The main points covered in the talk were the following:-
Generating rotating charged black hole[string] solutions in d=5[6].
The talk started with a review of the bosonic Lagrangians in d=10 for massless string modes, then those for d=6[5] IIA theory on K3[xS^1], and lastly string/string duality of d=6 IIA(K3) to d=6 heterotic(T^4). The importance for rotating configurations of the "Chern-Simons" terms was stressed; these are couplings [e.g. in d=6 IIA(K3)] involving the NS-NS 2-form potential with the R-R gauge field strengths. The presence of the Chern-Simons terms is dictated by supersymmetry.
Next it was explained how combinations of string/string duality and heterotic(tori) "twisting" were used to obtain new black hole solutions of the IIB(K3xS^1) theory from previously-known solutions with more symmetry. Twisting procedures on tori were outlined as being straightforward combinations of Kaluza-Klein reduction along symmetry directions, symmetry transformations in the lower-dimensional theory, and then lifting back up to the original theory. It was noted that solutions obtained by using these algebraic methods can always be checked by plugging them in to the relevant differential equations of motion.
The resulting black strings/holes carried three charges, two rotation parameters, and the extremal ones preserved one of a possible four supersymmetries and were hence BPS-saturated. The black hole entropy, which was calculated using the classical Bekenstein-Hawking prescription, was exhibited, and it was noted that it was independent of the moduli fields. In addition, it was pointed out that the two angular momentum parameters of the black hole/string were found to be required to be equal and opposite, J_1=-J_2, in the extremal BPS limit.
Counting degeneracy of states for spinning D-branes.
In the second part of the talk, there was first a review of making a black string in the d=6 IIB picture by wrapping d=10 D5-branes on K3, not wrapping D1-branes, and adding momentum. Then, after Strominger-Vafa, it was noted that one could take the K3 to be small "relative to" the circle around which the string gets wrapped to go to d=5. Therefore, the effective field theory of the D-brane configuration in the d=5 target space lived on two dimensions, S^1xR[time], and so 2-d statistical field theory was used to count states. For one-quarter-BPS states, i.e. those of interest, this meant considering a configuration in the 2-d theory which was in [say] the right-moving vacuum and had arbitrary left-moving excitations.
Next came an exposition of how the Vafa-Witten conjecture relates the spacetime helicity of the D-brane configuration to its charges under the left- and right-handed U(1) currents in the 2-d superconformal algebra [recall that K3 is hyperkahler]. It was explained that the two macroscopic angular momenta of the 1/4-BPS state were carried by excited left-movers, and were built up from microscopic units. From bosonizing of the currents, it was shown how rotation [equivalently the U(1) charges] costed oscillator degeneracy and therefore reduced the entropy.
Lastly, precise agreement between the D-brane state-counting entropy in the large-charge regime and the Bekenstein-Hawking black hole entropy was exhibited, in tune with expectations from the adiabatic argument for BPS states. In addition, because for the 1/4-BPS states the spins of the D-brane configuration were carried [largely] by the left-movers, agreement was found in the large-charge regime with the J1=-J2 observation made earlier using classical solution-generating methods.
Nonextremal case
Finally, it was mentioned that precise agreement was also found for slightly nonextremal black strings, even though no adiabatic argument motivated that result. There is more to be learned here!
Q/A
[Tseytlin:] Did our results also apply to toroidal
compactifications?
Yes, methods for compactification to T^5 rather
than K3xS^1 carried over very straightforwardly.