Towards a classification of topological defects in $K3$ sigma models

TL;DR

This paper advances the understanding of topological defects in K3 sigma models, revealing their classification, fusion properties, and conditions for their existence, with implications for moonshine phenomena and string theory compactifications.

Contribution

It generalizes the notion of symmetries to categories of topological operators in K3 models and characterizes their fusion and quantum dimensions, linking to moonshine and automorphism classifications.

Findings

At generic points, the defect category is trivial, generated by the identity.

Infinite or continuum defects can occur in certain K3 models.

Defects have integral quantum dimension at attractor points for BPS D-brane configurations.

Abstract

Given a $K3$ surface, a supersymmetric non-linear K3 sigma model is the internal superconformal field theory (SCFT) in a six dimensional compactification of type IIA superstring on $\mathbb{R}^{1,5} \times K3$. These models have attracted attention due to the discovery of Mathieu moonshine phenomena for the elliptic genera of K3 surfaces, and have played a pivotal role in extending Mukai's theorem on classification of symplectic automorphisms of $K3$ surfaces. We report on recent progress (arXiv:2402.08719 [hep-th]) in characterizing topological defects in $K3$ models, generalizing the notion of symmetries to categories of topological operators supported on arbitrary codimension submanifolds with possibly non-invertible fusion rules. Taking advantage of the interpretation of Mukai lattice as the D-brane charge lattice, we present a number of general results for the category of topological defect lines preserving the superconformal algebra and spectral flow, obtained by studying their fusion with boundary states. While for certain K3 models infinitely many simple defects, and even a continuum, can occur, at generic points in the moduli space the category is actually trivial, i.e. it is generated by the identity defect. Furthermore, if a K3 model is at the attractor point for some BPS configuration of D-branes, then all topological defects have integral quantum dimension. We also introduce a conjecture that a continuum of topological defects arises if and only if the K3 model is a (possibly generalized) orbifold of a torus model. These general results are confirmed by the analysis of significant examples. We also point out the connection to recent studies of topological defects in the Conway moonshine module theory (arXiv:2412.21141 [hep-th],arXiv:2504.18619 [hep-th]).