# The algebraic structure of gravitational scrambling

## TL;DR

This paper develops an algebraic framework called the modular-twisted product to describe gravitational scrambling, connecting early and late-time operator algebras, and extends it to higher dimensions with localized boundary excitations.

## Contribution

It introduces the modular-twisted product algebra for gravitational scrambling, unifying previous algebraic structures and generalizing to higher dimensions with boundary excitations.

## Key findings

- The modular-twisted product interpolates between free and tensor-product algebras.
- Including the Hamiltonian yields a Type II$_inity$ von Neumann algebra with finite entropies.
- The framework applies to higher dimensions with localized boundary excitations.

## Abstract

We introduce a new algebraic framework to describe gravitational scrambling, including the semiclassical limit of any out-of-time-order correlation function that is built out of operator insertions separated by approximately the scrambling time. In two dimensions, the scrambling algebra, which we call a modular-twisted product, is defined in terms of two copies of the Leutheusser-Liu half-sided modular inclusion of von Neumann algebras; these describe early- and late-time operators respectively. In limits where the separation between insertions is taken to be either significantly greater or smaller than the scrambling time, the modular-twisted product reduces, respectively, to free- and tensor-product algebras that were previously studied in [arXiv:2209.10454]. In a sense, the modular-twisted product interpolates between these two products. Including the Hamiltonian in the scrambling algebra leads to a Type II$_\infty$ von Neumann algebra with finite renormalized entropies that interpolate between single-QES and multi-QES phases. We also describe how to generalize the modular-twisted product algebra to higher dimensions, including spatially localized boundary excitations.