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Non-invertible defects from the Conway SCFT to K3 sigma models II: duality and Fibonacci defects

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

TL;DR

The article advances a unified program to classify and realize non-invertible topological defect lines in the Conway moonshine module $V^{f\natural}$ and in K3 NLSMs, focusing on duality defects for Tambara–Yamagami categories and on defects with irrational quantum dimensions (notably Fibonacci-type lines). It develops a lattice-endomorphism framework tied to the Leech and Leech-derived structures, constructs explicit $\mathcal{N}=1$-preserving $Z_2$ and $Z_3$ defects in both theories, and computes defect-twined genera that match across the two sides, in line with a proposed categorical correspondence. The work extends to non-cyclic groups and to non-rational settings via orbifold and Gepner-model analyses, including Fibonacci and $Rep(S_3)$ defects, and it presents concrete K3 realizations for several TY categories, offering evidence for a broader V^{f\natural}–K3 defect correspondence. Collectively, these results deepen the connection between moonshine, Leech-lattice automorphisms, and generalized symmetries in non-rational CFTs and their geometric sigma-model avatars. The paper thus provides both explicit defect constructions and a solid conjectural framework for mapping generalized symmetries between holomorphic VOAs and geometric K3 theories, with concrete predictions for defect spectra in yet-unconstructed K3 models.

Abstract

We continue the study, initiated in [hep-th:2504.18619], of topological defect lines (TDLs) in the Conway module $V^{f \natural}$ and K3 non-linear sigma models (NLSMs). In the case of $V^{f \natural}$, we fully classify the potential $N=1$ (and $N=4$)--preserving duality defects for cyclic Tambara--Yamagami categories TY$(\mathbb{Z}_N)$, noting a curious relation to genus zero groups of monstrous moonshine. We use the correspondence with Leech lattice endomorphisms, discovered in [hep-th:2504.18619], to construct a number of non-trivial examples of TDLs in $V^{f \natural}$, including examples of irrational quantum dimension. In particular, we fully classify and construct defects for the TY$(\mathbb{Z}_2)$ and TY$(\mathbb{Z}_3)$ cases, and provide examples of duality defects for TY$(\mathbb{Z}_2\times \mathbb{Z}_2)$ and Fibonacci fusion categories as well. In the case of K3 NLSMs, we describe a duality defect of irrational quantum dimension $\sqrt{2}$ for the category TY$(\mathbb{Z}_2, -1)$ in a particular torus orbifold, which exists on a 16-dimensional slice of the moduli space. We also provide a detailed analysis of spectral flow--preserving TDLs in Gepner models of K3, of independent interest, and use this to construct non-invertible defects for Fibonacci and $Rep(S_3)$ categories in particular examples. Finally we provide evidence for our conjecture in [hep-th:2504.18619] that special subcategories of such TDLs in $V^{f \natural}$ correspond to $N=(4,4)$ and spectral flow--preserving defect lines in a corresponding K3 NLSM. In particular, we compute defect--twined elliptic genera for all non-invertible defects constructed in this article, demonstrating that for each defect found in a K3 NLSM, there is a corresponding defect in $V^{f \natural}$ with coincident twining genus, and making a prediction for a number of TDLs in K3 NLSMs yet to be found.

Non-invertible defects from the Conway SCFT to K3 sigma models II: duality and Fibonacci defects

TL;DR

The article advances a unified program to classify and realize non-invertible topological defect lines in the Conway moonshine module and in K3 NLSMs, focusing on duality defects for Tambara–Yamagami categories and on defects with irrational quantum dimensions (notably Fibonacci-type lines). It develops a lattice-endomorphism framework tied to the Leech and Leech-derived structures, constructs explicit -preserving and defects in both theories, and computes defect-twined genera that match across the two sides, in line with a proposed categorical correspondence. The work extends to non-cyclic groups and to non-rational settings via orbifold and Gepner-model analyses, including Fibonacci and defects, and it presents concrete K3 realizations for several TY categories, offering evidence for a broader V^{f\natural}–K3 defect correspondence. Collectively, these results deepen the connection between moonshine, Leech-lattice automorphisms, and generalized symmetries in non-rational CFTs and their geometric sigma-model avatars. The paper thus provides both explicit defect constructions and a solid conjectural framework for mapping generalized symmetries between holomorphic VOAs and geometric K3 theories, with concrete predictions for defect spectra in yet-unconstructed K3 models.

Abstract

We continue the study, initiated in [hep-th:2504.18619], of topological defect lines (TDLs) in the Conway module and K3 non-linear sigma models (NLSMs). In the case of , we fully classify the potential (and )--preserving duality defects for cyclic Tambara--Yamagami categories TY, noting a curious relation to genus zero groups of monstrous moonshine. We use the correspondence with Leech lattice endomorphisms, discovered in [hep-th:2504.18619], to construct a number of non-trivial examples of TDLs in , including examples of irrational quantum dimension. In particular, we fully classify and construct defects for the TY and TY cases, and provide examples of duality defects for TY and Fibonacci fusion categories as well. In the case of K3 NLSMs, we describe a duality defect of irrational quantum dimension for the category TY in a particular torus orbifold, which exists on a 16-dimensional slice of the moduli space. We also provide a detailed analysis of spectral flow--preserving TDLs in Gepner models of K3, of independent interest, and use this to construct non-invertible defects for Fibonacci and categories in particular examples. Finally we provide evidence for our conjecture in [hep-th:2504.18619] that special subcategories of such TDLs in correspond to and spectral flow--preserving defect lines in a corresponding K3 NLSM. In particular, we compute defect--twined elliptic genera for all non-invertible defects constructed in this article, demonstrating that for each defect found in a K3 NLSM, there is a corresponding defect in with coincident twining genus, and making a prediction for a number of TDLs in K3 NLSMs yet to be found.
Paper Structure (41 sections, 3 theorems, 456 equations, 16 tables)

This paper contains 41 sections, 3 theorems, 456 equations, 16 tables.

Table of Contents

  1. Introduction and outlook
  2. Summary of previous results
  3. Topological defects in $V^{f \natural}$
  4. Topological defects in K3 non-linear sigma models
  5. The Conjecture
  6. Duality defects and Tambara-Yamagami categories in $V^{f\natural}$
  7. General properties of TY categories
  8. Duality defects in $V^{f\natural}$
  9. Spin selection rules
  10. Cyclic orbifolds of $V^{f\natural}$ and the monster group
  11. Duality defects for $\mathbb{Z}_2$
  12. The lattice endomorphism associated with the $\mathbb{Z}_2$-duality defect
  13. Free fermion construction of the $\mathbb{Z}_2$ duality defect
  14. Duality defect partition functions and twining genera
  15. A K3 model with $\mathbb{Z}_2$ duality defect preserving $\mathcal{N}=(4,4)$
  16. ...and 26 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 $Gr(\mathsf{Top})$

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Claim 4
  • Claim 5
  • Conjecture 6