• 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}