The partition function in the four-dimensional Schwarz-type topological half-flat two-form gravity

We derive the partition functions of the Schwarz-type four-dimensional topological half-flat two-form gravity model on K3-surface or T-4 up to on-shell one-loop corrections. In this model the bosonic moduli spaces describe an equivalent class of a trio of the Einstein-Kahler forms (the hyper-Kahler forms). The integrand of the partition function is represented by the product of some <(partial derivative)over bar>-torsions. <(partial derivative)over bar>-torsion is the extension of R-torsion for the de Rham complex to that for the <(partial derivative)over bar>-complex of a complex analytic manifold.