Holonomy Separation and Anyonic Hair in AdS${}_3$

Abstract

Three-dimensional Einstein gravity and Chern-Simons-Higgs matter both admit Chern-Simons descriptions, but their holonomies encode different physics. We prove that charge conjugation reverses the vortex winding $n\to -n$ while leaving the metric and the gravitational connections pointwise unchanged, so every closed gravitational holonomy character, the BTZ charges, the Bañados data and the leading geometric mixed-state hierarchy are exactly even in $n$. On a fixed marked Euclidean replica filling the same change of variables gives $Z_q[n;{\mathcal{J}}_{+}]=Z_q[-n;{\mathcal{J}}_{+}]$, so the uncharged Rényi hierarchy, the Faulkner-Lewkowycz-Maldacena one-loop correction, the renormalized generalized entropy and the quantum extremal surfaces coincide for the two orientations. The complementary sector is fixed exactly: with an Aharonov-Bohm angle $\mu$ conjugate to the entangling-region charge, $Z_q[n;\mu ]=Z_q[-n;-\mu ]$, so the symmetry-resolved spectrum obeys ${\mathcal{Z}}_q(\mathcal{Q};n)={\mathcal{Z}}_q(-\mathcal{Q};-n)$ and every odd charged cumulant reverses with the winding. The orientation is therefore stored entirely in the charge asymmetry of the entanglement spectrum, and is read at topological order by a linked charged matter line. The pair is a bulk logical qubit on which the neutral gravitational-replica algebra acts as a multiple of the identity. Exact BTZ cross sections show that the purification plateau and the entanglement wedge disconnection transition are orientation blind, the latter closing with the cutoff-independent jump $(c/3)\log (1+\sqrt{2})$. Numerical Bogomolny solutions verify odd flux and even energy and spin; used as linearized charge-matching data they give the even asymptotic shifts $\delta \hat{M}=16\pi G{v}^2|n|$ and $|\delta \hat{j}|=16\pi G{v}^2{n}^2/(mL)$ and, conditionally, a finite continuous winding envelope.

arXiv comment44 pages, 4 figures