Quivers with potentials for cluster varieties associated to braid semigroups

Let $C$ be a simply laced generalized Cartan matrix. Given an element $b$ of the generalized braid semigroup related to $C$, we construct a collection of mutation-equivalent quivers with potentials. A quiver with potential in such a collection corresponds to an expression of $b$ in terms of the standard generators. For two expressions that differ by a braid relation, the corresponding quivers with potentials are related by a mutation. The main application of this result is a construction of a family of $CY_3$ $A_\infty$-categories associated to elements of the braid semigroup related to $C$. In particular, we construct a canonical up to equivalence $CY_3$ $A_\infty$-category associated to quotient of any Double Bruhat cell $G^{u,v}/{\rm Ad} H$ in a simply laced reductive Lie group $G$. We describe the full set of parameters these categories depend on by defining a 2-dimensional CW-complex and proving that the set of parameters is identified with second cohomology group of this complex.