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.

在一般科学领域，数学和物理学是相邻的学科。纵观历史，两个学科不断交流思想和成果，互利共赢。传统上，信息可以双向流动——物理理论的要求可以借鉴现有的数学结构，或者指向新的数学结构，而数学结果可以证实物理直觉或建立物理理论的一些迄今为止未知的属性。体现这种混合的最年轻、可以说是最有前途的数学物理学分支之一是源于弦理论的分支：高能物理学家试图理解量子引力并统一基本相互作用，需要使用复杂的数学机器，相反，影响或直接贡献了纯数学的新结果。也许有些令人惊讶的是，这种相互作用发生在数学领域，而在 20 世纪的大部分时间里，这些领域与理论物理学的发展脱节，尤其是代数几何和低维拓扑。这些新的联系不仅丰富了这两个领域，而且还提供了一个可持续的信心来源，即弦理论正走在正确的轨道上，成为人类理解宇宙、粒子物理学和宇宙学的下一个里程碑。该研究计划重点关注数学弦理论领域未来进展的两个具体愿景：（1）将拓扑量子理论统一为包罗万象的理论本身的一种简化版本。解开绳结和链节的神秘面纱是这种统一最有可能首先实现的领域。 (2)与数论的联系。从三流形到分类，从朗兰兹对偶到镜像对称等领域的最新进展证明，多年来发现的许多相似之处将进一步延伸到这两个主题的核心。全球范围内一个庞大且快速发展的数学家和物理学家团体正在研究这两个学科的弦理论接口。世界各地多个跨学科研究中心、奖项和系列会议的建立有力地证明了公众兴趣的日益增长。近年来，加拿大大学做出了在这一领域进行扩张的战略决策，因为我们社会的声誉将从这两门最基本科学知识的进步中获益。