Barbero--Immirzi--Holst Lagrangian with Spacetime Barbero--Immirzi Connections

We carry out the complete variational analysis of the Barbero--Immirzi--Holst Lagrangian, which is the Holst Lagrangian expressed in terms of the triad of fields $(\theta, A, \kappa)$, where $\theta$ is the solder form/spin frame, $A$ is the spacetime Barbero--Immirzi connection, and $\kappa$ is the extrinsic spacetime field. The Holst Lagrangian depends on the choice of a real, non zero Holst parameter $\gamma \neq 0$ and constitutes the classical field theory which is then quantized in Loop Quantum Gravity. The choice of a real Immirzi parameter $\beta$ sets up a one-to-one correspondence between pairs $(A, \kappa)$ and spin connections $\omega$ on spacetime. The variation of the Barbero--Immirzi--Holst Lagrangian is computed for an arbitrary pair of parameters $(\beta, \gamma)$. We develop and use the calculus of vector-valued differential forms to improve on the results already present in literature by better clarifying the geometric character of the resulting Euler--Lagrange equations. The main result is that the equations for $\theta$ are equivalent to the vacuum Einstein Field Equations, while the equations for $A$ and $\kappa$ give the same constraint equation for any $\beta \in \mathbb{R}$, namely that $A + \kappa$ must be the Levi--Civita connection induced by $\theta$. We also prove that these results are valid for any value of $\gamma \neq 0$, meaning that the choice of parameters $(\beta, \gamma)$ has no impact on the classical theory in a vacuum and, in particular, there is no need to set $\beta = \gamma$.