Is every connection form $\omega$ the extremal of some functional $S[\omega]$ ?
Context: Palatini action $S_{Pal}$ of General Relativity is (assuming Cosmological constant $\Lambda=0)$: $$ S_{Pal}[e,\omega]=\int e\wedge e\wedge R[\omega].$$
Motion equations (varying this action) gives us Einstein Equation and $$ d_\omega (e\wedge e)=0 \implies d_\omega e=0$$ which is the same to say connection $\omega$ is torsion-free.
But there is the famous Ashtekar connection: $A^i_a=\Gamma^i_a +\gamma K^i_a $ (which has torsion) and this gave me some questions:
There's some action $S_A[\omega]$ such that Ashtekar connection $A$ is its extremal ?
There are results that guarantee (or impose restrictions) about the existence of a extremal connection on functionals $S[\omega]$ ? I'd be glad for any reference (book, article etc..) about this topic.
What should be the interpretation about some connection be the extremal of some action ?
That is a deep geometrical meaning about the Levi-Civita connection be the extremal of this functional $S_{Pal}$?