The Interaction of finite-type and Gromov-Witten invariants : BIRS, 2003
The Interaction of finite-type and Gromov-Witten invariants : BIRS, 2003
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有限类型和 Gromov-Witten 不变量的相互作用：BIRS，2003
DOI：
10.2140/gtm.2006.8
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
J. Bryan
中科院分区：
文献类型：
--
作者：
D. Auckly;J. Bryan
这是题为“有限类型与GromovWitten不变量的相互作用”研讨会的最终报告。这是2003年11月15日至20日在班夫国际研究站举办的为期五天的讲习班。五年前，这两个截然不同的领域之间没有互动。1998年，Gopakumar和Vafa提出了M-理论中GW不变量与BPS态的某些(整数)计数之间的关系[14]。随后的工作表明，O(−1)⊕O(−1)上的开弦理论等价于陈-西蒙斯理论[1，29，24，23]。这是一个介于Chern-Simons理论和开弦理论之间的对偶性，它建立在t‘Hooft将大N规范理论和弦理论联系起来的猜想之上。这些想法在理论物理文献中发展非常迅速。由此产生的论文表明，这些不同的数学领域之间存在着强烈的联系。研究有限类型不变量和Gromov-Witten不变量的数学界过去和现在基本上是不相交的。研讨会的目标是让从事有限类型不变量研究的人、从事格罗莫夫-维腾不变量研究的人以及能够解释将这两个领域联系在一起的物理学最新结果的物理学家聚集在一起。研讨会取得了压倒性的成功，必将对这两个领域的数学发展产生影响。我们确实成功地将这两个领域的数学家和一位物理学家聚集在一起，告诉我们为什么我们应该互动。我们预测，再过五年，这两个数学学科将不再是分开的，而是会有许多密切的数学联系。本次研讨会的许多参与者将为这一领域的数学发展做出贡献，而本次研讨会开始的数学互动将不仅仅局限于参与者。我们正在筹备一次研讨会，会议记录将向全世界数学界通报这一领域的一些令人兴奋的发展。现在，我们将简要介绍研讨会所涉主题的技术概述。研讨会上展示的第一个领域是有限类型不变量和Chern-Simons理论。有限类型不变量理论植根于数学物理、统计力学、算子代数、拓扑学和奇点理论。E.Witten首先将这些不变量的物理直觉解释为基于Chern-Simons不变量的拓扑量子场理论[39]。3流形中的链的不变量被描述为路径积分，
This is the final report on the workshop titled “The interaction of finite type and GromovWitten invariants.” This was a five-day workshop held at the Banff International Research station from November 15 November 20, 2003. Before five years ago, there was no interaction between these two distinct fields. In 1998 Gopakumar and Vafa suggested a relation between the GW invariants and certain (integer) counts of BPS states in M-theory [14]. Shortly thereafter work appeared showing that open string theory on O(−1) ⊕ O(−1) is equivalent to Chern-Simons theory [1, 29, 24, 23]. This is a duality between Chern-Simons theory and open string theory built on the conjecture of t’Hooft relating large N gauge theories and string theories. These ideas developed very quickly in the theoretical physics literature. The resulting papers suggest strong links between these distinct mathematical areas. The mathematical communities working on finite-type invariants and Gromov-Witten invariants were and still are largely disjoint. The goal of the workshop was to bring together people working on finite-type invariants, people working on Gromov-Witten invariants and physicists who could explain the recent results in physics linking these two areas. The workshop was an overwhelming success, and is sure to influence the development of mathematics in these two areas. We did succeed in bringing together mathematicians from these two areas and a physicist to tell us why we should interact. We predict that in another five years, the two mathematical disciplines will no longer be separate, but will have many close mathematical connections. Many of the participants at this workshop will contribute to this area of mathematical development, and the mathematical interaction started at this workshop will not just be limited to the participants. We are in the process of preparing a workshop proceedings that will communicate some of the exciting developments in this area to the world-wide mathematical community. We will now give a brief technical overview of the subject matter covered at the workshop. The first area represented at the workshop was finite-type invariants and Chern-Simons theory. The theory of finite type invariants has roots in mathematical physics, statistical mechanics, operator algebras, topology and singularity theory. The physical intuition for these invariants was first explained by E. Witten as a Topological quantum field theory based on the Chern-Simons invariant [39]. The invariant of a link in a 3manifold is described as the path integral,