Hi. My name is Andy O'Bannon. I'm a postdoc at the University of Cambridge in England. I was asked, along with a few other participants, to maintain a blog for the workshop. Since the talks will be available in various formats on the KITP website, the purpose of the blog is not simply to summarize talks, but to summarize discussions, debates, concepts, etc. that might shape the direction of research, but might never see the light of day in the published literature. In the short-term, such a blog will enable those who can't be here to participate vicariously, and will also help incoming participants catch up on what happened before their arrival. In the long-term, such a blog will provide a record available to the entire community. Being a blog, the perspective will necessarily be subjective. Plus I may not understand everything. Caveat emptor.

Wednesday, August 31, 2011

Second Group Discussion

Physics-wise, the discussion was about zero-temperature, finite-density states of matter in holography, with focus on low-energy physics. The simplest example is some asymptotically AdS space, say AdS4, with a non-trivial gauge field. In the far IR, the two basic emergent geometries are AdS2 and Lifshitz. The latter has a mild singularity: all curvature invariants are finite constants, but tidal forces diverge. Notice that z=1, which is AdS4, and also z = infinity, which is AdS2, are special cases that are smooth. Starting now, "Lifshitz" refers only to finite z.

Sean Hartnoll presented a basic conundrum. People have presented arguments that AdS2 cannot be stable, and indeed only describes intermediate energy scales (some relevant references: 1105.1772 and 1105.4621). In the extreme IR, AdS2 must be unstable. For example, if AdS2 really survives in the extreme IR, then the (integral of the) density of states would diverge (1105.1772), which doesn't make sense, so something else must emerge. Experience with holographic superconductors and electron stars suggests that Lifshitz emerges.

On the other hand, Lifshitz probably also doesn't survive in the far IR. I didn't entirely understand why. Two ingredients were involved: a running dilaton and higher-curvature terms. According to Shamit Kachru, in cases where the dilaton/coupling is small in the IR, Lifshitz may be present over a large region, but eventually in the extreme IR the higher-curvature terms become important. He said that in the running-dilaton systems that he knew, in the extreme IR the only reasonable solution that emerges after Lifshitz appears to be... AdS2. We are thus going in circles.

Nobody had any immediate ideas for an escape route. Chris Herzog made a comment, which I found highly pertinent but that wasn't pursued very much: all of the above assumes rotational and translational symmetry. The actual IR physics may break these, for example the ground state may be a striped phase.

Max Metlitski mentioned that in fermionic quantum critical systems, which involve gapless fermionic and bosonic degrees of freedom, the bosons and fermions have different values of z. His question was: does that happen in AdS/CFT? The answer seemed to be yes, although as far as I could tell, some confusion arose as to what bosons he had in mind. My understanding was that the bosons he meant would be, in the AdS/CFT context, the SU(N) gauge fields of the field theory. In that case, the AdS2 signals that these have z = infinity, while fermions can have various z. A crucial issue that arose here was how to define z: it should be defined in terms of a (pole in a ) Green's function, not a self-energy, since the Green's function is what tells us about propagating modes, and hence z.

Tom Faulkner asked whether Luttinger's theorem has a gauge-invariant definition. Eduardo Fradkin basically said no. According to him, the usual formulations of Luttinger's theorems are perturbative, in terms of propagators, and so are not obviously gauge-invariant. Tom and Eduardo did seem to agree that a Luttinger's theorem could be obtained using 2 k_F singularities in (gauge-invariant) current-current correlators, however.

Friday, Sept 2:

Hello all, I am Nabil Iqbal, a new postdoc here at the KITP. Along with Andy and some others, I have also asked to help maintain this blog; here then is my summary of Per Kraus’s talk on higher spin black holes and holography.

There has been a great deal of work on higher spin theories of gravity (a la Vasiliev); they sit somewhere between string theory and ordinary gravity, and may provide a soluble example of AdS/CFT. In (1+1) dimensions, there is a recent conjecture by Gaberdiel and Gopakumar that a certain generalization of the familiar minimal model CFTs (called from now on the “W_N minimal model coset CFT”) should be dual to a certain bulk higher spin theory in (2+1) dimensions. This CFT admits a sort of large N limit in which the central charge of the theory c scales like N and so can be taken large; from general principles we then expect this theory to have a semiclassical bulk dual. There is also an analog of a t’Hooft coupling \lambda; and in these models the coupling is restricted to lie between 0 and 1.

The bulk theory dual to this CFT is a version of Vasiliev’s higher spin theories. Per pointed out that these theories exist on AdS space and are actually essentially nonlocal on the scales of the AdS radius, and so there is useful flat space limit as the AdS radius is taken to be large. There was some discussion on how it is peculiar that despite this nonlocality the dual theory still appears to be an entirely local unitary CFT; Don Marolf pointed out that this may be because the boundary is in some sense “infinitely large” in units of the AdS radius.

Before plunging into the full theory that is dual to the W_N model, Per then pointed out some calculations in a simpler model. It’s well-known that in (2+1) dimensions Einstein gravity can be written as a Chern-Simons theory on two copies of SL(2,R). If you replace this SL(2) with an SL(3), you get a different theory that contains Einstein gravity as a subsector; this SL(3) theory is a higher spin theory containing a spin 3 field, and when linearized around AdS this reduces to Fronsdal’s theory. What is particularly interesting (to me) is that there is now a larger “exotic” gauge invariance involving the SL(3); these gauge invariances enlarge diffeomorphisms and act in an interesting way on the metric, meaning that in some sense geometry now appears “gauge-variant”. For example, on the same gauge-orbit one may find both a black hole and a traversable wormhole – despite having a very different causal structure they are somehow gauge-equivalent.

But this means that usual tricks for constructing black holes (e.g. demanding smooth horizons, etc.) no longer work, as they involve data that is no longer gauge-invariant. To get around this and construct black hole solutions, Per et. al had to develop a new technique: they demanded that holonomies of the Chern-Simons gauge fields around the horizon agreed with those of the ordinary BTZ black hole. Shamit Kachru pointed out and Per agreed that this means that this approach may miss solutions that are not connected to the BTZ black hole. Per et. al then turned on chemical potentials for the new spin 3 charge and were able to construct a new sort of black hole carrying this charge.

However all of this was in some sense warmup; the bulk theory that is conjectured to be dual to the W_N minimal model is actually a Chern-Simons theory based around an algebra called “hs[\lambda]”, with \lambda the t’Hooft coupling mentioned above. This model contains infinitely many higher spin fields (not just one spin 3 field) that are all nonlinearly coupled; nevertheless Per and collaborators managed to find a black hole solution in this model, working perturbatively in the spin 3 charge. They computed the entropy of this black hole from the bulk, obtaining an impressive-looking expression that depends on the higher spin charge.

Now, remarkably, at special values of the t’Hooft coupling the symmetry algebra of the coset theory simplifies; at \lambda = 0 it is free fermions, and at \lambda = 1 it is free bosons! (I couldn’t quite understand whether it is the full theory that is equivalent, or just the symmetry algebra.) In any case, using (the right number of) free bosons/fermions, the field theoretical entropy was computed at \lambda = 1/0; they find perfect agreement between gravity and CFT.

I found this remarkable; in particular, I am not sure how to think about this idea that the entropy of the black hole is related to the entropy of free degrees of freedom. Does this just mean that there is a generalization of the Cardy formula that applies to the higher spin/enhanced symmetry theories, or does it mean that I can really understand the black hole microstates as free degrees of freedom? (Eric Perlmutter, one of the co-authors of the paper, tells me it is likely the former.)

Wednesday, August 31, 2011

Second Group Discussion

Physics-wise, the discussion was about zero-temperature, finite-density states of matter in holography, with focus on low-energy physics. The simplest example is some asymptotically AdS space, say AdS4, with a non-trivial gauge field. In the far IR, the two basic emergent geometries are AdS2 and Lifshitz. The latter has a mild singularity: all curvature invariants are finite constants, but tidal forces diverge. Notice that z=1, which is AdS4, and also z = infinity, which is AdS2, are special cases that are smooth. Starting now, "Lifshitz" refers only to finite z.

Sean Hartnoll presented a basic conundrum. People have presented arguments that AdS2 cannot be stable, and indeed only describes intermediate energy scales (some relevant references: 1105.1772 and 1105.4621). In the extreme IR, AdS2 must be unstable. For example, if AdS2 really survives in the extreme IR, then the (integral of the) density of states would diverge (1105.1772), which doesn't make sense, so something else must emerge. Experience with holographic superconductors and electron stars suggests that Lifshitz emerges.

On the other hand, Lifshitz probably also doesn't survive in the far IR. I didn't entirely understand why. Two ingredients were involved: a running dilaton and higher-curvature terms. According to Shamit Kachru, in cases where the dilaton/coupling is small in the IR, Lifshitz may be present over a large region, but eventually in the extreme IR the higher-curvature terms become important. He said that in the running-dilaton systems that he knew, in the extreme IR the only reasonable solution that emerges after Lifshitz appears to be... AdS2. We are thus going in circles.

Nobody had any immediate ideas for an escape route. Chris Herzog made a comment, which I found highly pertinent but that wasn't pursued very much: all of the above assumes rotational and translational symmetry. The actual IR physics may break these, for example the ground state may be a striped phase.

Max Metlitski mentioned that in fermionic quantum critical systems, which involve gapless fermionic and bosonic degrees of freedom, the bosons and fermions have different values of z. His question was: does that happen in AdS/CFT? The answer seemed to be yes, although as far as I could tell, some confusion arose as to what bosons he had in mind. My understanding was that the bosons he meant would be, in the AdS/CFT context, the SU(N) gauge fields of the field theory. In that case, the AdS2 signals that these have z = infinity, while fermions can have various z. A crucial issue that arose here was how to define z: it should be defined in terms of a (pole in a ) Green's function, not a self-energy, since the Green's function is what tells us about propagating modes, and hence z.

Tom Faulkner asked whether Luttinger's theorem has a gauge-invariant definition. Eduardo Fradkin basically said no. According to him, the usual formulations of Luttinger's theorems are perturbative, in terms of propagators, and so are not obviously gauge-invariant. Tom and Eduardo did seem to agree that a Luttinger's theorem could be obtained using 2 k_F singularities in (gauge-invariant) current-current correlators, however.

Friday, Sept 2:

Hello all, I am Nabil Iqbal, a new postdoc here at the KITP. Along with Andy and some others, I have also asked to help maintain this blog; here then is my summary of Per Kraus’s talk on higher spin black holes and holography.

There has been a great deal of work on higher spin theories of gravity (a la Vasiliev); they sit somewhere between string theory and ordinary gravity, and may provide a soluble example of AdS/CFT. In (1+1) dimensions, there is a recent conjecture by Gaberdiel and Gopakumar that a certain generalization of the familiar minimal model CFTs (called from now on the “W_N minimal model coset CFT”) should be dual to a certain bulk higher spin theory in (2+1) dimensions. This CFT admits a sort of large N limit in which the central charge of the theory c scales like N and so can be taken large; from general principles we then expect this theory to have a semiclassical bulk dual. There is also an analog of a t’Hooft coupling \lambda; and in these models the coupling is restricted to lie between 0 and 1.

The bulk theory dual to this CFT is a version of Vasiliev’s higher spin theories. Per pointed out that these theories exist on AdS space and are actually essentially nonlocal on the scales of the AdS radius, and so there is useful flat space limit as the AdS radius is taken to be large. There was some discussion on how it is peculiar that despite this nonlocality the dual theory still appears to be an entirely local unitary CFT; Don Marolf pointed out that this may be because the boundary is in some sense “infinitely large” in units of the AdS radius.

Before plunging into the full theory that is dual to the W_N model, Per then pointed out some calculations in a simpler model. It’s well-known that in (2+1) dimensions Einstein gravity can be written as a Chern-Simons theory on two copies of SL(2,R). If you replace this SL(2) with an SL(3), you get a different theory that contains Einstein gravity as a subsector; this SL(3) theory is a higher spin theory containing a spin 3 field, and when linearized around AdS this reduces to Fronsdal’s theory. What is particularly interesting (to me) is that there is now a larger “exotic” gauge invariance involving the SL(3); these gauge invariances enlarge diffeomorphisms and act in an interesting way on the metric, meaning that in some sense geometry now appears “gauge-variant”. For example, on the same gauge-orbit one may find both a black hole and a traversable wormhole – despite having a very different causal structure they are somehow gauge-equivalent.

But this means that usual tricks for constructing black holes (e.g. demanding smooth horizons, etc.) no longer work, as they involve data that is no longer gauge-invariant. To get around this and construct black hole solutions, Per et. al had to develop a new technique: they demanded that holonomies of the Chern-Simons gauge fields around the horizon agreed with those of the ordinary BTZ black hole. Shamit Kachru pointed out and Per agreed that this means that this approach may miss solutions that are not connected to the BTZ black hole. Per et. al then turned on chemical potentials for the new spin 3 charge and were able to construct a new sort of black hole carrying this charge.

However all of this was in some sense warmup; the bulk theory that is conjectured to be dual to the W_N minimal model is actually a Chern-Simons theory based around an algebra called “hs[\lambda]”, with \lambda the t’Hooft coupling mentioned above. This model contains infinitely many higher spin fields (not just one spin 3 field) that are all nonlinearly coupled; nevertheless Per and collaborators managed to find a black hole solution in this model, working perturbatively in the spin 3 charge. They computed the entropy of this black hole from the bulk, obtaining an impressive-looking expression that depends on the higher spin charge.

Now, remarkably, at special values of the t’Hooft coupling the symmetry algebra of the coset theory simplifies; at \lambda = 0 it is free fermions, and at \lambda = 1 it is free bosons! (I couldn’t quite understand whether it is the full theory that is equivalent, or just the symmetry algebra.) In any case, using (the right number of) free bosons/fermions, the field theoretical entropy was computed at \lambda = 1/0; they find perfect agreement between gravity and CFT.

I found this remarkable; in particular, I am not sure how to think about this idea that the entropy of the black hole is related to the entropy of free degrees of freedom. Does this just mean that there is a generalization of the Cardy formula that applies to the higher spin/enhanced symmetry theories, or does it mean that I can really understand the black hole microstates as free degrees of freedom? (Eric Perlmutter, one of the co-authors of the paper, tells me it is likely the former.)