New Vector Field and BRST Charges in 2-form Einstein Gravity

A new vector field is introduced into 2-form Einstein gravity in four dimensions to restore a large symmetry of its topological version. Two different expressions for the BRST charge are given in the system: one of them associated with a set of irreducible symmetries and the other with a set of on-shell reducible symmetries. ⋆ e-mail address: [email protected] † e-mail address: [email protected] Several approaches have been developed in recent studies of quantum gravity to clarify its various aspects. Among them, an approach based on a class of topological quantum field theories, called topological gravity, is expected to elucidate the global (topological) aspects of gravity. In Ref.[1], Lee and the present authors proposed a topological version of four-dimensional Einstein gravity. This topological version is obtained by modifying an alternative formulation of gravity enlightened by Capovilla et al.[2], in which anti-self-dual 2-forms are used as fundamental variables, instead of the metric. This formulation, which we call 2-form Einstein gravity, naturally leads to the canonical formalism discovered by Ashtekar [3]. Investigating the relation between 2-form Einstein gravity and its topological version, we found that, in the presence of a cosmological term, a unique quantum state in the latter turns out to be one of the physical states in the former, that is, one of the exact solutions of the Wheeler-DeWitt equation. The topological version is promoted back to Einstein gravity by adding a certain term in the action and, due to the term, a large topological symmetry is partially broken leaving diffeomorphism and local-Lorentz symmetries. The situation is similar to the one between the massless and massive U(1) gauge theories. As is well known, the gauge invariance of the former is explicitly broken by the mass term. We then encounter singularities in the zero-mass limit because there is no gauge prescription in the limiting action. To avoid the singularities, one needs the Stueckelberg field which restores the broken gauge invariance in the massive theory [4]. In this paper, we introduce a new vector field analogous to the Stueckelberg field into 2-form Einstein gravity to restore the large symmetry of its topological version. This new field will be useful for relating Einstein gravity to its topological