Week+Three+Talks

Hi, this is Andy O. I'm going to break all the rules and

1.) Blog about my own talk, 2.) Blog two months after the event.

The benefits are:

1.) I can answer many of the questions raised during the talk much better now, based on conversations and emails with many people: Jonathan Shock, Shu Lin, Kristan Jensen, Johanna Erdmenger, Sean Hartnoll, Hovhannes Grigoryan, Andreas Karch, Keun-young Kim, and Karl Landsteiner.

2.) I can make my first foray into [|open notebook science].

First, let me apologize to Andrei Parnachev. We were supposed to split a talk, with 30 minutes each. I went first and took a whole hour, and then walked off with the microphone, leaving Andrei with a grumpy laptop that didn't work right. I actually requested to hear him talk, so I felt like a jerk. Anyway, my apologies to him.

As for physics, let me first address a perennial question, raised also during my talk: does using D7-branes as probes ever make sense? I have talked to dozens of people about this very issue, including Michael Green, Rob Myers, Jeff Harvey, Carlos Nunez, Prem Kumar, Johanna Erdmenger, Jonathan Shock, Clifford Johnson, Nick Warner, Andreas Karch, and many others. The short answer is "yes." First I'll explain a few helpful points.

Number one is the deficit angle. In any theory of dynamical gravity, any codimension-two object with a nonzero tension will produce a deficit angle. That happens for cosmic strings in (3+1) dimensions and for D7-branes in (9+1) dimensions. Crucially, the size of the angular deficit is given just by dimensional analysis:

math \Delta \theta \propto G_N T

math

which is just Newton's constant times the tension of the object. Now, for a D7-brane in AdS5 x S5, we have (in units with the AdS radius set to one)

math G_N \propto g_s^2 \alpha'^4 \propto \frac{1}{N_c^2}, \qquad T \propto \frac{1}{g_s \alpha'^4} \propto \lambda N_c

math

so that we find

math \Delta \theta \propto G_N T \propto \frac{\lambda}{N_c} = g_{YM}^2 = g_s \rightarrow 0.

math

As long as the string, or Yang-Mills coupling, is small, we can ignore the deficit angle. When is the string coupling small? Recall the solution for the dilaton near a D7-brane,

math e^{-\phi} = g_s^{-2} =- \frac{1}{2\pi} \log z,

math

where z is the complex coordinate in the plane transverse to the D7-brane. Close to the D7-brane, meaning near z=0, the logarithm diverges to negative infinity. That is good: that means the dilaton and hence the string coupling is small near the D7-brane.

We thus have the following story. We start with the D3/D7 intersection. We then take the Maldacena limit with the D3-branes, and find some D7-branes in AdS5 x S5. Now, in fact the axio-dilaton will be running. That means the UV part of the geometry will not be AdS. The IR part of the geometry, however, will approximately be, namely in a region with a size controlled by Nf/Nc, where the string coupling is small and the deficit angle is not apparent. We thus do the following: we pick some scale, meaning some radial position, throw out everything above that position (the UV part of the geometry), approximate the remaining IR part of the geometry as pure AdS with constant dilaton and a probe D7-brane, and make up for the difference through the boundary conditions. As Shamit Kachru said during my talk, "you are only interested in some U(1) at the bottom of a throat." We don't care about the top of the throat, i.e. the UV physics. The upshot is: you CAN use D7-branes as probes, to first approximation, if they are outnumbered by very many D3-branes and if you only ask about low-energy physics.

Equivalent statements apply in the field theory. N=4 SYM by itself is conformal. Adding flavors makes the coupling run, and indeed produces a Landau pole. The beta function has a coefficient Nf/Nc, however, so if we make that small, then we can pick some scale, throw out the UV physics, and restrict ourselves to questions only about IR physics, meaning far below the Landau pole. In that regime the beta function is approximately zero. Equivalently, for small coupling we can say that we discard all diagrams with flavor loops. In that case the coupling doesn't run. The deficit angle corresponds to an anomaly that also disappears. (Notice that dropping those loops disturbs 't Hooft anomaly matching: if we drop those loops, the IR theory need not have the same anomaly as the UV theory, where we don't drop the loops).

In short, to quote Clifford Johnson, "It's like QED." Yes, the theory has a Landau pole, but as long as we only ask about IR physics, we don't care. Essentially 100% of the papers using probe D7-branes only ask about physics far below the Landau pole, and all of them give results consistent with the field theory, for example [|the supersymmetric meson spectrum].

Also like QED, we can confront the Landau pole and ask about the UV completion. In the UV the dilaton will eventually diverge. We're in type IIB, however, so we can do an S-duality and make the dilaton small. In the field theory, we're on the other side of the Landau pole, and we switch the degrees of freedom we use to describe the system to some new, dyonic, degrees of freedom (some (p,q)-strings) that are weakly coupled. At least, in principle we can. Carlos Nunez told me that doesn't work in practice. He said he tried doing that, and found that some components of the metric become negative, which is clearly unphysical. He said ultimately you must lift the system to M-theory to obtain a good UV completion.

Now let's move to other topics.

In my talk I gave a brief review of extremal AdS-Reissner-Nordstrom (AdS-RN), and the fact that it has nonzero entropy at zero temperature, which suggests that the system may be unstable. For example, if a charged scalar is present, and has sufficiently large charge and small mass, it may condense, sucking the charge out of the horizon and eventually forming a domain wall. The new ground state has no horizon and hence zero entropy. Veronika Hubeny asked an obvious question, though one often overlooked: doesn't that violate area-increase theorems for black holes? Sean Hartnoll answered this, although in fact the answer appeared in the very last paragraph of a paper by Horowitz and Roberts, [|0908.3677]. The black hole area-increase theorems apply for fixed energy and varying temperature. For the holographic superconductors, we usually work with fixed temperature and varying energy, hence we evade the area-increase theorems. The point is that we are supposed to imagine fixing the temperature, and then lowering the temperature through the transition and then eventually to zero temperature, where we will see that the black hole with scalar hair has lower energy than extremal AdS-RN.

Mukund Rangamani noted that my explanation of why hydrodynamics fails in extremal AdS-RN was woefully incomplete. What I said was correct, that the mean free path diverges as the temperature drops, but the real issue is that new light degrees of freedom emerge at low temperatures. Indeed, presumably these produce the nonzero entropy at zero temperature. They appear in correlators via branch cuts. Peskin and Schroeder section 7.1 will explain to you how a branch cut in a spectral function corresponds to a continuum of multi-particle states. What effective theory describes these is unclear. That was the original motivation for holographic Wilsonian RG: just integrate stuff out directly on the gravity side and see what the effective theory is. That is straightforward to do for free fields, but harder for gravity itself, so the question remains unanswered.

The other major issue in my talk was the presence of the AdS2 on a probe brane with flux. To recapitulate, the statement is that if you introduce the probe D7-brane and turn on its worldvolume gauge field so that in the field theory you have some charge density in the flavor sector, then in the bulk every single fluctuation of a worldvolume field will see AdS2 deep in the bulk. In other words, probe branes have AdS2 too, just like AdS-RN.

That raised many questions. The first was: does the open string metric exhibit the AdS2? After all, that is the metric these fluctuations should "see." The answer is no. Several people, including myself, Keun-young Kim, Tom Faulkner, and Martin Ammon, all computed the open string metric and looked at its form deep in the bulk. What you find is an ugly mess with no simple scaling symmetries. As Keun-young and Andreas Karch pointed out to me, however, the open string metric is not the only important object. We must also define an open string coupling, which in our case is running. Recall [|Seiberg and Witten]'s expression for the open string coupling (adapted from their eq. 2.44)

math g_s^o = g_s \left ( \frac{\textrm{det}(g + (2\pi\alpha')F)}{\textrm{det} g}\right)^{1/2}

math

The precise statement seems to be that the open string metric and open string coupling conspire precisely to produce AdS2. To me that makes sense. If you did allow the D7-brane to back-react, you probably would obtain an extremal black hole with a near-horizon AdS2. In the field theory, the correlators must all exhibit local quantum criticality in the IR. If you take a probe limit, that can't just disappear, so you must still find AdS2 in the flavor sector, which means on the D7-brane. (Carlos Hoyos also suggested that perhaps in the open string sector you could switch from "string frame" to "Einstein frame," in which case the Einstein-frame metric would presumably exhibit AdS2.)

The next question: if we have AdS2, do we see branch cuts in correlators? Here the answer is subtle, and took about six weeks to answer, with some work from Kristan Jensen, Shu Lin, and Jonathan Shock. The first important statement is that in the IR we obtain a neutral, massless scalar in AdS2. That is true for all D7-brane fluctuations. A neutral, massless scalar in AdS2 is a very special case where in fact the IR Green's function exhibits no branch cut whatsoever. The retarded Green's function is simply

math G_R(\omega) = i \omega

math

which clearly has no branch cut at all. Jonathan and Shu did see a branch cut, however, but only for timelike momenta. In fact, they are precisely the branch cuts you expect at zero density, for a dimension-three operator (the one they were studying) in a (3+1)-dimensional CFT: see appendix A.2.2 of [|hep-th/0205051], eq. A.23. In other words, they are the same branch cuts you obtain in pure AdS5. They have nothing to do with AdS2. The probe D7-brane seems to be very special in this respect.

Sean Hartnoll also asked the following question: in AdS-RN, for a fluctuation with finite momentum, the momentum appears in AdS2 as part of the mass. Does that happen for the probe D7-brane? The answer is no. In the near-horizon limit, the terms involving momentum scale with the wrong powers of r to describe a mass in AdS2. The same applies for the angular momentum that the fluctuation has on the S5 (the KK momentum).

Mukund also asked whether the AdS2 is related to the finite entropy at zero temperature. Our usual intuition is that the entropy is associated with a horizon, and so is the AdS2. The main issue here is that for the probe D7-brane the entropy comes from open string degrees of freedom, not closed string degrees of freedom. It is order Nc, not order Nc^2, and it is not associated with any horizon, i.e. is not really tied to geometry, but to the flux on the brane (open string degrees of freedom). The open string metric, for example, exhibits no horizon. ([|An electric field produces one], but that is a different story.) Indeed, as noted by [|Karch, Kulaxizi, and Parnachev], for very large quark mass the entropy of the D7-brane is simply the entropy of a single string times some number of strings on the order of Nc. What is curious is that a single string, or in field theory language a single heavy quark, has an entropy of order \root{lambda}. Anyway, the entropy and AdS2 are connected only in the sense that whenever one appears, so does the other. At the moment I will not commit to any deeper connection.

A number of open issues remain for the probe D7-brane at finite density, for example the physical meaning of the "R-spin diffusion" mode. I will leave those for future research.