A discrete formulation for three-dimensional winding number

Abstract

For a smooth map $g: X \to U(N)$, where $X$ is a three-dimensional, oriented, and closed manifold, the winding number is defined as $W_3 = \frac{1}{24\pi^2} \int_{X} \mathrm{Tr}\left[(g^{-1}dg)^3\right]$. We present a discrete formulation to compute $W_3$ based on the concept of $\theta$-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 grids, and a modified flux that strictly ensures integer quantization.