Real Chern-Simons wave function

We examine the status of Chern-Simons (or Kodama) state from point view a formulation gravity that uses only real connection and metric variables action. may package into complex self-dual Ashtekar (and will do so to make contact with previous work), but operation is essentially cosmetic can be undone at any step or even bypassed altogether. The action remain (real) Einstein-Cartan action, forgoing addition usual Holst Nieh-Yan) term an imaginary coefficient. It then found constraints are solved by modification which pure phase (in Lorentzian theory, we stress), containing fully gauge-invariant part functional. Thus, for ``real theory'' nonpathological regards most egregious criticisms facing its ``nonreal'' cousin, solving theory. A straightforward also solution in quasitopological theories based on Euler invariant, cosmological constant, $\mathrm{\ensuremath{\Lambda}}$, dynamical. In case it enough shift factor $\mathrm{\ensuremath{\Lambda}}$ wave function inside spatial integral. trick works quasi-Euler theory critical coupling previously identified literature. does not apply quasi-Pontryagin