Equivariant unoriented topological field theories and G-extended Frobenius algebras
For a finite group G, we define G-equivariant unoriented topological quantum field theories and G-extended Frobenius algebras and prove an equivalence between the categories of these two structures. This gives an equivariant version of the equivalence of unoriented topological quantum field theories and extended Frobenius algebras due to Turaev-Turner. Further, for the weighted projective space P(1,n), we study the virtual orbifold K-theory and its related virtual Adams operations. By applying the non-Abelian localization of Edidin-Graham, we obtain natural generators for the virtual orbifold K-theory. We express the virtual line elements in terms of these generators, and use the virtual line elements to give a presentation of the virtual orbifold K-theory. For a particular crepant resolution of T*P(1,n), we show the usual K-theory of the resolution is isomorphic to a summand of the virtual orbifold K-theory. This gives an example of the K-theoretic version of the hyper-Kahler resolution conjecture and generalizes a result of Edidin-Jarvis-Kimura.