Determine the semiclassical saddles of the gravitational index

Determine the complete set of semiclassical supersymmetric, non-extremal gravitational saddles that satisfy the prescribed boundary conditions and have finite on-shell action, thereby resolving the outstanding saddle-determination problem for the gravitational index.

Background

The gravitational index is defined by a gravitational path integral with boundary conditions chosen to implement a supersymmetric index. Its semiclassical evaluation requires identifying complexified supersymmetric, non-extremal supergravity solutions with finite on-shell action that obey those boundary conditions.

The paper constructs and analyzes broad families of candidate saddles, including multi-center five-dimensional configurations and their type IIB decoupling limits. However, the construction does not constitute a complete determination of all semiclassical saddles contributing to the gravitational index, which is why the authors frame the general saddle-classification problem as open.

References

A major open question regarding the gravitational index, which has attracted considerable attention, is the determination of its semiclassical saddles.

— Decoupling saddles of the gravitational index and the Farey tail  (2608.25438 - Cassani et al., 26 Aug 2026) in Section 1, Introduction

It is possible to identify ranges of the parameters for which this inequality holds, thereby constraining the ring dipole charges $q_1I$. %, even though it is unclear to us how such restrictions may be motivated on mathematical or physical grounds.

— Decoupling saddles of the gravitational index and the Farey tail  (2608.25438 - Cassani et al., 26 Aug 2026) in Section 5, 'Comments on black ring saddles'