Why gravity codes renormalization of conformal field theories

[marcus]

http://arxiv.org/abs/1305.6315
Why gravity codes the renormalization of conformal field theories
Henrique Gomes, Sean Gryb, Tim Koslowski, Flavio Mercati, Lee Smolin
(Submitted on 27 May 2013)
We give a new demonstration that General Relativity in d+1 dimensions with negative or positive cosmological constant codes the renormalization group behaviour of conformal field theories (CFT) in d dimensions. This utilizes Shape Dynamics, which is a conformally invariant theory known to be equivalent to General Relativity. A key result of Shape Dynamics is that the evolution of observables under local conformal transformations and spatial diffeomorphisms is shown to be equivalent to many fingered time, i.e., d+1-dimensional spacetime diffeomorphisms. This relationship explains why the renormalization group flow of a CFT is governed by a geometry with d+1-dimensional spacetime diffeomorphism invariance.
25 pages

See point 3 on page 4. An interesting challenge to the Maldacena Conjecture. Potentially undermines by removing supporting evidence.

 

[ohwilleke]

So, shorter version (assuming that I have understood) - the AsD/CFT correspondence can be established in a way that is in many ways more general and less string theory specific than the conventional proof and that illustrates more clearly why this is the case, at the expense of not being quite as comprehensive in some respects. The real bummer of the paper though (at the end) is that the paper says the math for the case we really care about (real GR) is so ugly that even a mother couldn't love it.

 

[marcus]

So, shorter version (assuming that I have understood) - the AsD/CFT correspondence can be established in a way that is in many ways more general and less string theory specific than the conventional proof and that illustrates more clearly why this is the case, at the expense of not being quite as comprehensive in some respects. The real bummer of the paper though (at the end) is that the paper says the math for the case we really care about (real GR) is so ugly that even a mother couldn't love it.

That's an interesting comment! Can you be specific about where in the Conclusions section at the end they say that, in effect? I'd enjoy seeing how you interpret some of these. It's all interesting so I will quote the entire Conclusions section. Looks like 8 bullets. Maybe we can refer to them by number:
======quote http://arxiv.org/abs/1305.6315 pages 19, 20=====
4 Comments and conclusions
We make several comments on these results

• The reason why a conformal field theory on a d-dimensional manifold with a fixed metric is related to a gravitational theory in one higher dimension now becomes transparent. The correspondence maps the constraint generating d-dimensional spatial diffeomorphisms of the gravitational theory to the conservation law of the energy-momentum tensor of the CFT. The remaining constraint, the Hamiltonian constraint generating many fingered time in d + 1 dimensions is shown, through the correspondence with Shape Dynamics, to be equivalent to the generator of volume-preserving conformal transformations in that theory, plus a single global time-reparametrization constraint (14). The latter is mapped by the correspondence using Wald’s axioms to the trace anomaly of the energy-momentum tensor of the CFT.
Thus, we can now understand why the geometry of the renormalization group of a CFT is given by General Relativity, possibly coupled to matter fields, a diffeomorphism invariant theory in one higher dimension.

• We can conjecture a more general correspondence. Given a CFT we get an expansion of the form (1) with coefficients, c_r. We can define a theory with the gauge symmetries of Shape Dynamics by the corresponding expansion, Eq. (4), where we equate the values of the coefficients, c ̃_r = c_r. This gives us some gravitational theory with a possibly different global Hamiltonian constraint replacing (14). We can reconstruct HSD by its volume expansion. This gives what may be called a generalized Shape Dynamics theory which by construction is matched to the original CFT.
We can then conjecture that by gauge symmetry trading of the VPCT for many fingered time, this can be matched to some spacetime diffeomorphism invariant theory in d + 1 dimensions. This may not be General Relativity, but by construction will have spacetime diffeomorphism invariance. If this matching can be achieved there is a general correspondence between any CFT and some gravitational theory in one higher dimension.

• We have studied only the ⟨0| T^cd |0⟩ but it is possible that the renormalization group flow of other conserved currents can be explained by expanding Wald’s axioms to them, and then by coupling Shape Dynamics to suitable matter fields.

• The correspondence we show is more general than the conjectured AdS/CFT correspondence as stated originally by Maldacena, in that it is not restricted to supersymmetric theories and String Theory and the properties of 10 dimensional supersymmetry algebra play no role in establishing the general correspondence at this level. However supersymmetry certainly plays a role in specific examples of correspondences, such as those involving supersymmetric Yang-Mills theory.

• The correspondence we demonstrate here is however weaker than the original Maldacena conjecture in that we claim only the classical bulk gravitational theory plays a role. Rather then a conjectured equivalence between the Hilbert spaces and observables algebras of a bulk and boundary theory, we demonstrate only an equivalence between the expectation values of operators in the CFT and solutions of the classical field equations of the bulk theory.

• However, some of the evidence used to support the stronger Maldacena conjecture can now be explained by the weaker and more general correspondence we give here. Hence, only results that are not explicable by our weaker correspondence can count as evidence for the stronger Maldacena conjecture.

• In particular, Shape Dynamics does not, so far, directly explain the correspondence between pure General Relativity and N = 4 super-Yang–Mills theory in the N → ∞ limit except as a coincidence of the coefficients in the two expansions. This coincidence is well represented by the work of Skenderis et al, and also from a purely gravitational perspective [3, 7].

• We can also conjecture that the correspondence we have demonstrated here extends to a stronger correspondence between a quantization of Shape Dynamics and the quantum conformal field theory. The evidence we possesses for an extension of the correspondence into the quantum regime is the apparent absence of anomalies (for odd dimensions) of our spatial conformal transformations (see appendix A). In spite of their classical correspondence, the quantization of Shape Dynamics is unlikely to coincide with a quantization of General Relativity due to the very different structure of their constraints.
==endquote==

One hopeful thing I notice here is that although Shape Dynamics and GR agree at a classical level they may lead to different quantum theories! So we have more chance of being right. SD and GR are equally good gravity theories empirically but we may end up with two inequivalent quantum gravity theories, and thus a possibility of comparing what they say about the early universe.

For anyone new to the subject. The abbreviation VPCT in bullet #2 stands for "Volume Preserving Conformal Transformation". These are physically meaningless ("gauge") transformations in SD, which allows local conformal scale change as long as the OVERALL volume remains unchanged. Expanding one place means you have to shrink somewhere else.

 

[mitchell porter]

I am apriori skeptical of this paper. But first let me see what it says and what it doesn't say.

First, as the authors note several times, they have not produced an equivalence between a (quantum) CFT and a theory containing quantum gravity. What they claim is that the expectation values of one operator in the CFT (the stress-energy tensor) behave like gravitational field momenta in a classical theory of gravity. Specifically, that a set of coefficients appearing in equations of motion on both sides will be equal.

Or just that they can be equal. See page 5, immediately after equation 5: "with specific choices of matter fields, the coefficients... exactly match their counterparts..." Nothing is said anywhere about which matter fields - no specific CFT is ever exhibited - and in fact later on they seem to be saying that any CFT will define a corresponding gravitational theory (some generalized form of shape dynamics). (By the way, the generality of this claim, and the direction in which it works, is reminiscent of this paper, which claimed that any non-interacting QFT, not necessarily conformal, is equivalent to some, very messy, Vasiliev-type theory of quantum gravity; but that was a claim of a quantum-quantum equivalence, not a quantum-classical correspondence. Another paper aims for generality in the case of interacting field theories, but its result is much narrower.)

What is their actual argument? It's peculiar - they characterize the CFT axiomatically rather than constructively - that is, they say, let us suppose that its renormalized stress-energy tensor behaves in a certain way - and they introduce a never-before-seen axiom to do this. Then, on the other side of the argument, they appeal to special properties of the shape dynamics reformulation of general relativity. Together this apparently implies that the equations of motion for the CFT expectation values, and the equations of motion for the gravitational momenta (which are described using Hamilton-Jacobi formalism), will have the term-by-term structure which would at least make it possible for the coefficients to match up as desired.

So the first question for me, is whether this argument applies to the canonical examples of AdS/CFT, in which there is a known quantum theory on both sides, and the conjecture (supported by extensive calculational checks) is that they are the same theory. This argument can't justify the equivalence, but perhaps it does explain why the CFT produces a theory with gravity on the other side?

Along with the possibility that the argument doesn't work and the possibility that it does work, there is a third possibility, that there's something trivial about the argument. This comes back to the new "axiom" that they introduce in order to reason about the CFT. The axiom has been selected precisely in order to imply the desired structure in the equations of motion, but then they try to give it an independent justification.

Another thing I notice (page 4) is that they start with an anomaly (quantum breaking of symmetry) in the CFT. The classic discussions of the holographic renormalization group (like de Boer et al, which they cite) also study the conformal anomaly in the CFT. So this provides a point of comparison, whereby people (that know this topic better than I) can judge whether these ideas are new or deep or consistent with what came before.

 

[marcus]

Mitchell, great to have your reactions to the points made in the paper. I'm finding it difficult to link up what you're saying to specific statements in the conclusion section. Let me make it easier by NUMBERING the 8 summary conclusion points so you can refer to them by number. Could you indicate which numbers are the ones you disagree with and tell us what you think the trouble is on a point by point basis?
======quote http://arxiv.org/abs/1305.6315 pages 19, 20=====
Comments and conclusions
We make several comments on these results

1 The reason why a conformal field theory on a d-dimensional manifold with a fixed metric is related to a gravitational theory in one higher dimension now becomes transparent. The correspondence maps the constraint generating d-dimensional spatial diffeomorphisms of the gravitational theory to the conservation law of the energy-momentum tensor of the CFT. The remaining constraint, the Hamiltonian constraint generating many fingered time in d + 1 dimensions is shown, through the correspondence with Shape Dynamics, to be equivalent to the generator of volume-preserving conformal transformations in that theory, plus a single global time-reparametrization constraint (14). The latter is mapped by the correspondence using Wald’s axioms to the trace anomaly of the energy-momentum tensor of the CFT.
Thus, we can now understand why the geometry of the renormalization group of a CFT is given by General Relativity, possibly coupled to matter fields, a diffeomorphism invariant theory in one higher dimension.

2 We can conjecture a more general correspondence. Given a CFT we get an expansion of the form (1) with coefficients, c_r. We can define a theory with the gauge symmetries of Shape Dynamics by the corresponding expansion, Eq. (4), where we equate the values of the coefficients, c ̃_r = c_r. This gives us some gravitational theory with a possibly different global Hamiltonian constraint replacing (14). We can reconstruct HSD by its volume expansion. This gives what may be called a generalized Shape Dynamics theory which by construction is matched to the original CFT.
We can then conjecture that by gauge symmetry trading of the VPCT for many fingered time, this can be matched to some spacetime diffeomorphism invariant theory in d + 1 dimensions. This may not be General Relativity, but by construction will have spacetime diffeomorphism invariance. If this matching can be achieved there is a general correspondence between any CFT and some gravitational theory in one higher dimension.

3 We have studied only the ⟨0| T^cd |0⟩ but it is possible that the renormalization group flow of other conserved currents can be explained by expanding Wald’s axioms to them, and then by coupling Shape Dynamics to suitable matter fields.

4 The correspondence we show is more general than the conjectured AdS/CFT correspondence as stated originally by Maldacena, in that it is not restricted to supersymmetric theories and String Theory and the properties of 10 dimensional supersymmetry algebra play no role in establishing the general correspondence at this level. However supersymmetry certainly plays a role in specific examples of correspondences, such as those involving supersymmetric Yang-Mills theory.

5 The correspondence we demonstrate here is however weaker than the original Maldacena conjecture in that we claim only the classical bulk gravitational theory plays a role. Rather then a conjectured equivalence between the Hilbert spaces and observables algebras of a bulk and boundary theory, we demonstrate only an equivalence between the expectation values of operators in the CFT and solutions of the classical field equations of the bulk theory.

6 However, some of the evidence used to support the stronger Maldacena conjecture can now be explained by the weaker and more general correspondence we give here. Hence, only results that are not explicable by our weaker correspondence can count as evidence for the stronger Maldacena conjecture.

7 In particular, Shape Dynamics does not, so far, directly explain the correspondence between pure General Relativity and N = 4 super-Yang–Mills theory in the N → ∞ limit except as a coincidence of the coefficients in the two expansions. This coincidence is well represented by the work of Skenderis et al, and also from a purely gravitational perspective [3, 7].

8 We can also conjecture that the correspondence we have demonstrated here extends to a stronger correspondence between a quantization of Shape Dynamics and the quantum conformal field theory. The evidence we possesses for an extension of the correspondence into the quantum regime is the apparent absence of anomalies (for odd dimensions) of our spatial conformal transformations (see appendix A). In spite of their classical correspondence, the quantization of Shape Dynamics is unlikely to coincide with a quantization of General Relativity due to the very different structure of their constraints.
==endquote==

What I'm hoping is you will "talk down" in somewhat simpler language, assuming less background about the Maldacena Conjecture, and thereby include a wider audience. Which of these 8 points do you find dubious, and why? It could be educational, I think.

 

[mitchell porter]

The argument for (1) is the core of everything. It just looks suspicious to me - too simple, and perhaps skipping over crucial technicalities. My standard for the "right" way to talk about this aspect of AdS/CFT (GR from RG flow) is Jan de Boer's http://arxiv.org/abs/hep-th/0101026. Just examining at the level of buzzwords, you can see there is some overlap, but de Boer talks about some extra topics - the role of cutoffs in section 3, or the use of supersymmetry to prove that something is zero, in section 8. What Gomes et al offer instead are Wald's axiomatic method and shape dynamics, so perhaps it mostly boils down to whether these substitutes can play the role demanded of them.

 

[marcus]

The argument for (1) is the core of everything. It just looks suspicious to me - too simple, and perhaps skipping over crucial technicalities. My standard for the "right" way to talk about this aspect of AdS/CFT (GR from RG flow) is Jan de Boer's http://arxiv.org/abs/hep-th/0101026. Just examining at the level of buzzwords, you can see there is some overlap, but de Boer talks about some extra topics - the role of cutoffs in section 3, or the use of supersymmetry to prove that something is zero, in section 8. What Gomes et al offer instead are Wald's axiomatic method and shape dynamics, so perhaps it mostly boils down to whether these substitutes can play the role demanded of them.

Thanks! That's very helpful. So one thing I should do is get some understanding of Wald's axioms.

 

[atyy]

How about CFTs where the gravity dual is thought not to be GR, but higher spin gravity?

Like in this review http://arxiv.org/abs/1208.4036 ?

 

[mitchell porter]

A real string theorist had a look at this paper...

 

[MTd2]

He calls shape dynamic a kind of loopy formulation... He surely dislikes the thing...

 

[marcus]

He calls shape dynamic a kind of loopy formulation... He surely dislikes the thing...

SD and LQG are rival approaches that differ at fundamental level. So he may have no clear idea of Shape Dynamics or of the Loop approach or else could have allowed hostility and resentment to overcome basic ability to reason. Yes, he "surely dislikes". But what, exactly?

 

[marcus]

Aside from his confusion regarding the different research programs, Distler's crit of the SD paper could be valid and the kind of thing I was on the lookout for when opening this thread. Ironically his take-down of a rival SD paper would, if correct, be seen by Loop researchers as reassuring.