Twist Gap and Global Symmetry in Two Dimensions

TL;DR

This work shows that any compact, unitary 2d CFT with a $U(1)$ current has vanishing twist gap under the $\mathfrak{u}(1)\times\text{Virasoro}$ algebra, i.e., $2t_{\text{gap}}^{\mathfrak{u}(1)\times\text{Virasoro}}=0$. The authors derive this via two complementary mechanisms: (i) when the holomorphic current generates a compact $U(1)$, spectral flow from the vacuum produces a zero-twist charged state, enhancing the chiral algebra; (ii) in generic cases with two abelian factors related by irrational coefficients, independent spectral flows produce an infinite tower of charged primaries approaching zero twist. They illustrate with the $c=1$ compact boson and discuss implications for ${\cal N}=2$ theories with $c>3$, showing vanishing twist gaps under the unextended ${\cal N}=2$ Vira­so­ro algebra, with additional corollaries constraining the holomorphic $R$-symmetry and central charge. Overall, the results constrain the spectrum and chiral-algebra structure of broad classes of 2d CFTs with abelian symmetries and have implications for moduli and extended superconformal theories.

Abstract

We show that every compact, unitary two-dimensional CFT with an abelian conserved current has vanishing twist gap for charged primary fields with respect to the $\mathfrak{u}(1)\times$Virasoro algebra. This means that either the chiral algebra is enhanced by a charged primary field with zero twist or there is an infinite family of charged primary fields that accumulate to zero twist.

Figures (2)

• Figure 1: Using the operator-state correspondence, a state $|\phi^\eta\rangle$ in the twisted Hilbert space ${\cal H}_\eta$ is mapped to a non-local, point-like operator attached to a topological line defect $U_\eta$.
• Figure 2: Starting with a local operator $\phi(x)\in {\cal H}$, we can adiabatically turn on a topological line $U_\eta$ from $\eta=0$ (the trivial line, shown in dashed line) to a small but finite value of $\eta$. This implements the spectral flow map from a local operator $\phi(x)$ to a non-local one $\phi^\eta(x)$ living at the end of the line. When $\eta$ increases to 1 (corresponding to the $2\pi$ rotation of $U(1)$), the topological line becomes trivial, and we end up with another local operator $\phi^{\eta=1}(x)\in {\cal H}$. The whole process implements a spectral flow by one unit that maps a local operator $\phi(x)$ to another $\phi^{\eta=1}(x)$.