The twining-genus correspondence for Conway defects and K3 sigma models

Let $\Pi^{\natural} \subset \Lambda\otimes\mathbb{R}\subset V^{f\natural}_{tw}$ be a $4$-dimensional subspace, and let $\mathsf{Top}_{\Pi^{\natural}}$ be the full subcategory of topological defects whose action on $\Pi^{\natural}$ is scalar. Then there should be a nonlinear sigma model $\mathcal{C}$ on K3, an equivalence of tensor categories

$$F:\mathsf{Top}_{\Pi^{\natural}}\longrightarrow \mathsf{Top}^{K3}_{\mathcal{C}},$$

where $\mathsf{Top}^{K3}_{\mathcal{C}}$ consists of topological defects preserving $\mathcal{N}=(4,4)$ and spectral flow, such that the twining genera agree,

$$\phi^{\mathcal{L}}(V^{f\natural},\tau,z)=\phi^{F(\mathcal{L})}(\mathcal{C},\tau,z),$$

and there is an isometry of Hilbert spaces intertwining the actions of corresponding defects. Defect–K3 twining correspondence. For every $\mathcal{L}\in\mathsf{Top}_{\Pi^{\natural}}$, the three properties above hold: categorical equivalence, equality of twining genera, and an action-intertwining isometry $\varphi:V^{f\natural}_{tw}(1/2)\cong\mathcal{H}^{K3}_{\mathrm{RR},gr}$. This conjecture proposes a tensor-categorical and state-space correspondence between defects of the Conway SCFT and defects in suitable K3 sigma models. It is known in some special K3 models, while verification for other K3 models remains open.