Moduli Spaces, Quivers, and Duality

对偶性在数学和物理中都扮演着重要的角色。对偶性提供了一对典型的完全不同的理论中不同对象之间的对称对应。这一点很重要，因为在对偶性下，一种理论中的困难问题往往与对偶理论中的较简单问题相对应。两个重要且具有挑战性的二元性是起源于数论的朗兰兹对偶性和镜像对称性，前者是由理论物理学家最先发现的，现在在数学中也发挥着越来越重要的作用。这两种二元性都不是完全理解的。这两个二元性都将在本项目中进行研究。预计这项工作将在看似无关的主题之间建立新的意想不到的联系。这项研究项目将探索一种新的方法来研究与一大类范畴相关的模空间上的指数和与积分。在箭图表示的模空间的情况下，该项目旨在发展Kac猜想的一个新的证明以及一个新的公式，该公式包含一个势并且与Kontsevich和Soibelman引入的Donaldson-Thomas不变量密切相关。该项目还将研究带穿孔的射影代数曲线上抛物(向量)丛的模空间的类似计数问题。

Dualities play a fundamental role in both mathematics and physics. A duality provides a symmetric correspondence between various objects of a pair of typically quite different theories. This is important because a difficult problem in one theory often corresponds, under the duality, to a simpler problem in the dual theory. Two important and challenging dualities are Langlands duality, which has origins in number theory, and mirror symmetry, which was first discovered by theoretical physicists and is playing an increasingly important role in mathematics as well. Neither of these two dualities is fully understood. Both of these dualities will be studied in this project. It is anticipated that the work will establish new and unexpected links between seemingly unrelated topics. This research project will explore a new approach to the study of exponential sums and integrals over moduli spaces associated to a wide class of categories. In the case of the moduli space of quiver representations, the project aims to develop a new proof of the Kac conjecture as well as a new formula, which involves a potential and which is closely related to the Donaldson-Thomas invariants introduced by Kontsevich and Soibelman. The project will also investigate a similar counting problem for the moduli space of parabolic (vector) bundles on a projective algebraic curve with punctures.