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Non-invertible defects from the Conway SCFT to K3 sigma models I: general results

Roberta Angius, Stefano Giaccari, Sarah M. Harrison, Roberto Volpato

Abstract

We initiate the study of supersymmetry-preserving topological defect lines (TDLs) in the Conway moonshine module $V^{f \natural}$. We show that the tensor category of such defects, under suitable assumptions, admits a surjective but non-injective ring homomorphism into the ring of $\mathbb{Z}$-linear maps of the Leech lattice into itself. This puts strong constraints on possible defects and their quantum dimensions. We describe a simple construction of non-invertible TDLs from orbifolds of holomorphic (super)vertex operator algebras, which yields non-trivial examples of TDLs satisfying our main theorem. We conjecture a correspondence between four--plane--preserving TDLs in $V^{f\natural}$ and supersymmetry--preserving TDLs in K3 non-linear sigma models, which extends the correspondence between symmetry groups to the level of tensor category symmetry. We establish evidence for this conjecture by constructing non-invertible TDLs in special K3 non-linear sigma models.

Non-invertible defects from the Conway SCFT to K3 sigma models I: general results

Abstract

We initiate the study of supersymmetry-preserving topological defect lines (TDLs) in the Conway moonshine module . We show that the tensor category of such defects, under suitable assumptions, admits a surjective but non-injective ring homomorphism into the ring of -linear maps of the Leech lattice into itself. This puts strong constraints on possible defects and their quantum dimensions. We describe a simple construction of non-invertible TDLs from orbifolds of holomorphic (super)vertex operator algebras, which yields non-trivial examples of TDLs satisfying our main theorem. We conjecture a correspondence between four--plane--preserving TDLs in and supersymmetry--preserving TDLs in K3 non-linear sigma models, which extends the correspondence between symmetry groups to the level of tensor category symmetry. We establish evidence for this conjecture by constructing non-invertible TDLs in special K3 non-linear sigma models.

Paper Structure

This paper contains 25 sections, 3 theorems, 242 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Conway module $V^{f\natural}$
  3. Constructions
  4. Symmetries and twining functions
  5. The connection with K3 non-linear sigma models
  6. Topological defects in $V^{f\natural}$
  7. Generalities on topological defects
  8. Topological defects in supersymmetric VOAs
  9. Topological defects in $V^{f\natural}$ and main theorem
  10. The connection with topological defects in K3 non-linear sigma models
  11. Proof of theorem \ref{['th:main']}
  12. Topological defects in orbifold CFTs
  13. VOAs with finite groups of symmetries
  14. Topological defects from orbifolds by subgroups
  15. Generalizations to SuperVOAs and non-holomorphic CFTs
  16. ...and 10 more sections

Key Result

Theorem 1

Let $\mathsf{Top}$ be a tensor category of topological defects $\mathcal{L}$ of $V^{f\natural}$ containing all invertible defects $\mathcal{L}_g$, $g\in Co_0$, and such that all objects ${\cal L}\in \mathsf{Top}$ satisfy properties 1, 2 and 3 above. Then there is an embedding $\Lambda\hookrightarrow defines a surjective, non-injective ring homomorphism from the Grothendieck ring $Gr(\mathsf{Top})$

Figures (2)

  • Figure 1: Representation of the action of the linear operator $\hat{\mathcal{L}}$ on the Hilbert space of states on $S^1$ and on the corresponding set of local operators on $\hat{\mathbb{C}}$ as related by the conformal mapping.
  • Figure 2: The insertion of a line defect $\mathcal{L}$ along the Euclidian time direction on $S^1 \times \mathbb{R}$ generates the new $\mathcal{L}-$twisted Hilbert space $\mathcal{H}_{\mathcal{L}}$, which states are mapped in defect starting operator for the defect $\mathcal{L}$ in $\hat{\mathbb{C}}$.

Theorems & Definitions (5)

  • Theorem 1
  • Conjecture 4
  • Theorem 5
  • Lemma 1
  • proof