Cusp regularization with spin structures and gravitational anomalies

Abstract

The family of $SL(2,\mathbb Z)$ black holes has an accumulation point corresponding to a cusped geometry, which causes the sum over black holes to diverge. We extend cusp regularization to the sum over geometries compatible with a fixed boundary spin structure. For $c_L=c_R$, including the cusp as a counterterm cancels the leading divergence for each of the three even boundary spin structures. The required subtraction is the same in all three sectors and is two thirds of the bosonic one. We further show that gravitational anomalies cause cancellations between black hole contributions, removing the leading cusp divergence except at $c_L-c_R=0,\pm24$ in pure gravity and $c_L-c_R=0,\pm12$ in $\mathcal N=(1,1)$ supergravity. We also derive the spin structure dependent anomaly phases responsible for these cancellations.