Work in progress

I am trying to look at higher derivative corrections to ashoke's entropy function. My idea is whether one can reverse the usual line of argument starting from the knowledge of the terms, to arrive at the entropy function.
I intend to first assume some coefficients for the higher derivative terms and fix them by requiring that the entropy come out right.

This is a little tricky partly because of field redefinitions. Firstly, AdS X S spaces are symmetric, which makes the coefficients not ALL determinable.

First fix the metric that appears in front of the field-strengths. And express all quantities in terms of this metric (this is because, we would like the charges to be fixed - these are computed in terms of the metric and the field-strengths). Now we have no more freedom to perform field redefinitons of the metric at least.

Then, we require that the entropies match - along with the requirement that there be no additional graviton poles. This fixes the coefficients nicely to quadratic order (for heterotic large black holes with four charges).
This means the following: If we had started with some other choice for the higher derivative curvature terms, then by a suitable field redefinition, one could bring it to the above form. In this case, the metric changes, and hence what one calls charges will change. Thus keeping the charges the same, will (partially) fix the field redefinition freedom.

I guess that it would be possible to do better, if I find some rotating black holes for which the metric is no longer that of a symmetric space examples are Horowitz+Sen. In this case, I expect that i will be able to relax the requirement of having no additional graviton poles.

The key point in this assumption is that AdS times S is expected to be an exact solution, so that order by order in alpha' also, it is a solution. It may happen that for rotating black holes, the corresponding "near horizon" geometry is not as simple - thus thwarting any attempt at determining coefficients in this manner beyond quadratic order.