Chern–Simons Theory, the Orbifold Delta Function, and GUT-Models

There is a deep way to geometrically engineer Yang-Mills GUT models from a coupling of Chern-Simons theory to Heterotic string theory via B-model topological twisting and double T-dualizing on the base of the elliptic fibration of F-theory where the orbifold delta function plays an essential role. The topological gauge part of the SYM Chern-Simons Lagrangian is given by:

\displaystyle \begin{array}{l}\mathcal{L}_{G}^{T}=-\frac{1}{{4{{g}^{2}}}}\left( {\text{Tr}F_{{\mu \nu }}^{\alpha }-2{{g}^{2}}\alpha _{\mu }^{\alpha }{{D}^{2}}{{a}^{{\mu \alpha }}}} \right.\\\left. {-2a_{\mu }^{\alpha }\tilde{S}_{{10}}^{{{{{\left( a \right)}}^{{\exp \left( {i\phi A} \right)}}}}}{{F}^{{b\mu \nu }}}a_{\nu }^{\alpha }} \right)\end{array}

where \tilde{S}_{{10}}^{{\left( a \right)}} is the orbifold delta function:

\displaystyle \tilde{S}_{{10}}^{{\left( a \right)}}=\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)

with \phi the dilaton and (a,b) are terms derived from the D5-brane backreaction and such that varying the orbifold function with respect to the Type-IIB action induces orbifold-compactifications that locally inject 4-D gauge actions written as:

\displaystyle \mathcal{L}_{G}^{{Loc}}=\sum\limits_{{b=1}}^{{N-1}}{{\frac{1}{{2g_{b}^{2}}}}}{{\int{\text{d}}}^{2}}\theta {{W}^{\alpha }}{{W}_{\alpha }}{{\delta }^{2}}\left( {\left( {1-{{e}^{{ib\phi }}}} \right)z} \right)

Hence, the Ramond-Ramond coupling is given by:

\displaystyle {{S}_{{CS}}}=\frac{{{{T}_{p}}}}{2}\int\limits_{{{{\Sigma }_{{p+1}}}}}{{C\wedge \text{Tr}}}\left( {{{e}^{{F/2\pi }}}} \right)

and since for Type-IIB, p is odd, the potential for the Type-IIB theory compactified on a Calabi-Yau threefold X takes the form:

\displaystyle W=\int_{X}{{{{G}_{3}}}}\wedge {{\Omega }_{3}}+\sum\limits_{{i=1}}^{{{{h}^{{1,1}}}}}{{{{A}_{i}}}}\left( {\left( {{{e}^{{-\phi }}}+i{{C}_{0}}} \right),U} \right){{e}^{{-a\left( {{{e}^{{-\phi }}}{{\tau }_{i}}+i{{\rho }_{i}}} \right)}}}

where the translational, rotational, and Chern-Simons 3-form of gauge-class:

\displaystyle C=\text{Tr}\left\{ {A\wedge F} \right\}

are respectively:

\displaystyle {{C}_{T}}\equiv \frac{1}{{2{{l}^{2}}}}{{\vartheta }^{2}}\wedge {{\Gamma }_{\alpha }}

\displaystyle {{C}_{R}}\equiv {{\left( {-1} \right)}^{s}}{{T}^{{*\alpha }}}\wedge R_{\alpha }^{*}-\frac{1}{{3!}}{{\eta }_{{\alpha \beta \gamma }}}{{\Gamma }^{{*\alpha }}}

and

\displaystyle {{C}_{{TR}}}\equiv \frac{1}{l}\left( {{{\Gamma }^{{*\alpha }}}\wedge {{T}_{\alpha }}-\frac{{{{{\left( {-1} \right)}}^{s}}}}{2}{{\eta }_{{\alpha \beta \gamma }}}{{\Gamma }^{{*\alpha }}}\wedge {{\vartheta }^{\gamma }}} \right)

which are derived by varying the Lagrangian density:

\displaystyle {{\mathcal{L}}_{{MB}}}\left( {{{\vartheta }^{\alpha }},\Gamma _{\alpha }^{*}} \right)={{\theta }_{T}}{{C}_{T}}+{{\theta }_{R}}{{C}_{R}}+{{\theta }_{{TR}}}{{C}_{{TR}}}

with respect to {{{\vartheta }^{\alpha }}} and {\Gamma _{\alpha }^{*}}. This yields us the crucial NS-NS field equations:

\displaystyle -{{\theta }_{{TR}}}R_{\alpha }^{*}-\frac{{{{\theta }_{T}}}}{l}{{T}_{\alpha }}=l\Sigma _{\alpha }^{\dagger }

\displaystyle -{{\left( {-1} \right)}^{s}}{{\theta }_{{TR}}}{{T}_{\alpha }}-\frac{{{{\theta }_{\text{Y}}}}}{{2l}}{{\eta }_{\alpha }}-{{\theta }_{R}}R_{\alpha }^{*}=l\tau _{\alpha }^{*}

Noting that the Einstein-Hilbert terms in the metaplectic Riemann-Cartan formalism constitute systolic algebraic 1-forms as well as a super-Lie-algebraic dual of the Lorentz connection:

 \displaystyle {{\Gamma }^{{\beta \gamma }}}=\Gamma _{j}^{{\beta \gamma }}d{{x}^{i}}

more precisely:

\displaystyle \left\{ {\begin{array}{*{20}{c}} {{{\vartheta }^{\alpha }}=e_{i}^{\alpha }d{{x}^{i}}} \\ {\Gamma _{\alpha }^{*}=\frac{1}{2}{{\eta }_{{\alpha \beta \gamma }}}{{\Gamma }^{{\beta \gamma }}}} \end{array}} \right.

it follows that the corresponding dual-field strength is the 2-form Kähler torsion:

\displaystyle T_{K}^{\alpha }\equiv d{{\vartheta }^{\alpha }}-{{\left( {-1} \right)}^{s}}{{\eta }^{{\alpha \beta }}}\wedge \Gamma _{\beta }^{*}

with curvature form:

\displaystyle 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 }^{*}

The Poisson-Lie duality allows us to add Chern-Simons forms, and by gauging the super-Poincaré group, we get the desired Mielke-Baekler theory that solves for the Einstein-Cartan Lagrangian:

\displaystyle {{\mathcal{L}}_{{EC}}}\equiv \frac{\chi }{l}{{\vartheta }^{\alpha }}\wedge {{R}^{*}}=-\chi {{C}_{{TR}}}-\frac{\chi }{l}\text{d}\left( {\Gamma _{\alpha }^{*}\wedge {{\vartheta }^{\alpha }}} \right)

Now, combining the Chern-Simons VEV equations:

\displaystyle -{{\theta }_{{TR}}}R_{\alpha }^{*}-\frac{{{{\theta }_{T}}}}{l}{{T}_{\alpha }}=l\Sigma _{\alpha }^{\dagger }

and:

\displaystyle -{{\left( {-1} \right)}^{s}}{{\theta }_{{TR}}}{{T}_{\alpha }}-\frac{{{{\theta }_{\text{Y}}}}}{{2l}}{{\eta }_{\alpha }}-{{\theta }_{R}}R_{\alpha }^{*}=l\tau _{\alpha }^{*}

by modularity, we get the torsion and Riemann-Cartan curvature, respectively:

\displaystyle {{T}_{\alpha }}=\frac{{2\kappa }}{l}{{\eta }_{\alpha }}

\displaystyle R_{\alpha }^{*}=\frac{\rho }{{{{l}^{2}}}}{{\eta }_{\alpha }}

where \kappa is the Picard constant:

\displaystyle \kappa ={{\theta }_{{TR}}}{{\theta }_{T}}/2A

and:

\displaystyle \rho =-\theta _{T}^{2}/A

is the CS-Witten term. Now, coupling to matter fields, we get the torsion condition:

 \displaystyle {{T}_{\alpha }}-\frac{{2\kappa }}{l}{{\eta }_{\alpha }}=\frac{2}{A}l\left( {{{\theta }_{{TR}}}\tau _{\alpha }^{*}-{{\theta }_{R}}l\Sigma _{\alpha }^{\dagger }} \right)

and the Riemann-Cartan form reduces to:

\displaystyle R_{\alpha }^{*}-\frac{\rho }{{{{\rho }^{2}}}}{{\eta }_{\alpha }}=\frac{2}{A}\left( {{{\theta }_{{TR}}}l\sum\limits_{\alpha }{{-{{\theta }_{T}}\tau _{\alpha }^{*}}}} \right)

which yields the 4-D action for A:

\displaystyle \int{{\left[ {dA} \right]}}\int{{\left[ {d\alpha } \right]}}\,{{e}^{{\tilde{S}_{{10}}^{{\left( a \right)}}}}}=\int{{\left[ {dA} \right]}}\,{{e}^{{-\tilde{S}_{{10}}^{{\left( a \right)}}}}}

In order to show the CS-H Yang-Mills GUT construction modulo a Teichmüller orbifold, note that by the F/M-theory duality, flux-compactification yields moduli-stabilization via double-Higgsing and solving the Yukawa coupling integral-equation:

\displaystyle \int_{S}{{\left[ W \right]}}=\frac{{{{N}_{s}}}}{{{{\rho }^{2}}}}m_{*}^{4}\int{{\text{tr}}}\left( {F\wedge \Phi } \right)

Now, taking the Hodge dual gives us the Hodge-Fukaya form:

\displaystyle *dW\wedge {{\left( {d\int{{\left\langle \Phi \right\rangle }}} \right)}^{*}}