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String Theory Demysti ed
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5. In type II B theory the GSO projection can be written as: (a) (b) (c) (d)
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Heterotic String Theory
In Chap. 10, we learned that there are two string theories that treat the left- and right-moving sectors differently. These theories are called heterotic string theories for this reason. The modes are treated as follows: The left-moving sector is bosonic. The right-moving sector is supersymmetric. This idea sounds nonsensical because we have learned that bosonic string theory lives in a world of 26 space-time dimensions, while superstring theories live in a world of 10 space-time dimensions. The reason for doing such a radical thing is the following. First, we already know that bosonic string theories by themselves are awed because they do not incorporate fermions. On the other hand, type II superstrings do not incorporate nonabelian gauge symmetries. This means that the standard model cannot be described by superstring theory as formulated with those theories alone. So type II theories seem to leave us with a description of the universe
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String Theory Demysti ed
that lacks electroweak theory and QCD, an unacceptable situation since the real world does include these interactions. One way around this problem is to add charges to the ends of the strings, something we will discuss in Chap. 15, but in this chapter we consider a more successful and elegant approach. The idea of heterotic strings was originally proposed by Gross, Harvey, Martinec, and Rohm. They proposed a theory of closed superstrings with decoupled left- and right-moving modes that preserves the best aspects of both theories, producing a theory which is large enough and sophisticated enough to incorporate the features we know the theory must have to include the standard model. By making the rightmoving modes super-symmetric We are able to include fermions in the theory. We keep tachyons out of the theory, so it has a stable vacuum. We incorporate nonabelian gauge theory in the left-moving modes. This is done by adding Majorana-Weyl fermions A to the left-moving sector, without adding supersymmetry. We must eliminate the extra 16 dimensions from the 26-dimension contribution of the bosonic theory. This can be understood by discarding the view that the extra 16 dimensions are space-time dimensions. First, note that The right-moving modes are supersymmetric. So, there are 10 bosonic elds X among the right-moving modes. We keep 10 bosonic elds X from the left-moving modes to match up with the right-moving modes. Since 26 = 10 + 16, we need to cancel the remaining 16 contributions from the unwanted X present in the left-moving sector. Since the A are spinors, we need 32 of them to enable the desired calculation, hence we take A = 1,..., 32 . The symmetry group for the A is SO(32) when all of the A have the same boundary condition. This is the SO(32) heterotic theory.
The Action for SO(32) Theory
We can write down the action as follows: It will include a bosonic contribution for left- and right-moving modes for 10 dimensional space-time. It will include fermionic spinors to add supersymmetry to the 10 dimensional space-time. These will only be rightmovers. It will include a contribution from the left-moving A spinors.
CHAPTER 12 Heterotic String Theory
The rst two pieces are familiar, we use then Majorana-Weyl fermions which are space-time vectors like those used for worldsheet supersymmetry. So, we have S1 = 1 a 2 d ( a X X 2i + ) 4 (12.1)
The right-moving sector must incorporate supersymmetry. This is done in the same way as in Chap. 7, we incorporate the following transformations:
X = i
= X
We will add a second action to incorporate the A . This piece is similar to the fermionic piece used in S1 but now we are considering left-moving modes and we have to include all 32 A . So the action is
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