define them is to assume that space-time is asymptotically flat and use the asymptotic behavior of the field strengths: $$F_{0i}^a \sim {q_{el}^a\over r^3} x^i~~\quad
\tilde{F}_{0i}^a \sim {q_{mag}^a\over r^3} x^i ~.$$ The allowed charges are controlled by the asymptotic values of the moduli ($\rho \sim \rho^{(0)}$ and $M_{ab} \sim
M_{ab}^{(0)}$) and a pair of vectors $\alpha^a, \beta^a$ belonging to the Narain lattice, which is an even self-dual Lorentzian lattice of signature (6,22). The formulas are $$q_{el}^a = {1\over \rho_2^{(0)}} M_{ab}^{(0)} (\alpha^b + \rho_1^{(0)}
\beta^b), \quad q^a_{mag} = L_{ab}\beta^b ~.$$ These formulas automatically incorporate the Dirac–Schwinger–Zwanziger–Witten quantization rules ([*i.e.*]{}, the quantization condition for dyons in the presence of a $\theta$ angle). States in the perturbative string spectrum carry electric charge only and therefore have $\beta^a =0$.

The 4d Type II Superstring
--------------------------

The type II (A or B) superstring compactified on $T^6$ is approximated at low-energy by $N=8$ supergravity. The classical theory has a non-compact symmetry group $E_{7,7}$. The natural conjecture for the duality group in this case is the discrete subgroup $E_7(\ZZ)$, which is defined as the intersection of the continuous $E_{7,7}$ group and the discrete group $Sp(28;\ZZ)$.[@hull94] Written in the 56-dimensional fundamental representation, it is evident that $E_{7,7}$ is a 