Index saddle for supersymmetric F1-P black ring

TL;DR

This work constructs the gravitational index saddle for the supersymmetric F1--P black ring by taking a carefully chosen supersymmetric limit of a non-extremal, doubly spinning solution, and reformulates the saddle as a three-center Bena–Warner configuration. Despite possessing a finite-area horizon, the two-derivative index vanishes, and the horizon is singular along two circles where higher-derivative corrections become important. A scaling analysis in the thin-ring regime shows the corrected index reproduces the microscopic dependence on charges through $nw - JQ$, up to an overall constant $K$ that cannot be fixed by scaling alone. The results reinforce the utility of index saddles for protected microscopic data even when the classical geometry is singular, and point to the necessity of including higher-derivative effects and potential connections to bubbling/multi-centered saddles in a broader framework.

Abstract

We construct the index saddle for the supersymmetric F1--P black ring. Our construction proceeds by taking a supersymmetric limit of a non-supersymmetric doubly spinning F1--P black ring. We express the resulting saddle as a three-center Bena--Warner solution. The black ring saddle possesses a finite-area event horizon, yet the two-derivative index vanishes. The solution is singular on certain subspaces of the horizon, where higher-derivative corrections are expected to become important. We argue that, once such corrections are taken into account, the solution can yield a finite result. In particular, we present a scaling analysis showing that the index agrees with the microscopic result, up to an overall numerical constant that cannot be fixed by the scaling argument alone. This analysis applies only within a restricted region of parameter space, whose full significance is not yet fully understood.