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.