Here e S 1 (·) denotes the S 1 -equivariant Euler class. As in the cases we have studied earlier, the left hand side of the above formula indicates that when V = L is a line bundle, we should compare the Euler **data** Q d = ϕ ! π * e(V d ), to the Euler **data** given by
P d = c 1 (L),d m=0 (c 1 (L) − mα).
What is left is to develope uniqueness and mirror transformations, which we are unable to achieve at this moment, though they can be easily axiomized. Now let us look at K-theory formula, which can be proved by using equivariant localization in K-theory. First following the same idea, we get the explicit formula as follows: