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

In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral ${\int }_X{\tau }^{\ast }{e}^{H\cdot t}\cap {\mathbf{1}}_d$ for a flag manifold $X={\text{Fl}}_{r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_I}({\mathbb{C}}^n)$ via an extended mirror principle diagram. This method turns the required localization computation on the augmented moduli stack ${\bar{\mathcal{M}}}_{0,0}({\text{CP}}^1\times X)$ of stable maps to a localization computation on a hyper-Quot-scheme $\text{HQuot}({\mathcal{E}}^n)$. In this article, the detail of this localization computation on $\HQuot({\cal E}^n)$ is carried out. The necessary ingredients in the computation, notably, the $S^1$-fixed-point components and the distinguished ones $E_{(A;0)}$ in $\text{HQuot}({\mathcal{E}}^n)$, the $S^1$-equivariant Euler class of $E_{(A;0)}$ in $\text{HQuot}({\mathcal{E}}^n)$, and a push-forward formula of cohomology classes involved in the problem from the total space of a restrictive flag manifold bundle to its 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 on the Hori-Vafa conjecture are given in the end.