Classification of real and imaginary modules of quantum affine algebras in monoidal categorifications of affine cluster algebras

Recently, Kashiwara-Kim-Oh-Park introduced a wide family of monoidal categories of finite-dimensional representations of quantum affine algebras, which provide monoidal categorifications of cluster algebras. In this paper, we prove that, for types $ADE$, some of these categories provide monoidal categorifications of cluster algebras of affine type. Moreover, by means of the combinatorial theory of affine type cluster algebras, we give a complete classification of real and imaginary simple modules in these categories. In particular, we show that, in these cases, the conjecture asserting that real simple modules correspond exactly to cluster monomials holds.