Topological Defect Lines and Renormalization Group Flows in Two Dimensions

TL;DR

The paper develops a comprehensive framework of topological defect lines (TDLs) in 2d CFTs, unifying global-symmetry invertible lines and Verlinde lines within fusion-category language without requiring braiding. It derives crossing kernels, pentagon identities, and spin-selection rules, and uses Ocneanu rigidity to constrain RG flows, often determining IR TQFTs from UV data. Through detailed analyses of diagonal and non-diagonal RCFTs, Ising-family models, Potts-type theories, and WZW/coset constructions, it demonstrates powerful constraints on whether flows can end in unique gapped vacua or must exhibit vacuum degeneracy, and it elucidates the role of anomalies and orbifolds in shaping IR physics. The results provide a rigorous, nonperturbative diagnostic toolkit for predicting IR behavior from UV TD L data, with explicit applications to a broad class of 2d CFTs and their massive deformations.

Abstract

We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the 't Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.

Figures (63)

• Figure 1: Fusion of a pair of TDLs ${\cal L}_1$ and ${\cal L}_2$ wrapping the spatial loop on the cylinder.
• Figure 2: An admissible configuration of TDLs with endpoints (purple dots), joined by T-junctions (black dots).
• Figure 3: A correlation functional (on the plane), where the TDLs are joined by a T-junction (black dot) with the order of lines specified (last leg marked by the "$\times$"), and ending on defect operators (purple dots). It is a linear function on the junction vector space $V_{{\cal L}_1,{\cal L}_2,{\cal L}_3}$.
• Figure 4: Conjugation map in a correlation function. Here, $O_{i}$ with $i=1,2,3$ denote defect operators in ${\cal H}_{{\cal L}_{i}}$, and $\overline O_{i}\equiv \iota(O_{i})$ denote their conjugates, which are defect operators in ${\cal H}_{\overline{\cal L}_{i}}$. Similarly, $v$ is a junction vector in $V_{{\cal L}_1,{\cal L}_2,{\cal L}_3}$, and $\overline v \equiv \iota(v) \in V_{\overline{\cal L}_3,\overline{\cal L}_2,\overline{\cal L}_1}$.
• Figure 5: Left: Cutting a TDL graph along the gray circle and inserting a complete basis of states in ${\cal H}_{{\cal L}_1, {\cal L}_2}$ is equivalent to replacing the defect operators $\Psi_1$ and $\Psi_2$ by their OPE, which is a defect operator in ${\cal H}_{{\cal L}_1, {\cal L}_2}$. Right: A similar cut-and-insert procedure, where the defect operators inside the circle are all junction vectors. The graph inside the smaller gray circle implements the map $V_{{\cal L}_1, {\cal L}_2, \overline{\cal L}_5} \otimes V_{{\cal L}_3, {\cal L}_4, {\cal L}_5} \to V_{{\cal L}_1, {\cal L}_2, {\cal L}_3, {\cal L}_4}$.