Orbifold Quantum Cohomology
Orbifold Quantum Cohomology
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发表时间：
2000-05
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通讯作者：
Weimin Chen;Y. Ruan
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作者：
Weimin Chen;Y. Ruan
这是关于辛或射影轨道的Gromov-Witten不变量和量子上同调理论的研究公告。我们的项目开始于1998年的夏天，我们最初的动机是研究复维三维中奇异缺缺下的量子上同调。在这种情况下，我们允许我们的三折线有末端奇点，它可以变形成辛折线。我们花了1998年下半年和1999年春季的大部分时间来发展轨道上的Gromov-Witten不变量的基础，包括关键的概念成分——好地图的概念。1999年4月，我们有幸见到了R. Dijkgraaf，他向我们解释了轨道弦理论和扭曲扇区的作用。扭曲扇区为我们的轨道量子上同调提供了精确的拓扑框架。我们的轨道量子上同调理论实际上是在1999年夏天完成的。在此，我们对轨道量子上同调的基础进行了概述，而将其在其他领域的应用，如双态几何等，留给未来的研究。我们工作的第一部分已经发表在[CR1]上。第二部分[CR2]将很快出现。我们要感谢dr . Dijkgraaf在我们项目的关键时刻让我们注意到轨道弦理论。我们也从与E. Zaslow的多次讨论中受益。最后，第二作者要感谢E. Witten关于轨道弦理论的许多有启发性的讨论。
This is a research announcement on a theory of Gromov-Witten invariants and quantum cohomology of symplectic or projective orbifolds. Our project started in the summer of 98 where our original motivation was to study the quantum cohomology under singular flops in complex dimension three. In this setting, we allow our three-fold to have terminal singularities which can be deformed into a symplectic orbifold. We spent the second half of 98 and most of spring of 99 to develop the foundation of Gromov-Witten invariants over orbifolds, including the key conceptual ingredient — the notion of good map. In the April of 99, we were lucky to meet R. Dijkgraaf who explained to us the orbifold string theory and the role of twisted sectors. The twisted sector provides the precise topological framework for our orbifold quantum cohomology. Our theory of orbifold quantum cohomology was virtually completed in the summer of 99. Here, we give an over-look of the foundation of orbifold quantum cohomology while we leave its applications in other fields such as birational geometry for future research. The first part of our work has already appeared in [CR1]. The second part [CR2] will appear shortly. We would like to thank R. Dijkgraaf for bringing orbifold string theory to our attention at the critical moment of our project. We are also benefited from many discussions with E. Zaslow. Finally, the second author would like to thank E. Witten for many stimulating discussions about orbifold string theory.