Definition of the complex for non-generic stability conditions

Construct the correct simplicial complex replacing \(\Sigma_{\tau}\) for non-generic stability conditions on a 2-Calabi–Yau category, when HN factors of spherical objects need not be direct sums of spherical objects.

Background

For generic stability conditions, semistable spherical objects are stable and the HN support construction yields a simplicial complex. For non-generic stability conditions, the paper notes that HN factors may fail to be direct sums of spherical objects, with a counterexample in type $D_4$. Consequently, the existing definition does not provide the desired support space.