The construction of the $p$-adic local Langlands correspondence for $mathrm{GL}_2(mathbf{Q}_p)$ uses in an essential way Fontaine's theory of cyclotomic $(varphi ,Gamma )$-modules. Here cyclotomic means that $Gamma = mathrm {Gal}(mathbf{Q}_p(mu_{p^infty})/mathbf{Q}_p)$ is the Galois group of the cyclotomic extension of $mathbf Q_p$. In order to generalize the $p$-adic local Langlands correspondence to $mathrm{GL}_{2}(L)$, where $L$ is a finite extension of $mathbf{Q}_p$, it seems necessary to have at our disposal a theory of Lubin-Tate $(varphi ,Gamma )$-modules. Such a generalization has been carried out, to some extent, by working over the $p$-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic $(varphi ,Gamma )$-modules in a different fashion. Instead of the $p$-adic open unit disk, the authors work over a character variety that parameterizes the locally $L$-analytic characters on $o_L$. They study $(varphi ,Gamma )$-modules in this setting and relate some of them to what was known previously.