Dirac Geometry and Moduli Spaces

狄拉克几何与模空间。 曲面上平坦丛的模空间一直是数学和物理学中深入研究的课题。在物理方面，它们出现在保形场理论、规范理论和弦理论中。从数学上讲，它们出现在辛几何、代数几何、纽结理论和表示论的各种背景下。密切相关的“多边形连杆”的模空间在几何力学和机器人学中发挥着重要作用。 从20世纪80年代Atiyah-Bott和Witten的工作开始，利用Poisson几何的技巧，特别是动量映射理论，研究了模空间。(在物理学中，泊松流形是量子理论的经典极限，而动量映射是对称性的生成器。)虽然这导致了对模空间结构的深入了解，但有趣的悬而未决的问题仍然存在。 在过去的几年里，狄拉克几何作为一种新的工具出现在模空间问题的研究中。狄拉克几何诞生于20世纪90年代，是泊松几何的深远推广。它最初的目的是为具有约束的经典机械系统提供一个几何框架，但事实证明它在数学和物理中有广泛的应用。这也被发现是通过非线性动量映射，即所谓的准哈密顿空间，对模空间的有限维方法的适当设置。从物理学的角度来看，狄拉克几何学进入了弦理论中的D-膜理论以及镜像对称性的方面。 这一研究将有助于从狄拉克几何的角度研究模空间和群值动量映射理论。它的预期应用是在数学物理中，例如D-膜理论和无限维哈密顿系统理论，以及在纯数学领域中，例如表示理论或指数理论。该项目包括一些副主题，它们将非常适合作为博士生的论文项目。

Dirac Geometry and Moduli Spaces. Moduli spaces of flat bundles over surfaces have been the subject of intensive investigation in mathematics and physics. On the physics side they appear in conformal field theory, gauge theory and string theory. Mathematically they arise in a variety of contexts in symplectic geometry, algebraic geometry, knot theory, and representation theory. The closely related moduli spaces of `polygonal linkages' play a role in geometric mechanics and robotics. Beginning with the work of Atiyah-Bott and Witten in the 1980s, the moduli spaces have been studied using techniques from Poisson geometry, and specifically the theory of momentum maps. (In physics, Poisson manifolds arise as classical limits from quantum theories, while momentum maps are the generators of symmetries.) While this has led to deep insights into the structure of the moduli spaces, interesting open questions remain. Over the past few years, Dirac geometry has emerged as a new tool in the study of moduli space problems. Introduced in the 1990s, Dirac geometry is a far-reaching generalization of Poisson geometry. Its original purpose was to provide a geometric framework for classical mechanical systems with constraints, but it turned out to have a wide range of applications in mathematics and physics. It was also found to be the appropriate setting for the finite-dimensional approach to moduli spaces via non-linear momentum maps, the so-called quasi-Hamiltonian spaces. From the physics perspective, Dirac geometry enters the theory of D-branes in string theory as well as aspects of mirror symmetry. This research will contribute to the theory of moduli spaces and group-valued momentum maps from the perspective of Dirac geometry. Its expected applications are in mathematical physics, for example the theory of D-branes and the theory of infinite-dimensional Hamiltonian systems, as well as in areas of pure mathematics such as representation theory or index theory. The project includes a number subtopics that will be well-suited as thesis projects for Ph.D. students.