However, string theory compactified on a two-torus can have exotic discrete symmetries. Because of the winding modes, the dimension of the relevant lattice is actually doubled. And higher-dimensional lattices admit less usual discrete symmetries such as "Z_7". If your compactification has such as symmetry, you may orbifold by it. Because you used a different action on the momenta and the windings, your orbifold will be a so-called asymmetric orbifold. In a stringy sense, it is still an orbifold of the lower-dimensional torus.

Obtaining lattices with otherwise forbidden symmetries by a projection from a higher-dimensional lattice is a well-known procedure.

**Figure 1**: Penrose tilings (drawn by an L.M. Pascal program).

The relevant lattice structures - and especially the idea that higher-dimensional lattices can have more diverse symmetries than the lower-dimensional ones - is identical in both cases (orbifolds and tilings). What's not identical is that the Penrose tilings fully cover the plane: almost every point belongs exactly to one of the diamond-shaped tiles that some people used in their bathroom. In the particular case above, there are only two types of tiles.

The tiling is quasiperiodic: it is not strictly periodic, but an arbitrarily large finite segment of the tiling is found in infinitely many copies spread all over the plane.

How do you construct the tiling above? Start with a five-dimensional space "R^5". Inside that space, you find the obvious lattice "Z^5". Look at all possible two-dimensional "1 x 1" squares with vertices in this lattice. Some of them will be orthogonally projected onto a particular two-dimensional plane "P" inside "R^5" which will give you the tiling; below, I specify which of them. There are "(5 choose 2) = 10" different directions in which the two-dimensional squares can be directed, and the projections only have two possible shapes, with internal angles 36+144 degrees or 72+108 degrees.

OK, what is the plane "P" onto which you project? Consider a "Z_5" action on the five-dimensional lattice "Z^5". It generator "g" acts as

- g (a,b,c,d,e) = (b,c,d,e,a).

This generator has five eigenvalues - the fifth roots of unity. The eigenvalue "lambda=1" corresponds to the vectors generated by "(1,1,1,1,1)". The remaining eigenvectors and eigenvalues are complex. But you can find a two-real-dimensional subspace "P" of the real space "R^5" where all vectors are combinations of the eigenvectors

- (1,e,e^2,e^3,e^4) and its complex conjugate

and a similar subspace "Q" inside "R^5" with vectors that are combinations of the eigenvectors

- (1,e^2,e^4,e^6,e^8) and its complex conjugate

where "e=exp(2.pi.i/5)". You will be projecting onto the plane "P" but you would get a similar picture if you chose "Q" instead of "P". Finally, I must say which two-dimensional squares will be projected and drawn on the plane "P". It will be those that are subsets (completely inside)

- U = P + CUBE = { p + c; p in P, c in CUBE }

where "CUBE" is the "[0,1]^5" unit cube inside "R^5". Here, "U" is a codimension zero, five-dimensional "band" around the "P" plane. One can prove that with these choices, you will cover the whole "P" plane without any holes or overlaps.

Similar constructions exist for other groups and other lattices, including cases where you project onto a three-dimensional (or higher-dimensional) hyperplane instead of the two-dimensional plane, and what you get are quasicrystals. They're not periodic but they can have a "Z_5" symmetry - something that is not possible for ordinary three-dimensional rigorous crystals but something that can be realized in Nature, as we know from experiments.

The fact that you cover the plane exactly once plays no role in the orbifold compactifications, as far as I know. But the lattice considerations are analogous or even isomorphic. The virtue of these quasicrystalline compatifications is that you can project most - or even all - scalar fields out of the spectrum without any need to stabilize them: the required lattices only have the symmetries needed for orbifolding at special points of the moduli space. Of course, these constructions are still perturbative string theory, so all of them still have the dilaton (or the string coupling constant).

I believe that these constructions are among the understudied topics in theoretical physics. I say so despite the fact that during the duality revolution, the asymmetric orbifolds became less popular because it is less clear how they fit into the M-theory/D-brane/F-theory/Calabi-Yau landscape than it is for the geometric compactifications. My impression that you never know what is the strong coupling dual of the asymmetric orbifolds such as the quasicrystalline compactifications has been falsified: there are cases in which you know the answer: the strong coupling dual can be found by using the SO(5,5,Z) U-duality of type IIA on a five-torus fiber-wise.

Despite the small amount of interest in the models, I find the asymmetric orbifolds and the related free fermionic constructions to be attractive phenomenologically. The free fermionic models are free CFTs to describe perturbative heterotic strings in which you try to fermionize everything that you can: you just keep four bosons "X^mu" on both sides, and fermionize the "E8 x E8" current algebra, the six compact dimensions, and of course all the fermionic partners "psi". Besides "X's", you will end up with 44 left-moving fermions (on the bosonic side) and 20 right-moving fermions (on the supersymmetric side) in the light-cone gauge.

Now you're free to impose arbitrary GSO projections and create the corresponding sectors, as long as the modular invariance is preserved. Because you can treat the left-movers and right-movers more or less independently, including those that came from the compact bosons, which is why these constructions are at least morally asymmetric orbifolds. There are many independent GSO projections, and the minimal set that leads you to a GUT-like gauge group is called the NAHE set, after its discoverers. (NAHE is "beautiful" in Hebrew and "naked" in Czech.) See, for example, this 21st century more recent preprint.

A very attractive feature of the models based on the free fermions, the NAHE set of GSO projections, and on additional projections is that they automatically predict three generations of quarks and leptons. I still feel that it is not an accident. From Occam's razor viewpoint, these models are very efficient. The worldsheet is a collection of free fields. They predict the right gauge group structure, the right representations for the fermions, the right number of generations, and quantum gravity. As far as physics as opposed to mathematical beauty and/or complexity goes, do we really need anything else?

The models have also predicted the correct top quark mass, by a calculation that was not quite right, but we have already discussed this achievement.

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