The mathematics of conformal field theory: a unified approach

共形场论的数学：统一方法

• 批准号: RGPIN-2022-04104
• 负责人: Cliff, Emily
• 金额: $ 1.38万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759317
• 关键词: mathematics; conformal; field; theory; unified

This research program aims to unite three mathematical disciplines, each using its own language to rigorously describe conformal field theory, an area of physics with important applications to string theory, statistical mechanics, and condensed matter physics. The three mathematical approaches to the theory are as follows. Vertex algebras encode the symmetries of a two-dimensional field theory, while chiral algebras encode the collisions between local operators in the theory, and factorization algebras encode the quantum observables of the theory. (Observables are simply things that one can measure in the system, such as mass or velocity of particles.) Mathematically, vertex algebras are algebraic objects, while chiral and factorization algebras are objects in different flavours of geometry.   Because each of the theories approaches the physics from a different perspective, and uses a different set of mathematical techniques, they each have their own strengths and weaknesses. Although each approach is insightful on its own, it is difficult for mathematicians trained in one approach to communicate with those working with the others because they use very different mathematical languages. By building bridges between the three perspectives and combining them, this research program will allow researchers in all three fields to harness the strengths of each approach while overcoming their weaknesses, and thus will lay the foundation for a still more powerful approach to this important subject. An expected outcome of the research program is a mathematical Rosetta Stone, enabling experts in the three disciplines to communicate effectively for the first time and to join forces in their work. In the long term, this will enable me, as well as other researchers, to use new tools to solve a range of long-standing problems. Students and post-doctoral researchers trained in this program will become familiar with the wide range of mathematical techniques needed to understand all three approaches, and will be equipped to continue research in any of these three fields or in a variety of related fields.

该研究计划旨在联合三个数学学科，每个学科都使用自己的语言来严格描述共形场论，这是一个在弦理论、统计力学和凝聚态物理中具有重要应用的物理学领域。该理论的三种数学方法如下。顶点代数编码二维场论的对称性，而手性代数编码理论中局部算子之间的碰撞，分解代数编码理论的量子可观测量。 （可观测量只是人们可以在系统中测量的东西，例如粒子的质量或速度。）在数学上，顶点代数是代数对象，而手性和因式分解代数是不同几何风格的对象。   由于每种理论从不同的角度研究物理学，并使用不同的数学技术，因此它们都有各自的优点和缺点。尽管每种方法本身都很有洞察力，但对于接受过一种方法训练的数学家来说，很难与其他方法的研究人员进行交流，因为他们使用非常不同的数学语言。通过在这三种观点之间建立桥梁并将其结合起来，该研究计划将使所有三个领域的研究人员能够利用每种方法的优点，同时克服其弱点，从而为这一重要课题的更强大的方法奠定基础。 该研究计划的预期成果是一块数学罗塞塔石碑，使三个学科的专家首次能够有效地沟通并在工作中通力合作。从长远来看，这将使我以及其他研究人员能够使用新工具来解决一系列长期存在的问题。在该计划中接受培训的学生和博士后研究人员将熟悉理解所有三种方法所需的广泛数学技术，并将有能力继续在这三个领域中的任何一个或各种相关领域进行研究。