Supersymmetry for brane diagrams and bow varieties

We provide combinatorial and numerical criteria to characterize affine type A bow diagrams giving rise to a non-empty bow variety. The key idea is to prove that such diagrams correspond to supersymmetric brane systems in type IIB string theory, allowing us to reformulate the problem in purely combinatorial terms. To achieve this, we characterize supersymmetry for affine type A brane systems (and, by extension, for types B, C, and D) using Hanany--Witten transitions. This leads to a finite-step algorithm that decides whether a given affine type A bow or brane diagram is supersymmetric, which consists in checking a finite set of inequalities, so providing a numerical criterion for non-emptiness. Finally, we provide a different perspective by introducing a further criterion in terms of weights of affine Lie algebras. Along the way, we also prove that increasing dimension vectors between two consecutive x-points or arrows in a bow diagram (not necessarily of type A) preserves the properties of generating non-empty bow varieties.