The Stringy K-theory of orbifolds and the Chern character

We introduce K-theoretic versions of the Fantechi-Goettsche ring of a variety with a group action and the Chen-Ruan cohomology of a smooth complex orbifold which we call stringy K-theory. Our definition is a generalization of a construction due to Givental and Y. P. Lee and it differs from the orbifold K-theory of Adem-Ruan. We also introduce a stringy Chern character isomorphism Ch taking stringy K-theory to the (even dimensional) stringy cohomology ring. Ch requires correction terms to the usual Chern character to preserve the stringy multiplications. The result follows from a new, simple formula for the obstruction bundle which exorcises complex curves from the definitions of the stringy K-theory, and Chen-Ruan orbifold cohomology. Consequently, a K-theoretic version of Ruan’s conjectures holds for a crepant resolution of the symmetric product of a projective surface with trivial first Chern class.