Twisted K-theory and moduli spaces

扭曲 K 理论和模空间

• 批准号: 203124-2011
• 负责人: Meinrenken, Eckhard
• 金额: $ 3.42万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569868
• 关键词: Twisted; theory; moduli; spaces

Moduli spaces of flat bundles over surfaces have been intensively studied in both mathematics and physics. On the physics side they appear in conformal field theory, gauge theory and string theory, while mathematically they arise in a variety of contexts in symplectic geometry, algebraic geometry, knot theory, and representation theory. Developments in quantum field theory in the 1990's led to predictions about the geometry and topology of these moduli spaces, some of which were later confirmed by rigorous mathematical arguments. In 1998, a new construction of the moduli spaces was discovered, involving the so-called q-Hamiltonian spaces. In contrast to earlier approaches, this construction is entirely finite-dimensional, but it involves a new type of `nonlinear moment map'. The quantization of such q-Hamiltonian spaces was developed quite recently. The technique of `twisted K-theory' used in this quantization procedure had gained popularity in theoretical physics, and specifically the `D-branes on group manifolds' considered in string theory are very closely related to q-Hamiltonian spaces. The proposed research will study open questions concerning the quantization procedure for q-Hamiltonian spaces, and explore its applications. We want to work out explicit formulas for the quantization of certain classes of moduli spaces, which had been out of reach with the more traditional techniques. We also plan to develop a general framework for nonlinear moment maps, unifying and generalizing the various existing settings. Apart from its intrinsic mathematical interest, this research will contribute to our understanding of 2-dimensional gauge theory, and of moduli spaces arising in mathematics and physics.

曲面上平坦丛的模空间在数学和物理学中有着广泛的研究。在物理学方面，它们出现在共形场论、规范理论和弦理论中，而在数学上，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中。20世纪90年代量子场论的发展导致了对这些模空间的几何和拓扑的预测，其中一些后来被严格的数学论证所证实。 1998年，模空间的一种新构造被发现，涉及所谓的q-哈密尔顿空间。与以前的方法相比，这种结构是完全有限维的，但它涉及到一种新型的“非线性矩映射”。这种q-哈密顿空间的量子化是最近才发展起来的。在量子化过程中使用的“扭曲K理论”技术在理论物理学中很受欢迎，特别是在弦理论中考虑的“群流形上的D膜”与q-哈密顿空间非常密切相关。 本研究将探讨q-哈密尔顿空间量子化过程的相关问题，并探讨其应用。我们希望为某些类模空间的量子化给出明确的公式，这是传统技术所无法达到的。我们还计划开发一个非线性矩映射的一般框架，统一和推广现有的各种设置。除了其内在的数学兴趣，这项研究将有助于我们了解 二维规范理论，以及数学和物理学中出现的模空间。