The Kramers–Wannier Symmetry and $S$-Duality in the Two-Dimensional $g\Phi ^4$ Theory

We show that the exact beta function of the two-dimensional $g\Phi ^4$ theory possesses two dual symmetries. These are the Kramers–Wannier symmetry $d(g)$ and the strong-weak-coupling symmetry, or the $S$-duality $f(g)$, connecting the strong- and weak-coupling domains lying above and below the fixed point $g^*$. We obtain explicit representations for the functions $d(g)$ and $f(g)$. The $S$-duality transformation $f(g)$ allows using the high-temperature expansions to approximate the contributions of the higher-order Feynman diagrams. From the mathematical standpoint, the proposed scheme is highly unstable. Nevertheless, the approximate values of the renormalized coupling constant $g^*$ obtained from the duality equations agree well with the available numerical results.