Relative Anomaly in (1+1)d Rational Conformal Field Theory

TL;DR

This work addresses how 't Hooft anomalies arise in symmetry-enriched (1+1)d RCFTs and develops a bulk–boundary framework by mapping to symmetric gapped boundaries of (2+1)d doubled SET phases. By formulating a precise correspondence with a G-crossed braided tensor category description and using an exactly solvable lattice model, the authors derive an explicit formula for the relative 't Hooft anomaly in terms of algebraic data (F,U,eta,v) describing the bulk and boundary, enabling computation from RCFT symmetry actions. The paper provides extensively worked examples (e.g. SU(2)_k, minimal models, WZW theories) and analyzes LSM-type anomalies, illustrating constraints on when a given RCFT can be realized with onsite symmetry or as an edge of a 2+1d SPT. These results offer a practical toolkit to diagnose and compare symmetry-enriched RCFTs and their bulk realizations, with potential implications for anomaly matching, boundary gapping, and the classification of (2+1)d SET/SPT phases.

Abstract

We study 't Hooft anomalies of symmetry-enriched rational conformal field theories (RCFT) in (1+1)d. Such anomalies determine whether a theory can be realized in a truly (1+1)d system with on-site symmetry, or on the edge of a (2+1)d symmetry-protected topological phase. RCFTs with the identical symmetry actions on their chiral algebras may have different 't Hooft anomalies due to additional symmetry charges on local primary operators. To compute the relative anomaly, we establish a precise correspondence between (1+1)d non-chiral RCFTs and (2+1)d doubled symmetry-enriched topological (SET) phases with a choice of symmetric gapped boundary. Based on these results we derive a general formula for the relative 't Hooft anomaly in terms of algebraic data that characterizes the SET phase and its boundary.

Figures (4)

• Figure 1: A symmetric gapped boundary between $\mathcal{B}\boxtimes \overline{\mathcal{B}}$ and SPT$_{[\omega]}$, equivalent to the trivialization of SPT$_{[\omega]}$ by $\mathcal{B}$.
• Figure 2: An illustration of the strip construction. The (blue) dot on the lower edge denotes local operators on the edge, which form the chiral algebra $\mathcal{V}_L$. The thick line across the strip represents a primary operator, which can be thought of as an anyon tunneling between the two edges. The dashed line denotes a gapped domain wall inside the bulk of the strip.
• Figure 3: Top panel: illustration of a three-way junction of TDLs which define the space $V^{\mathbf{g,h}}_\mathbf{gh}$. Bottom: definition of the 3-cocycle $\omega(\mathbf{g,h,k})$.
• Figure 4: The $m$ move in a left $\mathcal{C}$-module $\mathcal{M}$. The dashed lines represent boundary edges.