Conformal invariance in water-wave turbulence

We experimentally investigate the statistics of zero-height isolines in gravity wave turbulence as physical candidates for conformal invariant curves. We present direct evidence that they can be described by the family of conformal invariant curves called stochastic Schramm-L\"owner evolution (or SLE$_{\kappa}$), with diffusivity $\kappa = 2.88(8)$. A higher nonlinearity in the height fields is shown to destroy this symmetry, though scale invariance is retained.