By Van Proeyen A.
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Extra resources for Introduction to string theory
6) where x9 = x9L + x9R and α09 = α pL = 2 α n 1 + √ wR , 2R 2α α ˜ 09 = α pR = 2 α n 1 − √ wR . 5), while the first one follows from the sum ˜ 0 constraint. 5) are invariant under the exchange R ↔ α , R n ↔ w. 8) In other words, we can exchange compactification radius R with radius α /R if we exchange the winding modes with the quantized momentum modes. This mode exchange is the basis of the duality known as T-duality. Notice that if the √ compactification radius R is much smaller than the string scale α , then the compactification radius after the winding and momentum modes are exchanged is much larger than the string scale.
It turns out that there are different possibilities leading to different superstring theories. Supersymmetry can be seen as a fermionic version of general coordinate transformations. The analogue of the conformal group is a ‘superconformal group’. We saw that in the bosonic case, the conformal group has an infinite number of generators, represented by the Ln . The fermionic partners are also an infinite number of generators. In the bosonic case we introduced first a metric on the worldsheet hαβ and then fixed it to ηαβ , but were left with constraints that were ++ and −− components of the energy momentum tensors.
But there is one exception that one has to treat specially: the ψ0 mode (for the R case). Their anticommutation relations are similar to the ‘gamma matrices’ of a Clifford algebra. Therefore the vacuum can not be a single state, but should be a spinor on which these gamma matrices can act. This will give rise to the fermions in the spectrum (in spacetime) as we will see below. As mentioned, these modes contribute to the Virasoro generators, and to the analogous fermionic operators: Lm = µ : αnµ αm−n : + 14 1 2 n∈Z µ µ α−n ψr+n Gr = µ µ (2r + m) : ψ−r ψm+r : +aν δm , r∈Z+ν R : aν = 1 D, 16 N S : aν = 0 .