Reconstructing the base field from imaginary multiplicative chaos

We show that the imaginary multiplicative chaos exp(i beta Gamma) determines the gradient of the underlying field Gamma for all log-correlated Gaussian fields with covariance of the form -log|x-y|+g(x,y) with mild regularity conditions on g, for all d >= 2 and for all beta is an element of(0, root d). In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable with respect to its imaginary chaos.