The mathematics of conformal field theory: a unified approach

这项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保形场理论，保形场理论是物理学的一个领域，在弦理论、统计力学和凝聚态物理中有重要的应用。这一理论的三种数学方法如下。顶点代数编码二维场理论的对称性，手征代数编码理论中局部算符之间的碰撞，因式分解代数编码理论的量子能观。(可观测性仅仅是人们可以在系统中测量的东西，比如粒子的质量或速度。)从数学上讲，顶点代数是代数对象，而手征和因式分解代数是几何中不同风格的对象。由于每种理论从不同的角度接近物理，使用不同的数学技巧，所以它们都有自己的优点和缺点。尽管每种方法本身都很有洞察力，但受过一种方法训练的数学家很难与那些与其他方法一起工作的人交流，因为他们使用的是非常不同的数学语言。通过在这三个视角之间搭建桥梁并将它们结合在一起，这一研究计划将使这三个领域的研究人员能够利用每种方法的优势，同时克服它们的弱点，从而为更有力地解决这一重要课题奠定基础。该研究项目的预期成果是数学罗塞塔·斯通，使这三个学科的专家能够首次有效地沟通，并在工作中联合起来。从长远来看，这将使我和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。在这个项目中培训的学生和博士后研究人员将熟悉理解所有这三种方法所需的广泛的数学技术，并将具备继续在这三个领域中的任何一个或在各种相关领域进行研究的能力。

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.