Other Constructions

During the past few years several other constructions of chiral 4D strings in four dimensions, have been found in terms of explicit CFT's. We described before how the use of free field CFT's lead us naturally to orbifold compactifications. We can also use the property of these 2D field theories for which there is an equivalence between fermions and bosons. Since the bosonic fields in 2D are the coordinates in target space, by fermionizing them we lose the geometrical interpretation, but it is a consistent string model as long as we keep the 4D spacetime coordinates as bosons. If we fermionize all the extra coordinates and choose nontrivial boundary conditions on the fermions we can get nontrivial 4D strings [24], which do not have to have a geometrical interpretation in terms of compactifications!. Many models have been studied using this approach which, in many cases, are equivalent to orbifold compactifications at some particular value of the radius. Some quasi-realistic models have been studied with much detail using this approach. This includes models with three families and standard model gauge group $SU(3)\times SU(2)\times U(1)$ as well as a version of $SU(5)\times U(1)$ known as flipped $SU(5)$ [25]. In some cases again, the models reproduce many of the nice features of the standard model. Nevertheless, as in the case of orbifolds, there is not a totally realistic model yet.

A related approach uses bosonization in the opposite direction , i.e., it bosonizes all of the fermions of the (supersymmetric) right moving sector (including the ghost system needed for consistent quantization of the 2D theory) [26]. This is the so called covariant lattice approach which in some way generalizes the Narain lattice of toroidal compactifications. Again many of these models are equivalent to orbifolds at a particular radius. In particular some of the three generation orbifold models mentioned before have been explicitly reproduced using this approach [28].

A probably more general construction goes into the name of Gepner-Kazama-Suzuki models [27]. They depart from free field CFTs and construct more general CFTs by using cosets $G/H$ to describe the CFT of the `internal' dimensions. This construction includes (products of) statistical mechanics models such as the Ising and Potts models and their supersymmetric generalizations. One of the salient features of these constructions is that for the models with $(2,2)$ supersymmetry in the worldsheet, it can be shown explicitly that they do correspond to particular points of Calabi-Yau compactifications, despite of their original non-geometric construction. This was found by using their realization in terms of Landau-Ginzburg 2D effective field theories. Generalization of these constructions to $(0,2)$ supersymmetric models have also been achieved and a large class of models exist [29], including some close to the standard model.

We have then seen that there are several formalisms for constructing chiral 4D strings. Many models can be built using different approaches. Each formalism has advantages and disadvantages in terms of the level of generality and for performing explicit calculations for the low energy effective theory.