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Elliptic cohomology is a natural big brother to ordinary cohomology and K-theory. In contrast to the geometric objects that provide representatives for cohomology and K-theory classes (leading to many applications), as yet there is no such geometric description of elliptic cohomology. A long-standing conjecture is that a suitable space of 2-dimensional supersymmetric quantum field theories would afford such a geometric model. Furthermore, 2-dimensional supersymmetric gauge theories are expected to provide equivariant refinements with connections to loop group representation theory. This talk will explain some recent progress in joint work with Arnav Tripathy.