terms of compactification from 10d, 21 of the scalars originate from the metric, 15 from the two-form $B_{\mu\nu}$, and 96 from the 16 $U(1)$ gauge fields that form the Cartan subalgebra of $E_8 \times E_8$ or $SO(32)$.

The toroidally compactified heterotic string also contains two additional scalar fields – called the axion $\chi$ and the dilaton $\phi$ – which belong to the $N = 4$ supergravity multiplet. The dilaton is the 10d dilaton shifted by a function of the other moduli such that the exponential of its vev gives the 4d coupling constant. The 4d axion is the scalar field that is dual to the two-form $B$ in 4d. The supergravity theory that contains these fields is precisely the one studied by Cremmer, Ferrara, and Scherk. They showed that $\chi$ and $\phi$ parametrize the homogeneous space $SL(2, \RR)/U(1)$. Actually, in the quantum theory, only the discrete $S$ duality subgroup $SL(2,\ZZ)$ is a symmetry, and the moduli space is $${\cal M}_S  = SL(2,\ZZ)\backslash SL(2, \RR)/U(1) .$$ To see how this works, let us introduce a complex scalar field $$\rho = \chi + i e^{-2\phi} = \rho_1 + i \rho_2 ~$$ whose vev is $<\rho > = {\theta/ 2\pi} + { i/
