  Factorization algebras play a central role in the formulation of quantum
field theories given by Kevin Costello and his collaborators. In this paper, we
propose a concrete construction of a topological operator in their formulation.
We focus on a shift symmetry of a one-dimensional massless scalar theory. And
we give some discussions of $\mathbb{Z}$-gauging of it, i.e., compact scalar
theory and its $\theta$-vacuum.
