Why the need for the Barbero-Immirzi field? Let me briefly explain. We saw that LQG in the Holst formulation faces the serious problem that unless the Barbero-Immirzi parameter is promoted to a field, the three-dimensional action with the Barbero-Immirzi parameter:

cannot admit a 4-D uplifting of the reduced 3-D gauge-free spacetime compactified action:

to the 4-D the Holst action:

and that is because the total 3-D action with the Barbero-Immirzi parameter:

is invariant under rescaling symmetry and translational symmetry, which destroy the time-gauge accessibility of the theory and 4-D-uplifting. Let us see whether and how promoting the Barbero-Immirzi parameter to a field and using the Nieh–Yan topological invariant can ameliorate our crises. In Lagrangian Holst theory, a Hilbert–Palatini action can always be generalized to contain the Holst term and promotes the Barbero–Immirzi parameter to a field via:

I showed that in 4-D spacetime, the general relativistic starting point for canonical loop quantum gravity is given by:

with the dynamical variables are the tetrad one-form fields:

and the $SL\left( {2,\mathbb{C}} \right)$-valued connection $\omega _\mu ^{IJ}$ whose curvature is:

Hence, we have the two-form:

with:

allowing us to write down the Holst action as:

The Ashtekar-Barbero connection enters the picture in the following way: the phase space is parametrized by an $\widetilde {S{U_{a\lg }}}(2)$-valued connection and its conjugate triad field: and that is exactly the Ashtekar-Barbero connection! Moreover, the compactness of the gauge group ensures that the quantization leads to a mathematically exact kinematical Hilbert space. A good angle at seeing how the Barbero-Immirzi parameter comes in is via gauge-free spacetime compactification which is equivalent to a 3-D dimensional reduction of the 4-D Holst action to:

with $\mu = 0,1,2$ the three-dimensional spacetime index and

is the local volume form on

With

One then recovers the total three-dimensional action with the Barbero-Immirzi parameter:

A major problem for LQG is that the full theory does not imply that the above total three-dimensional action implies three-dimensional gravity, as it ought given compactification and time-gauging: that would be a no-go theorem for up-lifting LQG to 4-D spacetime, and LQG theory would simply be false.