Orientifolds, Brane Coordinates and Special Geometry

TL;DR

This work analyzes four-dimensional gauged supergravity arising from Type IIB flux compactifications on $K3\times T^2/\mathbb{Z}_2$ with D3-D7 branes, focusing on moduli stabilization and Minkowski vacua. Moduli dynamics are governed by ${\mathcal{N}}=2$ special Kähler and quaternionic geometries, with the scalar potential determined by the period matrix ${\Scr N}$ and the symplectic embedding of holomorphic sections; careful choice of this embedding reproduces known flux vacua and allows transitions to ${\mathcal{N}}=1$ or ${\mathcal{N}}=0$ vacua. The paper shows that in ${\mathcal{N}}=2$ vacua certain flux gaugings fix the axion-dilaton and $T^2$ structure while leaving D3/D7 moduli, whereas in ${\mathcal{N}}=1$ vacua the D7 moduli freeze and D3 moduli may persist, with the SUSY status depending on flux ratios. A key bridge is established between the ${\mathcal{N}}=2$ special geometry and the Born-Infeld action for brane moduli, highlighting the need for nonpolynomial corrections and introducing solvable-Lie-algebra coordinates to obtain a manageable, supersymmetric description of the bulk-brane moduli space. The results align with prior flux-vacua analyses and provide a framework for incorporating open-string moduli into flux-stabilization schemes, with potential relevance to KKLT-like scenarios and inflationary model-building.

Abstract

We report on the gauged supergravity analysis of Type IIB vacua on K3x T2/Z2 orientifold in the presence of D3-D7-branes and fluxes. We discuss supersymmetric critical points correspond to Minkowski vacua and the related fixing of moduli, finding agreement with previous analysis. An important role is played by the choice of the symplectic holomorphic sections of special geometry which enter the computation of the scalar potential. The related period matrix N is explicitly given. The relation between the special geometry and the Born--Infeld action for the brane moduli is elucidated.