Tuesday, September 05, 2006

Orbifolds of six-dimensional tori

First, a few unphysical comments. Many physicists, especially in the U.S., may have observed that their article is likely to get a slightly higher attention of the peers if it appears at the top or near the top of the arXiv listings.

In a wild random community, this would lead to an increased concentration of texts written by silly but self-confident authors. I think it is fair to say that in the context of hep-th, this effect leads to an increased percentage of important, impressive, and perfectionist papers near the top.




So far this mechanism works well and I hope that no irresponsible reader of these lines has an access to the arXiv. This night seems to be no exception. There are two impressive papers by D. Lüst, S. Reffert, E. Scheidegger, S. Stieberger - two gifts from German string theory:

Be ready for 170+ pages in total plus 256 bonus pages. The first paper focuses on the KKLT physics and the Kähler potential (for the Kähler moduli), among other physical quantities. The second paper is more mathematical and kind of cute. Both papers share a certain mathematical framework.

Free two-dimensional conformal field theories are probably the simplest ones we have - at least for particle physicists (the statistical mechanicians could find the Ising model and minimal models easier). They describe strings propagating in locally flat spaces or their generalizations. They are easier to be dealt with and they enjoy the same physically attractive and realistic properties as more general Calabi-Yau spaces. The orbifolds of tori and their resolutions can be viewed as a subclass of the Calabi-Yau manifolds that is friendly to those who want to make explicit calculations. It is also conceivable that they have more important special physical properties.

In the conventional physical scenarios of perturbative string theory, 10-4=6 dimensions must be compactified. Assuming locally flat geometries, the compact manifold must be an orbifold of a six-torus. Such an orbifold must be an orbifold by a finite group. As far as I can imagine, it must be an Abelian one. This finite group must be a symmetry of the torus - or equivalently, it must the symmetry of the six-dimensional lattice Gamma that defines the torus via

  • T^6 = R^6 / Gamma

If you view this "R^6" space as a Cartan subalgebra, "Gamma" may be interpreted as a root lattice of a group, at least if you scale its elements appropriately. The corresponding group is a product of factors chosen from the set SU(2), SU(3), SU(4), SU(5), SU(6), SU(7), SO(5), SO(8), SO(9), SO(10), G2, F4, E6. Note that some groups are omitted - such as SO(7) - try to guess why. Others are equivalent to an element of the list. The total rank must equal six and the authors classify all possibilities.

Truth to be said, you can't get all lattices as these simple products. You will also need two kinds of a generalized Coxeter twist to obtain three choices or so. The discrete symmetries that you may use to orbifold these tori are products "Z_M x Z_N" where the numbers can be between 1 and 8 or they may be 12. Again, the possibilities are listed.

There are new closed string moduli in the twisted sectors stuck near the fixed points of the orbifold whose vev resolves the orbifolds. Such a resolution creates new cycles in the homology and the mathematical paper tells you what the topologies and intersection numbers of these cycles are. Recall that a complex codimension 1 submanifolds - four-cycles, in this case - are called divisors. Their topology is given. Possible orientifolds of the previous constructions are also constructed.

The physical paper gives remarkably explicit formulae for the Kähler potential entering into the supersymmetric spacetime action for the Kähler moduli. They pay a special attention to the Calabi-Yaus without any complex structure moduli. Surprisingly, this condition expressed as "h^{2,1}=0" is satisfied by quite many elements in their list. They study carefully what is needed for the KKLT construction and argue that these conditions prohibit the "h^{2,1}=0" orientifolds. In the later parts of the physics paper, they discuss the stabilization of those cases that can be stabilized, via fluxes and racetrack superpotential.

1 comment:

  1. Dear Lubos,
    thank you very much for this recommendation!


    Lubos, is your work schedule form by Eurostring workshop Brno 2006 ?

    http://dumbell.physics.muni.cz/string/program.html

    (I am looking forward to see especially Hull, Horava, Schnabl lectures.)


    Please what is your opinion about no-go theorem for moduli stabilization?

    Thank you very much obliged.
    Salute, Planckeon

    ReplyDelete