  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 \emph{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 our 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, we work over a
character variety, that parameterizes the locally $L$-analytic characters on
$o_L$. We study $(\varphi,\Gamma)$-modules in this setting, and relate some of
them to what was known previously.
