Extension of moduli and gauged linear sigma models

Some period spaces (or parameter spaces) of different geometric objects coincide or are embedded in each other like in matroska thanks to relating constructions (such as Jacobians or Kummer construction). E.g. one can make a matroska out of moduli of six-tuples of points on P^1, genus two curves, abelian surfaces, cubic surfaces, K3 surfaces and cubic fourfolds. Under such extensions of moduli sometimes it is possible to generalize formulations (but not proofs) of known theorems to larger class of objects. I will speak about one kind of such extensions, which is a particular case of meta-problem: relate a category of sheaves on moduli space of objects in a category and the original category. To establish such a relation I will consider slightly more general geometric data of so-called gauged linear sigma models and their transformation under variation of stability condition (renormalization group flow).