Holst action

In the field of theoretical physics, the Holst action[1] 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,

${\displaystyle S={\frac {1}{2}}\int ee_{\ I}^{\alpha }e_{\ J}^{\beta }(F_{\alpha \beta }^{\ \ \ IJ}-\alpha \ast F_{\alpha \beta }^{\ \ \ IJ})\equiv {\frac {1}{2}}\int ee_{\ I}^{\alpha }e_{\ J}^{\beta }(F_{\alpha \beta }^{\ \ \ IJ}-{\frac {\alpha }{2}}\epsilon _{\;\;\;KL}^{IJ}F_{\alpha \beta }^{\ \ \ KL})}$

where ${\displaystyle e_{\ I}^{\alpha }}$ is the tetrad, ${\displaystyle e}$ its determinant (the space-time metric is recovered from the tetrad by the formula ${\displaystyle g_{\alpha \beta }=e_{\alpha }^{I}e_{\beta }^{J}\eta _{IJ}}$ where ${\displaystyle \eta _{IJ}}$ the Minkowski metric), ${\displaystyle F_{\alpha \beta }^{\ \ \ IJ}}$ the curvature considered as a function of the connection ${\displaystyle A_{\alpha \beta }^{\ \ \ IJ}}$:

${\displaystyle F_{\alpha \beta }^{\ \ \ IJ}={A_{\alpha \beta }}^{IJ}=2\partial _{[\alpha }{A_{\beta ]}}^{IJ}+2{A_{[\alpha }}^{IK}{A_{\beta ]K}}^{J}}$,

${\displaystyle \alpha }$ a (complex) parameter, and where we recover the Palatini action when ${\displaystyle \alpha =0}$. It only works in 4D. To be torsion free means the covariant derivative defined by the connection ${\displaystyle A_{\alpha \beta }^{\ \ \ IJ}}$ when acting on the Minkowski metric ${\displaystyle \eta _{IJ}}$ vanishes, implying the connection is anti-symmetric in its internal indices ${\displaystyle I,J}$.

As with the first order tetradic Palatini action where ${\displaystyle e_{\ I}^{\alpha }}$ and ${\displaystyle A_{\alpha \beta }^{\ \ \ IJ}}$ are taken to be independent variables, variation of the action with respect to the connection ${\displaystyle A_{\alpha \beta }^{\ \ \ IJ}}$ (assuming it to be torsion-free) implies the curvature ${\displaystyle F_{\alpha \beta }^{\ \ \ IJ}}$ be replaced by the usual (mixed index) curvature tensor ${\displaystyle R_{\alpha \beta }^{\ \ \ IJ}}$ (see article tetradic Palatini action for definitions). Variation of the first term of the action with respect to the tetrad ${\displaystyle e_{\ I}^{\alpha }}$ 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.