  We prove the vertex-reinforced jump process (VRJP) is recurrent in two
dimensions for any translation invariant finite range initial rates. Our proof
has two main ingredients. The first is a direct connection between the VRJP and
sigma models whose target space is a hyperbolic space $\mathbb{H}^n$ or its
supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of
well-known relations between the Gaussian free field and the local times of
simple random walk. The second ingredient is a Mermin--Wagner theorem for these
sigma models. This result is of intrinsic interest for the sigma models and
also implies our main theorem on the VRJP. Surprisingly, our Mermin--Wagner
theorem applies even though the symmetry groups of $\mathbb{H}^n$ and
$\mathbb{H}^{2|2}$ are non-amenable.
