N=4 Supergravity Lagrangian for Type IIB Orientifold on T⁶/Z₂ in Presence of Fluxes and D3-Branes

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We derive the Lagrangian and the transformation laws of N=4 gauged supergravity coupled to matter multiplets whose sigma-model of the scalars is SU(1,1)/U(1)x SO(6,6+n)/SO(6)xSO(6+n) and which corresponds to the effective Lagrangian of the Type IIB string compactified on the T⁶/Z₂ orientifold with fluxes turned on and in presence of n D3-branes. The gauge group is T¹2 x G where G is the gauge group on the brane and T¹2 is the gauge group on the bulk corresponding to the gauged translations of the R-R scalars coming from the R-R four--form. The N=4 bulk sector of this theory can be obtained as a truncation of the Scherk-Schwarz spontaneously broken N=8 supergravity. Consequently the full bulk spectrum satisfies quadratic and quartic mass sum rules, identical to those encountered in Scherk-Schwarz reduction gauging a flat group. This theory gives rise to a no scale supergravity extended with partial super-Higgs mechanism.