$S^1$-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram

摘要

In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed tocompute the integral $\int_{X}\tau^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ for aflag manifold $X=\Fl_{r_1, ..., r_I}({\Bbb C}^n)$ via an extended mirrorprinciple diagram. This method turns the required localization computation onthe augmented moduli stack $\bar{\cal M}_{0,0}(\CP^1\times X)$ of stable mapsto a localization computation on a hyper-Quot-scheme $\HQuot({\cal E}^n)$. Inthis article, the detail of this localization computation on $\HQuot({\calE}^n)$ is carried out. The necessary ingredients in the computation, notably,the $S^1$-fixed-point components and the distinguished ones $E_{(A;0)}$ in$\HQuot({\cal E}^n)$, the $S^1$-equivariant Euler class of $E_{(A;0)}$ in$\HQuot({\cal E}^n)$, and a push-forward formula of cohomology classes involvedin the problem from the total space of a restrictive flag manifold bundle toits base manifold are given. With these, an exact expression of$\int_{X}\tau^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ is obtained. Comments onthe Hori-Vafa conjecture are given in the end.