  We numerically show that the statistical properties of the shortest path on
critical percolation clusters are consistent with the ones predicted for
Schramm-Loewner evolution (SLE) curves for $\kappa=1.04\pm0.02$. The shortest
path results from a global optimization process. To identify it, one needs to
explore an entire area. Establishing a relation with SLE permits to generate
curves statistically equivalent to the shortest path from a Brownian motion. We
numerically analyze the winding angle, the left passage probability, and the
driving function of the shortest path and compare them to the distributions
predicted for SLE curves with the same fractal dimension. The consistency with
SLE opens the possibility of using a solid theoretical framework to describe
the shortest path and it raises relevant questions regarding conformal
invariance and domain Markov properties, which we also discuss.
