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:
![\[\begin{array}{c}{{\hat V}_D} = \frac{1}{{2{\mathop{\rm Re}\nolimits} \left( {{f_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}{\phi _j}\frac{{\partial K}}{{\partial {\phi _j}}} + M_P^2\sum\limits_j {{q_{hj}}} } } \right)^2}\\ = \frac{\pi }{{{\mathop{\rm Re}\nolimits} \left( {{T_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}\frac{{{{\left| {{\phi _j}} \right|}^2}}}{s} - {\xi _h}} } \right)^2}\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-c9ec4ecd434d4bae8432fcfd42e4db82_l3.png "Rendered by QuickLaTeX.com")
with the 
-charge:
![\[{q_{hj}} = \frac{1}{{l_s^4}}\int_{{D_h}} {{{\hat D}_j}} \wedge {F^G}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-d9a6470f082b417ab3c612efdfb5010a_l3.png "Rendered by QuickLaTeX.com")
and 
the gauge flux that yields the Fayet-Iliopoulos term:
![\[\begin{array}{l}\frac{{{\xi _h}}}{{M_P^2}} = \frac{{{e^{ - \phi /2}}}}{{4\pi \mathcal{V}}}\frac{1}{{l_s^4}}\int_{{D_h}} {J \wedge {F^G}} = \frac{1}{{2\pi }}\sum\limits_j {\frac{{{q_{hj}}}}{\mathcal{V}}} \\ = - \sum\limits_j {{q_{hj}}} \frac{{\partial K}}{{\partial {T_j}}}\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-9bf658cb1cfe270f94e785956f4612d7_l3.png "Rendered by QuickLaTeX.com")
where the D-brane partition function for closed strings is given by:
![\[P_{{\rm{int}}}^{Dp} \equiv Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}} \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-4e2e5a5bba4fe487b08d9cf299490ba1_l3.png "Rendered by QuickLaTeX.com")
with a non-Abelian D-term:
![\[D_{\hat A}^K = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {{{\rm{A}}^\prime }(\Gamma )/{{\overline {\rm{A}} }^\prime }({\rm{N}})} {\rm{ }}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-141d8fc6e5b16009b962f488dae87cb2_l3.png "Rendered by QuickLaTeX.com")
and
![\[\sqrt {{\rm A}'(\Gamma )/\bar {\rm A}'({\rm N})} \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-cc99a646dae8f86bac0d17f3aadb590c_l3.png "Rendered by QuickLaTeX.com")
is the first Pontryagin class-term, and 
is the flat space Kähler form:
![\[J = \underbrace {\frac{i}{2}{\rm{dx}} \wedge {\rm{d\bar x + }}\frac{i}{2}{\rm{dy}} \wedge {\rm{d\bar y}}}_{ = :\omega } + 2i{\rm{dz}} \wedge {\rm{d\bar z}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-417733ad7b8551d8b3173be9dd53eec2_l3.png "Rendered by QuickLaTeX.com")
where 
is given by:
![\[\begin{array}{*{20}{c}}{S_{cld}^s = - \frac{1}{{4\pi {\alpha ^\dagger }}}\int_{\partial E_S^5} {{d^2}\sigma d} \Omega {{\left( {{\phi _{Inst}}} \right)}^2}\sigma \sqrt { - \gamma } \left( {\phi \left( {{{\bar X}^\mu }} \right)} \right.R}\\{ + {\gamma ^{\alpha \beta }}{\partial _\alpha }{X^\mu }{g_{\mu \nu }}\left( {{{\bar X}^\nu }} \right) + \frac{1}{{\sqrt { - \gamma } }}{\varepsilon ^{ - {c_{2n}}/{Y_k}\left( {{{\cos }^2}\varphi } \right)}}{\partial _\beta }{{\bar X}^\nu }{b_{\mu \nu }}{{\left( {\bar X} \right)}^2}}\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-72921c00690470d9b037a6d7da26d27b_l3.png "Rendered by QuickLaTeX.com")
Then the non-Abelian profiles for and 
must satisfy the 7-brane functional equations of motion. Non-Abelian generalisation of:
![\[{W^O} = \int_{{\Sigma _5}} {\rm{P}} \left[ {{\Omega _0} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-dae9836ecb7fabd81a8b028f24f71b89_l3.png "Rendered by QuickLaTeX.com")
![\[{D^K} = \int_{\tilde S} {\rm{P}} \left[ {{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-d4d38e53690548e0a7a54e69a63262f6_l3.png "Rendered by QuickLaTeX.com")
are built up as follows. Write locally:
![\[{\Omega _0} \wedge {e^B} = d\gamma \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-79ef0318f4db9a46f9edc47f5d1f154f_l3.png "Rendered by QuickLaTeX.com")
and localize the integral in:
as:
![\[\int_{\tilde S} {P\left[ \gamma \right]} \wedge {e^{\lambda F}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-6a432af182af18e0c1c9623cffa9b836_l3.png "Rendered by QuickLaTeX.com")
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:
![\[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-b5900aae1c1b81a3090cd360339db009_l3.png "Rendered by QuickLaTeX.com")
![\[{D^K} = \int_{\tilde S} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}} \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-6b99954d970e9ebaa35da9fc96c862df_l3.png "Rendered by QuickLaTeX.com")
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:
![\[\Phi \equiv \phi \frac{\partial }{{\bar \partial z}} + \bar \phi \frac{{\bar \partial }}{{\partial \bar z}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-d4065431891e85033d6aeaee0318212f_l3.png "Rendered by QuickLaTeX.com")
where 
is a matrix in the complexified adjoint representation of 
and 
its Hermitian conjugate. Thus, locally, we have:
![\[\gamma \equiv z{\rm{d}}x \wedge {\rm{d}}y\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-7f1ee30e3e29b12aba65c6baa12e8f82_l3.png "Rendered by QuickLaTeX.com")
with:
![\[\iota \Phi \gamma = 0\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-c15a856d36345fabe5c2279e23710ddc_l3.png "Rendered by QuickLaTeX.com")
a Kähler coordinate expansion of 
and gives us, after inserting it in:
the following:
![\[\begin{array}{l}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = \\{\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-8a7c66201e6214e5a9404169ffaf3f77_l3.png "Rendered by QuickLaTeX.com")
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:
![\[\begin{array}{l}{g_{ab}}\left( \xi \right) = {g_{\mu \nu }}\left( {x\left( \xi \right)} \right)\frac{{\partial {x^\mu }}}{{\partial {\xi ^a}}}\frac{{\partial {x^\nu }}}{{\partial {\xi ^b}}}\\ = {\eta _{ab}} + {\partial _a}{x^\mu }{\partial _b}{x_\mu } + \vartheta \left( {{{\left( {\partial x} \right)}^4}} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-e81bde468b31c8219c65a749e71b3fd4_l3.png "Rendered by QuickLaTeX.com")
we can write the Dirac-Born-Infeld action as:
![\[\begin{array}{c}{S_{DBI}} = - \frac{{{T_p}{{\left( {2\pi \alpha '} \right)}^2}}}{{4{g_s}}}\int {{d^{p + 1}}} \xi \left( {{F_{ab}}{F^{ab}} + \frac{2}{{{{\left( {2\pi \alpha '} \right)}^2}}}{\partial _a}{x^m}{\partial ^a}{x_m}} \right)\\ - \frac{{{T_p}}}{{{g_s}}}{V_{p + 1}} + \vartheta \left( {{F^4}} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-94ceb9803a2cb28862871fcc819700b3_l3.png "Rendered by QuickLaTeX.com")
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:
![\[{S_{YM}} = - \frac{1}{{4g_{YM}^2}}\int {{{\rm{d}}^{10}}} x\left[ {Tr\left( {{F_{\mu \nu }}{F^{\mu \nu }}} \right) + 2{\rm{i}}Tr\left( {\bar \psi \,{\Gamma ^\mu }{D_\mu }\psi } \right)} \right]\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-325857195823707ba142f0a06de0d2c9_l3.png "Rendered by QuickLaTeX.com")
with:
![\[g_{YM}^2 = \frac{{{g_s}}}{{\sqrt {\alpha '} }}{\left( {2\pi \sqrt {\alpha '} } \right)^{p - 2}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-6009fb9895e1ea158308f2a7a318eaed_l3.png "Rendered by QuickLaTeX.com")
and the action is invariant under the supersymmetric transformations:
![\[\left\{ {\begin{array}{*{20}{c}}{{\delta _\varepsilon }{A_\mu } = \frac{{\rm{i}}}{2}\bar \varepsilon \,{\Gamma ^\mu }\psi }\\{{\delta _\varepsilon }\psi = \frac{1}{2}{F_{\mu \nu }}\left[ {{\Gamma ^\mu },{\Gamma ^\nu }} \right]\varepsilon }\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-459615d12d523cabc21a413f873fb44a_l3.png "Rendered by QuickLaTeX.com")
with 
the infinitesimal Majorana-Weyl spinor. By double-gauging, we get our desired Dp-brane action:
![\[\begin{array}{l}{S_{Dp}} = - \frac{{{T_p}{g_s}{{\left( {2\pi \alpha '} \right)}^2}}}{4}\int {{{\rm{d}}^{p + 1}}} \xi \left( {{F_{ab}}{F^{ab}} + 2{D_a}{\Phi ^m}{D^a}{\Phi _m}} \right.\\ + \sum\limits_{m \ne n} {{{\left[ {{\Phi ^m},{\Phi ^n}} \right]}^2}} \left. { + {\rm{fermions}}} \right)\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-ea8441394a75356b23d67d92af934542_l3.png "Rendered by QuickLaTeX.com")
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 
:
![\[{S_{RR}} = - \frac{1}{4}\int\limits_X {G \wedge * G} - \sum\limits_i {\frac{{{\mu _i}}}{2}\int\limits_X {{\delta _{{\Sigma _i}}} \wedge \left( {C - G \wedge Y_i^{\left( 0 \right)}} \right)} } \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-7ebd65a8c28285e65f3aec6eb643be65_l3.png "Rendered by QuickLaTeX.com")
Hence, the gauged supergravity action is derivable as:
![\[{\delta _G}{S_{RR}} = - \sum\limits_{i,j} {\frac{{{\mu _i}{\mu _j}}}{2}} \int\limits_X {{\delta _{{\Sigma _i}}} \wedge } \,\,{\delta _{{\Sigma _j}}} \wedge {\left( {{Y_i} \wedge {{\bar Y}_j}} \right)^{\left( 1 \right)}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-69246605eca291fca45916f9e660ac29_l3.png "Rendered by QuickLaTeX.com")
with:
![\[{\delta _G}C = {\sum\limits_i {{\mu _i}{\delta _{{\Sigma _i}}} \wedge {{\bar Y}_i}} ^{\left( 1 \right)}}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-723dc582be3f0f7265c7ee1c327ff483_l3.png "Rendered by QuickLaTeX.com")
and 
is the Ramond-Ramond potential, thus yielding the Chern-Simons action:
![\[{S_{CS}} = \frac{{{T_p}}}{2}\int\limits_{{\Sigma _{p + 1}}} {C \wedge {\rm{ch}}\left( F \right)} \wedge \sqrt {\frac{{\hat A\left( {{R_T}} \right)}}{{\hat A\left( {{R_N}} \right)}}} \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-0d28857a8be6146ec36cc75a365b6029_l3.png "Rendered by QuickLaTeX.com")
The non-Abelian D-term thus takes the form:
![\[D = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {\tilde A\left( {\tilde T} \right)/\tilde A\left( {\tilde N} \right)} \]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-9617a7f8ae1fdc8193b82c3dbdd19a12_l3.png "Rendered by QuickLaTeX.com")
In the local patch on the C-manifold, we take the flat-space-Kähler-form:
![\[J = \underbrace {\frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar x + \frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar y}_{ = :\omega } + 2i{\rm{d}}z \wedge {\rm{d}}\bar z\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-719e7d63efe47a17c5e22d2b783d44d4_l3.png "Rendered by QuickLaTeX.com")
and decompose the Kähler-background B-field as:
![\[B \equiv B\left| {_{\tilde S}} \right. + {B_{z\overline z }}{\rm{d}}z \wedge {\rm{d}}\bar z\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-7cacf4a4f3cdce35f099206256eb5d15_l3.png "Rendered by QuickLaTeX.com")
with:
![\[\tilde F = \lambda F - B\left| {_{\tilde S}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-6d427a2f397144a5634227a9c4ff5127_l3.png "Rendered by QuickLaTeX.com")
thus giving us:
![\[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\rm{P}} \right.} \left[ J \right] \wedge \tilde F + \frac{{i\lambda }}{2}\left( {{\iota _\Phi }{\iota _\Phi }J} \right)\\ - \left( {{{\tilde F}^2} - {\omega ^2}} \right) - i\lambda \left( {{\iota _\Phi }{\iota _\Phi }B} \right)\omega \wedge \tilde F\\ - \omega \wedge {\rm{P}}\left[ {{B_{z\overline z }}{\rm{d}}z \wedge {\rm{d}}\bar z} \right]\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-a0fe25124bde774846e7c3141c746fd2_l3.png "Rendered by QuickLaTeX.com")
with the Abelian pull-back 
to 
given by:
Hence we have:
![\[\left\{ {\begin{array}{*{20}{c}}{{\iota _\Phi }{\iota _\Phi }J = 2i\left[ {\phi ,\bar \phi } \right]}\\{{\iota _\Phi }{\iota _\Phi }{J^3} = 6i\left[ {\phi ,\bar \phi } \right]{\omega ^2}}\end{array}} \right.\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-3f57c52a0887d876c27057c1c7ba9498_l3.png "Rendered by QuickLaTeX.com")
Now: realize that ![2i\left[ {\phi ,\bar \phi } \right]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-ad750d2282fbfd6d2bb0c666a2e5c3a0_l3.png "Rendered by QuickLaTeX.com")
is a zero-form and ![6i\left[ {\phi ,\bar \phi } \right]{\omega ^2}](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-b264689ab2e8b8d85c8d9e7e8fdd23b4_l3.png "Rendered by QuickLaTeX.com")
does not have transverse-legs to 
, and thus the pull-back 
has a trivial action. So, after solving:
![\[{\rm{P}}\left[ J \right] = \omega + 2i{\lambda ^2}\left( {{\rm{D}}\phi } \right) \wedge \left( {{\rm{\bar D}}\bar \phi } \right)\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-1b7e7862ca3915f2622c1ea2271f7698_l3.png "Rendered by QuickLaTeX.com")
the D-term equations amount to 
with:
![\[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\omega \wedge \tilde F} \right.} + {\lambda ^2}{\rm{D}}\phi \wedge \overline {{\rm{D}}\phi } \\ \wedge \left( {2i\tilde F - {B_{z\overline z }}\omega } \right) + \lambda \left[ {\phi ,\bar \phi } \right]\\\left. {\left( {{\omega ^2} - {{\tilde F}^2} - i{B_{z\overline z }}\omega \wedge \tilde F} \right)} \right\}\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-92ef74f7300fefeb087fd6471cca8ad1_l3.png "Rendered by QuickLaTeX.com")
and with the 
-field vanishing on the sheave of the C-manifold, one gets a reduction to:
![\[\begin{array}{l}D = \lambda \int_{\tilde S} {S\left\{ {\omega \wedge F + 2i{\lambda ^2}} \right.} {\rm{D}}\phi \wedge \overline {{\rm{D}}\phi } \wedge F\\ + \left[ {\phi ,\bar \phi } \right]\left. {\left( {{\omega ^2} - {\lambda ^2}{F^2}} \right)} \right\}\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-5dfda56951072117a2ae248db9b68de4_l3.png "Rendered by QuickLaTeX.com")
which yields a non-Abelian -corrected Chern-Simons action for a stack of D7-branes with both terms at leading order in
-corrected Chern-Simons action for a stack of D7-branes with both terms at leading order in entailing that for matrix algebras:
![\[{\Im ^G} \subset GL\left( {n,\mathbb{C}} \right)\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-3b9c245916b88a311d8ad1f3f11292dc_l3.png "Rendered by QuickLaTeX.com")
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:
![\[\begin{array}{l}D = \lambda {\int_{\tilde S} {\rm{P}} _{ab}}\left[ J \right] \wedge F = \\\lambda \int_{\tilde S} {\left( {\omega + 2i{\lambda ^2}\partial \phi \wedge \overline {\partial \phi } } \right)} \wedge F\end{array}\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-61e524c54fb70f29933f6e98df30d853_l3.png "Rendered by QuickLaTeX.com")
Hence, the -corrections are given entirely by the abelian pull-back of the Kähler-form to
-corrections are given entirely by the abelian pull-back of the Kähler-form to ![\[{{\rm{P}}_{ab}}\left[ J \right] \equiv \left( {\omega + 2i{\lambda ^2}\partial \phi \wedge \overline {\partial \phi } } \right)\]](https://www.georgeshiber.com/wp-content/ql-cache/quicklatex.com-c0ad03a640782d01963affe74ed9e48b_l3.png "Rendered by QuickLaTeX.com")
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.
