Quantum vertex $F((t))$-algebras and their modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we develop a theory of what we call (weak) quantum vertex $\F((t))$-algebras with $\F$ a field of characteristic zero and $t$ a formal variable, and we give a conceptual construction of (weak) quantum vertex $\F((t))$-algebras and their modules. As an application, we associate weak quantum vertex $\F((t))$-algebras to quantum affine algebras, providing a solution to a problem posed by Frenkel and Jing. We also explicitly construct an example of quantum vertex $\F((t))$-algebras from a certain quantum $\beta\gamma$-system.