Noninvertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra

TL;DR

The paper develops a refined, noninvertible symmetry-resolved Affleck-Ludwig-Cardy (ALC) framework for 1+1d CFTs by leveraging boundary tube algebras and the symmetry topological field theory (SymTFT). It derives a universal high-temperature behavior for symmetry-resolved annulus partition functions and the leading/subleading contributions to symmetry-resolved entanglement entropy, with a closed-form projector expressed through generalized half-linking numbers. The critical double Ising model is analyzed to reveal an $H_8$ Hopf algebra symmetry of the ground-state entanglement Hamiltonian on an interval, illustrating the formalism and connecting representation data to explicit entropy corrections. The work also clarifies how boundary conditions at entangling surfaces influence SREE and reconciles results with prior literature, highlighting the role of SymTFT in organizing noninvertible symmetries. Overall, the results provide concrete universal predictions for SREE in theories with fusion-category symmetries and demonstrate the rich interplay between boundary data, topological lines, and entanglement in 1+1d CFTs.

Abstract

We derive a refined version of the Affleck-Ludwig-Cardy formula for a 1+1d conformal field theory, which controls the asymptotic density of high energy states on an interval transforming under a given representation of a noninvertible global symmetry. We use this to determine the universal leading and sub-leading contributions to the noninvertible symmetry-resolved entanglement entropy of a single interval. As a concrete example, we show that the ground state entanglement Hamiltonian for a single interval in the critical double Ising model enjoys a Kac-Paljutkin $H_8$ Hopf algebra symmetry when the boundary conditions at the entanglement cuts are chosen to preserve the product of two Kramers-Wannier symmetries, and we present the corresponding symmetry-resolved entanglement entropies. Our analysis utilizes recent developments in symmetry topological field theories (SymTFTs).

Figures (3)

• Figure 1: The SymTFT picture of boundary conditions and boundary-changing local operators.
• Figure 2: Boundary lasso opeartors in the double Ising CFT. Here, $g \in \mathbb{Z}_2 \times \mathbb{Z}_2$ and $s,s' = 0,1$ label topological point junctions. All the topological lines are self-dual and we do not draw arrows on them.
• Figure 3: Boundary $\widetilde{F}$-symbols for the fusion category $\mathcal{C}_{\mathsf{TY}} \cong \mathrm{Rep}(H_8)$ acting on the (strongly) symmetric boundary condition $B$. Here, $\omega$ is a $\mathbb{Z}_2 \times \mathbb{Z}_2$ group 2-cocycle given by $\omega(a^{m_1} b^{n_1}, a^{m_2} b^{n_2}) = (-1)^{n_1 m_2}$, and $\sigma_i$'s are the Pauli matrices.