Flux Attractors and Generating Functions.

Abstract

We use the flux attractor equations to study IIB supergravity compactifications with 3-form fluxes. We show that the attractor equations determine not just the values of the complex structure moduli and the axio-dilaton, but also the masses of those moduli and the gravitino. We then show that the flux attractor equations can be recast in terms of derivatives of a single generating function. A simple expression is given for this generating function in terms of the D3 tadpole and gravitino mass, with both quantities considered as functions of the fluxes. For a simple prepotential, we explicitly solve the attractor equations. We also discuss a thermodynamic interpretation of this generating function, and possible implications for the landscape.

Having solved the flux attractor equations for 3-form fluxes, we add generalized fluxes to the compactifications and study their effects. We find that when we add only geometric fluxes, the compactifications retain their no-scale structure, and minimize their scalar potential when the appropriate complex flux is imaginary self-dual (ISD). These minima are still described by a set of flux attractor equations, which can be integrated by a generating function. The expressions for the vector moduli are formally identical to the case with 3-form fluxes only, while some of the hypermoduli vii are determined by extremizing the generating function. We work out several orbifold examples where all vector moduli and many hypermoduli are stabilized, with VEVs given explicitly in terms of fluxes.