Sunday, September 30, 2012

An idea for gauge/gravity dualities

Change of plans. Converting TeX to HTML and the post editing involved is a drag, so I'm not going to finish the series about asymptotic geometry etc. Instead I wrote a new paper on the subject! The preprint is posted on arXiv here, but I haven't submitted it yet(I will probably revise a bit). Here's a short summary.

In the paper I try to generalize the successful results of gauge gravity duality for $AdS_3$ gravity to higher dimensions (or at least motivate the attempt). This was the big idea by Brown and Henneaux (link to paper at which was later refined in here (and in other people's works, but I don't want to cite a lot of sources). The idea is that a three dimensional gravity theory that approaches an anti-de Sitter space at a spherical boundary at infinity is equivalent (or dual) to a two dimensional conformal field theory on the boundary, which turns out to be a 2D Liouville theory. Brown and Henneaux discovered this by studying the asymptotic symmetries of $AdS_3$ gravity, which they discovered to actually tend to conformal transformations at the boundary. The asymptotic symmetries are here diffeomorphisms which preserve the asymptotically $AdS_3$ conditions. Moreover, they discovered that the algebra of canonical generators of these asymptotic symmetries is the Virasoro algebra with a classical, nonvanishing central charge! 

What's perhaps even more interesting is that this kind of approach does not work in higher dimensions. The reason is that the boundary of $AdS_{d+1}$ is a $d$ dimensional flat space (Minkowskian or Euclidean, depending on if you're talking about real $AdS$ spaces or the Euclidean versions, i.e. hyperbolic spaces) and because the conformal transformations in dimensions $d>2$ forms a finite dimensional group. The infinite dimensionality of the asymptotic symmetries will be the key!

As I explain in the paper, the key to understanding how to generalize the extremely successful 3D case to higher dimensions is the observation that (loosely speaking) the 3D $AdS$ space is in fact the Lie group $SL(2)$. Indeed, my hypothesis was that instead of studying higher dimensional $AdS$ spaces, one should study the geometry of the Lie groups $SL(N)$ (and their homogeneous spaces). In the first article of the subject, (published in JMP),I discovered that the asymptotic symmetries on the boundary of the five dimensional homogeneous space $SL(3)/SO(3)$ form an infinite dimensional Lie algebra, which implies the generalization of the duality (the paper was a bit of a "tour de force" and not very elegant...). This is completely different to the higher dimensional $AdS$ spaces, where the asymptotic symmetries form a finite dimensional Lie algebra. It will then be possible to extend the $SL(3)/SO(3)$ asymptotic symmetries to near boundary symmetries to obtain a similar dual theory as the Liouville theory in the 3D AdS gravity case.

Anyway, I hope the stuff makes sense... I wrote this piece quite quickly. The intro in the paper is probably better!