Bifundamental wrt two gravities

[atyy]

In http://arxiv.org/abs/1009.3094, Nickel and Son say "Hydrodynamics, therefore, is a theory of a Goldstone boson, bifundamental with respect to two gravities."

What does that mean?

 

[Physics Monkey]

This paper is an attempt to break down the correlation function producing machinery of holographic duality into its most basic pieces. They are trying to formalize the idea that the region deep in the bulk corresponds to the IR and to understand how to carry the data contained there to the boundary in order to compute correlation functions.

Figure 1 in that paper shows the basic setup. They first consider an external U(1) gauge field A. This A couples to a conserved current J which is dual to gauge field in the bulk B. At this point A is a 4d gauge field and B is a 5d gauge field. Now they introduce a new slice in the bulk and study the partition function of the bulk theory as a function of the boundary conditions at the asymptotic boundary and the cutoff. This is the middle bit between cutoff and boundary in Figure 2. These boundary conditions consist of the value of A at infinity and the value of B restricted to the cutoff surface which they call a. They also introduce a scalar field, a "goldstone boson", that couples to both A and a (both 4d gauge fields) with charges 1 and -1. In the usual parlance of gauge theory we would say that the scalar is a bifundamental of U(1)_A x U(1)_a. This terminology is often used for describing the matter content of non-Abelian gauge theories and is summarized in "moose diagrams" like Figure 1.

That is background information for your question.

Now they consider the case of gravity. As before, one has a boundary metric G, and boundary stress T, and a dual metric H. G is 4d and H is 5d. Again, introduce a bulk surface and study the bulk action as a function of boundary conditions at the boundary and the cutoff. H restricted to the cutoff surface is g, a 4d metric. There are also scalar fields that are "charged" under both metrics. In fact, these scalars are a mapping from the coordinates on one boundary to the coordinates on the other boundary. In the U(1) case above, the scalar field could be understood as taking a field charged under A, removing the A charge, and adding some a charge. The gravitational case is similar. The coordinate mapping takes things (in the simplest case just vectors) "charged" under G to things "charged" under g (or vice versa). So in some fairly precise sense the scalar field is "bifundamental with respect to two gravities" (fundamental because the vector is the smallest non-projective representation).

Hope this helps.

 

[mitchell porter]

This all looks rather important!

I haven't read much about http://arxiv.org/abs/hep-th/0104005" .

First application that comes to mind: iteration of https://www.physicsforums.com/showthread.php?t=506476". The dS/dS correspondence somehow seems to be the reverse of this bigravity goldstone idea. Here, you have an AdS boundary, and further in, a horizon. In dS/dS, you have a cosmological horizon, and then further in, a special timelike hypersurface where the graviton becomes massive.

But either way, you're starting to turn holography into a modularized process. The bulk is divided into regions, and then you collapse a region onto its boundary. Leaping ahead, this reminds me very much of the role that volumes of polytopes play in the new motivic picture of amplitudes. It's quite logical to suppose that a "holographic cohomology" will provide the most general picture of this process.

Second application that comes to mind: understanding the relationship between 4d gauge theories and 6d strongly coupled theories. Here I link to a http://arxiv.org/abs/hep-ph/0207164" .)

Third idea that comes to mind: I wonder if Evenbly's "branching MERA" can be understood as a generalization of deconstruction.

 

[atyy]

As before, one has a boundary metric G, and boundary stress T, and a dual metric H. G is 4d and H is 5d. Again, introduce a bulk surface and study the bulk action as a function of boundary conditions at the boundary and the cutoff. H restricted to the cutoff surface is g, a 4d metric.

Thanks for the background and explanation! I don't understand why there is gravity at the boundary. I thought the boundary in AdS/CFT scenarios had no gravity?

 

[Physics Monkey]

Gravity at the boundary has the status of a background field. It just corresponds to putting the cft on a non-fluctuating curved background.

However, gravity on the cutoff surface is fluctuating, so we must sum over it in principle. In Son et al's paper they use large N to approximate this sum via saddle point.

The same thing is true for the U(1) story. The boundary gauge field is a non-fluctuating background field.

 

[atyy]

Gravity at the boundary has the status of a background field. It just corresponds to putting the cft on a non-fluctuating curved background.

However, gravity on the cutoff surface is fluctuating, so we must sum over it in principle. In Son et al's paper they use large N to approximate this sum via saddle point.

The same thing is true for the U(1) story. The boundary gauge field is a non-fluctuating background field.

Ok, thanks!