Holomorphic Vector Field and Topological Sigma Model on CP^1 World Sheet

Abstract

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold M with potential term induced from Killing vector on M. One of the well-known fixed-point theorem is the Bott residue formula which represents intersection number of Chern classes of holomorphic vector bundles on a Kahler manifold M as sum of contributions from fixed point sets of a holomorphic vector field K on M. In this paper, we derive the Bott residue formula by using topological sigma model (A-model) that describes dynamics of maps from CP^{1} to M, with potential term induced from the vector field $K$. Our strategy is to restrict phase space of path integral to maps homotopic to constant maps. As an effect of adding a potential term to topological sigma model, we are forced to modify BRST symmetry of the original topological sigma model. Our potential term and BRST symmetry are closely related to the idea used in the paper by Beasley and Witten where potential terms induced from holomorphic section of a holomorphic vector bundle and corresponding supersymmetry are considered.