Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond

TL;DR

This work develops a framework to identify and analyze fusion-category symmetries in quantum field theories, focusing on 1+1D and applying it to c=1 conformal field theories. By combining generalized modular bootstrap, RCFT techniques, and gauging constructions, the authors uncover a rich zoo of non-invertible topological defect lines, including continuous families, duality defects, and Tambara–Yamagami categories, across the circle and orbifold branches of the c=1 moduli space. They prove a Noether-type theorem linking continuous fusion-category symmetries to defect currents and derive RG-flow constraints that prohibit trivial gapped IR phases when such anomalous symmetries are preserved. The analysis spans compact free bosons, Z_2 orbifolds, and exceptional SU(2)_1 orbifolds, yielding explicit TY data (ω, ε) and duality actions on local operators, alongside detailed partition-function checks at rational points and in special limits like KT and Dirac points. The results suggest a broad universality of fusion-category symmetries and hint at rich structures in higher-dimensional QFTs, motivating further mathematical development of enriched, parameterized defect categories.

Abstract

We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to $c = 1$ CFTs and uncover a rich structure. We find that even those $c = 1$ CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.

Figures (13)

• Figure 1: The F-move, that relates correlation functions containing two topologically distinct networks of TDLs. Here ${\mathcal{K}}^{\mathcal{L}_1 \mathcal{L}_4}_{\mathcal{L}_2 \mathcal{L}_3}(\mathcal{L}_5,\mathcal{L}_6)$ are the F-symbols (also known as crossing kernels or $6j$ symbols).
• Figure 2: The moduli space of $c=1$ CFTs and fusion category symmetries.
• Figure 3: The three-point function $\langle O_a O_b O_c\rangle$ connected by TDLs $\mathcal{L}_{a,b,c}$ and equivalently the CFT partition function on a pair of pants decorated by the TDLs. Here $v$ is a index denoting the fusion channel (topological junction), which we henceforth suppress from the notation and diagrams. The arrows denote orientations of the TDLs. Strictly speaking, the correlation functions depend on a co-orientation, i.e. a choice of normal direction for each TDL, but since we will only work on orientable surfaces, we can choose an orientation to make them equivalent.
• Figure 4: The lasso diagram that depicts the two-point function of defect operators $O_b$ and $O_c$ joined by a web $X$ of TDLs $\mathcal{L}_{a,b,c,d}$ and equivalently the partition function of the CFT on a cylinder decorated by the same TDL web $X$. They compute the matrix element of the TDL acting on the defect Hilbert spaces $(\widehat{\mathcal{L}}_a)_X:\mathcal{H}_b \to \mathcal{H}_c$.
• Figure 5: The first diagram on the left defines the twisted partition functions $Z^{\mathcal{L}_3}_{\mathcal{L}_1\mathcal{L}_2}$. The second diagram is obtained from a modular $S$-transform. After a F-move, it is related to a combination of $Z^{\mathcal{L}_k}_{\mathcal{L}_2\bar{\mathcal{L}}_1}$ for each simple $\mathcal{L}_k$ that appear in the fusion product $\mathcal{L}_2\bar{\mathcal{L}}_1$.