  We construct a global geometric model for complex analytic equivariant
elliptic cohomology for all compact Lie groups. Cocycles are specified by
functions on the space of fields of the two-dimensional sigma model with
background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also
consider a theory of free fermions valued in a representation whose partition
function is a section of a determinant line bundle. We identify this section
with a cocycle representative of the (twisted) equivariant elliptic Euler class
of the representation. Finally, we show that the moduli stack of $U(1)$-gauge
fields carries a multiplication compatible with the complex analytic group
structure on the universal (dual) elliptic curve, with the Euler class
providing a choice of coordinate. This provides a physical manifestation of the
elliptic group law central to the homotopy-theoretic construction of elliptic
cohomology.
