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This paper is the first of a series aiming at proving rigorously theanalyticity and the Borel summability of generic quartic bosonic and fermionicvector models (generalizing the O(N) vector model) in diverse dimensions. Bothnon-relativistic (Schr\"odinger) and relativistic (Klein-Gordon and Dirac)kinetic terms are considered. The 4-tensor defining the interactions isconstant but otherwise arbitrary, up to the symmetries imposed by thestatistics 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 forfermions in d = 1. These results are established using the loop vertexexpansion (LVE) and have applications in different fields, including datasciences, condensed matter and string field theory. In particular, thisestablishes the Borel summability of the SYK model both at finite and large N.

Constructive expansion for quartic vector fields theories. I. Low dimensions