T-Branes, the Chern-Simons Action and the Kähler Pull-Back

T-branes are supersymmetric intersecting brane configurations such that the non-Abelian Higgs field \Phi 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:

    \[\begin{array}{c}{{\hat V}_D} = \frac{1}{{2{\mathop{\rm Re}\nolimits} \left( {{f_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}{\phi _j}\frac{{\partial K}}{{\partial {\phi _j}}} + M_P^2\sum\limits_j {{q_{hj}}} } } \right)^2}\\ = \frac{\pi }{{{\mathop{\rm Re}\nolimits} \left( {{T_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}\frac{{{{\left| {{\phi _j}} \right|}^2}}}{s} - {\xi _h}} } \right)^2}\end{array}\]

with the U\left( 1 \right)-charge:

    \[{q_{hj}} = \frac{1}{{l_s^4}}\int_{{D_h}} {{{\hat D}_j}} \wedge {F^G}\]

and {F^G} the gauge flux that yields the Fayet-Iliopoulos term:

    \[\begin{array}{l}\frac{{{\xi _h}}}{{M_P^2}} = \frac{{{e^{ - \phi /2}}}}{{4\pi \mathcal{V}}}\frac{1}{{l_s^4}}\int_{{D_h}} {J \wedge {F^G}} = \frac{1}{{2\pi }}\sum\limits_j {\frac{{{q_{hj}}}}{\mathcal{V}}} \\ = - \sum\limits_j {{q_{hj}}} \frac{{\partial K}}{{\partial {T_j}}}\end{array}\]

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

    \[P_{{\rm{int}}}^{Dp} \equiv Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}} \]

with a non-Abelian D-term:

    \[D_{\hat A}^K = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {{{\rm{A}}^\prime }(\Gamma )/{{\overline {\rm{A}} }^\prime }({\rm{N}})} {\rm{ }}\]