Three problems on Gromov-Witten invariants of algebraic varieties

DMS-0303614在Ciocan-Fontanine上，PI提出了一个基于导模空间理论的程序，用于计算某些Calabi-Yau三重空间的更高亏格Gromov-Witten不变量，例如射影4-空间中的五次不变量。虽然亏格零不变式的计算已经相当成功，从而证明了物理学家在这种情况下基于镜像对称的预测，但到目前为止，在更高的亏格中还没有取得任何进展。主要的障碍是缺乏对高亏格稳定映射的模空间的虚拟基本类的具体理解。在最近的工作中，PI和Kapranov发展了一种通过微分分次(Dg)流形来实现虚拟类的方法。DG-流形是代数几何模空间的衍生形式。PI和Kapranov在稳定映射的模空间上构造了这样的结构。Dg-point观点的更大灵活性恢复了一些促进亏格零计算的特征。最终目的是证明物理学家伯沙德斯基、切科蒂、乌古里和瓦法所预言的更高的“镜像定理”。该程序的一个关键部分涉及到在先前方法所不能达到的情况下构造虚拟类的一种新颖的结构。这是代数几何中非常独立的兴趣，因为它允许将虚拟类的理论扩展到所有的模空间，而不仅仅是它目前应用的(相当小的)子集。Ciocan-Fontanine还建议通过还原代数群G建立Git商X//G的亏格零Gromov-Witten理论与相关的阿贝尔商X/T的关系，其中T是G中的最大扭矩。受PI及其合作者最近在X是复向量空间，G是一般线性群的情况下所得到的结果的启发。它们符合与X//G和X//T相关的物理理论如何联系的期望，并将引起物理学家和代数几何的兴趣。PI提出的第三个问题是给出复Grassmannians的三点、亏格为零的Gromov-Witten不变量的一个组合公式的证明。这是《量子舒伯特微积分》中唯一尚未解决的悬而未决的问题。这是代数几何领域的研究，它是现代数学中最古老的分支之一。近年来，代数几何的方法和基础，特别是模空间的研究，被用于弦理论，这是理论物理中非常活跃的部分。弦理论的发展引发了这两个群体研究人员之间卓有成效的互动，并导致了许多意想不到的新现象的发现和研究。格罗莫夫-威腾不变理论和镜面对称理论是特别引人注目的例子。

DMS-0303614Ionut Ciocan-Fontanine The PI proposes to pursue a program based on thetheory of derived moduli spaces towards calculating higher genusGromov-Witten invariants of certain Calabi-Yau threefolds, such as thequintic in projective 4-space. While computations of the genus zeroinvariants have been quite successful, leading to proofs of physicists'Mirror Symmetry - based predictions in that case, there has been noprogress so far in higher genus. The main obstacle has been a lack ofconcrete understanding of the virtual fundamental classes of the modulispaces of higher genus stable maps. In recent work, the PI and Kapranovhave developed an approach to virtual classes via differential-graded (dg)manifolds. Dg-manifolds appear as derived versions of algebro-geometricmoduli spaces. The PI and Kapranov constructed such a structure on themoduli spaces of stable maps. The greater flexibility of the dg-pointof view restores some of the features that facilitated the genus zerocomputations. The ultimate goal is to prove the higher genus ``mirrortheorem'', as predicted by the physicists Bershadsky, Cecotti, Ooguri,and Vafa. A crucial part of the program involves a novel construction ofa virtual class in situations outside the reach of earlier approaches.This is of great independent interest in algebraic geometry, as it allowsto extend the theory of virtual classes to all moduli spaces, as opposedto just the (rather small) subset to which it currently applies.Ciocan-Fontanine also proposes to establish a relationship between thegenus zero Gromov-Witten theory of a GIT quotient X//G by a reductivealgebraic group G, and that of the associated abelian quotient X//T,where T is a maximal torus in G. Precise conjectures are made on what therelationship is, inspired by results obtained recently by the PI and hiscollaborators in the case when X is a complex vector space and G is thegeneral linear group. They are consistent with expectations on how thephysical theories associated to X//G and X//T are related and will be ofinterest to physicists, as well as to algebraic geometers.The third problem proposed by the PI is to give a proof of a combinatorialformula for the three-point, genus zero Gromov-Witten invariants of complexGrassmannians. This is the only outstanding unsolved problem remaining in``Quantum Schubert Calculus.''This is research in the field of algebraic geometry, which is one of theoldest branches of modern mathematics. In recent years, the methods andideas of algebraic geometry, especially the study of moduli spaces, havebeen employed in string theory, a very active part of theoretical physics.Developments in string theory have sparked a fruitful interaction betweenthe two communities of researchers and have led to the discovery and studyof many unexpected new phenomena. The theory of Gromov-Witten invariantsand Mirror Symmetry are particularly striking examples.