Geometric realizations of Ringel-Hall algebras of continuous quivers of type $A$

Lusztig introduced the geometric realizations of quantum groups associated to finite quivers and defined their canonical bases. Sala and Schiffmann introduced the Ringel-Hall algebra of line and realized it as the direct limit of Ringel-Hall algebras of finite quivers of type $A$. In this paper, we shall give geometric realizations of Ringel-Hall algebras of continuous quivers of type $A$ via the geometric realizations of Lusztig by using the method of approximation given by Sala and Schiffmann.