Hagedorn, black holes, and Brownian motion

Most of us seem to agree that the most interesting hep-th paper today is one by

• Albion Lawrence and Martin Kruczenski.

It's true that no one seems to be able to say exactly what is the new stuff - probably because no one has read the paper in detail so far - but there is a lot of interesting material in it anyway. Let me describe some of the crucial ideas for this business.

They study the behavior of strings near the Hagedorn transition. As you know, the partition sum diverges above this critical temperature because the number of excited string states at a certain level increases faster than what the Maxwell-Boltzmann exponential factor is able to suppress; alternatively, the partition sum diverges because a thermal scalar emerges - it is a tachyonic winding string wrapped on the thermal (Euclidean periodic time) circle in spacetime; its winding number equals one. These two pictures are related by the modular transformation on the worldsheet (tau goes to -1/tau). Kruczenski and Lawrence insist on the traditional Atick-Witten rules of the game and don't try to believe some recent "Hagedorn revolution" papers that we discussed a short time ago.

When the tachyon dominates, the partition sum may be calculated using the path integral over the histories of this particular field in spacetime; these are the random walks or, equivalently, the Brownian motion. I believe it's a well-known fact that the highly excited strings - which are inevitably long (more precisely, one long string with a gas of small strings is the typical configuration) - are a dual description of the thermal scalar near the Hagedorn temperature.

Albion and Martin focus on the typical size of these chaotic strings and random walks and they argue that the characteristic radius goes like the square root of the energy in string units (recall the basic rules of the Brownian motion pointed by Einstein 100 years ago). Then they make a jump to the long strings in the AdS space and argue that there exists a transition to the black hole - something that has been advocated by various authors. We will only see the new argument once we look at the paper in detail.

You know, there exists a nice paradigm that a black hole may be thought of as a condensate of the thermal scalar. This is most clearly manifested in the work by Adams et al. In their case, a circle in spacetime with antiperiodic boundary conditions for the fermions carries a scalar field that becomes tachyonic if the radius shrinks below a critical value; the final state of the condensation of this scalar is a spacetime with a different topology in which the handle is cut into two pieces and the closed strings are banned from the region that used to connect the two sides. This picture can also be interpreted in the case when the circle is viewed as the thermal circle: the condensation of the thermal scalar then produces a black hole. Note that it is not true that you get the black hole "immediately"; you must roll over the configuration space and increase the condensate to actually change the topology of the spacetime, and this fact makes many connections between these two pictures difficult to calculate.

Together with Allan, we also tried to interpolate between the tachyonic perturbative instability and the Witten's non-perturbative instability of the Kaluza-Klein compactification but it is pretty hard to extract any quantitative checks from such a nice picture especially because the Witten bubble is a "large" condensate of the tachyon.

Albion and Martin also study the excited strings in the anti de Sitter space and advocate their connection with the anti de Sitter black holes. One of the paradigms that their work seems to support - or at least paradigms that they independently believe to be relevant - have been summarized in various intriguing papers by Lenny Susskind around 1994, for example in this work by Uglum+Susskind. They argued that the black hole entropy can be explained using a long string that is chaotically wound around the black hole horizon. I also believe that these ideas must be morally correct - well, I've rediscovered them independently, too. It is even plausible that this picture, once it's put on firm ground, will confirm Samir Mathur's ideas that the black hole interior is very different than we thought. But no one seems to have a clear picture yet.