Thursday, April 13, 2006

Flux compactifications of M-theory and F-theory

Today we had an oral exam, some minor progress in the calculations of the black hole corrections, and I attended Cumrun Vafa's class which is always a good opportunity to refresh one's knowledge of various things.

He started with the Dijkgraaf-Vafa correspondence, and finished with flux compactifications. I will write comments about Dijkgraaf-Vafa later, but let me start with the following:

Flux compactifications

As the Becker sisters explained, the compactification of M-theory on Calabi-Yau four-folds (which are eight-real-dimensional which leaves three large spacetime dimensions) actually requires nonzero values of the four-form field strength G4. It is because the eleven-dimensional action contains the terms of the form
  • S = int C3 /\ ( G4 /\ G4 - I8(R) ) + ...

The first term is a tree-level Chern-Simons term needed for classical supersymmetry of eleven-dimensional supergravity while the second term depending on the Riemann tensor R may be viewed as a one-loop correction. Note that the one-loop terms are often determined independently of the UV details of physics and M-theory is no exception.

The equations of motion obtained by varying with respect to C3 tell you, after you integrate the corresponding eight-form over the whole eight-manifold, that

  • Int G4 /\ G4 = chi/24

where "chi" is the Euler character. Whenever the Euler character is nonzero, you are forced to turn on the G4 field strength - otherwise the equations of motion simply cannot be satisfied. This is a very natural occurence of the field strength: in this case, it is actually required for supersymmetry to be preserved - while in most cases, we add fluxes in order to break supersymmetry; the field strength appears in the equation for the covariantly constant spinors as a kind of torsion term. Incidentally, the equation above may be modified if you add M2-branes - their number must be added to the Euler character.

This M-theory picture leads to three large spacetime dimensions which you may view as too little. However, one can get to four large spacetime dimensions if you add with 12 dimensions - this is what F-theory where F stands for Father of VaFa, according to taste, allows you - and compactify this F-theory on the same Calabi-Yau four-folds. Because the twelve dimensions of F-theory are not really full dimensions that can be simultaneously decompactified, the Calabi-Yau four-fold cannot be arbitrary. It must be an elliptic manifold - which means that it must be possible to write it as an elliptic fibration over a six-real-dimensional base space.

What is an elliptic fibration? It is a fiber bundle whose fiber is an elliptic curve - a fancy complex-geometric term for the two-dimensional torus. (Rational curves are spheres and hyperelliptic curves are Riemann surfaces of higher genera.) In other words, this eight-real-dimensional manifold must be representable as a two-torus attached to every point of a six-real dimensional base. The complex structure "tau" of the torus then plays the role of the dilaton-axion complex scalar field of type IIB supergravity (or type IIB string theory). The SL(2,Z) S-duality group is manifest because it is the group of large diffeomorphisms of the two-torus.

The F-theory on the four-fold is dual to M-theory on the same manifold. If you compactify one large dimension on the F-theory side on a circle, you obtain type IIB on a circle times the four-fold which is T-dual to type IIA on another (T-dual) circle multiplied with the same four-fold which goes at strong coupling to M-theory on the four-fold.

F-theory on the four-fold typically has a lot of moduli and some points on the moduli space can be described as orientifolds of type II string theory.

The story from M-theory can be directly translated to the type IIB language using the duality dictionaries. The eight-dimensional integral becomes a six-dimensional integral over the base space of the four-fold. The eight-form becomes a six-form

  • G /\ G ... becomes ... H(RR) /\ H(NS)

where the two fields H are three-form field strengths from the NS-NS and R-R sector, respectively. The Euler character "chi" remains the Euler character of the whole Calabi-Yau four-folds, and it is now D3-branes instead of the M2-branes, still filling all the large dimensions, whose number can be added to the Euler character to shift it.

The dilaton superfield and the complex structure is stabilized by the perturbative Gukov-Vafa-Witten superpotential while the Kähler moduli are stabilized by nonperturbative effects - the overall size of the manifold is the most difficult one to stabilize. All these things lead to supersymmetric anti de Sitter vacua, the KKLT vacua, and breaking of SUSY that is expected to lift the anti de Sitter vacuum to a semi-realistic de Sitter vacuum is often achieved by adding anti-D3-branes and other things. Once you do all these things, following the example of KKLT, you find out that there are many integer flux choices for the fluxes which leads you to a large number of possibilities i.e. to the infamous anthropic landscape.

At this point, things become philosophical and uncertain. Cumrun gave the students a homework problem to show that the usual wisdom

  • there are no exactly stable non-supersymmetric configurations in string theory

is flawed which would be a major step in getting rid of the anthropic principle: the huge semi-realistic landscape is a landscape of metastable vacua. What a task!

You may even call it an intellectual student loan consolidation incentive if you found the terminology of intellectual student loan refinance acceptable.

Of course, they are also solving some easier problems. :-)

1 comment:

  1. Dear Lubos,
    thank you very much for this lucid delineation.

    "Cumrun gave the students a homework problem to show that the usual wisdom:
    there are no exactly stable non-supersymmetric configurations in string theory
    is flawed"

    Nonperturbative physics imposes to integrate over moduli space of the wrapping instanton -- notwithstanding it is infeasible because of Cantorian blossom of the octonionic vacuum tree where every bloom is one from the elements of the U-dual modular group.

    There are uncountable many landscape decay channels (viz structure of the Weyl fundamental chambers) -- so the generic nonSUSY vacua do not have moduli space.

    An irreducible representation of the U-dual root lattice is the multi-flux warped octonionic black hole throat which equal to 0 (= Z_top = O matrix valued Hagedorn tachyon = off schell duality cascade = infinite-order orbifold of the generalized conifold).

    Ergo, dear Lubos, I conjecture that Cumrun's homework is imperfect ;-)

    Salute, Planckeon