Spectral sequences and vacua in N = 2 $$ mathcal{N}=2 $$ gauged linear quantum mechanics with potentials

We study the behaviour of supersymmetric ground states in a class of one-dimensional $$ mathcal{n}=2 $$ abelian gauged linear sigma models, including theories for which the target space is a complete intersection in projective space, and more generally, models with an interaction term introduced by herbst, hori and page in which the vacua correspond to elements of hypercohomology groups of complexes of sheaves. combining physical insights from recent work by hori, kim and yi with the use of spectral sequences, we propose a way to reconcile the non-linear sigma model description, valid deep within a geometric phase, with the effective coulomb branch description, valid near a phase boundary. this leads to a physical interpretation of the hypercohomology groups from the perspective of the coulomb branch, as well as an interpretation for the spectral sequences used to compute them.