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

Chien-Hao Liu Kefeng Liu Shing-Tung Yau

Algebraic Geometry mathscidoc:1912.43647

arXiv preprint math/0401367, 2004.1
In [LL-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral \int_ {X} ^{st} 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 \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -fixed-point components and the distinguished ones \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n) $, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -equivariant Euler class of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d 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} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d is obtained. Comments on the Hori-Vafa conjecture are given in the end.
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@inproceedings{chien-hao2004s^1-fixed-points,
  title={S^1-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram},
  author={Chien-Hao Liu, Kefeng Liu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204722802387211},
  booktitle={arXiv preprint math/0401367},
  year={2004},
}
Chien-Hao Liu, Kefeng Liu, and Shing-Tung Yau. S^1-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram. 2004. In arXiv preprint math/0401367. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204722802387211.
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