T-branes are supersymmetric intersecting brane configurations such that the non-Abelian Higgs field that describes D-brane deformations is not diagonalisable and satisfies nilpotency conditions where the worldvolume flux has non-commuting expectation values and their worldvolume adjoint Higgs field is given a VEV that cannot be captured by its characteristic polynomial, and thus derive their importance from the fact that heterotic string compactifications are dual to T-branes in F-theory. Let’s probe their dynamics. Starting with the D-term potential:

with the -charge:

and the gauge flux that yields the Fayet-Iliopoulos term:

where the D-brane partition function for closed strings is given by:

with a non-Abelian D-term:

and

is the first Pontryagin class-term, and is the flat space Kähler form:

where is given by:

Then the non-Abelian profiles for and must satisfy the 7-brane functional equations of motion. Non-Abelian generalisation of:

are built up as follows. Write locally:

and localize the integral in:

as:

thus,

the non-Abelian generalisation of and have both the form of the D7-brane Chern-Simons action and hence satisfy the T-brane equation of motion

So effectively, we have a Kähler-equivalence of the derivatives in the pull-back with gauge-covariant ones, yielding:

with the inclusion of the complex Higgs field , and represents the symmetrization over gauge indices.

In this local description, the Higgs field is given by:

where is a matrix in the complexified adjoint representation of and its Hermitian conjugate. Thus, locally, we have:

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

Fixing our induced Dp-brane worldvolume metric:

we can write the Dirac-Born-Infeld action as:

which is a Higgsed gauge theory in dimensions with scalar fields. Thus, by dimensional reduction, this action is equivalent to a Yang-Mills gauge theory in 10-spacetime-dimensions with action:

with:

and the action is invariant under the supersymmetric transformations:

with the infinitesimal Majorana-Weyl spinor. By double-gauging, we get our desired Dp-brane action:

Crucially, note that the theory contains intersecting D2-D4-branes, since in the Casimir representation, the open string worldsheet boundary is a vertex vacuum connection coupled to a closed string state. This is the worldsheet-state correspondence in F-theory. Hence, the n-th loop open string Casimir force is equivalent to the n-th tree-level closed string charge exchange between two D-branes. It follows that the complete action of the Ramond-Ramond D-brane is an integral over the full space :

Hence, the gauged supergravity action is derivable as:

with:

and is the Ramond-Ramond potential, thus yielding the Chern-Simons action:

The non-Abelian D-term thus takes the form:

In the local patch on the C-manifold, we take the flat-space-Kähler-form:

and decompose the Kähler-background B-field as:

with:

thus giving us:

with the Abelian pull-back to given 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 for 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 yield the D-term equations:

Hence, the -corrections are given entirely by the abelian pull-back of the Kähler-form to

establishing a deep Wess–Zumino-based connection between T-branes and Abelian gauge field theory. This has vast implications for building realistic N/M-stack intersecting Dp-brane theories embedding the Standard Model of particle physics as well as that of cosmology.

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