K3 x T^2/Z_2 orientifolds with fluxes, open string moduli and critical points

TL;DR

This work extends the ${\Scr N}=2$ gauged supergravity framework for type IIB compactifications on ${K3 \times T^2/\mathbb{Z}_2}$ to include D3 and D7 open-string moduli and background fluxes. By introducing a trilinear prepotential $F(s,t,u,x^k,y^r)=s t u - 1/2 s x^k x^k - 1/2 u y^r y^r$, the authors show that the moduli space is symmetric only when either brane sector is absent, while the full space with both brane sectors activated is no longer symmetric and requires careful selection of the symplectic embedding for the scalar potential. They analyze ${\Scr N}=2$ and ${\Scr N}=1$ supersymmetric critical points, finding that in ${\Scr N}=2$ vacua the open-string moduli remain unfixed while the axion-dilaton and $T^2$ complex structure are stabilized; in ${\Scr N}=1$ vacua the D7 moduli are frozen while D3 moduli survive as flat directions, with further obstructions and no-scale behavior depending on the gauging data. Extending to generalized gaugings that couple brane vectors, they show that supersymmetric vacua require brane coordinates to vanish and that the ${\Scr N}=2,1,0$ classification aligns with whether $g_0$ and $g_1$ vanish or are equal, respectively. The results illuminate how fluxes and open-string moduli shape moduli stabilization and have implications for inflationary model-building in string compactifications.

Abstract

We extend the four-dimensional gauged supergravity analysis of type IIB vacua on $K3\times T^2/Z_2$ to the case where also D3 and D7 moduli, belonging to N=2 vector multiplets, are turned on. In this case, the overall special geometry does not correspond to a symmetric space, unless D3 or D7 moduli are switched off. In the presence of non--vanishing fluxes, we discuss supersymmetric critical points which correspond to Minkowski vacua, finding agreement with previous analysis. Finally, we point out that care is needed in the choice of the symplectic holomorphic sections of special geometry which enter the computation of the scalar potential.