This lecture identifies the two consistent, modular-invariant choices of internal c=16 conformal field theory for the heterotic string (32 free fermions with GSO projection either all together or split into two groups of 16) as, after bosonization, the even self-dual lattices Gamma_16 and Gamma_8 x Gamma_8 — the unique Euclidean even self-dual lattices in 16 and 8+8 dimensions — and shows these lattices coincide with the root-plus-weight lattice structure of SO(32) and E8 x E8 respectively, with every lattice vector of length-squared 2 corresponding to a massless spacetime vertex operator generating a non-abelian gauge symmetry. This belongs to string compactification / worldsheet conformal field theory and Lie algebra representation theory, establishing that the heterotic string's spacetime gauge group is forced by worldsheet consistency (modular invariance, GSO projection) to be exactly SO(32) or E8 x E8.
This lecture identifies the two consistent, modular-invariant choices of internal c=16 conformal field theory for the heterotic string (32 free fermions with GSO projection either all together or spl…