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:

where is the orbifold delta function:

with the dilaton and 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:

Hence, the Ramond-Ramond coupling is given by:

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

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

are respectively:

and

which are derived by varying the Lagrangian density:

with respect to and . This yields us the crucial NS-NS field equations:

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:

more precisely:

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

with curvature form:

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:

Now, combining the Chern-Simons VEV equations:

and:

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

where is the Picard constant:

and:

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

and the Riemann-Cartan form reduces to:

which yields the 4-D action for :

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:

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

and realizing that the elliptic fibration induces a Calabi-Yau potential as a polynomial in that yields via Kaluza-Klein reduction and -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:

where is the 3-flux form:

and is the Chern-Simons 3-form given by:

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:

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

and

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:

on the orbifolded fibration to:

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

The GUT is hence achieved by solving the monodromy action-equation on the elliptic-curve line-bundle by -cusp D7-brane intersections on the base with divisors defining the elliptic singularities, where 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 :

giving us a K3-class model over an integral basis of 2-cycles and where the corresponding period integrals elliptic over 1-cycles:

factoring in the base modulus cusp -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 maps the fundamental region to the 7-plane geometry via D7-brane backreaction as a function of the Type-IIB Jacobi-term of the -plane for:

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:

and:

satisfy the Picard-Fuchs differential equations, and by coupling the Einstein-Cartan Lagrangian:

to the Hodge-Fukaya form:

we get the B-twist courtesy of the Yukawa coupling integral-equation:

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

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

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