Microscopic and macroscopic counting of BPS states of four-dimensional superstring theories
ABSTRACT :
We study various aspects of counting degeneracy of supersymmetric blackholes in four dimensions. We consider states with both N=4 and N=2 supersymmetry. The degeneracy of N=4 supersymmetric black holes have been widely studied. The generating functions for the degeneracy of black hole microstates turn out to be modular forms and hence there are rich mathematical structures which emerges for the N=4 BPS states. We also see there are unexpected connections of these modular forms with certain simple groups. There also exists beautiful Lie algebraic structures for these modular forms possibly indicating some unexplored symmetries of the BPS states. We discuss some of these connections and show that our study leads to some of the new features. While these aspects of N=4 supersymmetric black holes are extensively studied, not so much structures are known for N=2 supersymmetric black holes making it harder to compute the degeneracy. We briefly discuss some semi-classical analysis of the macroscopic degeneracy of small black holes in this theory.