# Dynamics for holographic codes

## TL;DR

This paper introduces a framework for incorporating dynamics into holographic codes by utilizing the semicontinuous limit of Hilbert spaces and representing relevant symmetry groups, creating a toy model of AdS/CFT.

## Contribution

It develops a method to define dynamics for holographic codes using the semicontinuous limit and group representations, linking bulk geometry with boundary symmetries.

## Key findings

- Realization of bulk Hilbert space as a subspace with discrete geometry
- Implementation of dynamics via Thompson's group T
- Toy model of AdS/CFT called Pt/T correspondence

## Abstract

We describe how to introduce dynamics for the holographic states and codes introduced by Pastawski, Yoshida, Harlow and Preskill. This task requires the definition of a continuous limit of the kinematical Hilbert space which we argue may be achieved via the semicontinuous limit of Jones. Dynamics is then introduced by building a unitary representation of a group known as Thompson's group T, which is closely related to the conformal group in 1+1 dimensions. The bulk Hilbert space is realised as a special subspace of the semicontinuous limit Hilbert space spanned by a class of distinguished states which can be assigned a discrete bulk geometry. The analogue of the group of large bulk diffeomorphisms is given by a unitary representation of the Ptolemy group Pt, on the bulk Hilbert space thus realising a toy model of the AdS/CFT correspondence which we call the Pt/T correspondence.