SO(2) Duality, Type IIB SuperGravity and the Super-D3-Brane Action

In this post, I shall analyze certain relations between holomorphic properties of Dp-brane actions and SO(2)-duality of type IIB supergravity. Specifically, I will show that the super D3-brane action:

    \[S = {S_{DBI}} + {S_{WZ}}\]

in Type IIB SUGRA background satisfies the Gaillard-Zumino duality condition and exhibits exact self-duality. Dp-branes are p + 1 dimensional Ramond-Ramond charged dynamical hypersurfaces {\Sigma _{p + 1}} that open strings end on and admit perturbative worldsheet description in terms of open strings satisfying Dirichlet boundary conditions in p + 1 dimensions. Naturally, for 4-D spacetime physics, D3 branes are especially important for string-phenomenology due to mirror symmetry on Calabi-Yau 3-folds where they holomorphically wrap Fukaya super-Lagrangians. The D3-brane effective action in the NS5-brane geometry, given that it satisfies D3-brane self-duality and Poincaré invariance, is given by:

    \[{S_{D3}} = {g_s}{{\rm T}_3}\left[ { - \int {{d^4}} \varsigma \,{e^{ - \Phi }}\sqrt { - \,{\rm{det}}\left( {{G_{\mu \nu }} + {F_{\mu \nu }}} \right)} + i\int {\left( {{C_{\left[ 4 \right]}} + F \wedge {C_{\left[ 2 \right]}}} \right)} } \right]\]


    \[{{\rm T}_3} = \frac{1}{{{g_s}\,l_s^{\left[ 3 \right]}{{\left( {2\,\pi \,{l_s}} \right)}^2}}}\]

{{\rm T}_3} being the D3-brane tension, and {C_{\left[ 4 \right]}}, {C_{\left[ 2 \right]}} are the RR-4 and RR-2 exterior forms, and generally, the DBI action is:

    \[{S_{DBI}} = - {T_D}\int {{d^{p + 1}}} \sigma {e^{ - \phi }}\sqrt { - \det \left( {\mathcal{G} + F} \right)} \]

and the D-brane WZ action is given by:

    \[\int_{{\Sigma _{p + 1}}} {{C^{RR}}} \wedge {e^F}\]

In order for the effective action to be integrable with fields in 2nd-quantized form, one must work under the Gaillard-Zumino Condition: that is – using 8-loop counterterms with superspace torsion:

    \[{S^8} \sim {k^{14}}\int_{T8} {{d^4}} x\,{d^{32}}\theta {B_{er}}\,{\rm E}\,{T_{ijk\alpha }}\left( {\chi ,\theta } \right){T^{ * \dagger ijk\alpha }}\left( {\chi ,\theta } \right){{\rm T}_{mn{l^\alpha }}}{T_{{\vartheta _i}(\chi ,\theta )}}^{ * \dagger mnl}\]

where {T_{ijk\alpha }} is the superfield torsion. One starts with a Lagrangian:

    \[{L_G}\left( {{F_{\mu \nu }},{g_{\mu \nu }},{\Phi ^A}} \right) = \sqrt { - {g_s}} L\left( {{F_{\mu \nu }},{g_{_{\mu \nu }}},{\Phi ^A}} \right)\]

in D = 4, with a dependence on a U(1) gauge field strength {F_{\mu \nu }}, metric {g_{\mu \nu }}, and matter field {\Phi ^A}. So, we now have:

    \[{K^{*\dag \mu \nu }} = \frac{{\partial {L_G}}}{{\partial {F_{\mu \nu }}}}\]

    \[\frac{{\partial {F_{\alpha \beta }}}}{{\partial {F_{\mu \nu }}}} = \left( {\delta _\alpha ^\mu {\mkern 1mu} \delta _{{\beta ^{{\mkern 1mu} {\mkern 1mu} - }}}^\nu - \delta _\beta ^\mu {\mkern 1mu} \delta _\alpha ^\nu } \right)\]

and the Hodge dual components for the tensor {K_{\mu \nu }} are given by:

    \[K_{\mu \nu }^{ * \dagger } = \frac{1}{2}{\eta _{\mu \nu }}^{\rho \sigma }{k_{\rho \sigma }}\]

    \[K_{\mu \nu }^{ * \dagger } = - {K_{\mu \nu }}\]

The Gaillard-Zumino condition is an infinitesimal duality transformation of F and K and fermionic transformation given by:

    \[\delta \left( {{\Gamma _k}} \right) = \left( {\begin{array}{*{20}{c}}\alpha &\beta \\\gamma &\delta \end{array}} \right)\left( {\begin{array}{*{20}{c}}F\\K\end{array}} \right)\]

    \[\delta \,{\Phi ^A} = {\xi ^A}(\Phi )\]

    \[\delta {g_{\mu \nu }} = 0\]

Now, the Lagrangian must transform as:

    \[\delta {L_G} = \frac{1}{4}\left( {\gamma F\,{F^{ * \dagger }} + \beta K\,{K^{ * \dagger }}} \right)\]

and one has an SO(2) transformation given by \delta F = \lambda k, \delta K = - \lambda F, and so the Lagrangian is given by:

    \[\delta {L_G} = \frac{1}{2}\,\frac{{\partial L}}{{\partial F}}\delta {F_{\mu \nu }} + \frac{{\partial L}}{{\partial {\phi ^A}}} = \frac{\lambda }{2}{\widetilde K^{\mu \nu }}{K_{\mu \nu }} + {\delta _\Phi }L\]

and by D3-brane self-duality, it follows that:

    \[\frac{\lambda }{4}\left( {F \cdot {F^{ * \dagger }} + K \cdot {K^{ * \dagger }}} \right) + {\delta _\Phi }{L_G} = 0\]