  This paper presents a comprehensive analysis of the spectral properties of
the connection Laplacian for both real and discrete tori. We introduce novel
methods to examine these eigenvalues by employing parallel orthonormal basis in
the pullback bundle on universal covering spaces. Our main results reveal that
the eigenvalues of the connection Laplacian on a real torus can be expressed in
terms of standard Laplacian eigenvalues, with a unique twist encapsulated in
the torsion matrix. This connection is further investigated in the context of
discrete tori, where we demonstrate similar results.
  A significant portion of the paper is dedicated to exploring the convergence
properties of a family of discrete tori towards a real torus. We extend
previous findings on the spectrum of the standard Laplacian to include the
connection Laplacian, revealing that the rescaled eigenvalues of discrete tori
converge to those of the real torus. Furthermore, our analysis of the discrete
torus occurs within a broader context, where it is not constrained to being a
product of cyclic groups. Additionally, we delve into the theta functions
associated with these structures, providing a detailed analysis of their
behavior and convergence.
  The paper culminates in a study of the regularized log-determinant of the
connection Laplacian and the converging results of it. We derive formulae for
both real and discrete tori, emphasizing their dependence on the spectral zeta
function and theta functions.
