Modular chaos, operator algebras, and the Berry phase

TL;DR

This paper explores how the modular Berry phase in quantum operator algebras encodes geometric and spacetime information, extending previous holographic results to a broader algebraic context.

Contribution

It reformulates modular Berry transport for general von Neumann algebras, introduces a zero mode projection via conditional expectation, and links Berry phases to emergent spacetime structures.

Findings

Modular Berry phase encodes spacetime geometry information.

Zero curvature indicates emergence of local Poincaré algebra.

Large N limit reveals an emergent symplectic form.

Abstract

Modular Berry transport associates a geometric phase to a zero mode ambiguity in a family of modular operators. In holographic settings, this phase was shown to encode nontrivial information about the emergent spacetime geometry. We reformulate modular Berry transport for arbitrary von Neumann algebras, including giving a precise definition of the zero mode projection in terms of a conditional expectation. For a certain class of state perturbations, we demonstrate that the modular Berry phase gives rise to an emergent symplectic form in the large $N$ limit, extending related results in the context of subregion/subalgebra duality. We also show that the vanishing of the Berry curvature for modular scrambling modes signals the emergence of a local Poincar\'e algebra, which plays a key role in the quantum ergodic hierarchy. These results provide an intriguing relation between geometric phases, modular chaos and the local structure of spacetime.