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

with:

\iota \Phi \gamma = 0

a Kähler coordiante 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

T-Branes and F-Theory: Towards GUT-Model-Construction

In this series, I will study T-branes as they relate to F-theory in the context of GUT-model-construction. Aside: the book on the cover is a great read. Note that D-branes are crucial since they allow us to build various 4-D string theory vacua and dictate the phenomenology of compactifications as well. T-branes are non-Abelian deformation of intersecting D-brane systems in the corresponding compactification manifold. Without loss of generality, I will work in type IIB string theory compactified on a Calabi-Yau threefold {X_3} quotiented by an orientifold action stacks of D3-branes and D7-branes. The BPS conditions are functionals:

{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}}

and in 4-D are interpreted as a superpotential and D-term for each D7-brane.

and generally, the second quantized integral of the D-brane partition function for closed strings is given by:

P_{{\rm{int}}}^{Dp} \equiv \not 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{ }}

and

\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}}