Friday, November 04, 2005

A Bridge to Higher Spin Theory and AdS/CFT

Last weekend I was pleasantly surprised when a friend phoned me and said he was in my neighbourhood and thought we could go for a drink. Unfortunately there were scheduled works on a bridge between where he was and where I live; I thought the bridge would be raised and he replied asking if that meant we were topologically disconnected... This was somewhat similar to how I felt yesterday at our weekly seminar, since the topic seemed to be "topologically disconnected" from my own small area of understanding. So in these comments I propose to try to build a small bridge to the beginnings of the subject matter discussed in the talk.

So Paul Heslop from DAMTP in the real Cambridge came to talk to the King's group under the title "On the higher spin/gauge theory correspondence" which is based on his work with M. Bianchi and F. Riccioni. Paul's talk was based mostly around their paper "More on La Grande Bouffe: towards higher spin symmetry breaking in AdS". Don't be put off by "La Grande Bouffe" it is the name of a film and it refers to an aspect of the theory where a higher spin field "eats" the entire chain of lower spin fields to acquire mass, these chains can be very long. The film of the same name, we were told by an anonymous member of our department, is about "three men and one woman" and the men all eat so much that they die! Via IMDB the plot is described as:
Four successful middle-aged men Marcello, a pilot; Michel, a television executive; Ugo, a chef; and, Philippe, a judge go to Philippe's villa to eat themselves to death.
It has 7.2 stars from 1225 reviews. So don't be put of by the term, which I believe is due to Massimo Bianchi.

The conjectured correspondence is that "the massless higher spin theory is holographically dual to a free gauge theory on the boundary". We can summarise it best using a picture given in Paul Heslop's talk:
If you wish to learn about the correspondence then you can read through the following papers of Bianchi et al:

  • "Higher Spin Symmetry and N=4 SYM" by Niklas Beisert, Massimo Bianchi, Jose F. Morales and Henning Samtleben
  • "Higher spin symmetry (breaking) in N=4 SYM theory and holography" by Massimo Bianchi
  • "Higher Spins and Stringy AdS5xS5" by Massimo Bianchi

    However since there is a gap in my understanding, I thought it would be constructive to find a brief paragraph or two to express the formative ideas behind Vasiliev's higher spin theory, if I can. However a better option would be to listen and see pictures from a talk by Vasiliev here, where you will hear him commence by describing the totally symmetric massless free fields of Fronsdal (1978) and de Wit and Freedman (1980) where the bosonic case contains a field with s symmetrised indices which is double-traceless (i.e. if you contract two pairs of its indices it is zero). There is a uniquely associated action which is chosen by requiring that some action containing terms up to second order in derivatives of the field is gauge invariant. There are some familiar gauge invariance principles for s=1 (Maxwell) and s=2 (gravity) fields and the essence of higher spin theory is the question of whether there is a unifying gauge symmetry that exists for other higher spin fields as well.

    What are the connections with strings and supergravity I hear you cry? Well Vasiliev goes on to say (in the video above) that while there is a limit on the spin of the massless particles in a d-dimensional theory (s=d-2) there is also a corresponding limit on the number of supersymmetries (e.g. he draws a parallel between the s<=2 and N<=8 in 4-dimensions). He says that this is "practically equivalent" to the limit on the number of dimensions of supergravity (d=11) and says that if one wishes to consider theories beyond supergravity then one might start by wondering what happens when one includes massless spin fields of spins higher than those in D=11 sugra. Further motivation comes from the Stueckelberg symmetries of the superstring which are similar to spontaneously broken symmetries of higher spin gauge symmetries. As well as from the work of Sundborg and Witten, arguing for a nonlinear theory with infinite higher spin fields in the bulk.

    So there you have it a small bridge to travel over and go and study higher spin gauge theory. A couple of further papers to help you on your way towards the triangle above are:

  • "Higher Spin Gauge Theories"(be warned this is a link to a 205 page pdf), Volume 1 of the proceedings of the Solvay Workshop, a compilation of work in various areas related to higher spin theories.
  • "Notes On Higher Spin Symmetries" by Andrei Mikhailov

    It transpired last weekend that the scheduled roadworks were not happening, the bridge was down, and just like this week's seminar experience my friend and I were topologically connected after all, and we went and had a merry afternoon in Greenwich.