Mathematical Structures^1234.3 Prepared Comments
4.3.1 Renata Kallosh: Stabilization of moduli in string theory
Stabilization of moduli is necessary for string theory to describe the effective 4-
dimensional particle physics and cosmology. This is a long-standing problem. Re-
cently a significant progress towards its solution was achieved: a combination of flux
compactification with non-perturbative corrections leads to stabilization of moduli,
with de Sitter vacua [l]. Such vacua with positive cosmological constant can be
viewed as the simplest possibility for the string theory to explain the observable
dark energy. This is a prerequisite for the Landscape of String Theory [a]. Main
progress was achieved in type I1 string theories, the heterotic case still remains
unclear, which is a serious problem for particle physics models related to string
theory.
In type IIB string theory flux compactification with Wfluz = JG3 A 023 leads
to stabilization of the dilaton and complex structure moduli [3], whereas gaugino
condensation and/or instanton corrections Wnon--pert = Ae-(vol+i“) stabilize the
remaining Kahler moduli. The basic steps here are: i) Using the warped geometry
of the compactified space and nonperturbative effects one can stabilize all moduli in
anti-de-Sitter space. ii) One can uplift the AdS space to a metastable dS space by
adding anti-D3 brane at the tip of the conifold (or D7 brane with fluxes [4]). More
recently there was a dramatic progress in moduli stabilization in string theory and
various successful possibilities were explored [5]- [6].
Examples of new tools include the recently discovered criteria, in presence of
fluxes, for the instanton corrections due to branes, wrapping particular cycles [7].
This has allowed one of the simplest models with all moduli stabilized. We have
found that M-theory compactified on K3xK3 is incredibly simple and elegant [6].
Without fluxes in the compactified 3d theory there are two 80-dimensional quater-
nionic Kahler spaces, one for each K3. With non-vanishing primitive (2,2) flux,
(2,O) and (0,2), each K3 becomes an attractive K3: one-half of all moduli are fixed.
40 in each K3 still remain moduli and need to be fixed by instantons. There are
20 proper 4-cycles in each K3. They provide instanton corrections from M5-branes
wrapped on these cycles: Moduli space is no more ...
All cases of moduli stabilization in black holes and in flux vacua which are due
to fluxes in string theory can be described by the relevant attractor equations, the
so-called “new attractors” [S].
It is known for about 10 years that in extremal black holes the moduli of vector
multiplets are stabilized near the horizon where they become fixed function of fluxes
(p,q) independently of the values of these moduli far away from the black hole
horizon. This is known as a black hole attractor mechanism [9].
Stabilization of moduli is equivalent to minimization of the black hole potentialtfiZ = %4) , Efiz = t(p, 4). (1)
VBH = 1O2l2 + (^1212) (2)