Holomorphic vector field and topological sigma model on ℂp¹worldsheet

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold M with potential terms induced from a Killing vector on M.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 M as the 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 the topological sigma model (A-model) that describes dynamics of maps from ℂP1 to M, with potential terms 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 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.