Recently, there appeared a paper by Giacomo Pollari, A Nieh-Yan-like topological invariant in General Relativity, where the action for gravity looks like:
$$S_g=S_{EHP}+S_{HO}+S_{PO}+S_{GB}+S_{NY}+S_{ET}$$
and where
$$S_{EHP}=\alpha_0\int e^I\wedge e^J \wedge \star R_{JI}$$
is the Einstein-Hilbert-Palatini action term,
$$S_{HO}=\alpha_1\int e^I\wedge e^J \wedge R_{JI}$$
is the Holst action
$$S_{PO}=\alpha_2\int R^{IJ}\wedge R_{JI}$$
is the Pontryagin action
$$S_{GB}=\alpha_3\int R^{IJ}\wedge \star R_{JI}$$
is the Gauss-Bonnet term
$$S_{NY}=\alpha_4\int d(e^I\wedge T_I)$$
is the topological Nieh-Yan action, and the new exotic (dual) torsion term reads
$$S_{ET}=\alpha_5\int d(e^I\wedge \star T_I)$$
The above whole action is suggestively dual symmetric under the following transformations:
$$\alpha_0 \star R_{IJ}\leftrightarrow \alpha_1 R_{JI}$$
(interchanging EHP with HO)
$$\alpha_3 R^{IJ}\leftrightarrow \alpha_3 \star R_{JI}$$
(interchanging the Gauss-Bonnet term with itself)
$$\alpha_4 T_I\leftrightarrow \alpha_5 \star T_I$$
(interchanging the Nieh-Yan topological term with the exotic dual torsion term)
provided we supplement a further seventh term to the action, the (double)-dual term to the Pontryagin action
$$S_{DP}= \alpha_6 \int\star R^{IJ}\wedge \star R_{JI}$$
such as changing $R^{IJ}$ by its dual $\star R^{JI}$ the whole action is totally symmetry after substituting $R$ and $T$ by their duals, provided
$$\alpha_2 R^{IJ}\wedge R_{JI}\leftrightarrow \alpha_6 \star R^{IJ}\wedge \star R_{JI}$$
(that interchanges the Pontryagin action with its double dualized action, $S_{DP}$).
With these transformations the complete action seems to be invariant under the above duality. However, to be more precise, is the doubly dual of the Pontrjagin action a sensible meaningul action? Furthermore, what could be the physical interpretation of this duality transformation and the meaning of every action term? I thought this question because I can find out natural the the GB and PO terms are "self-dual" to theirselves (being topological), and also I can understand somehow torsional topological NY being dual, but I can not figure out why EHP should be "dual" to the Holst action after this duality. Has it any sense?
