# Great title (and a nice looking paper)

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The Gravity Dual of the Ising Model
http://arxiv.org/abs/1111.1987

A quite interesting paper. Basically they define a way to sum over geometries in 3d pure gravity, they show the sum is finite in some special cases, and they find that for some choice of parameters out pops the 2d ising model, the simplest of all CFTs.

Thus AdS/CFT is completely general (as many people, myself included, have long suspected/hoped). In the words of Hudson: "That's it man, game over man, game over!"

I would like to understand their story better.

## Answers and Replies

mitchell porter
I would like to understand their story better.
Once upon a time there were two theories, general relativity and quantum mechanics. General relativity described things that were very, very big [storyteller throws arms out wide], and quantum mechanics described things that were very, very small [storyteller holds up thumb and forefinger, almost touching but not quite]. One day, quantum mechanics was telling general relativity all about the partition function of the two-dimensional Ising model at the conformal critical point. "There are just three primary fields at the conformal point", said quantum mechanics, "and the partition function is the sum of the squares of their characters"...

One thing I want to understand is, how this relates to Witten's duality between pure gravity in AdS3, and the CFT with monster symmetry. Presumably it has something to do with the exceptional symmetries which can show up in spin systems (E8 in Ising model, E6 in Potts model).

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They only did the 2D Ising model, so presumably it's not quite game over yet?

The O(N) model supposedly gives non-Einstein gravity, yet the Ising model gives Einstein gravity? Does this imply that http://arxiv.org/abs/0909.0518" [Broken] has been falsified?

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negru
Yeah what do you mean AdS/CFT is completely general

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Once upon a time there were two theories, general relativity and quantum mechanics. General relativity described things that were very, very big [storyteller throws arms out wide], and quantum mechanics described things that were very, very small [storyteller holds up thumb and forefinger, almost touching but not quite]. One day, quantum mechanics was telling general relativity all about the partition function of the two-dimensional Ising model at the conformal critical point. "There are just three primary fields at the conformal point", said quantum mechanics, "and the partition function is the sum of the squares of their characters"...

One thing I want to understand is, how this relates to Witten's duality between pure gravity in AdS3, and the CFT with monster symmetry. Presumably it has something to do with the exceptional symmetries which can show up in spin systems (E8 in Ising model, E6 in Potts model).

This story also has a picture that goes with it, one I made for a talk a while back.

#### Attachments

• holocompact.pdf
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They only did the 2D Ising model, so presumably it's not quite game over yet?

The O(N) model supposedly gives non-Einstein gravity, yet the Ising model gives Einstein gravity? Does this imply that http://arxiv.org/abs/0909.0518" [Broken] has been falsified?

Well, they mentioned a few other models as well. Certainly a full and precise look up table isn't there yet, but progress has been made on the O(N) model in higher dimensions that strongly suggests these higher spin gauge theories are the dual. Although again, these theories are quite mysterious, at least to me. The gravity theories are very complicated in higher dimensions so conservation of evil is still in force.

2d is very special. In 2d holomorphy and modular invariance and so on are strong enough to defeat conservation of evil so that all sides of the story appear to be knowable. But I would emphasize that this is something we could have guessed many years ago.

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Yeah what do you mean AdS/CFT is completely general

If the 2d Ising model and $\mathcal{N} = 4$ SYM have duals then surely so does everything in between, the relevant axis being craziness.

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Alas, no further interest in this breakthrough it seems ...

I've not made much progress either due to other responsibilities, but I did start to wonder about primary operators. Although they already "know" the scaling dimensions in the sense that they are encoded in the partition function, I will be very interested to see if these dimensions can be obtained more explicitly. I've heard that it may be possible to see them by studying conical defects in the geometry. It's all a bit puzzling since the usual ads/cft dictionary would suggest that these primary operators should be dual to massive fields in the bulk, but where are they?

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And an answer comes quickly, albeit for a different class of theories:

Conical Defects in Higher Spin Theories
http://arxiv.org/abs/1111.3381

This seems pretty neat.

qsa
What about electromagnetic force can it also be linked to an Ising model type.

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What about electromagnetic force can it also be linked to an Ising model type.

You mean holographically? Or otherwise? Not that I'm aware of.

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So I just yesterday heard a talk by Alex Maloney on this subject, and I'm prepared to try to sketch some further details of their approach.

They want to compute the partition function of gravity in 2+1 dimensions with an asymptotic boundary given by a torus with modular parameter $\tau$. Alex gave the usual "we don't know what this means and how to do it" speech. Their approach is the following: define the path integral via an extension of semiclassical analysis.

Step 1: find all classical solutions of the EOMs.

Step 2: "sum" all perturbative corrections to each saddle.

The semi-classical partition function is then the sum over classical solutions dressed with the infinite set of perturbative corrections. Two important points about this approach.

1. It is not clear that the semi-classical partition function is the same as the full partition function. They simply assume that this is true, or better, they simply see what the semi-classical partition function looks like and check against known CFT partition functions. In practice, their criterion is to make sure that the numbers appearing in the expansion of the partition function in powers of $e^{-\beta}$ are integers (hence potentially counting states).

2. Although Steps 1 and 2 sound hard, they can be carried in 2+1 dimensions because of the usual folklore that "gravity has no local degrees of freedom".

All classical solutions may be obtained from modular transformations of a single solution. For example, take "thermal Ads" and act by different modular transformations that map the torus boundary of thermal AdS onto the actual torus boundary. One can switch which cycle is contractible, etc.

All perturbative fluctuations may be summed by studying the spectrum of "boundary gravitons" (a really terrible name). Basically, not all diffeomorphisms go to zero fast enough at the boundary to be called gauge transformations. Thus we need to quantize the space of non-trivial diffeomorphisms, a task that sounds hard, but which is doable because these diffeomorphisms must form a representation of the Virasoro algebra. The central charge of the quantum algebra is $c = \frac{ 3 \ell}{2 G}$ where $\ell$ is the AdS radius.

Putting all these ingredients together, they find that for $G = 3 \ell$, the gravity partition function is that of the critical Ising model. They also find one other value of $\ell$ where the semiclassical partition function has a quantum interpretation which yields the tricritical Ising model. They study a variety of other "higher spin" theories as well.

1. The value of the cosmological constant is highly quantum in these examples i.e. it's of order the Planck length.
2. The holographic entanglement entropy formula for Einstein gravity works exactly right (this is surprising).
3. The Ising model in 1+1 is the example I studied relating MERA to AdS/CFT. This suggests it is time for more progress here.
4. It would be very interesting to see RG flows from tricritical Ising to critical Ising. Perhaps this could help explain the special values of the CC?
5. What happens when the CC is not one of these special values? Is the semi-classical result incomplete in these cases?

Alexander Maloney gave a nice talk on this at the Perimeter earlier this year.
http://pirsa.org/12010164/

He explicitly addresses the (non)-connection to Chern-Simons theory - he thinks it fails at the non-perturbative level (21:30).

An interesting idea at the end of the talk is also in his paper "Three dimensional AdS gravity possesses BTZ black holes which are separated from the AdS ground state by a gap. These black holes correspond to states in the dual CFT with weight larger than c/24, i.e. they lie in the “Cardy regime” of the dual CFT. Indeed, this is why it is possible to describe the BTZ entropy using the Cardy formula of the dual CFT [55]. Thus the critical and tricritical Ising models are the only two diagonal minimal model CFTs with the property that all primary states can be interpreted as black holes. ... All other minimal models have primaries which may be interpreted as matter fields, and hence may be dual not to pure gravity but rather to something more complicated."

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The discussion about matter degrees of freedom reminds me about the discussion about bulk matter at the end of the AdS/MERA paper. At that time I remember thinking that maybe the reason it works is that in gravity, the spacetime geometry and energy define each other.

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It's neat if black holes are somehow primary operators, however, I still wonder how to reproduce the correlation functions of the Ising model. The usual dictionary doesn't seem to work since there are no bulk fields specified which are dual to the primary operators in question.

Hmmm, so it's not game over then, and ABJM/MERA is going to be a safer bet?

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I don't know. The fact that it is (maybe) the Ising model has a lot going for it. We definitely have the MERA for that.

I'm still optimistic about the defect as primary operator approach, but we'll have to see if it can work out. It is reasonable in that particle worldlines in 3d gravity lead to conical defects. Somehow we would like to see that only special defects make sense in the theory.

Also, the leading entanglement entropy works out right, but if we had a way to compute the corrections that would be amazing.

Both. The bulk AdS radius is $\ell = G/3$ (this is the value that gives the right partition function) and the length of a minimal curve (ending on an interval of length L) is $2 \ell \ln{(L/\epsilon)}$ which gives an entropy of $S = \frac{2 \ell \ln{(L/\epsilon)}}{4 G} = \frac{c}{3} \ln{(L/\epsilon)}$ with $c = \frac{1}{2} = \frac{3 \ell}{2 G}$. The last equality is the old result of Brown and Henneaux.