  We perform a study of the moduli space of framed torsion free sheaves on
Hirzebruch surfaces by using localization techniques. After discussing general
properties of this moduli space, we classify its fixed points under the
appropriate toric action and compute its Poincare' polynomial. From the
physical viewpoint, our results provide the partition function of N=4
Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole
entropy counting problems according to a conjecture due to Ooguri, Strominger
and Vafa.
