Three problems on Gromov-Witten invariants of algebraic varieties

PI提出了一个基于衍生模空间理论的程序，用于计算某些Calabi-Yau三倍的高属gromov - witten不变量，例如射影4-空间中的五次。虽然对属零不变量的计算已经相当成功，导致了物理学家基于镜像对称的预测的证明，但到目前为止，在高等属中还没有任何进展。主要的障碍是缺乏对高属稳定映射模空间的虚拟基本类的具体理解。在最近的工作中，PI和kapranov通过微分梯度（dg）流形开发了一种虚拟类的方法。dg流形表现为代数-几何模空间的派生版本。PI和Kapranov在稳定映射的模空间上构造了这样一个结构。dg-point的更大的灵活性恢复了一些促进归零计算的特征。最终目标是证明高属的“镜像定理”，正如物理学家Bershadsky、Cecotti、Ooguri和Vafa所预测的那样。该计划的一个关键部分涉及在早期方法无法达到的情况下构建虚拟类的新结构。这在代数几何中具有很大的独立意义，因为它允许将虚类理论扩展到所有模空间，而不仅仅是它目前应用的（相当小的）子集。Ciocan-Fontanine还提出建立一个约化代数群G的GIT商X//G的格零Gromov-Witten理论与相关的阿别商X//T的格零Gromov-Witten理论之间的关系，其中T是G中的极大环面。在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下，对这种关系进行了精确的推测。他们与如何与X//G和X//T相关的物理理论的期望是一致的，物理学家将感兴趣，以及代数几何家。PI提出的第三个问题是给出复格拉斯曼型的三点零格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”中唯一未解决的问题。这是代数几何领域的研究，代数几何是现代数学最古老的分支之一。近年来，代数几何的方法和思想，特别是对模空间的研究，已被应用于弦理论，这是理论物理中一个非常活跃的部分。弦理论的发展激发了两个研究群体之间富有成效的互动，并导致了许多意想不到的新现象的发现和研究。格罗莫夫-威滕不变量理论和镜像对称就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.