Musings

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)$$\int_M (F_3-\tau H_3)\wedge \Omega$$

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

2. They then guess at the structure of the the nonperturbative superpotential for $\rho$. With $\rho$ 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 $\rho$ 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 $X$ of $G_2$-holonomy also has a flux-induced superpotential

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

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

The critical points of $W_2$ are flat (complexified) $G$-connections on $Q$ and on the space of critical points, we can write $W_2=c_1+i c_2$ for some constants $c_{1,2}$. The combination $W_1+W_2$ 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 $c_2$ is large. Unfortunately, this excludes the familiar candidates for $Q$, like $S_3$ or $S^3/\mathbb{Z}_n$ (which have “known” heterotic duals). $Q$ 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.