It has been proposed that AdS/CFT arises because of the entanglement structure of quantum field theories, e.g. see the discussion which occurred right here. Until now I have been skeptical of the idea, on the grounds that so much else is going on in the duality. A CFT only has a gravity dual if it has the right properties, or so I thought.

But I've just started looking into Vasiliev's "higher spin gravity" or "higher spin gauge theory", and eventually I ran across "Holographic dual of free field theory", which claims to construct a dual Vasiliev-type theory for every free QFT! Looking at the paper's only figure, I began to wonder if the tensor-network MERA diagrams could somehow be interpreted as Witten diagrams (which are the boundary-to-bulk Feynman diagrams used in AdS/CFT). Or perhaps the "discrete version of anti de Sitter space" which the MERA diagrams can provide, could be the background space on which a "discrete Witten diagram" is drawn. Suddenly I'm not so sure about my skepticism. Does anyone have an argument either way?


2 Answers 2


I will now answer yes to my own question - there is a relationship - though I can't prove it yet. I was thinking about the primordial example of AdS/CFT - N=4 Yang-Mills - and I was wondering how to interpolate between flat space and AdS space (exposition here) in the string picture; that is, between the situation in which there is no geometric back-reaction due to the stack of branes, and the situation in which the stack of branes forms a black hole. AdS space shows up as the geometry around the branes in the latter case, the geometry at the bottom of the black hole's "throat". The throat is infinitely deep, so the space at the bottom is decoupled from the space outside the black hole, and one is left with a pure AdS geometry. Yet this lies on a continuum with the flat space case; how can this be?

It suddenly occurred to me that the AdS geometry is already present in the flat space geometry - it is the "geometry" created by RG flow in the worldvolume theory! That's the emergent geometry showing up in MERA, including its continuum version; this must be "holographic renormalization" by another name. In the paper I cited, on constructing a Vasiliev dual to a free field theory, they explicitly write down equations of motion on the RG geometry. However you approach it, this must be how you interpolate smoothly between flat space and AdS space - the AdS space goes from being the RG geometry of the worldvolume theory, to the physical geometry, while the physical space spanned by the branes recedes to the boundary of AdS.

Some of the conceptual details remain to be worked out, of course. :-) But I imagine that will be done soon. Perhaps the real open question is still the relationship between string theory and Vasiliev theory.

  • $\begingroup$ It suddenly occurred to me that the AdS geometry is already present in the flat space geometry - it is the "geometry" created by RG flow in the world-volume theory! Right. This is the traditional interpretation of the radial dimension in AdS. But, congratulations if you happened to realize this independently. $\endgroup$
    – user346
    Apr 27, 2011 at 9:01
  • 1
    $\begingroup$ this is the sort of post that makes spending time in Physics.SE worth it. +1 $\endgroup$
    – lurscher
    Apr 27, 2011 at 20:30

There are several proposals about the emergence of holographic gravity duals to QFT, based on the spatial structure of quantum correlations such as the discussion that you mention as well as the very interesting ideas posed by Van Raamsdonk.

Related to your last question, in Entanglement Renormalization and Holography what is postulated is that the full structure of the MERA tensor network (defined by the structure of quantum entanglement in the boundary theory) happens to define "a discrete version" of AdS space. In this proposal, the points in the MERA bulk, represent points in a discretized version of the Anti de Sitter space, while the links between these points account for the structure of the causal cones of operators defined on sites on the boundary. The causal cone of a block of sites is defined as the set of sites, disentanglers, and isometries that can affect the chosen block.

In fact, if one analyses how MERA computes the entanglement entropy of a block of L sites on the boundary of the tensor network (the original quantum many body system), one finds an striking similarity with a proposal for the computation of entanglement entropy in theories with a "right" gravity dual posed by Ryu & Takayanagi.


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