The topological B model as a twisted spinning particle

Abstract

The B-twisted topological sigma model coupled to topological gravity is supposed to be described by an ordinary field theory: a type of holomorphic Chern-Simons theory for the open string, and the Kodaira-Spencer theory for the closed string. We show that the B model can be represented as a particle theory obtained by reducing the sigma model to one dimension, and replacing the coupling to topological gravity by a coupling to a twisted one-dimensional supergravity. The particle can be defined on any Kähler manifold—it does not require the Calabi-Yau condition—so it may provide a more generalized setting for the B model than the topological sigma model. The one-loop partition function of the particle can be written in terms of the Ray-Singer torsion of the manifold, and agrees with that of the original B model. After showing how to deform the Kähler and complex structures in the particle, we prove the independence of this partition function on the Kähler structure, and investigate the origin of the holomorphic anomaly. To define other amplitudes, one needs to introduce interactions into the particle. The particle will then define a field theory, which may or may not be the Chern-Simons or Kodaira-Spencer theories.