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February 18, 2003

Moduli-Fixing in M-theory

So I thought I’d say some more about the relation between Bobby Acharya’s paper on moduli-fixing in M-theory (which I’ve blogged about before) and the work of Kachru et al that I wrote about here.

Recall that the latter proceed in three steps

  1. The flux-induced superpotential (in the Type-IIB orientifold description)

    (1) M(F 3τH 3)Ω\int_M (F_3-\tau H_3)\wedge \Omega

    fixes the complex structure and the string coupling, leaving the Kahler modulus, ρ\rho (assume just one), as a flat direction.

  2. They then guess at the structure of the the nonperturbative superpotential for ρ\rho. With ρ\rho fixed, we end up with a supersymmetric solution in 4D anti-de Sitter space.
  3. They introduce supersymmetry-breaking in the form of anti-D3 brane(s). This contribution to the potential for ρ\rho has its coefficient fine-tuned so as to raise the previous anti-de Sitter minimum to slightly above zero, producing a non-supersymmetric metastable solution with a small positive cosmological constant.

M-theory compactified on a manifold XX of G 2G_2-holonomy also has a flux-induced superpotential

(2)W 1=18π 2 X(C2+iϕ)GW_1= \frac{1}{8\pi^2}\int_X \left(\textstyle{\frac{C}{2}}+i\phi\right)\wedge G

where ϕ\phi is the G 2G_2 structure. In addition, Bobby argues that if XX is fibered over a 3-manifold QQ, with the generic fiber having an ALE singularity corresponding to the simply-laced gauge group GG, there’s a further contribution to the superpotential that looks like a complex Chern-Simons term

(3)W 2=18π 2 QTr(𝒜d𝒜+23𝒜𝒜𝒜)W_2=\frac{1}{8\pi^2}\int_Q Tr ( \mathcal{A}\wedge d\mathcal{A} +\textstyle{\frac{2}{3}}\mathcal{A}\wedge\mathcal{A}\wedge\mathcal{A} )

where 𝒜=A+iB\mathcal{A}=A+iB. AA is the GG gauge connection on QQ and BB is a 1-form in the adjoint of GG (the twisted version of the 3 scalars in the 7D gauge multiplet).

The critical points of W 2W_2 are flat (complexified) GG-connections on QQ and on the space of critical points, we can write W 2=c 1+ic 2W_2=c_1+i c_2 for some constants c 1,2c_{1,2}. The combination W 1+W 2W_1+W_2 lifts all the flat directions, producing, as above, a supersymmetric solution in 4D anti-de Sitter space.

Bobby argues that the supergravity computation that led to this is reliable provided c 2c_2 is large. Unfortunately, this excludes the familiar candidates for QQ, like S 3S_3 or S 3/ nS^3/\mathbb{Z}_n (which have “known” heterotic duals). QQ must be a hyperbolic 3-manifold (yuck!).

Anyway, we have achieved points 1 and 2 above with no fudging whatsoever. This puts us in comparatively better shape to understand step 3. If we can introduce supersymmetry-breaking in the M-theory formulation, we might actually be able to say something reliable about the resulting de Sitter vacuum.

Posted by distler at February 18, 2003 9:57 AM

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