Covariant Origin of the $U(1)^3$ model for Euclidean Quantum Gravity

The utility of the U(1)$^3$ model as a test laboratory for quantum gravity has recently been emphasized in a recent series of papers due to Varadarajan et al. The simplification from SU(2) to U(1)$^3$ can be performed simply by hand within the Hamiltonian formulation by dropping all non-Abelian terms from the Gauss, spatial diffeomorphism and Hamiltonian constraints respectively. However, one may ask from which Lagrangian formulation this theory descends. For the SU(2) theory it is known that one can choose the Palatini action, Holst action or (anti-)selfdual action (Euclidian signature) as starting point all leading to equivalent Hamiltonian formulations. In this paper we systematically analyse this question directly for the U(1)$^3$ theory. Surprisingly, it turns out that the Abelian analog of the Palatini or Holst formulation is a consistent but topological theory without propagating degrees of freedom. On the other hand, a twisted Abelian analog of the (anti-)selfdual formulation does lead to the desired Hamiltonian formulation. A new aspect of our derivation is that we work with 1. half-density valued tetrads which simplifies the analysis, 2. without the simplicity constraint (which admits one undesired solution that is usually neglected by hand) and 3. without imposing the time gauge from the beginning. As a byproduct we show that also the non-Abelian theory admits a twisted (anti-)selfdual formulation. Finally we also derive a pure connection formulation of Euclidian GR including a cosmological constant by extending previous work due to Capovilla, Dell, Jacobson and Peldan which may be an interesting starting point for path integral investigations and displays (Euclidian) GR as a Yang-Mills theory with non-polynomial Lagrangian.