$ \mathcal{N} = 4 $ superconformal algebra and the entropy of hyperKähler manifolds

Abstract

We study the elliptic genera of hyperKähler manifolds using the representation theory of $ \mathcal{N} = 4 $ superconformal algebra. We consider the decomposition of the elliptic genera in terms of $ \mathcal{N} = 4 $ irreducible characters, and derive the rate of increase of the multiplicities of half-BPS representations making use of Rademacher expansion. Exponential increase of the multiplicity suggests that we can associate the notion of an entropy to the geometry of hyperKähler manifolds. In the case of symmetric products of K3 surfaces our entropy agrees with the black hole entropy of D5-D1 system.