A discrete formulation for three-dimensional winding number

For a smooth map g: X \to U(N) g : X → U ( N ) , where X X is a three-dimensional, oriented, and closed manifold, the winding number is defined as W_3 = \frac{1}{24\pi^2} ∫t_{X} \mathrm{Tr}\big[(g^{-1}dg)^3\big] W 3 = 1 24 π 2 ∫ t X T r [ ( g − 1 d g ) 3 ] . We present a discrete formulation to compute W_3 W 3 based on the concept of \theta θ -gaps. Our approach provides a robust scheme that is directly applicable even to systems with accidental or symmetry-enforced degeneracies. Furthermore, we define two versions of the discrete flux: a simple unmodified flux that is highly practical and almost always quantized for fine grcids, and a modified flux that strictly ensures integer quantization.