Topological subregions in Chern Simons theory and topological string theory

TL;DR

This paper introduces a topological approach to defining subregions in Chern-Simons theory, using quantum groups and diagrammatic calculus to obtain finite entanglement entropy, with implications for topological string theory.

Contribution

It develops a purely topological framework for subregion entanglement in Chern-Simons theory using quantum group functions and $q$-deformation, avoiding UV divergences.

Findings

Defined a topological notion of subregion using quantum group functions.

Developed a diagrammatic calculus for $q$-deformed entanglement entropy.

Produced finite entanglement entropy via $q$-deformation, linking to topological string theory.

Abstract

The standard, gapped entanglement boundary condition in Chern Simons theory breaks the topological invariance of the theory by introducing a complex structure on the entangling surface. This produces an infinite dimensional subregion Hilbert space, a non-trivial modular Hamiltonion, and a UV-divergent entanglement entropy that is a universal feature of local quantum field theories. In this work, we appeal to the combinatorial quantization of Chern Simons theory to define a purely topological notion of a subregion. The subregion operator algebras are spaces of functions on a quantum group. We develop a diagrammatic calculus for the associated $q$-deformed entanglement entropy, which arise from the entanglement of anyonic edge modes. The $q$-deformation regulates the divergences of the QFT, producing a finite entanglement entropy associated to a $q$-tracial state. We explain how these ideas provide an operator algebraic framework for the entanglement entropy computations in topological string theory \cite{Donnelly:2020teo,Jiang:2020cqo, wongtopstring}, where a large- $N$ limit of the $q$-deformed subregion algebra plays a key role in the stringy description of spacetime.