# Holst action

$S={\frac {1}{2}}\int ee_{\ I}^{\alpha }e_{\ J}^{\beta }(F_{\alpha \beta }^{\ \ \ IJ}-\alpha \ast F_{\alpha \beta }^{\ \ \ IJ})\equiv {\frac {1}{2}}\int ee_{\ I}^{\alpha }e_{\ J}^{\beta }(F_{\alpha \beta }^{\ \ \ IJ}-{\frac {\alpha }{2}}\epsilon _{\;\;\;KL}^{IJ}F_{\alpha \beta }^{\ \ \ KL})$ As with the first order tetradic Palatini action where $e_{\ I}^{\alpha }$ and $A_{\alpha \beta }^{\ \ \ IJ}$ are taken to be independent variables, variation of the action with respect to the connection $A_{\alpha \beta }^{\ \ \ IJ}$ (assuming it to be torsion-free) implies the curvature $F_{\alpha \beta }^{\ \ \ IJ}$ be replaced by the usual (mixed index) curvature tensor $R_{\alpha \beta }^{\ \ \ IJ}$ (see article tetradic Palatini action for definitions). Variation of the first term of the action with respect to the tetrad $e_{\ I}^{\alpha }$ gives the (mixed index) Einstein tensor and variation of the second term with respect to the tetrad gives a quantity that vanishes by symmetries of the Riemann tensor (specifically the first Bianchi identity), together these imply Einstein's vacuum field equations hold.
The canonical 3+1 Hamiltonian formulation of the Holst action with $\alpha =i$ happens to correspond to Ashtekar variables which formulates (complex) GR as a special type of Yang-Mills gauge theory. The action was seen simply to be the Palatini action with the curvature tensor replaced by its self-dual part only (see article self-dual Palatini action).
The canonical 3+1 Hamiltonian formulation of the Holst action for real $\alpha$ was shown to have a configuration variable which is still a connection, and the theory still a special kind of Yang-Mills gauge theory, but has the advantage that it is real, as is then the corresponding gauge theory (so we are dealing with real General Relativity). This Hamiltonian formulation is the classical starting point of loop quantum gravity (LQG) which imports non-perturbative techniques from lattice gauge theory. The parameter defined by $\beta :=1/\alpha$ is usually referred to as the Barbero-Immirzi parameter The Holst action finds application in most recent versions of spin foam models, which can be considered path integral versions of LQG.