I last introduced T-branes, which are non-Abelian deformation of intersecting D-brane systems in the corresponding compactification manifold. Then I showed that we have a Kähler-equivalence of the derivatives in the pull-back with the gauge-covariant ones, which gave us:
with the inclusion of the complex Higgs field , and representing the symmetrization over gauge indices.
Locally, the Higgs field is given by:
where is a matrix in the complexified adjoint representation of and its Hermitian conjugate. Thus, I could derive:
a Kähler coordiante expansion of and 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