T-Branes and F-Theory: α’- Corrections and the D-Term-Equations

I last introduced T-branes, which are non-Abelian deformation of intersecting D-brane systems in the corresponding compactification manifold. Then I showed that we have a Kähler-equivalence of the derivatives in the pull-back with the gauge-covariant ones, which gave us:

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

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

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

Locally, 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, I could derive:

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


    \[\iota \Phi \gamma = 0\]

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

Aside: the book on the post cover is excellent.

However, the D-term in:

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

is in need of \alpha '-corrections, since it is evaluable as:

    \[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\lambda P\left[ J \right]} \right.} \wedge F - \frac{{i\lambda }}{6}{\iota _\Phi }{\iota _\Phi }{J^3} + \\\frac{{i{\lambda ^3}}}{2}{\iota _\Phi }{\iota _\Phi }J \wedge F \wedge F - {\rm{P}}\left[ {J \wedge B} \right] \wedge F\\\left. { + i{\lambda ^2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge B} \right) \wedge \frac{{i\lambda }}{2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge {B^2}} \right)} \right\}\end{array}\]

and the non-Abelian D-term has 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, one takes the flat-space Kähler-form as having the form:

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

Then, we 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} as determined 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 to:

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

which yields a non-Abelian \alpha '-corrected Chern-Simons action of a stack of D7-branes with both terms at leading order in \lambda

entailing that for matrix algebras:

    \[{\Im ^G} \subset GL\left( {n,\mathbb{C}} \right)\]

they are the matrix products in the fundamental representation of {\Im ^G}

and so the \alpha '-corrections on D-terms with the gauge flux F diagonalization yields

the D-term equations

    \[\begin{array}{l}D = \lambda {\int_{\tilde S} {\rm{P}} _{ab}}\left[ J \right] \wedge F = \\\lambda \int_{\tilde S} {\left( {\omega + 2i{\lambda ^2}\partial \phi \wedge \overline {\partial \phi } } \right)} \wedge F\end{array}\]

Deep upshot: the \alpha '-corrections are given entirely by the abelian pull-back of the Kähler-form J to \tilde S

    \[{{\rm{P}}_{ab}}\left[ J \right] \equiv \left( {\omega + 2i{\lambda ^2}\partial \phi \wedge \overline {\partial \phi } } \right)\]

And this has a deep physical interpretation which can be extracted from: ‘The energy-momentum tensor and D-term of Q-clouds‘.…