2001/11/24 by Bong H. Lian, Lian, Bong H., Chien-Hao Liu +7 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th #math.AG
paper · pdf · doi:10.48550/arxiv.math/0111256
39 pages with 6 figures
arxiv created 2001/11/24 · openalex publication_date 2001/11/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe the S1-action on the Quot-scheme \Quot(\cal En) associated to the trivial bundle \cal En=CP1×\smallBbb Cn. In particlular, the topology of the S1-fixed-point components in \Quot(\cal En) and the S1-weights of the normal bundle of these components are worked out. Mirror Principle, as developed by three of the current authors in the series of work [L-L-Y1, I, II, III, IV], is a method for studying certain intersection numbers on a stable map moduli space. As an application, in Mirror Principle III, Sec 5.4, an outline was given in the case of genus zero with target a flag manifold. The results on S1 fixed points in this paper are used here to do explicit Mirror Principle computations in the case of Grassmannian manifolds. In fact, Mirror Principle computations involve only a certain distinguished subcollection of the S1-fixed-point components. These components are identified and are labelled by Young tableaus. The S1-equivariant Euler class eS1 of the normal bundle to these components is computed. A diagrammatic rule that allows one to write down eS1 directly from the Young tableau is given. From this, the aforementioned intersection numbers on the moduli space of stable maps can be worked out. Two examples are given to illustrate our method. Using our method, the A-model for Calabi-Yau complete intersections in a Grassmannian manifold can now also be computed explicitly. This work is motivated by the intention to provide further details of mirror principle and to understand the relation to physical theory. Some related questions are listed for further study.