  We systematically study the interesting relations between the quantum
elliptic Calogero-Moser system (eCM) and its generalization, and their
corresponding supersymmetric gauge theories. In particular, we construct the
suitable characteristic polynomial for the eCM system by considering certain
orbifolded instanton partition function of the corresponding gauge theory. This
is equivalent to the introduction of certain co-dimension two defects. We next
generalize our construction to the folded instanton partition function obtained
through the so-called "gauge origami" construction and precisely obtain the
corresponding characteristic polynomial for the doubled version, named the
elliptic double Calogero-Moser (edCM) system.
