Holomorphic vector field and topological sigma model on ℂP1 worldsheet

Abstract

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold [Formula: see text] with potential terms induced from a Killing vector on [Formula: see text].3. One of the well-known fixed-point theorems is the Bott residue formula9 which represents the intersection number of Chern classes of holomorphic vector bundles on a Kähler manifold [Formula: see text] as the sum of contributions from fixed point sets of a holomorphic vector field [Formula: see text] on [Formula: see text]. In this paper, we derive the Bott residue formula by using the topological sigma model (A-model) that describes dynamics of maps from [Formula: see text] to [Formula: see text], with potential terms induced from the vector field [Formula: see text]. 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 the topological sigma model, we are forced to modify the 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 Witten2 where potential terms induced from holomorphic section of a holomorphic vector bundle and corresponding supersymmetry are considered.