Notes on GLSMs for Supermanifolds and Their Mirrors

In this paper, we revisit the A-twisted gauged linear sigma models (GLSMs) whose geometric phases are complex K\"ahler supermanifolds. For abelian models without superpotentials we propose an explicit orbifold description of the non-geometric (Landau-Ginzburg) point, and give a systematic rule for the nontrivial R-charge assignments at that point. We then study topological super Landau-Ginzburg models, derive chiral ring relations and genus-$g$ correlation functions, and use these formulas to test a Hori-Vafa-type mirror proposal for supermanifolds.