Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/6

0 Comments & 1 Trackback

Read the post From One, Many
Weblog: Musings
Excerpt: Been meaning to make some comments about this, but just hadn't gotten around to it. Lenny Susskind has taken the...
Tracked: January 5, 2008 11:52 AM

Post a New Comment