  This paper is the first of a series aiming at proving rigorously the
analyticity and the Borel summability of generic quartic bosonic and fermionic
vector models (generalizing the O(N) vector model) in diverse dimensions. Both
non-relativistic (Schr\"odinger) and relativistic (Klein-Gordon and Dirac)
kinetic terms are considered. The 4-tensor defining the interactions is
constant but otherwise arbitrary, up to the symmetries imposed by the
statistics of the field. In this paper, we focus on models of low dimensions:
bosons and fermions for d = 0, 1, and relativistic bosons for d = 2. Moreover,
we investigate the large N and massless limits along with quenching for
fermions in d = 1. These results are established using the loop vertex
expansion (LVE) and have applications in different fields, including data
sciences, condensed matter and string field theory. In particular, this
establishes the Borel summability of the SYK model both at finite and large N.
