Wednesday, March 15, 2006

Heterotic TOE and the vacuum energy

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The previous blog article about the heterotic minimal supersymmetric standard models is here.

One of the usual complaints against the traditional line of superstring model building is that there is no huge landscape of such vacua, which is why you can't solve the cosmological constant problem.

This is a pseudoargument because there exists no quantitative evidence that the cosmological constant problem can't be resolved in a scientific way.

Burt Ovrut and Volker Braun propose their picture here:
Inside the visible E_8 on their favorite pure MSSM Calabi-Yau manifold, they use their latest pure MSSM bundle that leads to the correct minimal MSSM spectrum. The corresponding bundle for the hidden E_8 that would cancel the anomalies has been known to be unstable. Instead, Ovrut and Braun use a trivial bundle for the hidden E_8 and cancel the anomalies by M5-branes and anti-M5-branes that live in the bulk, in between the two domain walls.

If you only used M5-branes but no anti-M5-branes, you would obtain a supersymmetric AdS vacuum whose cosmological constant is of order -10^{-16} M_{Pl}^4 - which is large and has a wrong sign - as they show in two simplified but nevertheless pretty complex models. However, when you use both M5-branes as well as anti-M5-branes, you may obtain positive and perhaps even semi-realistic values of the vacuum energy as long as you adjust the Kähler moduli to appropriate values. At these values, the visible bundle is slope-stable, too, which is a good news.

At the beginning of their paper, they review their best and minimal model. They advocate the opinion that gaugino condensation in the hidden sector, even though it is popular, is not a phenomenologically attractive way to break SUSY in heterotic M-theory. This is why they decide that M5-branes and anti-M5-branes whose curves are fixed by the anomaly cancellation are a better way to break SUSY "explicitly" (of course that the breaking is still "spontaneous" from the viewpoint of the full string/M-theory).

In their simplified analysis, they neglect the bundle moduli but show the stabilization of the dilaton, complex structure moduli, and Kähler moduli.


Imagine that you believe that they are converging to the theory of everything. What are the relevant scales?
  • the coupling alpha_{GUT} = 1/25 near the Planck scale
  • the Planck scale is around the usual 10^{19} GeV
  • the Calabi-Yau radius scale is around the GUT scale, 10^{16} GeV - the GUT and Calabi-Yau scales coincide because the momentum modes at the Calabi-Yau scale that feel the Wilson lines can be interpreted as the GUT-breaking Higgses
  • the eleventh dimension is longer, 1/R is around 10^{14} GeV that can already be called an intermediate scale, which is why the theory is five-dimensional above this scale and below the GUT scale
This sounds great to a conservative ear. Where does the smallness of the cosmological constant come from? As you know, the cosmological constant, about 10^{-120} in Planck units, is often viewed as a reason to adopt new forms of religion because it is so incredibly tiny.

In Ovrut and Braun's framework, the realistic values of the cosmological constant may be obtained from very reasonable numbers. The tiniest number they need to use is "Vscript_5bar" - the volume of a one-complex dimensional surface on which the anti-M5-brane is wrapped. Recall that one needs to get a cosmological constant that is 120 orders of magnitude below the Planck scale, so you might think that you need a ridiculously small two-cycles on which the anti-M5-brane is wrapped.

However, their calculation shows that you only need a two-cycle whose area is about 10^{-7} in the Calabi-Yau area unit (the unit of area is the third root of the volume of the Calabi-Yau manifold). This more or less reasonable number leads to the observed value of the cosmological constant.

Snow has returned to Cambridge...


  1. I haven't read the paper I have to admit, but maybe somebody could provide me with an indication why having both M5 and anti-M5 branes does not lead to an instability leading to tachyons in the low energy theory form the M5 moduli. Or do they wrap different holomorphic curves in the CY?

    This leads me to a different question: Assume I have a M5-anti-M5 pair which wraps different holomorphic curves. Could they annihilate? Would there be a remnant as in the case of D-anti-D there would probably be a a lower dimensional D-brane remnant

  2. Dear Robert,

    if you wrapped a brane and an antibrane on the same curve, surely they could annihilate completely.

    On the other hand, although I have not checked their particular example, it is quite clear that there are examples of holomorphic cycles on complex manifolds such that the branes can't geometrically annihilate.

    If the total class on which the brane is wrapped has no holomorphic representative, it is probably a generic fact that the minimal are representative of the class is disconnected.

    Take del Pezzo 2, a CP2 with two blowups that give you non-intersecting new 2-cycles E1,E2. Each of them has a CP1 (spherical) holomorphic representative. E1 does not intersect E2.

    It seems pretty clear that both E1+E2 as well as E1-E2 will have a minimal representative composed from two spheres. The latter case, for example, can be interpreted as a brane-antibrane pair wrapped on two different curves. In the Calabi-Yau case, the sign determines whether you break SUSY. In complex geometry, the choice which cycle is holomorphic (and its minus is antiholomorphic) is pretty much canonical, up to an overall sign convention.

    In the geometrical case (only shapes of branes determine your state, no other light degrees of freedom), you know that the total homology (charge) must be conserved, and if you admit that this must mean that there are still visually some branes wrapped on the cycle(s), you won't be able to find a more economical configuration which means that the brane-antibrane configuration is stable.

    This argument would break down if there were an instability that also demands a topology change for the Calabi-Yau manifold or some other process involving some light degrees of freedom that were neglected above.

    Note that there was no real tachyon in my E1-E2 example because the two curves did not intersect, and the strings stretched between E1 and E2 were heavy.