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: $ 4.44万
• 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: 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=589180
• 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.

在一般的科学领域，数学和物理是相邻的学科。纵观历史，这两门学科交流了思想和成果，相互受益。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，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性质。