dirichlet and Newmann. Newmann boundary conditions effectively allow the string end to move arbitrarily. Dirichlet conditions restrict movement of the string end to some p-dimensional surface in the bulk space, that is called Dp-brane. From the point of view of external space a D-brane wrapped around internal torus looks as a point particle.

Interestingly, that in the spectrum of supergravity compactified to 3 dimensions one meets exotic states whose higher dimensional origin is not obvious. In string theory these correspond to the so-called exotic branes. A fascinating feature of these branes is that they typically generate non-geometric backgrounds, i.e. such configurations of the space-time metric and p-form fields that cannot be globally described in geometric terms [1].

Dual dynamics of string theory

Before proceeding with the dual dynamics itself I would like to mention very good pedagogical reviews of the Double Field Theory appeared recently [1,2].

The so-called Duff's procedure implies a non-trivial feature of T-duality transformations: the hidden symmetry between equations of motion and Bianchi identities. It was suggested to understand left-moving and right-moving coordinates of a closed string as independent [1,2, for review of T-duality see 3]. And to collect them to a set of ordinary and dual coordinates. The so-called dual Lagrangian governs the dual dynamics. The key point is, equations of motion for the dual coordinates appear to be equivalent to Bianchi identities for the ordinary coordinates and vice versa. A relation between these coordinates leads to the notion of the generalised metric.

T-duality

The subject of dualities in physics has become a very popular topic these days. The examples that immediately come to mind are: AdS/CFT correspondence, AGT duality, mirror symmetry, non-perturbative dualities in string and M-theory. The latter are better known as T and U-dualities and are briefly reviewed in this post.

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.

Official string theory site

Found very interesting "official" superstring theory website http://superstringtheory.com. One can read there about string theory on basic and advanced level. Actually, the advanced level is basic for a PhD student and very understandable.

Nice reading during breakfast or leasure time.

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.