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:

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

where S_{cld}^s in:

    \[P_{{\rm{int}}}^{Dp} \equiv \not Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}} \]

is given by:

    \[S_{cld}^s = - \frac{1}{{4\pi {\alpha ^\dagger }}}\int_{\partial E_S^5} {{d^2}\sigma d} \Omega {\left( {{\phi _{INST}}} \right)^2}\sigma \sqrt { - \gamma } \left( {\phi \left( {{{\bar X}^\mu }} \right)} \right.{R_{icci}} + {\gamma ^{\alpha \beta }}{\not \partial _\alpha }{X^\mu }{g_{\mu \nu }}\left( {{{\bar X}^\nu }} \right) + \frac{1}{{\sqrt { - \gamma } }}{\varepsilon ^{ - {c_{2n}}/{Y_k}\left( {{{\cos }^2}\varphi } \right)}}{\not \partial _\beta }{\bar X^\nu }{b_{\mu \nu }}{\left( {\bar X} \right)^2}\]

In this, part one, I will show that the 7-brane superpotential for F-theory and the corresponding F-term conditions are purely topological and in no need for \alpha '-corrections. 

Recalling that a central class of T-brane configurations feature a Higgs field \Phi and a set of non-commuting generators {E_i} as well as a non-primitive worldvolume flux of the form:

    \[F = - i\partial \bar \partial fP\]

which is a solution to the D-brane equation, and P is the Cartan generator of the gauge group G and f is a function of the 7-brane coordinates that solves the D-brane equation. The central problem of this series of posts is that this Abelian profile for F needs the \alpha '-correction for 4-D compactificational Calabi-Yau consistency conditions.

First, note that the Kähler form J and:

    \[{\Omega _0} = {e^{\phi /2}}\Omega \]

a holomorphic (3,0)-form in {X_3}, satisfy:

    \[\frac{1}{6}{J^3} = - \frac{i}{8}\Omega \wedge \bar \Omega \]

with F = dA the worldvolume flux and \lambda = 2\pi \alpha '. Hence,

the pull-back on the D7-brane worldvolume is

    \[{\rm{P}}{\left[ {{V_\mu }{\rm{d}}{z^\mu }} \right]_\alpha } = {V_\alpha } + \lambda {V_i}{\not \partial _\alpha }{\Phi ^i}\]

where \alpha is a coordinate on \tilde S.

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:

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

as:

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

thus,

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:

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

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

In the next post, I shall get into T-brane and \alpha '-corrections