Generalized Tube Algebras, Symmetry-Resolved Partition Functions, and Twisted Boundary States

We introduce a class of generalized tube algebras which describe how finite, non-invertible global symmetries of bosonic 1+1d QFTs act on operators which sit at the intersection point of a collection of boundaries and interfaces. We develop a 2+1d symmetry topological field theory (SymTFT) picture of boundaries and interfaces which, among other things, allows us to deduce the representation theory of these algebras. In particular, we initiate the study of a character theory, echoing that of finite groups, and demonstrate how many representation-theoretic quantities can be expressed as partition functions of the SymTFT on various backgrounds, which in turn can be evaluated explicitly in terms of generalized half-linking numbers. We use this technology to explain how the torus and annulus partition functions of a 1+1d QFT can be refined with information about its symmetries. We are led to a vast generalization of Ishibashi states in CFT: to any multiplet of conformal boundary conditions which transform into each other under the action of a symmetry, we associate a collection of generalized Ishibashi states, in terms of which the twisted sector boundary states of the theory and all of its orbifolds can be obtained as linear combinations. We derive a generalized Verlinde formula involving the characters of the boundary tube algebra which ensures that our formulas for the twisted sector boundary states respect open-closed duality. Our approach does not rely on rationality or the existence of an extended chiral algebra; however, in the special case of a diagonal RCFT with chiral algebra $V$ and modular tensor category $\mathscr{C}$, our formalism produces explicit closed-form expressions - in terms of the $F$-symbols and $R$-matrices of $\mathscr{C}$, and the characters of $V$ - for the twisted Cardy states, and the torus and annulus partition functions decorated by Verlinde lines.

💡 Research Summary

The paper develops a comprehensive framework for describing how finite, possibly non‑invertible global symmetries act on operators located at the junction of multiple boundaries and interfaces in 1+1‑dimensional quantum field theories. The authors begin by recalling the ordinary tube algebra Tube(C), which encodes the action of a fusion category C of topological line operators on twisted‑sector local operators via “lasso” configurations. They then generalize this construction to the situation where the operators sit at the intersection of several boundaries or interfaces. To this end they introduce a (C, C)‑bimodule category I that captures the left and right fusion of symmetry lines onto a collection of (generally non‑topological) line defects. The data of I—fusion coefficients, F‑symbols, and a middle associator—allow one to define generalized lassos that convert a junction operator attached to one line into another. The algebra generated by these generalized lassos is the generalized tube algebra TUBE(I). Crucially, TUBE(I) depends only on the bimodule structure of I, not on the detailed microscopic realization of the lines.

The second major ingredient is the symmetry topological field theory (SymTFT) TV_C, a 2+1‑dimensional topological theory whose bulk encodes the symmetry category C. One places a Dirichlet (symmetry‑preserving) boundary B_reg on one side of an interval and the physical boundary e_Q on the other, thereby “inflating’’ the original 1+1‑d QFT Q into a slab of TV_C. In this picture, symmetry lines are confined to B_reg, while operators of Q become bulk anyons (objects of the Drinfeld center Z(C)) terminating on the physical boundary. Junction operators at boundary–interface intersections are labeled by generalized half‑linking numbers, denoted Ω‑symbols, which are precisely the structure constants of the tube algebra.

Using the bulk–boundary correspondence, the authors show that the extended Hilbert space H_I = ⊕_{i∈Irr(I)} H_i carries a left action of TUBE(I) and a right action of the bulk anyons. This yields a Schur–Weyl type duality: the decomposition of H_I into irreducible TUBE(I) modules is governed by quantum characters, i.e. traces of TUBE(I) elements weighted by the half‑linking numbers. These characters obey a generalized Verlinde formula, \