Connection Laplacian on discrete tori with converging property

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.