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:

…

## Social Networks