String Theory and M-Theory

where
7.2 Fermionic construction of the heterotic string 263
¯λ (2)
0
= λ17 0 λ 18
0 . . . λ 32
0 . (7.48)
This rule eliminates one of the two spinors from each of the AP and PA
sectors. Therefore, their surviving contribution to the massless spectrum
is
(128, 1) ⊕ (1, 128). (7.49)
Each of the left-moving multiplets (7.49) is tensored with the right-moving
vector multiplet and therefore contributes additional massless vectors. To
understand what this means let us focus on the massless vector fields. The
massless spectrum contains vector fields that transform as (120, 1) + (128, 1)
as well as ones that transform as (1, 120) + (1, 128). The only way this can
make sense is if these 248 states form the adjoint representation of a Lie
group.
Here is where E8 enters the picture. This Lie group is the largest of the
five exceptional compact simple Lie groups in the Cartan classification. It
has rank eight and dimension 248. Moreover, it contains an SO(16) subgroup
with respect to which the adjoint decomposes as 248 = 120 + 128.
This is exactly the content that we found, so it is extremely plausible that
the heterotic theory with the projections described here gives a consistent
supersymmetric string theory in ten dimensions with E8 × E8 gauge symmetry.
This suggests that there exists a consistent heterotic string theory with
E8 × E8 gauge symmetry. First indications appeared already from the
anomaly analysis in Chapter 5, where this gauge group is one of the two
possibilities that was singled out. The GSO-like projections introduced here
are a straightforward generalization of those that gave the SO(32) heterotic
theory (as well as those of the RNS string), and they give precisely the
necessary massless spectrum.
EXERCISES
EXERCISE 7.1
Consider left-moving currents
J a (z) = 1
2 T a ABλ A (z)λ B (z),