Random Conformal Weldings at criticality

We construct a family of random Jordan curves in the plane by welding together two disks on their boundaries using a random homeomorphism. This homeomorphism arises from a random measure whose density, in a generalized sense, is the exponentiated Gaussian Free Field at criticality. We also introduce a representation of the Gaussian Free Field in terms of vaguelets, which may be of separate interest. The result extends a theorem of Astala, Jones, Kupiainen and Saksman (\cite{AJKS09}) to criticality.