Mathematical Sciences/GIG: A Team Approach to String Theory

这项活动旨在促进哈佛数学系丘成桐和哈佛物理系Cumrun Vafa领导的研究小组之间的跨学科合作。 Vafa的小组在沿着几个方面进一步加深了我们对弦理论非微扰方面的理解：将Seiberg-Witten结果推导为弦理论中预势模依赖性的低能极限;研究D-膜物理并将BPS状态的计数与拓扑问题联系起来，包括K3上的瞬子;将IIB型弦的SL（2，Z）不变性几何化，并假设一个十二维理论（“F理论”），其中包含许多有趣的弦对偶;发展几何理解非阿贝尔规范对称性和限制弦理论;并通过弦理论和弦对偶性用计数态描述黑洞的Beckenstein-Hawking熵。最后一个结果也许是弦理论在量子引力方面最惊人的发现。 丘氏群在理解镜像对称性和弦对偶性的几个方面起到了重要作用：用枚举几何解释BPS态计数;研究Calabi-Yau流形上周期积分的微分方程;解决大半径极限问题，这对于将镜像对称性应用于枚举几何至关重要;将镜像流形描述为超对称子流形的模空间;和分类K3纤维化，这是异质II型对偶的关键。 该GIG奖将资助数学和弦理论（物理）的博士后研究员和访问学者。 弦理论在过去的一年里经历了一场革命，但如果没有新的数学结果，弦理论将枯萎-因为许多理论需要严格的证明！ 数学的步伐需要加快，以适应最近涌入的这些非凡的想法。 这对数学来说也是一个激动人心的时刻。 大量的确凿证据，物理推理为严肃的调查提供了指导。 同样，弦论物理学也渴望新的数学，这两个群体拥有丰富的智力资源。 数学和物理学都将大大受益于充分利用这一跨学科的合作。

This activity is intended to foster the interdisciplinary collaboration between the research groups headed by Shing-Tung Yau in the Harvard Mathematics Department and Cumrun Vafa in the Harvard Physics Department. Vafa's group has furthered our understanding of the non-perturbative aspects of string theory along several fronts: deriving the Seiberg-Witten results as a low-energy limit of the moduli dependence of the prepotential in string theory; studying D-brane physics and relating the counting of BPS states to topological questions, including instantons on K3; geometrizing the SL(2,Z) invariance of the Type IIB string and postulating a twelve-dimensional theory (``F-theory"), which contains many interesting string dualities; developing a geometric understanding of non-abelian gauge symmetries and confinement in string theory; and describing the Beckenstein-Hawking entropy of black holes in terms of counting states via string theory and string duality. This last result is perhaps the most striking revelation in quantum gravity from string theory. Yau's group has been instrumental in understanding several aspects of mirror symmetry and string dualities: interpreting BPS state counting in terms of enumerative geometry; investigating the differential equations of period integrals on Calabi-Yau manifolds; solving the large radius limit problem in general, which is crucial for applying mirror symmetry to enumerative geometry; describing mirror manifolds as the moduli spaces of supersymmetric submanifolds; and classifying K3 fibrations, which are the key to heterotic-Type II duality. This GIG Award will sponsor postdoctoral research fellows and visiting scholars in mathematics and string theory (physics). String theory has undergone a revolution during the past year but will languish without fresh mathematical results --- for the many conjectures need to be justified with rigor! The pace of mathematics needs to quicken in order to accommodate this recent influx of remarkable ideas. T his is a very exciting time for mathematics, as well. Hard conjectures abound, with physical reasoning providing guidance for serious investigation. Likewise, the physics of string theory is hungry for new math. Within these two groups there is a wealth of intellectual resources. Mathematics and physics will both greatly benefit from full exploitation of this interdisciplinary collaboration.