Let us see how the Yukawa couplings among 4-D fermionic fields can be derived from the F-theory superpotential and relate them to the tree-level superpotential. This is of utmost importance since D7/D3-brane-phenomenology of 4-D F-theory can be promoted to M-theory in light of the F/M-theory duality and the compactness of Calabi-Yau 4-folds. Start with a Kähler coordinate expansion of which gives us, after inserting it in:
which is the exact 7-brane superpotential for F-theory and the integrand is independent of , entailing that the F-term conditions are purely topological and in no need for -corrections.
However, the D-term in:
is in need of -corrections, since it is evaluable as:
and the non-Abelian D-term has the form:
With our target Calabi-Yau 4-fold and Lie algebra , for: