Vertex Operator Algebras and Mathematical Conformal Field Theory

点击翻译按钮获取中文摘要

Huang and Lepowsky intend to solve a range of problems related tovertex (operator) algebra theory and conformal field theory, and theirrelations with and applications to a variety of areas of mathematicsand physics. They intend to use new and evolving algebraic, geometricand analytic ideas and methods. Huang proposes to continue to developan algebraic, analytic and geometric theory underlying conformal fieldtheory and to apply the results obtained to the solution of problemsin algebra, geometry and topology. Lepowsky intends to continue todevelop vertex operator algebra theory and its relations withconformal field theory, infinite-dimensional algebraic structures,number theory and the theory of combinatorial identities. Huang andLepowsky also propose to solve problems related to the remarkablecircle of ideas called ``monstrous moonshine,'' to geometricuniformizations and to logarithmic tensor product theory. The variedtopics discussed in this proposal are in fact deeply connected withone another, and the solutions of certain of these problems areexpected to be useful in the analysis and solution of others. Along-term goal of Huang and Lepowsky is to contribute to the deeperdevelopment of a mathematical theory that will enhance the conceptualunderstanding of many known and still-unknown connections among manybranches of mathematics and theoretical physics. Huang and Lepowskywill continue to be very active in their teaching, writing, organizingof conferences and seminars, and their other activities designed toincrease and enhance the understanding in the mathematical communityof the very rich field in which they work. They will also continue totry to convey to members of the theoretical physics community, throughwriting and their speaking, their enthusiasm for the richness andutility of their approach to vertex operator algebra theory andmathematical conformal field theory.The theory of vertex operator algebras arose naturally in the study ofcertain infinite-dimensional symmetries and in the solution offundamental problems relating the ``Monster,'' a certain finitesymmetry group, with number theory, and this theory is fundamental toa wide range of problems in both mathematics and theoretical physics.Vertex operator algebras are crucial ingredients in the mathematicalstudy of conformal field theory, a physical theory that arose in bothcondensed matter physics and string theory, which in turn is aphysical theory attempting to unify all the interactions in theuniverse. Conformal field theory is in the process of being rapidlydeveloped into a rich and beautiful mathematical theory. Thisdevelopment has already led to new ideas, surprising results and thesolutions of many mathematical problems, and it is expected to lead tomany more. The mathematical development of conformal field theory hasalso been applied to the study of physical phenomena such as disordersytems, condensed matter physics on surfaces with boundaries andcertain higher-dimensional physical objects called D-branes in stringtheory.