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Flux Attractors and Generating Functions.

O'Connell, Ross C.

O'Connell, Ross C.

2010

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
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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.