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.

• The paper demonstrates that non-Abelian Berry curvature diagnoses chaos in BPS microstates with exact degeneracies.
• It employs random matrix theory, showing eigenvalue statistics similar to the GUE ensemble in supersymmetric contexts.
• Numerical analyses, particularly of the ℕ=2 SYK model, validate the chaotic dynamics and spectral features of black hole microstates.

Introduction

The concept of chaos in quantum systems has been extensively studied, particularly in the context of energy level repulsion and random matrix statistics. This essay explores the chaotic nature of Berry curvature for BPS microstates, drawing insights from several technical papers available on arXiv.

Berry Curvature in Quantum Systems

Berry curvature is a fundamental concept in quantum mechanics, capturing the geometric phase acquired over adiabatic evolution of quantum states. In systems where energy levels are degenerate, the Berry curvature matrix provides a measure of this phase, influenced by the system's parameters. This matrix's eigenvalues can sometimes hint at chaotic behavior, similar to eigenvalue statistics in random matrix theory.

Chaos in Berry Curvature for BPS Microstates

In the context of supersymmetric black holes, detecting chaos poses unique challenges due to exact degeneracies of energy levels. The papers (2604.23287), "Chaos of Berry curvature for BPS microstates," propose that non-Abelian Berry curvature can diagnose chaotic properties distinct from ordinary level statistics. For supersymmetric black holes, ordinary methods fail due to exact energy level degeneracies. By examining non-random nature of Berry curvature for horizonless geometries, these works suggest that random matrix-like statistics could emerge for states associated with supersymmetric black holes.

Numerical Evidence and Theoretical Predictions

Numerous studies offer numerical evidence supporting the chaotic behavior of Berry curvature in specific BPS systems. For instance, analysis of the $\mathcal{N}=2$ SYK model highlights the non-trivial topology and large degeneracies in BPS states, with Berry curvature showing random matrix statistics akin to generalized unitary ensembles (GUE). This chaotic nature is further reinforced by spectral form factor calculations and eigenvalue repulsion observations.

Implications and Future Directions

The findings from these papers provide a robust framework for understanding chaos in BPS microstates. Implications extend to broader theories, including $\mathcal{N}=4$ SYM and D1/D5 systems, where similar Berry curvature analyses could reveal chaotic dynamics. Future work may explore these concepts in non-BPS black holes, providing deeper insights into entropic dynamics and the role of Berry phase in quantum gravity.

Conclusion

This exploration of Berry curvature chaos in BPS microstates offers profound insights into quantum chaotic dynamics in systems with significant degeneracies. Utilizing tools from random matrix theory and Berry phase analyses, researchers can probe deeper into the chaotic nature of quantum systems, expanding our understanding of quantum chaos in complex supersymmetric landscapes. By embracing the nuanced behavior of Berry curvature, future research can further elucidate the intricate dance between quantum chaos and supersymmetric structures.