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:

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


    \[\sqrt {{\rm A}'(\Gamma )/\bar {\rm A}'({\rm N})} \]

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

    \[J = \underbrace {\frac{i}{2}{\rm{dx}} \wedge {\rm{d\bar x + }}\frac{i}{2}{\rm{dy}} \wedge {\rm{d\bar y}}}_{ = :\omega } + 2i{\rm{dz}} \wedge {\rm{d\bar z}}\]

where S_{cld}^s is given by:

    \[\begin{array}{*{20}{c}}{S_{cld}^s = - \frac{1}{{4\pi {\alpha ^\dagger }}}\int_{\partial E_S^5} {{d^2}\sigma d} \Omega {{\left( {{\phi _{Inst}}} \right)}^2}\sigma \sqrt { - \gamma } \left( {\phi \left( {{{\bar X}^\mu }} \right)} \right.R}\\{ + {\gamma ^{\alpha \beta }}{\partial _\alpha }{X^\mu }{g_{\mu \nu }}\left( {{{\bar X}^\nu }} \right) + \frac{1}{{\sqrt { - \gamma } }}{\varepsilon ^{ - {c_{2n}}/{Y_k}\left( {{{\cos }^2}\varphi } \right)}}{\partial _\beta }{{\bar X}^\nu }{b_{\mu \nu }}{{\left( {\bar X} \right)}^2}}\end{array}\]

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

    \[{W^O} = \int_{{\Sigma _5}} {\rm{P}} \left[ {{\Omega _0} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]

    \[{D^K} = \int_{\tilde S} {\rm{P}} \left[ {{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]

are built up as follows. Write locally:

    \[{\Omega _0} \wedge {e^B} = d\gamma \]

and localize the integral in:

    \[{W^O} = \int_{{\Sigma _5}} {\rm{P}} \left[ {{\Omega _0} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]


    \[\int_{\tilde S} {P\left[ \gamma \right]} \wedge {e^{\lambda F}}\]


the non-Abelian generalisation of {W^0} and {D^K} 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:

    \[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]

    \[{D^K} = \int_{\tilde S} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}} \]

with \iota \Phi the inclusion of the complex Higgs field \Phi, and S represents the symmetrization over gauge indices.

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

    \[\Phi \equiv \phi \frac{\partial }{{\bar \partial z}} + \bar \phi \frac{{\bar \partial }}{{\partial \bar z}}\]

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

    \[\gamma \equiv z{\rm{d}}x \wedge {\rm{d}}y\]


    \[\iota \Phi \gamma = 0\]

a Kähler coordinate expansion of \gamma and gives us, after inserting it in:

    \[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]

the following:

    \[\begin{array}{l}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = \\{\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}\end{array}\]

which is the exact 7-brane superpotential for F-theory and the integrand is independent of \lambda, entailing that the F-term conditions are purely topological and in no need for \alpha '-corrections

Fixing our induced Dp-brane worldvolume metric:

    \[\begin{array}{l}{g_{ab}}\left( \xi \right) = {g_{\mu \nu }}\left( {x\left( \xi \right)} \right)\frac{{\partial {x^\mu }}}{{\partial {\xi ^a}}}\frac{{\partial {x^\nu }}}{{\partial {\xi ^b}}}\\ = {\eta _{ab}} + {\partial _a}{x^\mu }{\partial _b}{x_\mu } + \vartheta \left( {{{\left( {\partial x} \right)}^4}} \right)\end{array}\]

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

    \[\begin{array}{c}{S_{DBI}} = - \frac{{{T_p}{{\left( {2\pi \alpha '} \right)}^2}}}{{4{g_s}}}\int {{d^{p + 1}}} \xi \left( {{F_{ab}}{F^{ab}} + \frac{2}{{{{\left( {2\pi \alpha '} \right)}^2}}}{\partial _a}{x^m}{\partial ^a}{x_m}} \right)\\ - \frac{{{T_p}}}{{{g_s}}}{V_{p + 1}} + \vartheta \left( {{F^4}} \right)\end{array}\]

which is a Higgsed U\left( 1 \right) gauge theory in p + 1 dimensions with 9 - p scalar fields. Thus, by dimensional reduction, this action is equivalent to a U\left( 1 \right) Yang-Mills gauge theory in 10-spacetime-dimensions with action:

    \[{S_{YM}} = - \frac{1}{{4g_{YM}^2}}\int {{{\rm{d}}^{10}}} x\left[ {Tr\left( {{F_{\mu \nu }}{F^{\mu \nu }}} \right) + 2{\rm{i}}Tr\left( {\bar \psi \,{\Gamma ^\mu }{D_\mu }\psi } \right)} \right]\]


    \[g_{YM}^2 = \frac{{{g_s}}}{{\sqrt {\alpha '} }}{\left( {2\pi \sqrt {\alpha '} } \right)^{p - 2}}\]

and the action is invariant under the supersymmetric transformations:

    \[\left\{ {\begin{array}{*{20}{c}}{{\delta _\varepsilon }{A_\mu } = \frac{{\rm{i}}}{2}\bar \varepsilon \,{\Gamma ^\mu }\psi }\\{{\delta _\varepsilon }\psi = \frac{1}{2}{F_{\mu \nu }}\left[ {{\Gamma ^\mu },{\Gamma ^\nu }} \right]\varepsilon }\end{array}} \right.\]

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

    \[\begin{array}{l}{S_{Dp}} = - \frac{{{T_p}{g_s}{{\left( {2\pi \alpha '} \right)}^2}}}{4}\int {{{\rm{d}}^{p + 1}}} \xi \left( {{F_{ab}}{F^{ab}} + 2{D_a}{\Phi ^m}{D^a}{\Phi _m}} \right.\\ + \sum\limits_{m \ne n} {{{\left[ {{\Phi ^m},{\Phi ^n}} \right]}^2}} \left. { + {\rm{fermions}}} \right)\end{array}\]

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 X:

    \[{S_{RR}} = - \frac{1}{4}\int\limits_X {G \wedge * G} - \sum\limits_i {\frac{{{\mu _i}}}{2}\int\limits_X {{\delta _{{\Sigma _i}}} \wedge \left( {C - G \wedge Y_i^{\left( 0 \right)}} \right)} } \]

Hence, the gauged supergravity action is derivable as:

    \[{\delta _G}{S_{RR}} = - \sum\limits_{i,j} {\frac{{{\mu _i}{\mu _j}}}{2}} \int\limits_X {{\delta _{{\Sigma _i}}} \wedge } \,\,{\delta _{{\Sigma _j}}} \wedge {\left( {{Y_i} \wedge {{\bar Y}_j}} \right)^{\left( 1 \right)}}\]


    \[{\delta _G}C = {\sum\limits_i {{\mu _i}{\delta _{{\Sigma _i}}} \wedge {{\bar Y}_i}} ^{\left( 1 \right)}}\]

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

    \[{S_{CS}} = \frac{{{T_p}}}{2}\int\limits_{{\Sigma _{p + 1}}} {C \wedge {\rm{ch}}\left( F \right)} \wedge \sqrt {\frac{{\hat A\left( {{R_T}} \right)}}{{\hat A\left( {{R_N}} \right)}}} \]

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

    \[D = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {\tilde A\left( {\tilde T} \right)/\tilde A\left( {\tilde N} \right)} \]

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

    \[J = \underbrace {\frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar x + \frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar y}_{ = :\omega } + 2i{\rm{d}}z \wedge {\rm{d}}\bar z\]

and decompose the Kähler-background B-field as:

    \[B \equiv B\left| {_{\tilde S}} \right. + {B_{z\overline z }}{\rm{d}}z \wedge {\rm{d}}\bar z\]


    \[\tilde F = \lambda F - B\left| {_{\tilde S}} \right.\]

thus giving us:

    \[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\rm{P}} \right.} \left[ J \right] \wedge \tilde F + \frac{{i\lambda }}{2}\left( {{\iota _\Phi }{\iota _\Phi }J} \right)\\ - \left( {{{\tilde F}^2} - {\omega ^2}} \right) - i\lambda \left( {{\iota _\Phi }{\iota _\Phi }B} \right)\omega \wedge \tilde F\\ - \omega \wedge {\rm{P}}\left[ {{B_{z\overline z }}{\rm{d}}z \wedge {\rm{d}}\bar z} \right]\end{array}\]

with the Abelian pull-back \omega to {S_4} given by:

    \[J = \underbrace {\frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar x + \frac{i}{2}{\rm{d}}x \wedge {\rm{d}}\bar y}_{ = :\omega } + 2i{\rm{d}}z \wedge {\rm{d}}\bar z\]

Hence we have:

    \[\left\{ {\begin{array}{*{20}{c}}{{\iota _\Phi }{\iota _\Phi }J = 2i\left[ {\phi ,\bar \phi } \right]}\\{{\iota _\Phi }{\iota _\Phi }{J^3} = 6i\left[ {\phi ,\bar \phi } \right]{\omega ^2}}\end{array}} \right.\]

Now: realize that 2i\left[ {\phi ,\bar \phi } \right] is a zero-form and 6i\left[ {\phi ,\bar \phi } \right]{\omega ^2} does not have transverse-legs to \tilde S, and thus the pull-back {\rm{P}} has a trivial action. So, after solving:

    \[{\rm{P}}\left[ J \right] = \omega + 2i{\lambda ^2}\left( {{\rm{D}}\phi } \right) \wedge \left( {{\rm{\bar D}}\bar \phi } \right)\]

the D-term equations amount to D = 0 with:

    \[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\omega \wedge \tilde F} \right.} + {\lambda ^2}{\rm{D}}\phi \wedge \overline {{\rm{D}}\phi } \\ \wedge \left( {2i\tilde F - {B_{z\overline z }}\omega } \right) + \lambda \left[ {\phi ,\bar \phi } \right]\\\left. {\left( {{\omega ^2} - {{\tilde F}^2} - i{B_{z\overline z }}\omega \wedge \tilde F} \right)} \right\}\end{array}\]

and with the B-field vanishing on the sheave of the C-manifold, one gets a reduction …