Chaos of Berry curvature for BPS microstates

Abstract

We expect black hole microstates to differ in their chaotic properties from states associated with other geometries. For supersymmetric black holes, ordinary level statistics cannot diagnose this distinction, since their energy levels are exactly degenerate. We propose that there is an intrinsic probe of chaos, encoded in the mixing of the microstates under changes in the couplings of the theory, as determined by the non-Abelian Berry curvature of the BPS states under certain deformations. For states dual to horizonless geometries in holographic systems, such as 1/2-BPS states in the D1/D5 CFT and 1/4-BPS states in $\mathcal{N}=4$ SYM, we find that the Berry curvature for marginal deformations is non-random and often exactly zero at generic couplings. By contrast, for states dual to supersymmetric black holes, we show through computations in $\mathcal{N}=2$ super-JT gravity and explicit numerics in the $\mathcal{N}=2$ SYK model that the Berry curvature resembles a random matrix. We also uncover interesting topological features of the $\mathcal{N}=2$ SYK moduli space, as probed by Chern numbers. These results suggest that the Berry curvature sharply distinguishes black hole microstates from smooth horizonless states and provides a robust diagnostic of chaos in supersymmetric sectors.