Remarks on Entanglement for Fuzzy Geometry and Gravity

TL;DR

This paper proposes a new approach to defining and calculating entanglement entropy in fuzzy spaces using fermionic field theory, linking it to Chern-Simons forms and offering insights relevant to quantum gravity.

Contribution

It introduces a novel method for defining entanglement in fuzzy geometries via fermionic states and integrates this with a Chern-Simons framework, enhancing understanding of quantum geometry.

Findings

Entanglement entropy can be expressed through direct field integration in fuzzy spaces.

The derived EE formula parallels quantum Hall effect results.

Arguments support EE being described by a generalized Chern-Simons form.

Abstract

We consider defining a fuzzy space by a specific state in a fermionic field theory in terms of which all the observables for the space can be evaluated. This allows for a definition of entanglement for a fuzzy space by direct integration of the fields over a certain region. Even though the resulting formula for the entanglement entropy (EE) is similar to what has been used in the quantum Hall effect, our derivation provides a novel perspective. We also review and strengthen the arguments for the EE to be described by a generalized Chern-Simons form.