Non-invertible defects from the Conway SCFT to K3 sigma models. Part II. Duality and Fibonacci defects

We continue the study, initiated in [1], of topological defect lines (TDLs) in the Conway module Vf♮ and K3 non-linear sigma models (NLSMs). In the case of Vf♮, we fully classify the potential N = 1 (and N = 4) — preserving duality defects for cyclic Tambara-Yamagami categories TY(ℤN), noting a curious relation to genus zero groups of monstrous moonshine. We use the correspondence with Leech lattice endomorphisms, discovered in [1], to construct a number of non-trivial examples of TDLs in Vf♮, including examples of irrational quantum dimension. In particular, we fully classify and construct defects for the TY(ℤ2) and TY(ℤ3) cases, and provide examples of duality defects for TY(ℤ2 × ℤ2) and Fibonacci fusion categories as well. In the case of K3 NLSMs, we describe a duality defect of irrational quantum dimension 2 for the category TY(ℤ2, –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(S3) categories in particular examples. Finally we provide evidence for our conjecture in [1] that special subcategories of such TDLs in Vf♮ 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 Vf♮ with coincident twining genus, and making a prediction for a number of TDLs in K3 NLSMs yet to be found.