【摘要】
The study is motivated by constructing the two-dimensional quantum nonlinear sigma model with the nonflat target manifold via vertex algebraic methods. When the target manifold is not flat, the zero modes of the vertex operators act via covariant derivatives. Thus, the vertex operators are expected to be noncommutative. To understand such vertex operators, the first step is to understand the covariant derivatives. In this talk, I will report my work on the covariant derivatives of eigenfunctions along the parallel tensors. Surprisingly, over a Riemannian manifold with constant nonzero sectional curvature, all the covariant derivatives are scalar multiples of the eigenfunction. We suspect that over Kahler and Calabi-Yau manifolds, the covariant derivatives of an eigenfunction might span a finite-dimensional space.
