News From The Front, V: Microscopic Weekend Diversions

I’ve been spending the day so far as an administrator, and not a researcher, since I have to present the results of two committees’ deliberations at one of the big annual organizational meetings tomorrow here at the Aspen Center for Physics. So I’ve been gathering and arranging data in a presentable form. Enough. I will take a break and blog a tiny bit before turning to a truly riveting task – reviewing an introductory physics textbook for a publisher… (Sigh…it is not so easy to escape these things, even out of semester time.)

I had big plans to do a hike each day on the weekend, but physics intervened. I should explain a bit more. Earlier last week I eventually got around to following Nick’s suggestion from an earlier post to take a look at Andy Strominger’s Strings 2007 talk entitled “Search for the Holographic Dual of N Heterotic Strings”. It was the usual nice Strominger talk, where he motivates the physics very well, and presents interesting and clear thoughts on the problem in hand. I shall try to say a bit more about what it is about later on, but the general gist of it is that it is to do with understanding certain types of four dimensional black hole in string theory. As you may know, one of the extraordinarily successful results in string theory in the last decade (and slightly more) has been that we can understand one of the most central results of semi-classical quantum gravity -that they have an entropy and behave like thermodynamical objects (the work of Bekenstein and of Hawking from the early 70s)- in precise terms in the full theory of quantum gravity that string theory appears to present us with. This started with the work of Strominger and Vafa in 1996, that showed how to describe a large class of black holes as essentially made of extended objects called D-branes (about which I’ve spoken at length earlier¹).

Just to fill in the gaps roughly: Hawking’s result that black holes can radiate as thermodynamical objects comes from taking Einstein’s theory of General Relativity and combining it with Quantum Mechanics in a partial way. He could not really do much better since there was no proper quantum theory of gravity at the time, but even in that “semi-classical” treatment, he was able to derive this robust result showing the consequences of combining Quantum Mechanics and Gravity. Bekenstein went further and showed that if the holes radiate and behave like objects with a definite thermodynamical temperature, they ought to have other thermodynamical properties too, and he showed that they have an entropy which is proportional to their area. This made contact with the striking classical Relativity result that the area of black holes is generically non-decreasing – it is just the Second Law of Thermodynamics, which says the same thing about the entropy in Thermodynamics! The result relating the entropy [tex]S[/tex] and the area [tex]A[/tex] which is: [tex]S_{\rm bh}=\frac{A}{4}\cdot\frac{kc^3}{G\hbar}[/tex] (note the appearance of Newton’s Gravitational constant [tex]G,[/tex] Planck’s constant [tex]\hbar[/tex], Boltzman’s constant [tex]k[/tex], and the speed of light [tex]c[/tex] in one amazing formula) is the celebrated Bekenstein-Hawking area/entropy formula (note the ambiguity of the initials bh here). In thermodynamics, we understand the entropy of an object (any object – that warm cup of coffee you’re drinking will do) in terms of the underlying statistical description of the “microstates” that make it up. So you’d expect the same thing here. Problem is that you can’t really understand that precisely without a proper description of Quantum Gravity. The entropy here has a Quantum Mechanical description, and in order to identify the microstates, you need both the Gravity side and the Quantum Mechanics side to be working properly in concert. Sprinkling in a bit of Quantum Mechanics is just not going to do it. This is what you can do with string theory, as Strominger and Vafa showed so precisely that they could get exactly the right formula above. They did it by showing that the microstates were basically D-branes, at least certain very nice and accessible corner of the the description of the black holes.

That’s all well and good, but recent years have seen much more progress: For certain classes of black hole we can even understand how to derive in a very controlled fashion the corrections to the entropy formula, even in cases where the leading expression vanishes because the black holes are not even black holes in General Relativity. For those latter cases, which might be called “microscopic” black holes, GR breaks down (they are singular) and has nothing more to say on the subject. String theory has a lot to say, however, since it supplies a whole series of corrections to GR, both classical and quantum. Classical since even when what might be called quantum gravity corrections are turned off (or very small), there are still corrections which are important when you are considering physical scales that are comparable to the scale set by the fact that your basic object is an extended object – the string. So there are “classical” (in the sense of “not quantum”) corrections as well as “quantum” corrections that string theory can supply to any story involving gravity and other spacetime physics. So even for these microscopic black holes, one can make sense of the entropy – and a whole bunch of stringy corrections to it- in a rather lovely set of examples. There’s been a lot of work on this stuff in recent years by a number of groups, but I think a lot of the recent effort was kicked off by this paper by Atish Dabholkar (although you should read his paper to get more of the background on the work by others (such as the crucial paper of Lopes, Cardoso, de Wit and Mohaupt) that led up to his paper). So we’re well beyond the regime of working with physics that is “General Relativity sprinkled with a bit of Quantum Mechanics” for these problems, and there’s been an extraordinary set of connections made to other branches of string theory, such as something called “topological string theory”, as kicked off by this paper of Ooguri, Strominger and Vafa. (Jacques Distler has been talking about these latter matters a bit in much more technical terms over on his blog (e.g. here, here, and here)).

Anyway, Strominger’s talk was geared to asking a very simple question. You see, the original 1996 Strominger-Vafa success at understanding in string theory the quantum description of the Bekenstein-Hawking entropy of a class of black holes was itself a surprisingly robust computation, and Strominger later on showed that there was a connection between this unreasonably generous situation and something that we now recognise as an example of a more general phenomenon, called “Holography”, of which the celebrated AdS/CFT correspondence of Maldacena (which I’ve discussed in earlier post) is an example. In short, at the core of the string theory D-brane configurations that give rise to the various types of black hole in question, there lies a relatively simple universal geometry (it is always three-dimensional anti-deSitter space, [tex]{\rm AdS}_3[/tex]). This geometry (it is now understood) essentially guarantees that the entropy counting will work, since the phenomenon of holography tells you that the physics of gravity in this sort of spacetime is captured by a non-gravitational two dimensional field theory in which entropy computations are more routine. (It is an [tex]{\rm AdS}_3/{\rm CFT}_2[/tex] correspondence. It’s a baby version of the [tex]{\rm AdS}_5/{\rm CFT}_4[/tex] which has been exciting a lot of us recently because it allows us to possibly connect string theory results to experimental signatures in nuclear physics. I’ve spoken about that earlier too, and recall that Bee and Stefan did a nice post on that issue last year.)

That all works then because the black holes in the Strominger-Vafa story are made of D-branes of two types that intersect. The intersection gives rise to the black hole, and is a sort of macroscopic string-like object wound on a circle. It is the core of this string-like geometry that has an [tex]{\rm AdS}_3[/tex] geometry.

So when you see other examples of an unreasonably successful entropy counting in string theory, the earlier example is going to inspire you to ask whether or not holography is playing a role again. But in these new examples, it will have many new wrinkles. For a start, in the cases in question, the black holes are not made of D-branes, but are instead made of a certain sort of string called an “Heterotic” string. These very strings are wound on a circle and themselves become black holes! Furthermore, we’ve not seen controllable examples of holography on the heterotic string side of things before, so this will be quite new, if true. Finally, for the microscopic black holes in question, the relevant geometry at the core of the string (describing the wound string) is microscopic. A direct description is not available in General Relativity, and only by adding corrections to GR can one see that there is a real [tex]{\rm AdS}_3[/tex] (and [tex]{\rm AdS}_2[/tex] when wound) geometry living at the core of the string, but it is it tiny. There’s a couple of nice papers showing this, one by Dabholkar, Kallosh and Maloney, for the black holes and another more recent one about the string itself (the object that becomes the black hole when you curl it up) by Castro, Davis, Kraus, and Larsen.

To get the full description of this sort of “microscopic” geometry in this context, we’ll need the full power of string theory to even properly describe the geometry and the string physics going on in that geometry since it’s all happening on scales invisible in GR. Hopefully that explains Strominger’s talk title a bit now (you can think of the “N” as relating to how many times you wind a single heterotic string on a circle to make a black hole – the N turns out to be N units of magnetic charge).

So what does this all have to do with me? Well, I’d not really been thinking much about these issues in recent times, but I really liked Strominger’s talk, particularly because he was wondering how to construct the description of these microscopic geometries in heterotic string theory, which is an interesting problem in what’s called conformal field theory. Near the end of the talk, he outlined a conjecture that he and his collaborators (Lapan and Simons) had about the part of the problem that he described as “hard” – in technical terms, how to find a heterotic conformal field theory description of the accompanying microscopic two-sphere ([tex]S^2[/tex]) that is part of the geometry (it’s really [tex]{\rm AdS}_3\times S^2[/tex] and [tex]{\rm AdS}_2\times S^2[/tex] that are relevant to the string and the wound string that becomes the black hole.) They conjectured that it is a special case of a certain conformal field theory construction of Giddings, Polchinski and Strominger, from way back in 1993.

At this point I sat up. Of course it is. It is a beautiful idea! But I knew that these spheres are not “hard” to describe at all. I showed this in 1994 in the bulk of my postdoc work at the Institute for Advanced study. (This is all amusing in a way, given what I was saying about those very “old days” seeming to return, just a few blog posts ago.) Of course, nobody reads my papers ( 😉 ), so how would they know?

The upshot is this: Spheres are not hard to describe as heterotic string backgrounds, they’re easy, and I could immediately see how to extend their idea and put it all in a setting that might permit easier computations. So on Thursday I decided to write a paper (making sure to give Lapan, Simons and Strominger the credit for the initial suggestion, of course), and I sat in a cafe all Friday morning drafting it, finishing it up on Saturday.

So much for the hiking plans.

There’s more to the story, and I’ll tell it to you later, perhaps in a bit more detail for the more technically trained among you to get something from it. If you can’t wait, the paper is going to appear on the arXiv any minute now. [Update: Oh.. it is here.]

-cvj

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1. I’ve talked about D-branes before, but to recap, let me say this briefly. D-branes are extended objects, and in certain string theories called “type I and type II string theories”, they have a description as the places where the ends of fundamental strings can lie, as in the diagram to the right (red things: strings; blue things: D-brane). The term brane? Well, that’s easily explained: quoting myself from a while back:
   

   […] Strings are not the only important objects in the game, but extended objects of more dimensions called “branes” are also important. (The term comes from starting with “membrane” which is a two dimensional object, calling it a “2-brane” and then having the idea of a “3-brane”, “4-brane”, “1-brane” etc. Or just “brane” when you want to be non-specific.) It turns out that we need to consider these objects too. […]

   [return]

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