are 132 scalar fields that live on the Narain moduli space ${\cal M}_{6,22}$. Narain spaces ${\cal M}_{k,l}$ are defined by $${\cal M}_{k,l} = SO(k,l;\ZZ) \backslash SO(k,l) / SO(k) \times SO(l).$$ It is convenient to introduce this notation here, since we shall encounter various Narain spaces in the course of these lectures. Recall that $SO(k,l)$ is the noncompact form of $SO(k+l)$ that preserves a metric with $k$ plus signs and $l$ minus signs. The group $SO(k)\times SO(l)$ is its maximal compact subgroup and the quotient space $SO(k,l) /SO (k) \times  SO(l)$ is a homogeneous space of dimension $kl$. The discrete group $SO(k,l;\ZZ)$ is an infinite group consisting of all $SO(k,l)$ matrices with integer entries. When $l = k + 16$, it is the subgroup of $SO(k,l)$ that preserves a certain even self-dual lattice of signature ($k, l$) introduced by Narain. A homogeneous space is very smooth and well-behaved, but modding out by the discrete group introduces orbifold singularities, corresponding to the fixed points of the discrete group, in the moduli space. The $T$ duality group for the 4d heterotic string is $G_T = SO(6, 22; \ZZ)$, and the 132 scalar fields belong to 22 Abelian $N = 4$ gauge multiplets. In 