A de Rham model for complex analytic equivariant elliptic cohomology

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowskis theory when the group is connected and Devotos when the group is finite. We then construct Mathai--Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique equivariant refinement of Hopkins theorem of the cube construction of the ${rm MString}$-orientation of elliptic cohomology.