Trace-free Einstein gravity as two interacting constrained $BF$ theories

TL;DR

This work develops two fully diffeomorphism-invariant actions for trace-free Einstein gravity, framing the theory as two interacting constrained BF theories with complex variables and built-in reality conditions. The first action uses two constrained BF copies for a Husain-Kuchar-type model with an interaction term, while the second employs two copies of the chiral Plebanski action with an explicit constraint; in both cases the trace-free gravitational sector persists and the cosmological constant arises as an integration constant via the Bianchi identity. The authors demonstrate that the equations of motion of each action are equivalent to the trace-free Einstein equations in Lorentzian signature, and they clarify the role of the variables (such as $\Psi$, $\overline{\Psi}$, $\Phi$, $\overline{\Phi}$) as encoding parts of the curvature in different dual decompositions. These formulations open pathways to quantum gravity approaches, including spin-foam quantization, and motivate further Hamiltonian analyses that could connect to Ashtekar-type variables and loop quantum gravity techniques.

Abstract

A theory of gravity alternative to general relativity is trace-free Einstein gravity, which has the remarkable property that the cosmological constant emerges as an integration constant. In this paper, we report two fully diffeomorphism-invariant actions for trace-free Einstein gravity. They describe the theory as two $BF$ theories supplemented with some constraints. The first action comprises two copies of the constrained $BF$ theory for the Husain-Kuchař model plus an interaction term involving the fields that impose the constraints on the $B$ fields. The second action employs two copies of the chiral Plebanski action for general relativity plus an additional constraint. Both actions use complex variables, and naturally include one of the reality conditions imposed in the Plebanski formulation of general relativity. The new actions have the advantage of not involving any nondynamical fields or unimodular condition, and their only gravitational sector is trace-free Einstein gravity.