Gravitational instantons and moduli spaces in topological two-form gravity.

A topological version of four-dimensional (Euclidean) Einstein gravity which we propose regards anti-self-dual 2-forms and an anti-self-dual part of the frame connections as fundamental fields. The theory describes the moduli spaces of conformally self-dual Einstein manifolds for a cosmological constant Λ 6= 0 case and Einstein-Kählerian manifold with the vanishing real first Chern class for Λ = 0. In the Λ 6= 0 case, we evaluate the index of the elliptic complex associated with the moduli space and calculate the partition function. We also clarify the moduli space and its dimension for Λ = 0 which are related to the Plebansky’s heavenly equations. ∗On leave of absence from Tokyo Institute of Technology after April 1994.