Tuesday, May 31, 2005

Ashoke's heterotic black holes

We returned from a seminar at MIT. Ashoke Sen was checking the entropy of heterotic black holes. At weak coupling, such an object has to look like an excited heterotic string. A heterotic string is made of two types of important excitations:
• the left wing
• the right wing
Note that both wings are important. The left wing gives it its potential instability and tachyons and ugliness. The right wing, on the other hand, gives the heterotic string its supersymmetry, beauty, and stability. ;-)

The number of left-moving and right-moving excitations must match. If you imagine a compactification on a circle - times a 5-torus - you may still obtain supersymmetric states that satisfy
• N_R = 0, N_L = n.w+1
where "n" is the momentum along the special circle and "w" is the winding. Note that the difference between "N_L" and "N_R" is determined by the level-matching conditions, and in the presence of momenta and winding, it is not zero but rather "n.w+1". There are no right-moving excitations, and because supersymmetry is only carried by the right-movers, it is not broken.

At strong coupling, these supersymmetric states become black holes. Ashoke constructs the corresponding black hole solutions of four-dimensional supergravity by a special solution-generating method involving T-dualizing of a Schwarzschild solution, but others - such as Natalia Saulina - could find the solution directly. ;-)

The question is whether one obtains the right entropy. The entropy in the CFT is, according to Cardy's formula, something like "4.pi.sqrt(n.w)" for large values of the momentum and winding. The leading black hole entropy is "A/4G". So do they agree? The classical area of the horizon turns out to be zero. Ashoke Sen then argues that
• the stringy loop corrections are absent in the appropriate limit
• the alpha' corrections are relevant
• the alpha' corrections have the correct scaling with the charges by two scaling arguments
• the coefficient must be checked
• the coefficient is not "log(3)" as in loop quantum gravity but rather "4.pi"
• this number is hard to calculate because the number of higher derivative terms is huge, but surprisingly, if one only considers the F-terms - those that are calculable from topological string theory - one obtains the correct corrections, indicating that no other corrections contribute to the black hole entropy