Moduli Spaces in the Four-Dimensional Topological Half-Flat Gravity

We classify the moduli spaces of the four-dimensional topological half-flat gravity models by using the canonical bundle. For a K3-surface or T 4 , they describe an equivalent class of a trio of the Einstein-Kähler forms (the hyperkähler forms). We calculate the dimensions of these moduli spaces by using the Atiyah-Singer Index theorem. We mention the partition function and the possibility of the observables in the Witten-type topological half-flat gravity model case. I. Introduction Recently, Witten gave some gravitational versions of topological quantum field theories [1]. These theories are important as the effective theories of the ordinal gravity theories. For example, he pointed out the relation between the two-dimensional topo-logical gravity models and the string theory [1]. He also conjectured that N = 4 topological twisted supersymmetric Yang-Mills theory on four-dimensional manifold satifies S-duality and has a link with the bosonic string or two-dimensional rational conformal field theories [2]. They seem to give the new light on the non-perturbative effect of the string theories and the gravity theories. Since the work of Witten, there have been several attempts to construct four-dimensional topological gravity theories over different kind of the gravitational moduli spaces [3]-[5]. There two types of models have been proposed for the four-dimensional half-flat 2-form topological gravity. (A) Witten-type topological gravity model, which was given by Kunitomo [5] and (B) Schwarz-type topological gravity model which we proposed [6, 7]. The base of their formalism are given by ref. [8, 9]. The interesting relation between the half-flat gravity and 2-dim. conformal field theory is investigated by Park [10]. In the previous paper we showed that by taking the suitable gauge fixing condition and the limit of the coupling constant for (B), the bosonic part of the moduli spaces of (B) coincides with that of (A). These moduli spaces are those of the Einstein Kählerian manifolds with vanishing real first Chern class. The purpose of this letter is to calculate the dimensions of the moduli spaces by using Atiyah-Singer index theorem and discuss about the possibility of the topological invariants such as the partition function and observables. We concentrate our attention for K3-surface and T 4 cases mainly in this paper. Their moduli spaces are identified with the deformation of a trio of the Einstein-Kähler forms (the hyperkähler forms) which is related to the Plebansky's heavenly equations [11]. We also discuss the partition function for the Witten-type model case and mension …