I will discuss the monoidal categorification of the (quantum) cluster algebra structure on the algebra $\hat A(b)$ associated with an arbitrary positive braid $b$. Here $\hat A(b)$ is the subalgebra of the bosonic extension $\hat A$ generated by the PBW generators associated with $b$. The construction takes place inside the category of finite-dimensional modules over a quantum affine algebra. Starting from a complete duality datum $D$ together with an expression of $b$, we define affine cuspidal modules using braid symmetries. We then define the category $C^D(b)$ as the full monoidal subcategory generated by the affine cuspidal modules. The affine determinantial modules corresponding to an admissible chain of i-boxes give the initial monoidal seed, yielding the cluster algebra structure on $\hat A(b)$. Moreover, the Grothendieck ring of the monoidal category is isomorphic to the cluster algebra $\hat A(b)$, and in fact every cluster monomial corresponds to a real simple module.
This is a joint work with Masaki Kashiwara, Se-jin Oh, and Euiyong Park.