## Friday, December 01, 2006 ... //

### Attractor flow trees

Frederik Denef of Belgium spoke about

• the split attractor flows and BPS state counting
during the Duality Seminar. As you may know, Frederik is the main guy who discovered the split black hole solutions and we should start with some general background for all these considerations which will be the main part of this article because of the general audience of readers of this blog and because some of the newer work is work in progress.

Split attractor flows

You may imagine that the popular spherically symmetric black holes are analogous to atoms. Frederik's split flow black holes are molecules: you may imagine that they are bound states of the "atomic" black holes - where the atoms sit at the minimum of a potential - and he has found precise solutions for them.

He has shown that for every attractor flow tree, there exists exactly one multi-centered black hole solution, a black hole "molecule". Let's explain what is the attractor flow tree.

Start with type IIA string theory on a Calabi-Yau three-fold: at low energies, this gives N=2 supergravity in d=4. The charges are carried by the D0, D2, D4, D6-branes. Call the total charge of a black hole Gamma which belongs to the lattice of even-dimensional homology. It has been known for some time that the moduli from the vector multiplet approach special values - the attractor values - as you go closer to the horizon.

This simple insight that follows from the SUGRA equations of motion tells you how the spherically symmetric black holes solutions look like near the horizon, regardless of the values of the moduli at infinity. Every such a solution may be associated with a curve in the moduli space that tells you how the moduli "flow" as you approach the horizon: this flow is governed by a first-order differential equation (arising from the requirement of unbroken supersymmetry) and is analogous to RG flows in the parameter spaces of field theories.

Frederik's solutions are more general. He realized that a black hole with charges Gamma can decay at some moment - at some values of the moduli. If it decays to smaller objects with charges Gamma1, Gamma2, there exists a simple rule to see whether such a decay is kinematically possible or not. Compute the complex central charge Z - a scalar quantity that enters the right-hand side of the N=2 supersymmetry algebra - for the initial big object (root) as well as the two subobjects Z1, Z2 (branches). If the complex numbers Z1, Z2 point in the same direction for a given choice of Gamma1, Gamma2 i.e. if they only differ by the absolute value, it means that the big black hole with the charge Gamma and Z is just becoming unstable.

Because the condition constrains one phase only, it defines a real codimension 1 region in the moduli space - the so-called line of marginal stability.

Frederik's procedure to construct more general solutions goes as follows. You choose the total charge Gamma of your desired object - the charge measured at infinity - and you try to split it into Gamma1+Gamma2 in all possible ways. For each such a split, you can calculate the line of marginal stability in the moduli space. The moduli for the big Gamma object evolve according to the first order flow differential equations until (=outside a sphere in space where) you hit the line of marginal stability.

Once you reach the line, you split the flow into two separate flows that evolve according to their own differential equations appropriate for the charges Gamma1, Gamma2, respectively. You may allow Gamma1 and/or Gamma2 to be split itself later, and so forth. You try to generate all possible trees like that whose all branches end up at attractor points.

OSV conjecture

The OSV conjecture relates the topological partition function and the black hole entropy and allows you to calculate all perturbative subleading corrections to the Bekenstein-Hawking black hole entropy of these type IIA D-brane black holes on a Calabi-Yau manifold. By the partition function for the black holes we really mean an index Omega that is more easily calculable and whose value is protected by supersymmetry.

Frederik essentially argued that the total index should be a sum over all possible attractor flow trees and he is currently working on many formulae in this direction with Greg Moore who was recently speaking at Harvard, too. I can't tell you all the details because many of them are unpublished but he ended up with a possible entropy enigma: the split black holes have entropies that scale like the third power of the charge which becomes much larger than the second power of the charge for large black holes.

The OSV experts debated whether this cubic scaling can be properly obtained from the MSW conformal field theory via Cardy's formula and whether the correct coefficient of the cubic term can be calculated on both sides. A solution that would be satisfactory for everyone has not yet emerged.

Anthropic consequences

In the final portions of his talk, Frederik argued that these ideas could change the democratic power of different classes of vacua in the anthropic landscape. According to some measures, the flux vacua typically tend to break the supersymmetry at very high-energy scales because the number of vacua where it happens vastly exceeds those with a low-energy supersymmetry breaking scale and because the anthropic experts like democracy of vacua.

Frederik argued that this argument could be changed by including vacua that mimick the structure of type IIA black hole microstates but that can also be re-interpreted as F-theoretical vacua. If his argument is right, one can say that the cutting-edge anthropic reasoning in Europe could again predict a low-energy supersymmetry breaking scale instead of the high-energy supersymmetry breaking scale. But you never know whether this conclusion would be accepted by Stanford and Santa Cruz and whether it survives this month. ;-)