  The inclusion of the Gaussian-curvature term in the bulk of Polyakov-Kleinert
string action renders new boundary terms and conditions by Gauss-Bonnet
theorem. Within a leading approximation, the eigenmodes of smooth worldsheets
and the free-energy of a gas of open rigid strings appears to be altered at
second order in the coupling by the topological term . In analogy to the
topological $\theta$ term, the Gauss-Bonnet term is introduced into the
effective action with a complex coupling to implement signed energy shifts. We
investigate the rigid color flux-sheets between two static color sources near
the critical point in the light of the topologically induced shifts. The
Yang-Mills lattice data of the potential of static quark-antiquark $Q\bar{Q}$
in a heatbath is compared to the string potential. The Monte-Carlo data
correspond to link-integrated Polyakov-loop correlators averaged over SU(3)
gauge configurations at $\beta=6.0$. Substantial improvement in the fit
behavior is displayed over the nonperturbative source separation distance $0.2$
fm to $1.0$ fm. Remarkably, the returned coupling parameter of the topological
term from the fit exhibits a proportionality to a quantum number. These
findings suggest that the manifested modes are the winding number of a
topological particle on the string's worldsheet.
