Spin-Refined Partition Functions and CRT Black Holes

We investigate spin-refined partition functions in AdS/CFT using Euclidean gravitational path integrals. We construct phase diagrams for ZX = Tr ( e-β H X ) in various dimensions and for different choices of discrete isometry X, discovering rich structures at finite temperature. When X is a reflection, ZX counts the difference between the number of even- and odd-spin microstates. The high-temperature regime is universally dominated by CRT-twisted black holes in any dimension, and in odd spacetime dimensions we examine whether complex rotating black hole solutions could contribute to spin-refined observables. We also analyze the microcanonical ensemble. There the leading contribution almost always come from rotating black holes, showing that the two ensembles are not equivalent.