Entanglement versus entwinement in symmetric product orbifolds

TL;DR

This paper investigates the entanglement entropy of internal degrees of freedom in symmetric product orbifold CFTs, comparing entanglement and entwinement, and computing these quantities holographically for specific geometries.

Contribution

It introduces an algebraic perspective on reduced density matrices in orbifold CFTs and computes entanglement and entwinement in holographic duals, highlighting differences and challenges.

Findings

Single strand entanglement matches entwinement in holographic states.

Two-strand entropy differs from entwinement, indicating complexity in gauge invariance.

Algebraic analysis clarifies the nature of reduced states in orbifold theories.

Abstract

We study the entanglement entropy of gauged internal degrees of freedom in a two dimensional symmetric product orbifold CFT, whose configurations consist of $N$ strands sewn together into "long" strings, with wavefunctions symmetrized under permutations. In earlier work a related notion of "entwinement" was introduced. Here we treat this system analogously to a system of $N$ identical particles. From an algebraic point of view, we point out that the reduced density matrix on $k$ out of $N$ particles is not associated with a subalgebra of operators, but rather with a linear subspace, which we explain is sufficient. In the orbifold CFT, we compute the entropy of a single strand in states holographically dual in the D1/D5 system to a conical defect geometry or a massless BTZ black hole and find a result identical to entwinement. We also calculate the entropy of two strands in the state that represents the conical defect; the result differs from entwinement. In this case, matching entwinement would require finding a gauge-invariant way to impose continuity across strands.