# Root weights and states in orbifold compactifications

I have the following question regarding orbifold compactifications of the heterotic string:

What is the relation between a certain representation and the weights of the root lattice? I mean: take the following example from Uranga & Ibañez's book: I understand that the possible weights of the $$\left [ E_{6}\times SU(3) \right ]\times E_{8}{}'$$ are $$0,1$$ and $$2$$ and each of them should correspond to one state $$(\mathbf{27},\boldsymbol{3};\mathbf{1}'),(\mathbf{27},\mathbf{1};\mathbf{1}')$$ or $$(\mathbf{1},\mathbf{\bar{3}};\mathbf{1}')$$.

But which is this correspondece (and why)?

• You should study and understand weights for something simple like SU(3) before trying to understand something this complicated. – G. Smith Jan 17 at 23:46

$$\frac{1}{3}(1,1,2,0,...,0)\times(0,...,0)$$
The part after the $$\times$$ is the $$E_8’$$ part of the weight vector, corresponding to a 1-dimensional irrep. The part before must be a weight vector for $$E_6\times SU(3)$$ but I don’t know why it is written as a single vector or why it has a prefactor of 1/3.