The Barbero-Immirzi Field and the Nieh–Yan Topological Invariant

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:

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

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

    \[\begin{array}{*{20}{l}}{{S^{{\rm{Red}}}} = - \int_{{S^1}} {\rm{d}} {x^3}\int_{{M_3}} {{{\rm{d}}^3}} x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.}\\{{\varepsilon _{IJKL}}e_3^Ie_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }e_3^Ie_\mu ^J{F_{\nu \rho }}_{IJ}} \right)}\end{array}\]

to the 4-D the Holst action:

    \[\begin{array}{*{20}{l}}{{S_{4D}}\left[ {e,\omega } \right] = \int_{{{\tilde M}_4}} {{{\rm{d}}^4}} x{\varepsilon ^{\mu \nu \rho \sigma }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}}\\{e_\mu ^Ie_\nu ^JF_{\rho \sigma }^{KL}\left. { + \frac{1}{\gamma }e_\mu ^Ie_\nu ^J{F_{\rho \sigma }}_{IJ}} \right)}\end{array}\]

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

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

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:

    \[\begin{array}{l}{S^{{\gamma _f}}} = \int_{{M_4}} {dtL = - \frac{1}{2}} \int_{{M_4}} {{d^4}} x{\,^{(4)}}e\\e_a^\mu e_b^\nu {\left( {{R^{ab}}} \right._{\mu \nu }} - \frac{{{\gamma _f}}}{2}{\varepsilon ^{ab}}_{cd}\left. {{R^{cd}}_{\mu \nu }} \right)\end{array}\]

Loop Quantum Gravity: the Barbero-Immirzi Parameter

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

    \[\begin{array}{l}{S_{4{\rm{D}}}}\left[ {e',\omega } \right] = \int_{\tilde M} {\left( {\frac{1}{2}} \right.} {\rm{tr}}\left( {e \wedge e \wedge F} \right)\\\left. { + \frac{1}{\gamma }{\rm{tr}}\left( {e \wedge e \wedge * F} \right)} \right)\end{array}\]

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

    \[{e^I} = e_\mu ^I{\rm{d}}{x^\mu }\]

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

    \[F = {\rm{d}}\omega + \omega \wedge '\omega \]

Hence, we have the two-form:

    \[\begin{array}{l}{F^{IJ}} = \left( {{{\not \partial }_\mu }} \right.\omega _\nu ^{IJ} - {{\not \partial }_\nu }\omega _\mu ^{IJ} + \omega _\mu ^{IK}{\omega _\nu }{K^J}\\\left. { - \omega _\nu ^{IK}{\omega _\mu }{K^J}} \right){\rm{d}}{x^\mu } \wedge '{\rm{d}}{x^\nu }\end{array}\]

with:

    \[ * {F^{IJ}} = \frac{1}{2}{\varepsilon ^{IJ}}_{KL}{F^{KL}}\]

allowing us to write down the Holst action as:

    \[\begin{array}{*{20}{l}}{{S_{4D}}\left[ {e,\omega } \right] = \int_{{{\tilde M}_4}} {{{\rm{d}}^4}} x{\varepsilon ^{\mu \nu \rho \sigma }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}}\\{e_\mu ^Ie_\nu ^JF_{\rho \sigma }^{KL}\left. { + \frac{1}{\gamma }e_\mu ^Ie_\nu ^J{F_{\rho \sigma }}_{IJ}} \right)}\end{array}\]

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:

    \[\begin{array}{*{20}{l}}{{S^{{\rm{Red}}}} = - \int_{{S^1}} {\rm{d}} {x^3}\int_{{M_3}} {{{\rm{d}}^3}} x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.}\\{{\varepsilon _{IJKL}}e_3^Ie_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }e_3^Ie_\mu ^J{F_{\nu \rho }}_{IJ}} \right)}\end{array}\]

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

    \[{{\rm{d}}^3}x{\varepsilon ^{\mu \nu \rho }}\]

is the local volume form on

    \[{M_3}\]

With

    \[{x^I} \equiv e_3^I\]

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

    \[\begin{array}{l}S\left[ {e,x,\omega } \right] = {\int_{{M_3}} {\rm{d}} ^3}x{\varepsilon ^{\mu \nu \rho }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}\\{x^I}e_\mu ^JF_{\nu \rho }^{KL}\left. { + \frac{1}{\gamma }{x^I}e_\mu ^J{F_{\nu \rho }}_{IJ}} \right)\end{array}\]

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.