  The single-valued projection (sv) is a relation between scattering amplitudes
of gauge bosons in heterotic and open superstring theories. Recently we have
studied sv from the aspect of nonlinear sigma models [1], where the gauge
physics of open string sigma model is under the Wilson loop representation but
the gauge physics of heterotic string sigma model is under the fermionic
representation since the Wilson loop representation is absent in the heterotic
case. There we showed that the sv comes from a sum of six radial orderings of
heterotic vertices on the complex plane. In this paper, we propose a Wilson
loop representation for the heterotic case and using the Wilson loop
representation to show that sv comes from a sum of two opposite-directed
contours of the heterotic sigma model. We firstly prove that the Wilson loop is
the exact propagator of the fermion field that carry the gauge physics of the
heterotic string in the fermionic representation. Then we construct the action
of the heterotic string sigma model in terms of the Wilson loop, by exploring
the geometry of the Wilson loop and by generalizing the nonabelian Stokes's
theorem [2], [3], [4] to the fermionic case. After that, we compute some three
loop and four loop diagrams as an example, to show how the sv for $\zeta_2$ and
$\zeta_3$ arises from a sum of the contours of the Wilson loop. Finally we
conjecture that this sum of contours of the Wilson loop is the mechanism behind
the sv for general cases.
