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:

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

Deriving 4D De Sitter Space from T-Branes via D7-Brane Action

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:

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

where \alpha is a coordinate on \tilde S and 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:

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

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:

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