Connectomes as holographic states

Abstract

We use the topological quantum field theory description of states in Chern-Simons theory to discuss the relation between spacetime connectivity and entanglement, exploring the paradigm entanglement=topology. We define a special class of states in Chern-Simons with properties similar to those of holographic states. While the holographic states are dual to classical geometries, these connectome states represent classical topologies, which satisfy a discrete analog of the Ryu-Takayanagi formula and characteristic inequalities for the entanglement entropy. Generic states are linear combinations of connectomes, and the theory also has nonperturbative states which are global spacetime defects formed by a large number of quantum fluctuations. Topological presentation of quantum states and emergence of topology from entanglement may be useful for building a generalization to geometry, that is quantum gravity. Thinking of further quantum gravity comparisons we discuss replica wormholes and conclude that similar objects exist beyond gravitational theories. The topological theory perspective suggests that the sum over all wormholes is always factorizable, even though the individual ones might not be.