On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity

TL;DR

This work addresses the ambiguity of defining entanglement entropy in gauge theories and gravity by developing a relational, excitation-based framework that uses gluing/splitting, flatness constraints, and a fusion-basis description built from the Drinfel'd double. It shows how extended Hilbert space techniques can be generalized to recover magnetic- and electric-center boundary data and derives an explicit three-term entropy S_A = H(P(ρ_∂)) + ⟨ln dim ρ_∂⟩ + ⟨S_A(D^A(ρ_∂))⟩, linking to algebraic approaches and applying the construction to BF theory with defects and 3D gravity. The fusion basis, ribbon operators, and puncture-based excitation data enable a region specification that is independent of the underlying lattice and compatible with background independence, yielding regulator-independent entanglement in topological settings. The results illuminate how vacua (BF vs Ashtekar–Lewandowski) and boundary conditions govern distillable versus classical entanglement, and they lay groundwork for extending these ideas to quantum gravity and related topological field theories. Overall, the paper provides a robust, excitation-centered, background-independent method to define and compute entanglement entropy in gauge theories and gravity, with clear connections to continuum TQFT and 3D gravity physics.

Abstract

Entanglement entropy is a valuable tool for characterizing the correlation structure of quantum field theories. When applied to gauge theories, subtleties arise which prevent the factorization of the Hilbert space underlying the notion of entanglement entropy. Borrowing techniques from extended topological field theories, we introduce a new definition of entanglement entropy for both Abelian and non-Abelian gauge theories. Being based on the notion of excitations, it provides a completely relational way of defining regions. Therefore, it naturally applies to background independent theories, e.g. gravity, by circumventing the difficulty of specifying the position of the entangling surface. We relate our construction to earlier proposals and argue that it brings these closer to each other. In particular, it yields the non-Abelian analogue of the "magnetic centre choice", as obtained through an extended-Hilbert-space method, but applied to the recently introduced fusion basis for 3D lattice gauge theories. We point out that the different definitions of entanglement theory can be related to a choice of (squeezed) vacuum state.

Figures (8)

• Figure 1: The left panel represents a lattice of three plaquettes embedded on a two-sphere by closing it with an outer plaquette. For every plaquette, including the outer one, an open edge going from the lattice to a marked point can carry torsion degrees of freedom. The middle panel represents an equivalent description where the plaquettes are replaced by punctures. The solid lines now represent a minimal graph embedded on the corresponding punctured sphere. The right panel finally proposes another graphical representation where the topology of the punctured sphere is deformed to obtain pairs of pants. This final representation is the preferential one for the construction of the fusion basis.
• Figure 2: Example of lattice with six plaquettes embedded on the two-sphere ${\mathbb S}_2$ by introducing an outer plaquette. The wiggly lines represent the connected spanning tree $\mathcal{T}'$. For each plaquette we associate a $G$-holonomy defined as the product of holonomies going from the corresponding end node to the root following $\mathcal{T}'$ and a $H$-holonomy defined by going anti-clockwise around the face starting and ending at the end-point. For the upper-left face we have for instance $G_7 = h_7h_{11}h_{12}h_6h_0$ and $H_7 = h_7h_4^{-1}h_1h_{10}h_7^{-1}$.
• Figure 3: Example of action of a closed ribbon operator. The ribbon operator, represented by the doubled represented line, acts on the links which it crosses. The dashed region accounts for the presence of any punctures and any graph.
• Figure 4: Action of an open ribbon operator from nodes $n_1$ to $n_4$. The ribbon acts as a Wilson path operator on the holonomy parallel to the ribbon i.e.$g_3g_2g_1$ and as a translation operator on the holonomies crossed by the ribbon i.e.$h_2$ and $h_1$.
• Figure 5: For a given fusion tree we identify the region $A$ as a set of punctures and $B$ its complement. The splitting is performed by cutting a cylinder of the fusion tree. This cut requires the introduction of additional punctures. Note that the fusion tree employed here is the same as the one appearing in the alternative fusion states defined in Appendix \ref{['app_alt']}.