Topology, The Orbifold Delta Function, And 3 – D Gauge Analysis

All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced. March 14, 1824. On Mathematics. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 360.
In this rough-draft post, I will derive the 3-dimensional topological gauge part of the action of supergravity. In my last post, I derived the orbifold Delta function:

    \[\widetilde S_{10}^{(a)} = \frac{1}{N}\sum\limits_{b = 0}^{N - 1} {{e^{iba\,\phi }}} \delta \left( {{\omega _{\mu \nu \rho }} - {e^{ib\,\phi }}{\omega _{\mu \nu \sigma }}} \right)\]

Keep it in mind, as it will figure essentially at the end of this post. In 3-dimensions, the central gravitational variables in the metaplectic Riemann-Cartan formalism constitute systolic algebraic 1-forms as well as a super-Lie algebraic dual of the Lorentz connection:

    \[{\Gamma ^{\beta \gamma }} = \Gamma _j^{\beta \gamma }d\,{x^i}\]

More precisely:

    \[\left\{ {\begin{array}{*{20}{c}}{{\vartheta ^\alpha } = e_i^\alpha d\,{x^i}}\\{\Gamma _\alpha ^ * \equiv \frac{1}{2}{\eta _{\alpha \beta \gamma }}\,{\Gamma ^{\beta \gamma }}}\end{array}} \right.\]

Hence, the corresponding field strengths are the 2-forms of Kähler torsion:

    \[{T^\alpha } \equiv d{\vartheta ^\alpha } - {\left( { - 1} \right)^s}{\eta ^{\alpha \beta }} \wedge \Gamma _\beta ^ * \]

and curvature:

    \[R_\alpha ^ * = \frac{1}{2}{\eta _{\alpha \beta \gamma }}{R^{\beta \gamma }} \equiv d\,\Gamma _\alpha ^ * + \frac{{{{\left( { - 1} \right)}^s}}}{2}{\eta _{\alpha \beta \gamma }}\,\Gamma _\beta ^ * \wedge \Gamma _{\gamma '}^ * \]

Note, after connecting bivectors to vectors via Poisson-Lie-duality, one gets a linearized variability that clarifies the local symmetries of topological models. However, one must add Chern-Simons terms and by gauging the super-Poincaré group, we land in the desired Mielke-Baekler theory, which, in 3-dimensions, yields the Einstein-Cartan (EC) Lagrangian:

    \[{\not L_{EC}} \equiv \frac{\chi }{l}{\vartheta ^\alpha } \wedge {R^ * } = \, - \chi {C_{TL}} - \frac{\chi }{l}d\left( {\Gamma _\alpha ^ * \wedge {\vartheta ^\alpha }} \right)\]

with {C_{TL}} the Chern-Simons total divergence term. Now introduce vacuum angles:

    \[\left\{ {\begin{array}{*{20}{c}}{{\theta _T}}\\{{\theta _L}}\\{{\theta _{TL}} = \, - \chi }\end{array}} \right.\]

and we get a hint at what a purely topological 3-dimensional ‘gravity’ Lagrangian looks like:

    \[{\not L_{MB}}\left( {{\vartheta ^\alpha },\Gamma _\alpha ^ * } \right) = {\theta _T}{C_T} + \,\,{\theta _L}{C_L} + {\theta _{TL}}{C_{TL}}\]

with:

    \[\left\{ {\begin{array}{*{20}{c}}{{C_T} \equiv \frac{1}{{2{l^2}}}{\vartheta ^\alpha } \wedge {T_\alpha }}\\{{C_L} \equiv {{\left( { - 1} \right)}^s}{\Gamma ^{ * \alpha }} \wedge R_\alpha ^ * - \frac{1}{{3!}}{\eta _{\alpha \beta \gamma }}{\Gamma ^{ * \alpha }}}\\{{C_{TL}} \equiv \frac{1}{l}\left( {{\Gamma ^{ * \alpha }} \wedge {T_\alpha } - \frac{{{{\left( { - 1} \right)}^s}}}{2}{\eta _{\alpha \beta \gamma }}{\Gamma ^{ * \beta }} \wedge {\vartheta ^\gamma }} \right)}\end{array}} \right.\]

holding, and, respectively, are translational, rotational and Chern-Simons 3-forms of gauge-type C = {\rm{Tr}}\left\{ {A \wedge F} \right\} in Riemann-Cartan space-time. Varying:

    \[{\not L_{MB}}\left( {{\vartheta ^\alpha },\Gamma _\alpha ^ * } \right) = {\theta _T}{C_T} + \,\,{\theta _L}{C_L} + {\theta _{TL}}{C_{TL}}\]

with respect to {\vartheta ^\alpha } and {\Gamma ^{ * \alpha }} gives us the topological field equations:

- {\theta _{TL}}R_\alpha ^ * - \frac{{{\theta _T}}}{l}{T_\alpha } = l\sum _\alpha ^\dagger

- {\left( { - 1} \right)^s}{\theta _{TL}}{T_\alpha } - \frac{{{\theta _Y}}}{{2l}}{\eta _\alpha } - {\theta _L}R_\alpha ^ * = l\tau _\alpha ^ *

Now, note that the Chern-Simons term proportional to {\theta _T} induces in the second-order field equation a constant term that plays the role, in 4-dimensions, of the cosmological constant in Einstein’s equations. Combining the vacuum equations:

- {\theta _{TL}}R_\alpha ^ * - \frac{{{\theta _T}}}{l}{T_\alpha } = l\sum _\alpha ^\dagger

and:

- {\left( { - 1} \right)^s}{\theta _{TL}}{T_\alpha } - \frac{{{\theta _Y}}}{{2l}}{\eta _\alpha } - {\theta _L}R_\alpha ^ * = l\tau _\alpha ^ *

we get the torsion and Riemann-Cartan curvature:

    \[\left\{ {\begin{array}{*{20}{c}}{{T_\alpha } = \frac{{2\kappa }}{l}{\eta _\alpha }}\\{R_\alpha ^ * = \frac{\rho }{{{l^2}}}{\eta _\alpha }}\end{array}} \right.\]

with \kappa the Picard constant:

    \[\kappa = {\theta _{TL}}{\theta _T}/2A\]

and:

    \[\rho = \,\,\, - {\mkern 1mu} \,\theta _T^2/A\]

Now coupling matter fields gives us the following torsion:

    \[{T_\alpha } - \frac{{2\kappa }}{l}{\eta _\alpha } = \frac{2}{A}l\left( {{\theta _{TL}}\tau _\alpha ^* - {\theta _L}l\sum _\alpha ^\dagger } \right)\]

and the Riemann-Cartan curvature being:

    \[R_\alpha ^ * - \frac{\rho }{{{\rho ^2}}}{\eta _\alpha } = \frac{2}{A}\left( {{\theta _{TL}}l{\sum _\alpha } - \,{\theta _{T\tau _\alpha ^ * }}} \right)\]

Putting all of the above together and to derive an effective 3-dimensional action for A, we integrate out the high-momentum modes:

    \[\int {\left[ {dA} \right]} {\int {\left[ {d\alpha } \right]} ^{e\widetilde {\,S}_{10}^{(a)}}} = \int {\left[ {dA} \right]} \,{e^{i\widetilde S_{10}^{(a)}}}\]

where \widetilde S_{10}^{(a)} is the orbifold Delta functional term. Hence, for the fermions and scalars, we get:

{\rm{Det}}{\left( {\not D} \right)^{{n_{{L_{TC}}}}}}{\rm{Det}}{\left( {{D^2}} \right)^{ - n\phi /2}}

with:

    \[{\rm{det}}\left( {\not D} \right) = {\rm{det}}{\left( {\not D\not D} \right)^{1/2}} = {\rm{det}}\left( {{D^2} + \frac{1}{2}{D_\mu }D_\nu ^\lambda } \right)\left[ {{\gamma ^\mu },{\gamma ^\nu }} \right] = {\rm{det}}\left( {{D^2} + {{\not F}^{\mu \nu }}{\Im _{\mu \nu }}} \right)\]

and {\not F^{\mu \nu }} the super-field strength corresponding to A. Now finally, the topological gauge part of the action can be derived as:

    \[L_{{\rm{gauge}}}^T = \, - \frac{1}{{4{g^2}}}\left( {{\rm{Tr}}\,{\rm{\not F}}_{\mu \nu }^\alpha - 2{g^2}a_\mu ^\alpha {D^2}{a^{\mu \alpha }} - 2\,a_\mu ^\alpha \widetilde S{{_{10}^{(a)}}^{{e^{i\,\phi \,/A}}}}{{\not F}^{b\mu \nu }}a_\nu ^\alpha } \right)\]

to be marshaled in 3-dimensional Lagrangian topological analysis in upcoming posts.

Why should things be easy to understand? ~ Thomas Pynchon!

 …