BPS states from number theory to knot homology

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。表现这种混合的最年轻、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的努力，需要使用复杂的数学机器，反过来，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，在20世纪的大部分时间里，与理论物理发展脱节的数学领域，尤其是代数几何和低维拓扑学，一直在相互作用。新的联系不仅丰富了这两个领域，而且还提供了一个持续的信心来源，即弦理论正走在正确的轨道上，成为人类对宇宙、粒子物理学和宇宙学理解的下一个里程碑。本研究计划集中于数学弦论领域未来发展的两个具体愿景：(1)将拓扑量子理论统一为包揽一切的理论本身的一种简化版本。解开结和联系的奥秘是这种统一最有可能首先走到一起的领域；(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，这些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学科的核心。一个庞大且快速增长的世界数学家和物理学家社区正在研究这两个学科的弦论接口。世界各地建立了多个跨学科研究中心、设立奖项和举办系列会议，这有力地证明了公众对这一领域日益增长的兴趣。近年来，加拿大的大学已经做出了在这一领域扩张的战略决定，因为我们社会的声誉将从这两门最基础的科学的知识进步中获益。

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.