Toroidal compactifications of 11-D supergravity naturally induce exceptional symmetries in that can be realized as U-duality symmetries of M-theory upon Z-discretization and without Betti-truncations. Hence, exceptional field theory based on the modular group uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry groups of M-theory. By mirror symmetry, there ought to be a deep internal symmetry induced between M-theory and F-theory upon KK-reduction to Type-IIB SUGRA. In the formalism taking the 6-8 limit, the content of the theory is given by the action:

with:

and:

where the Chern-Simons-topological Lagrangian has covariant variational form:

with:

and the Yang-Mills field equation for the covariant field strength form is:

Thus, we can derive the Chern-Simons-type topological action:

with:

and:

and the covariant curvature form and holomorphic curvature form are, respectively:

and:

where the Ramond-Ramond gauge-coupling sector is given by the action:

and the Ramond-Ramond term being:

thus giving us the Type-IIB Calabi-Yau three-fold superpotential:

Before we can see the duality relations between M-theory and F-theory elliptic fibrational Standard-Model constructions, note that the topologically mixed Yang-Mills action:

where the corresponding Chern-Simons action is:

with the Ramond-Ramond coupling-term:

has variational action:

with:

Now, since 11-D SUGRA on a torus is equivalent to Type-IIB string-theory on a circle, the action of the modular group on the Type-IIB axio-dilaton…