T-branes and $\alpha'$-corrections

We study $\alpha'$-corrections in multiple D7-brane configurations with non-commuting profiles for their transverse position fields. We focus on T-brane systems, crucial in F-theory GUT model building. There $\alpha'$-corrections modify the D-term piece of the BPS equations which, already at leading order, require a non-primitive Abelian worldvolume flux background. We find that $\alpha'$-corrections may either i) leave this flux background invariant, ii) modify the Abelian non-primitive flux profile, or iii) deform it to a non-Abelian profile. The last case typically occurs when primitive fluxes, a necessary ingredient to build 4d chiral models, are added to the system. We illustrate these three cases by solving the $\alpha'$-corrected D-term equations in explicit examples, and describe their appearance in more general T-brane backgrounds. Finally, we discuss implications of our findings for F-theory GUT local models.