Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras

Let $\mathcal{C}$ be the category of finite-dimensional modules over a simply-laced quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. For any height function $\xi$ and $\mathrm{\ell }\in {\mathbb{Z}}_{\ge 1}$, we introduce certain subcategories ${\mathcal{C}}_{\mathrm{\ell }}^{\le \xi }$ of $\mathcal{C}$, and prove that the quantum Grothendieck ring $K_t({\mathcal{C}}_{\mathrm{\ell }}^{\le \xi })$ of ${\mathcal{C}}_{\mathrm{\ell }}^{\le \xi }$ admits a quantum cluster algebra structure. Using $F$-polynomials and monoidal categorifications of cluster algebras, we classify all real simple modules in ${\mathcal{C}}_1^{\le \xi }$ in terms of their highest $\mathrm{\ell }$-weight monomials, among them the families of type $D$ and type $E$ are new. For any $\mathrm{\ell }$, inspired by Hernandez and Leclerc’s work, we propose two conjectures for the study of real simple modules, and prove them for the subcategories ${\mathcal{C}}_{\mathrm{\ell }}^{\le \xi }$ whose Grothendieck rings are cluster algebras of finite type.