$\mathbb{Z}_N$ Symmetries, Anomalies, and the Modular Bootstrap

TL;DR

The paper develops a modular bootstrap approach to constrain (1+1)d bosonic CFTs with a discrete $\ ext{Z}_N$ global symmetry, incorporating 't Hooft anomalies via topological defect lines. It derives a universal upper bound on the lightest symmetry-preserving scalar operator, with anomaly-dependent refinements, and shows that for small N (N ≤ 4) there must exist a $\ ext{Z}_N$-invariant relevant or marginal operator in certain central-charge windows, with the endpoints saturated by pairings of WZW models embeddable into $(\mathfrak{e}_8)_1$. The results connect defect-line data, anomaly constraints, and modular invariance to map out robust, symmetry-preserving fixed-point structures and illustrate saturations by explicit CFTs like WZW models. Overall, the work narrows the landscape of possible robust gapless IR fixed points under a microscopic $\ ext{Z}_N$ symmetry and provides a quantitative framework for anomaly-informed spectral bounds in 2D CFTs.

Abstract

We explore constraints on (1+1)$d$ unitary conformal field theory with an internal $\mathbb{Z}_N$ global symmetry, by bounding the lightest symmetry-preserving scalar primary operator using the modular bootstrap. Among the other constraints we have found, we prove the existence of a $\mathbb{Z}_N$-symmetric relevant/marginal operator if $N-1 \le c\le 9-N$ for $N\leq4$, with the endpoints saturated by various WZW models that can be embedded into $(\mathfrak{e}_8)_1$. Its existence implies that robust gapless fixed points are not possible in this range of $c$ if only a $\mathbb{Z}_N$ symmetry is imposed microscopically. We also obtain stronger, more refined bounds that depend on the 't Hooft anomaly of the $\mathbb{Z}_N$ symmetry.

Figures (10)

• Figure 1: A topological defect line $\cal L$ wrapped around the spatial circle of a cylinder leads to an action $\widehat{{\mathcal{L}}}$ on the Hilbert space $\cal H$.
• Figure 2: As a $\mathbb{Z}_N$ TDL is swept past a local operator $\phi(x)$, the correlation function changes by a phase $e^{2\pi i Q \over N}$ where $Q=0,1,\cdots, N-1$ is the $\mathbb{Z}_N$ charge of $\phi$.
• Figure 3: By quantizing the system on a spatial circle with a topological defect line $\cal L$ inserted at a point in space, we define a defect Hilbert space ${\cal H}_{\mathcal{L}}$. Via the operator-state correspondence, the states in ${\cal H}_{\mathcal{L}}$ are mapped to operators living at the end of $\cal L$.
• Figure 4: Upper bounds $\Delta_\text{scalar}^{Q=0}$ on the lightest $\mathbb{Z}_3$-symmetric scalar operator in the spectrum of Virasoro primaries with anomaly $k$.
• Figure 5: Upper bounds $\Delta_\text{scalar}^{Q=0}$ on the lightest $\mathbb{Z}_4$-symmetric scalar operator in the spectrum of Virasoro primaries with anomaly $k$.