Dirac Geometry and Moduli Spaces

狄拉克几何与模空间。******平面上的平面束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，它们出现在共形场理论、规范理论和弦理论中。在数学上，它们出现在辛几何、代数几何、结理论和表示理论的各种背景中。密切相关的“多边形连杆”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，模空间已经使用泊松几何技术，特别是动量映射理论进行了研究。（在物理学中，泊松流形是量子理论的经典极限，而动量图是对称性的产生者。）虽然这导致了对模空间结构的深刻见解，但有趣的开放问题仍然存在。***在过去的几年中，狄拉克几何已经成为模空间问题研究的一个新工具。狄拉克几何在20世纪90年代被引入，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，但它在数学和物理中有着广泛的应用。它也被发现是通过非线性动量映射（所谓的准哈密顿空间）来接近模空间的有限维方法的适当设置。从物理学的角度来看，狄拉克几何进入了弦理论中的d膜理论以及镜像对称的各个方面。***本研究将从狄拉克几何的角度对模空间和群值动量映射理论做出贡献。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，如表示理论或指标理论。该项目包括许多子主题，非常适合博士生的论文项目

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.**