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.
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机译：在[L-L-Y1，III：第二部分5.4]基于镜像原理，开发了一种方法来计算aflag流形$ X = \的整数$ \ int_ {X} \ tau ^ {\ ast} e ^ {H \ cdot t} \ cap {\ mathbf 1} _d $ Fl_ {r_1，...，r_I}（{\ Bbb C} ^ n）$通过扩展镜像原理图。此方法将稳定映射的增强模堆栈$ \ bar {\ cal M} _ {0,0}（\ CP ^ 1 \ times X）$上的所需定位计算转换为超报价方案$ \上的定位计算HQuot（{\ cal E} ^ n）$。在本文中，将对$ \ HQuot（{\ calE} ^ n）$进行本地化计算的详细信息。计算中的必要成分，特别是$ S ^ 1 $定点分量和$ E _ {（A; 0）} $ in $ \ HQuot（{\ cal E} ^ n）$中的特有成分， $ E _ {（（A; 0）} $ in $ \ HQuot（{\ cal E} ^ n）$中的$ S ^ 1 $-等量Euler类，以及从总数中推论涉及该问题的同调类的推式给出了限制性标志歧管束与其基础歧管的空间。利用这些，获得$ \ int_ {X} \ tau ^ {\ ast} e ^ {H \ cdot t} \ cap {\ mathbf 1} _d $的精确表达式。最后对Hori-Vafa猜想作了评论。
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