Modular invariance of (logarithmic) intertwining operators


主讲人:黄一知 美国罗格斯大学(Rutgers University)教授




主讲人介绍:黄一知,美国罗格斯大学(Rutgers University)教授,国际上著名的顶点算子理论和理论物理学专家,主要研究兴趣是建立量子场理论的数学基础,及其在代数学,拓扑学,几何学,凝聚态物理和弦理论上的应用,他的代表性研究工作包括建立公理化的顶点算子代数的定义,顶点算子代数的张量范畴理论的研究,顶点算子代数框架下一般形式的Verlinde猜想的证明,并以此为基础证明了大量的重要定理等。黄一知教授出版学术专著一部,撰写和发表研究论文80余篇,多数发表在国际顶尖数学杂志上,如《Duke Mathematical Journal》,《Communications in Mathematical Physics》等,他引次数超过1600次。黄一知教授还是国际知名数学杂志《Communications in Contemporary Mathematics》的主编,《New York Journal of Mathematics》的编委等。

内容介绍:I will discuss a proof of a conjecture of almost twenty years on the modular invariance of (logarithmic) intertwining operators. Let V be a C_2-cofinite vertex operator algebra without nonzero elements of negative weights. The conjecture states that the vector space spanned by pseudo-q-traces shifted by -c/24 of products of (logarithmic) intertwining operators among grading-restricted generalized V-modules is a module for the modular group SL(2, Z). In 2015, Fiordalisi proved that such pseudo-q-traces are absolutely convergent and have the genus-one associativity property and some other properties. Recently, I have proved this conjecture completely. This modular invariance result gives a construction of C_2-cofinite genus-one logarithmic conformal field theories. We expect that it will play an important role in the study of problems and conjectures on C_2-cofinite logarithmic conformal field theories. The talk will start with the meaning of modular transformations and the definition of vertex operator algebras.