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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:

with:

a Kähler coordinate expansion of  and gives us, after inserting it in:

the following:

### 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

Aside: the book on the post cover is excellent.

However, the D-term in:

is in need of -corrections, since it is evaluable as:

and the non-Abelian D-term has the form:

In the local patch on the C-manifold, one takes the flat-space Kähler-form as having the form:

Then, we decompose the Kähler-background B-field as:

with:

thus giving us:

with the Abelian pull-back  to  as determined by:

Hence we have:

Now: realize that  is a zero-form and  does not have transverse-legs to , and thus the pull-back has a trivial action. So, after solving:

the D-term equations amount to  with:

and with the -field vanishing on the sheave of the C-manifold, one gets a reduction to:

### which yields a non-Abelian -corrected Chern-Simons action of a stack of D7-branes with both terms at leading order in

entailing that for matrix algebras:

### they are the matrix products in the fundamental representation of

and so the -corrections on D-terms with the gauge flux F diagonalization yields

### Deep upshot: the -corrections are given entirely by the abelian pull-back of the Kähler-form  to

And this has a deep physical interpretation which can be extracted from: ‘The energy-momentum tensor and D-term of Q-clouds‘.…