Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants

TL;DR

The paper develops a framework to study multi-boundary entanglement in Chern-Simons theory by constructing link-state wavefunctions from the path integral on link complements in $S^3$ and analyzing their entanglement across partitions of boundary tori. It provides a complete solution for the Abelian case where entanglement entropies are framed as gcd-based invariants of Gauss linking modulo $k$, and advances to the non-Abelian case $SU(2)_k$ where entanglement patterns reveal richer topological structures such as GHZ-like and W-like states in representative two- and three-component links. Key results include $S_{EE;m|n-m}=\ln(k^m/|\ker \boldsymbol{G}|)$ for Abelian links, a maximally entangled Hopf link in the non-Abelian setting, and explicit analyses showing how knotting and Brunnian properties influence entanglement and negativity. The work offers a topological, framing-independent entanglement perspective on link invariants and opens avenues for exploring connections to hyperbolic geometry, quantum gravity, and the entanglement cone in multi-boundary states.

Abstract

We consider Chern-Simons theory for gauge group $G$ at level $k$ on 3-manifolds $M_n$ with boundary consisting of $n$ topologically linked tori. The Euclidean path integral on $M_n$ defines a quantum state on the boundary, in the $n$-fold tensor product of the torus Hilbert space. We focus on the case where $M_n$ is the link-complement of some $n$-component link inside the three-sphere $S^3$. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level $k$ ($G= U(1)_k$) we give a general formula for the entanglement entropy associated to an arbitrary $(m|n-m)$ partition of a generic $n$-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod $k$) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod $k$). For $G = SU(2)_k$, we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a "W-like" entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have "GHZ-like" entanglement (i.e., tracing out one torus does lead to a separable state).

Figures (19)

• Figure 1: The spatial manifold $\Sigma_n$ for $n=3$ is the disjoint union of three tori. $M_n$ is a 3-manifold such that $\partial M_n = \Sigma_n$.
• Figure 2: The link complement (the shaded region) of a 3-component link (bold lines) inside the three-sphere. The white region indicates a tubular neighbourhood of the link which has been drilled out of the 3-sphere.
• Figure 3: (a) The meridian and longitude cycles on a torus $T^2$. (b) The state $|j\rangle$ corresponds to a Wilson line in the representation $j$ placed in the bulk of the solid torus.
• Figure 4: Three unlinked knots.
• Figure 5: The Hopf-link.