In the field of theoretical physics, the Holst action is an equivalent formulation of the Palatini action for General Relativity (GR) in terms of vierbeins (4D space-time frame field) by adding a part of a topological term (Nieh-Yan) which does not alter the classical equations of motion as long as there is no torsion,
S = 1 2 ∫ e e I α e J β ( F α β I J − α ∗ F α β I J ) ≡ 1 2 ∫ e e I α e J β ( F α β I J − α 2 ϵ K L I J F α β K L ) {\displaystyle 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})}
where e I α {\displaystyle e_{\ I}^{\alpha }} is the tetrad, e {\displaystyle e} its determinant (the space-time metric is recovered from the tetrad by the formula g α β = e α I e β J η I J {\displaystyle g_{\alpha \beta }=e_{\alpha }^{I}e_{\beta }^{J}\eta _{IJ}} where η I J {\displaystyle \eta _{IJ}} the Minkowski metric), F α β I J {\displaystyle F_{\alpha \beta }^{\ \ \ IJ}} the curvature considered as a function of the connection A α β I J {\displaystyle A_{\alpha \beta }^{\ \ \ IJ}} :
F α β I J = A α β I J = 2 ∂ [ α A β ] I J + 2 A [ α I K A β ] K J {\displaystyle F_{\alpha \beta }^{\ \ \ IJ}={A_{\alpha \beta }}^{IJ}=2\partial _{[\alpha }{A_{\beta ]}}^{IJ}+2{A_{[\alpha }}^{IK}{A_{\beta ]K}}^{J}} ,
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