Lecture II  2012年1月6日（周五）上午10:00

Conjectures by physicists on nonlinear sigma models are one

of the most influential sources of inspirations and motivations for many works in geometry in the past two or three decades. Unfortunately nonlinear sigma models are still not mathematically constructed.  In my talk, I will discuss the first step in a program to construct these models mathematically using Riemannian geometry and the representation theory of vertex operator algebras.

Given a Riemannian manifold M, for an open ball centered at a point on M with radius less than or equal to the injectivity radius at the point, using normal coordinates at points in the oepn ball, we construct a module for a Heisenberg algebra generated by the space of smooth functions such that on smooth functions, L(0) acts as the Laplacian . Using these modules, modules for certain subalgebras of the Heisenberg algebras and the gluing method, we construct a sheaf of weak modules for a sheaf of vertex operator algebras generated by the sheaf of smooth functions. In particular,we construct an L(0)-semisimple lower-bounded generalized module for the vertex operator algebra of global sections on M generated by an eigenfunction of the Laplacian on M.