Black hole partition function using the hybrid formalism of superstrings

The IIA superstring partition function ZIIA, on a Euclidean AdS2×S2×CY3 computes the modified elliptic genus ZBH of the associated black hole. The hybrid formalism of superstrings on AdS2×S2, defined as a sigma model on the coset supermanifold PSU(1,1|2)/U(1)×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 AdS2×S2×CY3. Instead of the kappa symmetry analysis used by Beasely et al. in Ref. , 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 AdS2 and the north and south poles of S2. This leads to the Ooguri, Strominger, and Vafa result ZBH=Z IIA=|Ztop|2, where |Ztop|2 is the square of the topological string partition function. © 2011 American Physical Society.