[Paper Review] Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond

Ryan Thorngren, Yifan Wang|arXiv (Cornell University)|Jun 23, 2021

Theoretical and Computational PhysicsPhysics and Astronomy72 references72 citations

TL;DR

The paper develops and applies the framework of fusion category symmetries generated by non-invertible topological defect lines in 1+1D, focusing on c=1 CFTs to reveal a rich spectrum of fusion categories and their constraints on RG flows.

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.

Motivation & Objective

• Introduce and survey fusion category symmetries in 1+1D via topological defect lines (TDLs).
• Identify and classify fusion category symmetries in c=1 CFTs using RCFT data, modular bootstrap, and gauging methods.
• Show that even finite-group-like c=1 CFTs can host continuous fusion category symmetries and establish a Noether-type relation to non-local currents.
• Derive constraints on renormalization group flows between 1+1D CFTs arising from fusion category symmetries.

Proposed method

• Define TDLs and fusion category data (fusion rules and F-symbols) and their action on defect Hilbert spaces.
• Apply generalized modular bootstrap using torus partition functions with twisted defects (Z_{L1 L2}^{L3}) and modular transformations (S, F-moves).
• Use Verlinde lines in RCFTs to relate simple TDLs to chiral algebra primaries and derive fusion via Verlinde formula.
• Employ gauging of finite groups (orbifolds) and Tambara-Yamagami categories to construct non-invertible symmetries.
• Analyze self-duality under gauging in Z2 orbifold and identify corresponding TY categories and Rep(D8)/Ising-2 structures.
• Leverage parafermions, Ising, and four-state Potts models as RCFT anchors for explicit TDL realizations.

Experimental results

Research questions

• RQ1 What fusion category symmetries can arise in c=1 CFTs, including irrational points on the moduli space?
• RQ2 How can generalized modular bootstrap and RCFT techniques uncover non-invertible topological defect lines in 1+1D theories?
• RQ3 When do c=1 CFTs exhibit continuous fusion category symmetries, and how are these related to non-local conserved currents (Noether theorem)?
• RQ4 How do gauging and duality defects organize the fusion category structure in circle and orbifold branches of the c=1 moduli space?
• RQ5 What constraints do fusion category symmetries impose on RG flows between 1+1D CFTs?

Key findings

• c=1 CFTs can host continuous fusion category symmetries even when only finite group-like symmetries are visible.
• A Noether-type theorem links continuous fusion category symmetry to non-local conserved currents in generality.
• Circle branch at R=√(2k) exhibits self-duality under Z_k gauging with duality TDLs; at R∈√2ℚ the symmetry enhances to a continuum parameterized by six (or four) parameters.
• Z2 orbifold branch shows self-dualities under gauging Z4 and Z2×Z2 subgroups, with TY and Rep(D8)/Rep(H8) structures and connections to Ising-2 and four-state Potts theories.
• Exceptional SU(2)1 orbifolds (A4, S4, A5) carry six-parameter continua of TDLs, linking to Verlinde lines and parent SU(2)1 data.
• Overall, fusion category symmetries provide new constraints and structures for RG flows and dualities across the c=1 moduli space.