Dynamics for holographic codes

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.Comment: 40 pages (revised version submitted to journal). See video of related talk: https://www.youtube.com/watch?v=xc2KIa2LDF

Thompson field theory

We introduce Thompson field theory, a class of toy models of conformal field theory in which Thompson's group T takes the role of a discrete analogue of the chiral conformal group. T and the related group F are discrete transformations of dyadic partitions of the circle and the unit interval, respectively. When vectors or tensors are associated with partitions, one can construct a direct limit Hilbert space, here called the semicontinuous limit, and F and T have unitary representations on this space. We give an abstract description of these representations following the work of Jones. We also show that T can be thought of as acting on the boundary of an equal-time Poincaré disk in AdS3. This defines a representation of T on the Hilbert space that contains all tree-like holographic states, as introduced by Pastawski, Yoshida, Harlow, and Preskill. It also establishes a bulk-boundary correspondence through Imbert's isomorphism between T and Penner's Ptolemy group. We further propose definitions of field operators and correlation functions for the discrete theory. Finally, we sketch new developments like particle creation and annihilation, as well as black holes and possible connections with topological quantum field theory

Religious transformations in the Middle Ages: towards a new archaeological agenda

The study of religious change in Europe between the collapse of the Roman Empire and the Reformation forms one of the cornerstones of medieval archaeology but has been riven by period, denominational and geographical divisions. This paper lays the groundwork for a fundamental rethink of archaeological approaches to medieval religions, by adopting a holistic framework that places Christian, pagan, Islamic and Jewish case studies of religious transformation in a long-term, comparative perspective. Focused around the analytical themes of ‘hybridity and resilience’ and ‘tempo and trajectories’, our approach shifts attention away from the singularities of national narratives of religious conversion towards a deeper understanding of how religious beliefs, practices and identity were renegotiated by medieval people in their daily lives