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

TL;DR

This paper develops a detailed localization computation on hyper-Quot-schemes to derive an exact mirror formula for flag manifolds, extending the mirror principle and providing tools for related enumerative geometry problems.

Contribution

It provides a comprehensive localization framework on hyper-Quot-schemes, including fixed-point analysis and push-forward formulas, to compute mirror symmetry integrals for flag manifolds.

Findings

Explicit $S^1$-fixed-point components identified

Euler classes of fixed-point components computed

Exact integral formula for flag manifolds derived

Abstract

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 $X=\Fl_{r_1, ..., r_I}({\Bbb C}^n)$ via an extended mirror principle diagram. This method turns the required localization computation on the augmented moduli stack $\bar{\cal M}_{0,0}(\CP^1\times X)$ of stable maps to a localization computation on a hyper-Quot-scheme $\HQuot({\cal E}^n)$. 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 $\HQuot({\cal E}^n)$, the $S^1$-equivariant Euler class of $E_{(A;0)}$ in $\HQuot({\cal E}^n)$, and a push-forward formula of cohomology classes 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.