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- There are five types of string theories (Figure 08) that are supersymmetric, ghost free, and anomaly free. Each type incorporated a pairing of bosonic and fermionic vibrational patterns, but the details of this pairing as well as numerous other properties of the resulting theories differ substantially.
- Type I - Type I theory was formulated by Green and Schwarz in 1980. This type contains both open and closed strings. The two spinor fields
_{R}^{}, and_{L}^{}have same chirality. Gauge invariance can be added into the theory by attaching charges at the end of open strings (the charges are distributed on closed strings). The gauge group must be SO(32) in order to cancel all anomalies. The - Type IIA - For closed strings, there are two ways to choose the chiralities for
_{R}^{}, and_{L}^{}. If we choose them to have opposite chirality, then we have the Type IIA string. This is the only theory that is non-chiral (and thus not corresponding correctly to the physics of the real world). In the zero-slope limit (of the Regge trajectory), when only the massless sector of the theory survives, the theory reduces to the point particle N = 2, D = 10 supergravity theory, where N is the number of supersymmetry generators creating 2^{N}helicity states (N = 8 seems to be the limit beyond which particles with spin greater than two has to be included and the theory becomes inconsistent). Type IIA string theory contains D-branes with 0, 2, 4, 6, and 8 spatial dimensions. - Type IIB - The Type IIB superstring has the same chirality for
_{R}^{}, and_{L}^{}. In the zero-slope limit, there does not exist any known covariant version of this theory. It seems that the type II string (both A and B) cannot describe the physical SU(3) x SU(2) x U(1) symmetry of the low-energy universe. By compactifying from ten dimensions to four dimensions, the type II string can introduce a wide array of symmetries, but none of them seems to fit the description of this world. Type IIB string theory contains D-branes with -1, 1, 3, 5, 7, and 9 spatial dimensions. - SO(32) Heterotic - This is a theory of closed string. The fields that describe the physical degrees of freedom of the string in its ten-dimensional universe can be divided or decomposed into two independent parts - the left- and right-movers. The right mover is described by Eq.(53a) for the bosonic degrees of freedom, plus Eqs.(53b), (53c) for the fermionic degrees of freedom. It moves clockwise in a 10-dimensional space-time. The left mover is described by the bosonic degrees of freedom in Eqs.(53d), (53e). It moves counter-clockwise in a 26-dimensional space-time. So the heterotic string constructions are a hybrid - a
*heterosis*- in which counter-clockwise vibrational patterns live in 26 dimensions and clockwise patterns live in 10 dimensions. Such combination is possible because the right- and left-movers are independent of each other. Since the extra 16 dimensions on the bosonic side are rigidly curled up (compactified), each of these movers behaves as though it really has 10 dimensions. The extra 16 left mover dimensions provide the gauge group for the resulting 10-dimensional theory. It is found that the possible gauge group consistent with gauge and gravitational anomaly cancellation are SO(32) or E_{8}X E_{8}. It is the latter possibility that has led to phenomenologically promising models. The compactified dimensions carry all the quantum numbers around the loop. The Heterotic theories don't contain D-branes. They do however contain a fivebrane soliton which is not a D-brane. The IIA and IIB theories also contain this fivebrane soliton in addition to the D-branes. - E
_{8}X E_{8}Heterotic - The E_{8}group is a large Lie group of rank 8 and dimension 248, much like SU(2), but has no such geometric interpretation as the rotation of vectors. As mentioned earlier, compactification of the extra 16 dimension in term of the Calabi-Yau manifold produces a symmetry breaking to the E_{6}group. It is proposed that it further breaks down into SU(3) X SU(2) X U(1). This is exactly the symmetry demanded by the grand unified theory. SU(3) is the symmetry of the quark theory, while SU(2) X U(1) is the symmetry of the electroweak interaction. The mechanism for this symmetry breaking remains to be discovered theoretically. The hope is that when a theory of the compactification process is finally developed, it will indicate the precise steps by which the original heterotic symmetry breaks down. It should also determine the exact symmetry patterns of the elementary particles and their individual masses. As for the other half of the E_{8}X E_{8}gauge group, some physicists hypothesize that this gauge group corresponds to two universes, each belonging to the smaller symmetry pattern E_{8}by itself. Thus, in addition to our own universe, there is a new, hypothetical universe, a shadow world as it were. Other than the gravitational force, each E_{8}group describes its own universe, its own pattern of particles and forces. The elementary particles in one group are effectively invisible, or hidden, when viewed from the other group. This hypothesis could provide an explanation for dark matter, which is unseen but is responsible for holding astronomical objects together by gravitational force.

## Figure 08 Types of Superstring Theory [view large image] |
strong coupling limit of the Type I string theory is identical to the weak coupling limit of the Heterotic-O theory. Type I string theory contains D-branes with 1, 5, and 9 spatial dimensions. |

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