The real Chern-Simons wave function

TL;DR

This paper demonstrates that a real-valued formulation of gravity admits a modified Chern-Simons wave function that is well-behaved and solves the constraints, addressing issues faced by the complex theory.

Contribution

It introduces a real connection and metric-based approach to the Chern-Simons wave function, making it non-pathological and applicable to quasi-topological theories with a dynamical cosmological constant.

Findings

The real Chern-Simons state is a pure phase in Lorentzian theory.

The state solves the constraints in a real connection formulation of gravity.

The modification applies to quasi-Euler theories with a critical coupling.

Abstract

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