Kramers-Wannier symmetry and strong-weak-coupling duality in the two-dimensional $\Phi^{4}$ field model

It is found that the exact beta-function $\beta(g)$ of the continuous 2D $g\Phi^{4}$ model possesses two types of dual symmetries, these being the Kramers-Wannier (KW) duality symmetry and the weak-strong-coupling symmetry $f(g)$, or S-duality. All these transformations are explicitly constructed. The $S$-duality transformation $f(g)$ is shown to connect domains of weak and strong couplings, i.e. above and below $g^{*}$ with $g^{*}$ being a fixed point. Basically it means that there is a tempting possibility to compute multiloop Feynman diagrams for the $\beta$-function using high-temperature lattice expansions. The regular scheme developed is found to be strongly unstable. Approximate values of the renormalized coupling constant $g^{*}$ found from duality symmetry equations are in good agreement with available numerical results.