Black hole partition function using the hybrid formalism of superstrings

The IIA superstring partition function ZIIA, on a Euclidean AdS(2) x S(2) x CY(3) computes the modified elliptic genus Z(BH) of the associated black hole. The hybrid formalism of superstrings on AdS(2) x S(2), defined as a sigma model on the coset supermanifold PSU(1; 1 vertical bar 2)/U(1) x U(1) with a Wess-Zumino term, together with Calabi-Yau and chiral boson conformal field theories, is used to calculate the partition function of IIA superstrings on the Euclidean attractor geometry AdS(2) x S(2) x CY(3). Instead of the kappa symmetry analysis used by Beasely et al. in Ref. [33], we use world-sheet superconformal invariance to construct a nilpotent Becchi, Rouet, Stora, Tyutin ( BRST) operator. The sigma model action is explicitly shown to be closed under this BRST operator. Localization arguments are then used to deform the world-sheet path integral with the addition of a BRST exact term, where contributions arise only from the center of AdS(2) and the north and south poles of S(2). This leads to the Ooguri, Strominger, and Vafa result Z(BH) = Z(IIA) = j vertical bar Z(top)vertical bar(2), where vertical bar Z(top)vertical bar(2) is the square of the topological string partition function.