Monodromic T-Branes And The $SO(10)_{GUT}$

T-branes, which are non-Abelian bound states of branes, were first introduced by Cecotti, Cordova, Heckman and Vafa \cite{Cecotti:2010bp}. They are the refined version of the monodromic branes that feature in the phenomenological F-theory models. Here, we will be interested in the T-brane corresponding to the $Z_3$ monodromy which is used to break the $E_8$ gauge group to obtain the $SO(10)_{GUT}$. This extends the results of \cite{Cecotti:2010bp} to the case of $Z_3$ monodromic T-branes used to break the $E_8$ gauge group to $SO(10)\times SU(3)\times U(1)$ and compute the Yukawa coupling with the help of the residue formula. We conclude that the Yukawa coupling, ${\bf{10}}_{H}\cdot {\bf{16}}_{M}\cdot {\bf{16}}_{M}$, is non-zero for $E_7$, in complete agreement with \cite{Cecotti:2010bp}, but is zero for $E_8$. Furthermore, the case of $Z_2$ monodromic T-branes used to break the $E_8$ gauge group to $E_{6}\times SU(2)\times U(1)$, nothing interesting can be deduced by evaluating the Yukawa coupling ${\bf{27}}_{H}\cdot {\bf{27}}_{M}\cdot {\bf{27}}_{M}$ which is dependent on whether the MSSM fermion and electroweak Higgs fields can be included in the same ${\bf{27}}$ multiplet of a three-family $E_6$ GUT or assign the Higgs fields to a different ${\bf{27}}_{H}$ multiplet where only the Higgs doublets and singlets obtain the electroweak scale energy.