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Defect Conformal Manifolds from Phantom (Non-Invertible) Symmetries

Andrea Antinucci, Christian Copetti, Giovanni Galati, Giovanni Rizi

TL;DR

The paper shows that defect conformal manifolds can arise in (1+1)d CFTs even without bulk continuous symmetries, by exposing phantom (non-invertible) symmetries that emerge in the folded theory. Using the folding trick, it identifies exactly marginal tilt operators associated with these phantom currents and demonstrates that the $g$-function remains constant along the defect manifold while the reflection coefficient $\mathcal{R}$ typically varies due to non-commutation with the individual bulk stress tensors. The Ising CFT serves as a detailed, controllable example where continuous families of non-invertible topological lines generate a rich defect-moduli space, and the framework naturally extends to RCFT interfaces and higher-dimensional free-field theories. Together, these results sketch a broad landscape of phantom-symmetry-driven DCFTs and suggest new constraints and structures for defect data, including potential generalizations to defect fusion and central charges. The work thus reframes defect moduli spaces as robust consequences of generalized symmetries and their breaking, offering a symmetry-based route to populate and constrain DCFTs beyond conventional symmetry protection.

Abstract

We explore a general mechanism that allows (1+1)d CFTs to have interesting interface conformal manifolds even in the absence of any continuous internal symmetry or supersymmetry. This is made possible by the breaking of an enhanced continuous symmetry, which is generically non-invertible, arising in the folded theory. We provide several examples and showcase the power of the symmetry-based approach by computing the evolution of the reflection coefficient along the defect conformal manifold. We also discuss higher-dimensional generalizations and we comment on no-go theorems.

Defect Conformal Manifolds from Phantom (Non-Invertible) Symmetries

TL;DR

The paper shows that defect conformal manifolds can arise in (1+1)d CFTs even without bulk continuous symmetries, by exposing phantom (non-invertible) symmetries that emerge in the folded theory. Using the folding trick, it identifies exactly marginal tilt operators associated with these phantom currents and demonstrates that the -function remains constant along the defect manifold while the reflection coefficient typically varies due to non-commutation with the individual bulk stress tensors. The Ising CFT serves as a detailed, controllable example where continuous families of non-invertible topological lines generate a rich defect-moduli space, and the framework naturally extends to RCFT interfaces and higher-dimensional free-field theories. Together, these results sketch a broad landscape of phantom-symmetry-driven DCFTs and suggest new constraints and structures for defect data, including potential generalizations to defect fusion and central charges. The work thus reframes defect moduli spaces as robust consequences of generalized symmetries and their breaking, offering a symmetry-based route to populate and constrain DCFTs beyond conventional symmetry protection.

Abstract

We explore a general mechanism that allows (1+1)d CFTs to have interesting interface conformal manifolds even in the absence of any continuous internal symmetry or supersymmetry. This is made possible by the breaking of an enhanced continuous symmetry, which is generically non-invertible, arising in the folded theory. We provide several examples and showcase the power of the symmetry-based approach by computing the evolution of the reflection coefficient along the defect conformal manifold. We also discuss higher-dimensional generalizations and we comment on no-go theorems.
Paper Structure (10 sections, 62 equations, 1 figure, 3 tables)

This paper contains 10 sections, 62 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. phantom symmetry in the (1+1)d Ising CFT
  3. Phantom Symmetry and Defect Conformal Manifolds
  4. g-function and reflection coefficient
  5. Examples
  6. Phantom Symmetry action and reflection coefficients
  7. Phantom Symmetry Multiplet
  8. Inclusion of global symmetry
  9. An example.
  10. Reflection and transmission coefficients along the conformal manifold

Figures (1)

  • Figure 1: The folding trick maps an interface $\mathcal{I}$ between $CFT_L$ and $CFT_R$ to a boundary condition $B_\mathcal{I}$ for the folded double $CFT_L \times \overline{CFT}_R$.