  We present the geodesic nature and quantization of geometric shift vector in
quantum systems, with the parameter space defined by the Bloch momentum, using
the Wilson loop approach. Our analysis extends to include bosonic phonon drag
shift vectors with non-vertical transitions. We demonstrate that the gauge
invariant shift vector can be quantized as integer values, analogous to the
Euler characteristic based on the Gauss-Bonnet theorem for a manifold with a
smooth boundary. We reveal intricate relationships among geometric quantities
such as the shift vector, Berry curvature, and quantum metric. Our findings
demonstrate that the loop integral of the shift vector in the quantized
interband formula contributes to the non-quantized component of the trace of
conductivity in the circular photogalvanic effect. The Wilson loop method
facilitates first-principles calculations, providing insights in the geometric
underpinnings of these interband gauge invariant quantities and shedding light
on their nonlinear optical manifestations in real materials.
