String Theory and M-Theory

5.3 Quantization of the GS action 167
supersymmetry generators can be expressed in terms of these oscillators,
and therefore the physical spectrum is guaranteed to be supersymmetric.
Type II superstring theories
Type II superstrings, on the other hand, have the following spectrum. The
ground state for the closed string is also massless and is given by the tensor
product of left- and right-movers. Since the ground state for the open string
is the 16-dimensional multiplet given by 8v + 8c, there are 256 = 16 × 16
states in the closed-string ground state. The resulting supermultiplets are
different for the type IIA and type IIB theories.
In the case of the type IIA theory one should form the tensor product of
two supermultiplets in which the spinors have opposite chirality
(8v + 8c) ⊗ (8v + 8s). (5.88)
This tensor product gives rise to the following bosonic fields:
8v ⊗ 8v = 1 + 28 + 35 and 8s ⊗ 8c = 8v + 56t, (5.89)
while the tensor products of 8v ⊗ 8s and 8v ⊗ 8c give rise to the corresponding
fermionic superpartners. The product of the two vectors 8v ⊗ 8v
decomposes into a scalar, an antisymmetric rank-two tensor and a symmetric
traceless tensor. The corresponding fields are the dilaton, antisymmetric
tensor and the graviton. The product of the two spinors of opposite chirality,
denoted ζ and χ, is evaluated by constructing the independent tensors
¯ζΓiχ and ¯ ζΓijkχ. (5.90)
These describe 8 + 56 = 64 fermionic states, which is the expected number.
Equation (5.89) describes the massless bosons of the ten-dimensional
type IIA theory. This is the same bosonic content that is obtained when
11-dimensional supergravity is dimensionally reduced to ten dimensions.
Furthermore, the fermions also match. This relationship has rather deep
significance, as it suggests a connection between the two theories. This is
explored in Chapter 8.
The spectrum of massless particles of the type IIB theory is given by the
tensor product of two supermultiplets in which the spinors have the same
chirality. The massless ground states are then given by
This gives rise to the following bosonic fields:
(8v + 8c) ⊗ (8v + 8c). (5.91)
8v ⊗ 8v = 1 + 28 + 35 and 8c ⊗ 8c = 1 + 28 + 35+. (5.92)