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

TL;DR

The paper develops a comprehensive framework connecting generalized tube algebras, symmetry actions, and junction operators in 1+1d QFTs through a 2+1d symmetry TFT (SymTFT). It introduces generalized half-linking numbers and a Verlinde-like character theory to compute symmetry-resolved torus and annulus partition functions, including twisted sectors and orbifolds, without requiring rationality. Representations of boundary tube algebras are linked to simple topological line junctions in the bulk, yielding a generalized Schur–Weyl duality and open-closed duality constraints that extend traditional RCFT methods to non-invertible categorical symmetries. The framework is illustrated in detail (e.g., Fibonacci symmetry) and is positioned to impact selection rules, degeneracy bounds, interface fusion, and RG flows in theories with fusion-category symmetries, providing tools beyond conventional group-theoretic approaches.

Abstract

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.

Figures (26)

• Figure 1: A twisted sector operator $\mathcal{O}\in\mathcal{H}_a$ of a 1+1d theory $Q$ decomposes into a triple $(y,\mu,\widetilde{\mathcal{O}})$, where $\mu\in Z(\mathcal{C})$ is a bulk anyon, and $\widetilde{\mathcal{O}}$ and $y$ are suitable junction operators.
• Figure 2: The SymTFT interpretation of a boundary $B$ of a QFT $Q$ with symmetry category ${\mathcal{C}}$. The boundary condition inflates into a triple $(\underline{B}, \mathcal{B}_{\mathrm{bdy}}, \widetilde{B})$.
• Figure 3: The irreducible representations of $\mathrm{Tube}(\mathcal{I}_1\vert\cdots\vert\mathcal{I}_n)$ are labeled by topological line junctions $\gamma$ between the two-dimensional topological interfaces $\mathcal{I}_1,\cdots,\mathcal{I}_n$. The generalized lassos appear on the topological boundary conditions of the SymTFT. Illustrated in the case of $n=3$. See Figure \ref{['fig:junctionsymTFT']} for a more detailed version of this figure.
• Figure 4: The symmetry basis annulus partition functions of a 1+1d QFT $Q$ with a $\mathbb{Z}_2$ symmetry and a symmetric boundary condition $B$. The length of the circle direction is $\delta$.
• Figure 5: A twisted boundary state.