The gravitational index and allowable complex metrics
Pietro Benetti Genolini, Sameer Murthy
TL;DR
The work demonstrates that for supersymmetric gravitational indices, the Kontsevich–Segal–Witten (KSW) criterion for allowable complex metrics, geometric regularity, and convergence of microscopic traces all coincide across asymptotically flat and asymptotically AdS4 settings. By analyzing the helicity index in AF$_4$, the topologically twisted index in AAdS$_4$, and the superconformal index in AAdS$_4$, the authors show that complex saddles contributing to the index are selected by the same physical constraints as those ensuring a well-defined microscopic counting, providing a unified criterion for including saddles in the gravitational path integral. In AF$_4$ the SUSY-bound saddles reduce to real Euclidean instantons and pass KSW; in TT$_4$ they similarly reduce to real instantons with matching trace convergence; in SCI$_4$ the saddles are complex but still obey a region in parameter space where all three criteria align, with explicit relations between gravity potentials and field-theory indices. This convergence strengthens the interpretation of complex saddles in gravitational index calculations as physically legitimate and bridges gravitational thermodynamics, localization, and holographic microstate counting, with potential extensions to non-minimal supergravity and string/M-theory uplifts.
Abstract
We study the Kontsevich-Segal-Witten criterion for allowable complex metrics, in the context of the gravitational path integral corresponding to the supersymmetric index. In various theories of supergravity in asymptotically flat and asymptotically AdS space, the exponential growth of states of the corresponding microscopic index in string theory is known to be captured by complex saddle points of this path integral. We compare the KSW criterion for these complex saddles against constraints from geometric consistency and the convergence of microscopic indices for the same saddles. In all the situations we consider, we find that the three criteria precisely agree with each other.