Bulk quantum corrections to entwinement

TL;DR

This work computes $1/N$ corrections to entwinement, a non-spatial entanglement in holographic CFTs dual to winding geodesics in AdS$_3$, by recasting entwinement as ordinary entanglement on a fictitious covering space and applying the FLM/QES framework. For pure-state conical defects, the leading $1/N$ correction vanishes for a single winding geodesic, with bulk entanglement effectively renormalizing the Newton constant via the covering parameter $m$, i.e. $G_N\to mG_N$; for thermal states, the covering theory has central charge $c/m$ and the background geometry is thermal AdS$_3$ or BTZ with $G_N\to mG_N$ and $\beta\to \beta/m$. Using the monodromy method, the authors derive explicit $1/N$ corrections to entwinement for small windings, revealing metric and scalar contributions with detailed dependence on the interval size and temperature, e.g. $S_{w,m}(L)=\frac{c}{3m}\log\left[\frac{\beta}{2\pi^2\epsilon_{UV}}\sinh\left(\frac{2\pi^2(L+w)}{\beta}\right)\right]+\tilde{q}^{2m}\left[8-\frac{16\pi^2(L+w)}{\beta}\coth\left(\frac{2\pi^2(L+w)}{\beta}\right)\right]+\cdots$. The work also analyzes universal finite-size corrections for large winding numbers, showing that leading terms remain geodesic-like with $\beta\to\beta/m$ and that subleading corrections depend on the seed theory gap and arithmetical properties of $m$. Overall, the results support the view that entanglement encodes bulk geometry beyond leading order and clarify when entwinement tracks geodesic lengths, even when semiclassical gravity breaks down.

Abstract

We determine $1/N$ corrections to a notion of generalized entanglement entropy known as entwinement dual to the length of a winding geodesic in asymptotically AdS$_3$ geometries. We explain how $1/N$ corrections can be computed formally via the FLM formula by relating entwinement to an ordinary entanglement entropy in a fictitious covering space. Moreover, we explicitly compute $1/N$ corrections to entwinement for thermal states and small winding numbers using a monodromy method to determine the corrections to the dominant conformal block for the replica partition function. We also determine a set of universal corrections at finite temperature for large winding numbers. Finally, we discuss the implications of our results for the "entanglement builds geometry" proposal.

Figures (5)

• Figure 1: The conical defect geometry (left) is obtained by partitioning a pure AdS$_3$ covering space (right) into $m$ wedges (shown here is the case $m=3$) which are identified with each other. Ryu-Takayanagi surfaces, that is geodesics with winding number zero, do not penetrate the entanglement shadow indicated in gray. However, the remaining geodesics with winding number $w>0$ shown in blue and purple probe the entire constant time slice of the conical defect.
• Figure 2: Degrees of freedom on the covering space of the conical defect for ordinary entanglement and for entwinement (shown here is the case $m=3$). The dashed lines correspond to the subsystem whose entanglement is computed.
• Figure 3: Degrees of freedom for entwinement at finite temperature. Top: for ordinary entanglement, long strands of all sizes contribute. We compute the entanglement w.r.t. a subsystem of $k$ intervals on a strand of length $k$. Bottom: for entwinement, only long strands of length $km$ contribute (here we depict $m=3$). We compute the entanglement w.r.t. a subsystem of $k$ intervals on a strand of length $km$.
• Figure 4: Winding geodesics in the BTZ black hole background. LHS: geodesics with winding number $w=0$ (in red) and $w=1$ (in blue). RHS: covering space interpretation: the length of geodesics with zero winding number in the covering space BTZ geometry with inverse temperature $\beta/m$ is the same as that of winding geodesics in the original BTZ geometry with inverse temperature $\beta$. Here, we depict $m=3$.
• Figure 5: From the CFT perspective, the covering space at finite temperature (right) is a torus whose space direction is $m$ times larger than the original torus (left). Equivalently, the modular parameter is $\tau/m$. On the right, the case $m=3$ is illustrated where the space direction can be split up into $3$ parts, each fully covering the torus on the left. The subsystem on the covering space relevant for entwinement is indicated by a dashed line. In Euclidean signature, the bulk covering space is a solid torus that fills in the boundary covering space (thermal AdS$_3$ or the BTZ black hole with inverse temperature $\beta/m$).