We present a comprehensive analysis of the spectral properties of the connection Laplacian for both real and discrete tori. 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. Additionally, we study the theta functions associated with these structures, providing a detailed analysis of their behavior and convergence. This is based on the joint work with Wang and Zhang.
