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

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dc.creatorLiu, Chien-Hao
dc.creatorLiu, Kefeng
dc.creatorYau, Shing-Tung
dc.date2004-01-27
dc.date.accessioned2026-07-07T05:04:53Z
dc.date.available2026-07-07T05:04:53Z
dc.descriptionIn [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral $\int_{X}τ^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ 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 $\int_{X}τ^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ is obtained. Comments on the Hori-Vafa conjecture are given in the end.
dc.description44 pages with 6 figures
dc.identifierhttps://arxiv.org/abs/math/0401367
dc.identifierhttp://arxiv.org/abs/math/0401367
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/69982
dc.subjectAlgebraic Geometry
dc.subjectHigh Energy Physics - Theory
dc.subject14N35; 81T30
dc.title$S^1$-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram
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