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)}^{*}}

and realizing that the elliptic fibration induces a Calabi-Yau potential as a polynomial in \Phi that yields via Kaluza-Klein reduction and \tilde{S}_{{10}}^{{\left( a \right)}}-backreaction PBS conditions on the string spectrum to derive the SYM Green’s function and then coupling the Hodge-Fukaya form to the Heterotic action in the Einstein-frame:

\displaystyle \begin{array}{l}S=\frac{1}{{2{\alpha }'}}\int{{{{d}^{{10}}}}}x\sqrt{{-{{g}_{{10}}}}}\left( {{{R}_{{10}}}-\frac{1}{2}{{\partial }_{A}}\phi {{\partial }^{A}}\phi -\frac{1}{{12}}{{e}^{{-\phi }}}} \right)\\{{\left( {{{H}_{{ABC}}}-\frac{{{\alpha }'}}{{16}}{{e}^{{\frac{\phi }{2}}}}\bar{\chi }{{\Gamma }_{{ABC}}}\chi } \right)}^{2}}\left( {\frac{{{\alpha }'}}{4}{{e}^{{-\frac{\phi }{2}}}}\text{tr}\left( {{{F}_{{AB}}}{{F}^{{AB}}}} \right)} \right.\\-\left. {{\alpha }'\text{tr}\left( {\bar{\chi }{{\Gamma }^{A}}{{D}_{A}}\chi } \right)} \right)\end{array}

where H is the 3-flux form:

\displaystyle H=dB-\frac{{{\alpha }'}}{4}\left( {{{\Omega }_{3}}\left( A \right)-{{\Omega }_{3}}\left( \omega \right)} \right)

and {{{\Omega }_{3}}} is the Chern-Simons 3-form given by:

\displaystyle {{\Omega }_{3}}\left( A \right)\equiv \text{tr}\left( {A\wedge dA+\frac{2}{3}A\wedge A\wedge A} \right)

gives an action that is isomorphic for the class of Lorentzian manifolds to the path integral of F-theory, and visually, it gives rise to the following picture:

 

where the Type-IIB action is:

\displaystyle {{S}_{{IIB}}}={{S}_{{NS-NS}}}+{{S}_{{RR}}}+{{S}_{{CS}}}

with the Neveu-Schwarz, Ramond-Ramond, and Chern-Simons actions are, respectively:

\displaystyle {{S}_{{NS-NS}}}=\frac{1}{{2\kappa _{{10}}^{2}}}\int{{{{d}^{{10}}}}}x\sqrt{{-G}}{{e}^{{-2\Phi }}}\left( {R+4{{\partial }_{\mu }}\Phi {{\partial }^{\mu }}\Phi -\frac{1}{2}{{{\left| {{{H}_{3}}} \right|}}^{2}}} \right)

\displaystyle {{S}_{{RR}}}=-\frac{1}{{4\kappa _{{10}}^{2}}}\int{{{{d}^{{10}}}}}x\sqrt{{-G}}\left( {{{{\left| {{{F}_{1}}} \right|}}^{2}}+{{{\left| {{{{\tilde{F}}}_{3}}} \right|}}^{2}}+\frac{1}{2}{{{\left| {{{{\tilde{F}}}_{5}}} \right|}}^{2}}} \right)

and

\displaystyle {{S}_{{CS}}}=-\frac{1}{{4\kappa _{{10}}^{2}}}\int\limits_{{\text{S}\text{.T}\text{.}}}{{{{C}_{4}}}}\wedge {{H}_{3}}\wedge {{F}_{3}}

giving us the CS-H Yang-Mills D3-brane GUT model:


The key is the topological coupling of solutions to 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)

on the orbifolded fibration to:

\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)

and using the variation-principle with respect to the CS action term:

\displaystyle {{S}_{{CS}}}=-\frac{1}{{4\kappa _{{10}}^{2}}}\int\limits_{{\text{S}\text{.T}\text{.}}}{{{{C}_{4}}}}\wedge {{H}_{3}}\wedge {{F}_{3}}

The GUT is hence achieved by solving the monodromy action-equation on the elliptic-curve line-bundle by N-cusp N-d D7-brane intersections on the base with divisors defining the elliptic singularities, where d is the number of degeneracies of the torus-modulus as a varying function of the RR-C-form and the dilaton, and integrating the exact holomorphic 2-form \Omega _{{K3}}^{{\left( {2,0} \right)}}:

\displaystyle {{\tilde{\omega }}_{i}}=\int_{{{{\gamma }_{i}}}}{{\Omega _{{K3}}^{{\left( {2,0} \right)}}}}\equiv \int_{{{{\gamma }_{i}}}}{{\frac{{dxdz}}{{{{\partial }_{y}}W\left( {x,y,z;\lambda } \right)}}}}

giving us a K3-class model over an integral basis of 2-cycles {{\gamma }_{i}} and where the corresponding period integrals elliptic over {{\mathbb{Z}}_{N}} 1-cycles:

\displaystyle {{\tilde{\omega }}_{i}}=\int_{{{{\gamma }_{i}}}}{{\Omega _{{N-1}}^{{\left( {1,0} \right)}}}}

factoring in the base modulus cusp \lambda-terms and summing over divisor points over resolutions of D-7-branes localized on the line degeneracies. Pictorially, we get our desired F/M-GUT as:

 

 

Note that the Hauptmodul-function \lambda \left( \tau \right) maps the fundamental region to the 7-plane geometry via D7-brane backreaction as a function of the Type-IIB Jacobi-term of the {{\mathcal{H}}_{0}}-plane for:

\displaystyle \left\{ {\begin{array}{*{20}{c}} {z=\lambda \left( \tau \right)\quad \quad A//z=0} \\ {z=\infty \quad \quad \quad C//z=\infty } \end{array}} \right.

guaranteeing moduli stabilization, and by the Heterotic/F-theory duality, topological mirror symmetry twists our fibration to a Calabi-Yau 3-fold such that the period-integrals:

\displaystyle {{\tilde{\omega }}_{i}}=\int\limits_{{{{\gamma }_{i}}}}{{\Omega _{{N-1}}^{{\left( {1,0} \right)}}}}

and:

\displaystyle {{\tilde{\omega }}_{i}}=\int\limits_{{{{\gamma }_{i}}}}{{\Omega _{{K3}}^{{\left( {2,0} \right)}}}}

satisfy the Picard-Fuchs differential equations, and by coupling 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)

to the Hodge-Fukaya form:

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

we get the B-twist courtesy of 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)

on the orbifolded fibration, and by mirror symmetry, the A-model deformation and the Fourier-Mukai orbifold-delta functional transform:

\displaystyle {{\tilde{G}}_{{RR}}}^{{or{{b}^{\delta }}}}=\int_{{{{T}^{n}}}}{{{{G}_{{RR}}}}}\wedge {{e}^{{\tilde{F}}}}*d\int_{{a\backslash {{T}^{n}}/{{\Lambda }_{{{{\tau }^{{T-IIB}}}}}}}}{{\tilde{S}_{{10}}^{{\left( a \right)}}}}\wedge \int_{{{{\gamma }_{i}}}}{{\Omega _{{N-1}}^{{\left( {n,0} \right)}}}}

yield us the desired GUT model containing the Standard Model as a proper sub-embedded gauge groupoid variety.

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