Completed K-theory and Equivariant Elliptic Cohomology

Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of $S^1$-equivariant $K$-theory for spaces. Several authors (cf [ABG],[KM],[L]) have suggested that an equivariant version ought to be related to the work of Freed-Hopkins-Teleman ([FHT1],[FHT2],[FHT3]). However, a first attempt at this runs into apparent contradictions concerning twist, degree, and cup product. Several authors (cf. [BET],[G],[K]) have solved the problem over the complex numbers by interpreting the $S^1$-equivariant parameter as a complex variable and using holomorphicity as the technique for completion. This paper gives a solution that works integrally, by constructing a carefully completed model of $K$-theory for $S^1$-equivariant stacks which allows for certain ``convergent" infinite-dimensional cocycles.