\lambda^2}$, where $\theta$ is the vacuum angle and $\lambda$ is the coupling constant. $N=4$ Yang–Mills theories have vanishing $\beta$ function, so that $\theta$ and $\lambda$ are well-defined independent of scale. In terms of $\rho$, the $SL(2,\ZZ)$ symmetry is realized by the non-linear transformations $$\rho \rightarrow {a \rho + b\over c\rho + d} ~, \quad \quad
\left(\begin{array} {cc} a & b  \\ c & d \end{array}\right)\, \in \, SL(2,  \ZZ) .$$ As usual, when instanton effects are taken into account, the continuous Peccei–Quinn symmetry $\chi \rightarrow \chi + b$, is broken to the discrete subgroup for which $b$ is an integer. This subgroup and the inversion $\rho \rightarrow - 1/\rho$ generate the discrete group $SL(2,\ZZ)$ or, when matrices are not distinguished from their negatives, $PSL(2,\ZZ)$. When $\theta =0$, a special case is inversion of the coupling constant $\lambda \rightarrow 1/\lambda$. In general, the $SL(2,\ZZ)$ symmetry of the theory is broken completely by any specific choice of vacuum. Only when the vev of $\rho$ is at one of the orbifold points of the moduli space does some unbroken symmetry ($\Ztwo$ or $\Zthree$) remain.

Mathematically, $S$ and $T$ duality are quite analogous in the 4d low-energy effective field theory, even though their implications 