Moduli Space of Topological 2-form Gravity

We propose a topological version of four-dimensional (Euclidean) Einstein gravity, in which anti-self-dual 2-forms and an SU(2) connection are used as fundamental fields. The theory describes the moduli space of conformally self-dual Einstein manifolds. In the presence of a cosmological constant, we evaluate the index of the elliptic complex associated with the moduli space. e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] Topological gravity is a field theoretic description of the moduli space of gravitational instantons. Such a theory was first considered by Witten [1], where a topological version of conformal gravity in four dimensions was presented as a gravitational analogue of topological Yang-Mills theory (TYMT) [2]. The moduli space of the theory, the space of conformally self-dual gravitational instantons, was investigated in detail by Perry and Teo [3]. Since the work of Witten there have been several attempts to construct such four-dimensional topological theories modeling different gravitational moduli spaces [4]-[8]. In Ref. [9], we proposed a topological version of four-dimensional (Euclidean) Einstein gravity with or without a cosmological constant. This topological version is obtained by modifying an alternative formulation of Einstein gravity developed by Capovilla et al. [10], in which anti-self-dual 2-forms and an SU(2) connection are used as fundamental fields, instead of the metric or the tetrad. With an appropriate choice of gauge condition, the BRST invariant quantum action of the topological theory becomes the classical Einstein action plus ghost terms which cancel out all local degrees of freedom. However there still remain zero-modes in the quantum action. The (finite) number of them is closely related to the dimension of the moduli space, which now consists of Einstein manifolds with self-dual Weyl tensor. When the cosmological constant is non-zero, the moduli space is up to orientation, identical with the one considered by Torre in which the Weyl tensor is anti-self-dual [4]. In his paper the dimension of the moduli space is found to be zero when the cosmological constant is positive, and the result is true also in our case. In the case of non-zero cosmological constant, we evaluate the index of an elliptic complex associated with our moduli space by applying the Atiyah-Singer index theorem. We also discuss the case of vanishing cosmological constant and mention the dimension of the moduli space on the K3 surface. We start with fundamental fields, a trio of 2-forms Σ, which transform under the chiral local-Lorentz representation (2, 0) of SU(2)L×SU(2)R, and a connection 1-form ω associated with the SU(2)L. It is shown that Σ k and ω are anti-self-dual