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

By George Shiber Wednesday, September 14, 2016Sunday, August 18, 2019 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Science, Physics, Super String Theory, Witten Theory D-Branes, F-Theory

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

2 Responses

T-Branes and F-Theory: α’- Corrections and the D-Term-Equations
Sunday, September 18, 2016

[…] 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: […]
Deriving 4D De Sitter Space from T-Branes via D7-Brane Action
Friday, December 16, 2016

[…] Building on my earlier work on T-branes and F-theory, here I will show how De Sitter space emerges from an expansion of the D7-brane action around a T-brane background in the presence of 3-form supersymmetry breaking fluxes: this is crucial since de Sitter space is unstable in quantum gravity, while a D7-brane action resolves the instability. "T-branes are a non-abelian generalization of intersecting branes in which the matrix of normal deformations is nilpotent along some subspace", and it is truly remarkable that "the simplest heterotic string compactifications are dual to T-branes in F-theory." Recalling that the pull-back on the D7-brane worldvolume is given by: […]