BPS states from number theory to knot homology

BPS 从数论到结同调性

• 批准号: SAPIN-2014-00030
• 负责人: Walcher, Johannes
• 金额: $ 4.52万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Subatomic Physics Envelope - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568006
• 关键词: BPS; states; number; theory; knot

In the general field of science, mathematics and physics are neighbouring disciplines. Throughout history, the two subjects have exchanged ideas and results, and benefitted from each other. Traditionally, information can flow in both directions -- requirements of physical theories can draw on an existing mathematical structure, or point to new ones, and mathematical results can confirm physical intuition or establish some hitherto unknown property of the physical theory. The youngest and arguably one of the most promising branches of mathematical physics which manifests this mixture is the one stemming from string theory: Efforts of high-energy physicists trying to understand quantum gravity, and to unify the fundamental interactions, have required the use of sophisticated mathematical machinery and, conversely, influenced or directly contributed new results in pure mathematics. Perhaps somewhat surprisingly, the interaction has been with areas of mathematics that, for large parts of the 20th century, have been disconnected from developments in theoretical physics, most notably algebraic geometry and low-dimensional topology. The new connections not only enrich both fields, but also provide a sustainable source of confidence that string theory is on the right track to becoming the next milestone in humanity's understanding of the Universe, particle physics, and cosmology. This research program is focused on two concrete visions for future progress in the field of mathematical string theory: (1) The unification of topological quantum theory as a kind of simplified version of the all-encompassing theory itself. Unravelling the mysteries of knots and links is the area in which this unification is most likely to first come together; and (2) The connections with number theory. The recent progress in areas ranging from three-manifolds to categorification to Langlands duality to mirror symmetry is evidence that the many parallels discovered over the years will extend much further to the heart of the two subjects. A large and rapidly growing worldwide community of mathematicians and physicists is working at the string-theoretic interface of the two disciplines. A strong testimony to the growing public interest is the establishment of several interdisciplinary research centres, prizes, and conference series around the world. Canadian Universities have made strategic decisions to expand in this area in recent years, as the reputation of our society stands to gain from the advancement of knowledge in those two most fundamental sciences.

在科学的一般领域，数学和物理学是相邻的学科。在整个历史上，这两个主题都交换了思想和结果，并彼此受益。传统上，信息可以在两个方向上流动 - 物理理论的要求可以利用现有的数学结构，或指向新的结构，而数学结果可以确认物理直觉或建立迄今为止迄今未知的物理理论特性。表现出这种混合物的数学物理学最年轻，可以说是最有前途的分支之一是弦理论的一种：高能量物理学家试图理解量子重力的努力，并统一基本的相互作用，并需要使用复杂的数学机械使用，并且相互影响，并在纯数学中产生了新的影响或直接影响纯数学的结果。也许有些令人惊讶的是，与数学领域的相互作用是在20世纪的大部分地区与理论物理学的发展脱节，最著名的是代数几何形状和低维拓扑。新的联系不仅丰富了这两个领域，而且还提供了一种可持续的信心来源，即弦理论正处于正确的轨道上成为人类对宇宙，粒子物理和宇宙学的理解中的下一个里程碑。该研究计划集中在数学字符串理论领域的未来进步的两个具体愿景上：（1）将拓扑量子理论统一作为全包含理论本身的简化版本。揭开结和链接的奥秘是该统一最有可能首先融合在一起的领域。 （2）数字理论的联系。从三个序列到分类再到兰兰二元性能对称对称性的地区的最新进展证明，多年来发现的许多相似之处将进一步扩展到这两个主题的核心。一个大型且迅速发展的数学家和物理学家社区正在研究这两个学科的弦学理论界面。对公众兴趣日益增长的强烈证词是在全球建立了几个跨学科研究中心，奖品和会议系列。近年来，加拿大大学已经做出了战略决策以在这一领域扩展，因为我们社会的声誉从这两种最基本的科学中的知识中得到了进步。