Self-duality under gauging a non-invertible symmetry

TL;DR

The paper reveals that a non-invertible fusion category symmetry $ extsf{Rep}(H_8)$ acting on the orbifold branch of the $c=1$ CFTs yields a radius-duality map $R o 2/R$, with the Ising$^2$ point ($R= frac{1}{ ext{sqrt}}{2}$) fixed under gauging. It constructs a new topological defect line $oldsymbol{ ext{D}}$ via half-space gauging, shows it preserves the $c=1$ Virasoro algebra but not the full extended chiral algebra, and demonstrates that $oldsymbol{ ext{D}}$ with $ extsf{Rep}(H_8)$ forms a $ ext{Z}_2$-extension of the original fusion category. By solving the pentagon equations, the authors find eight possible $ ext{Z}_2$-extensions, two of which are realized in Ising$^2$, and they derive spin selection rules to distinguish them. They also establish that the Monster$^2$ and Ising$ imes$Monster CFTs are invariant under gauging $ extsf{Rep}(H_8)$ at the torus level, suggesting broader applicability of the construction to other $c=1$ theories and related CFTs.

Abstract

We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of $c=1$ CFTs, it is known that the theory is self-dual under gauging a $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry, and has $\mathsf{Rep}(H_8)$ and $\mathsf{Rep}(D_8)$ fusion category symmetries as a result. We find that gauging the entire $\mathsf{Rep}(H_8)$ fusion category symmetry maps the orbifold theory at radius $R$ to that at radius $2/R$. At $R=\sqrt{2}$, which corresponds to two decoupled Ising CFTs (Ising$^2$ in short), the theory is self-dual under gauging the $\mathsf{Rep}(H_8)$ symmetry. This implies the existence of a topological defect line in the Ising$^2$ CFT obtained from half-space gauging of the $\mathsf{Rep}(H_8)$ symmetry, which commutes with the $c=1$ Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the $c=1$ Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the $\mathsf{Rep}(H_8)$ symmetry to form a bigger fusion category which is a $\mathbb{Z}_2$-extension of $\mathsf{Rep}(H_8)$. We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising$^2$ CFT. Finally, we show that the torus partition functions of the Monster$^2$ CFT and Ising$\times$Monster CFT are also invariant under gauging the $\mathsf{Rep}(H_8)$ symmetry.

Figures (4)

• Figure 1: Moduli space of $c=1$ CFTs Ginsparg:1987eb. The horizontal line is the circle branch consisting of free compact boson CFTs with radius $R$, and the vertical line is the orbifold branch obtained from gauging the charge conjugation symmetry of the circle branch theories. In addition, there are 3 isolated points (not shown). Along the orbifold branch, two theories at radii $R$ and $2/R$ are related by gauging the $\mathsf{Rep}(H_8)$ symmetry, and $R=\sqrt{2}$, corresponding to the Ising$^2$ CFT, is a fixed point under this gauging.
• Figure 2: Action of a topological defect line $\mathcal{L}$ on a local operator $\mathcal{O}$. We start with the line $\mathcal{L}$ wrapping around the local operator $\mathcal{O}$. After shrinking $\mathcal{L}$, $\mathcal{O}$ is transformed by $\mathcal{L}$ to another local operator $\mathcal{L}\cdot \mathcal{O}$.
• Figure 3: Under the state-operator correspondence, a state in the defect Hilbert space $\mathcal{H}_{\mathcal{L}}$ is mapped to a non-local operator $\mathcal{O}$ which is attached to $\mathcal{L}$.
• Figure 4: The lasso diagram describing the action of a topological defect line $\mathcal{L}_2$ on a non-local operator $\mathcal{O}$ attached to $\mathcal{L}_1$. To fully determine the action, we need to also specify the line $\mathcal{L}_3$ as well as two junctions $\mu,\nu$.