Elliptic genera from multi-centers

I show how elliptic genera for various calabi-yau threefolds be understood from supergravity localization using the quantization of the phase space of certain multi-center configurations. i present a simple procedure that allows for the enumeration of all multi-center configurations contributing to the polar sector of the elliptic genera — explicitly verifying this in the cases of the quintic in $$ mathrm{mathbb{p}} $$ 4, the sextic in $$ mathbb{w}mathrm{mathbb{p}} $$ (2,1,1,1,1), the octic in $$ mathbb{w}mathrm{mathbb{p}} $$ (4,1,1,1,1) and the dectic in $$ mathbb{w}mathrm{mathbb{p}} $$ (5,2,1,1,1). with an input of the corresponding ‘single-center’ indices (donaldson-thomas invariants), the polar terms have been known to determine the elliptic genera completely. i argue that this multi-center approach to the low-lying spectrum of the elliptic genera is a stepping stone towards an understanding of the exact microscopic states that contribute to supersymmetric single center black hole entropy in $$ mathcal{n} $$ = 2 supergravity.