Gromov-Witten Theory

DMS-03401275Tom Coats首席研究员将研究Gromov-Witten理论中的许多主题。这些话题被一个共同的主题联系在一起：一个关于Gromov的新视角--由Givental提出的Witten理论，它揭示了与辛线性代数、几何量子化和环群理论之间令人惊讶的联系。这项研究的一个目的是了解这个观点所揭示的结构的几何起源；第二个也是主要的目的是应用这个观点提供的新的计算工具：1.用于辛商的Gromov-Witten不变量的计算；2.用于高亏格Gromov-Witten不变量的计算；3.非半单Frobenius流形的结构理论；4.量子K理论和量子非常上同调中的各种结构和计算问题。为了理解弦理论的一些数学结果，Gromov-Witten不变量在过去的十年里得到了深入的研究。当一根弦穿过空间时，它扫出了一个曲面：空间的Gromov--Witten不变量描述了空间中满足某些条件的这些曲面的数量-例如，可以坚持这些曲面通过一些不同的点，或者它们与一条特定的曲线相交。Gromov-Witten不变量提供了有关空间形状的重要信息。它们在镜像对称中起着至关重要的作用，镜像对称是一个用数学形式表示两个不同弦理论等价性的思想圈。镜像对称意味着我们可以通过研究称为X的镜像的空间X上的微分方程来计算X中的曲面(即计算X的Gromov-Witten不变量)。这从数学和物理的角度来看都很有趣：一个多世纪以来，数学家们一直在研究曲面计数和求解微分方程，但没有意识到这些主题彼此之间没有任何关系。到目前为止，弦理论最令人信服的一些证据不是来自实验，而是来自对其数学预测的逐步验证。

DMS-03401275Tom CoatesThe principal investigator will study a number of topics in Gromov-Witten theory. These topics are linked by a common theme: a new perspective on Gromov--Witten theory, introduced by Givental, which reveals surprising connections with symplectic linear algebra, geometric quantization and the theory of loop groups. One objective of this research is to understand the geometric origin of the structures revealed by this point of view; the second and main objective is to apply the new computational tools that this perspective provides: 1. To the calculation of Gromov--Witten invariants of symplectic quotients; 2. To the computation of higher-genus Gromov--Witten invariants; 3. To the structure theory of non-semisimple Frobenius manifolds; and 4. To various structural and computational questions in quantum K-theory and quantum extraordinary cohomology.Gromov--Witten invariants have been studied intensively for the past decade in an effort to understand some of the mathematical consequences ofstring theory. As a string moves through space, it sweeps out a surface: the Gromov--Witten invariants of a space describe the number of these surfaces in the space which satisfy certain conditions --- for example,one could insist that the surfaces pass through a number of different points, or that they intersect with a particular curve. Gromov-Witten invariants give important information about the shape of a space. They play a vital role in Mirror Symmetry, a circle of ideas which expresses in mathematical form the equivalence of two different string theories.Mirror symmetry suggests that we can count surfaces in a space X (i.e. compute Gromov--Witten invariants of X) by studying differential equations on a space X' called the mirror of X. This is interesting from the point of view of both mathematics and physics: mathematicians had studied surface-counting and solving differential equations for more than a century without realising that the subjects had anything to do with one other, and some of the most convincing evidence for string theory to date comes not from experiment but from the gradual verification of its mathematical predictions.