Imaginary Geometry I: Interacting SLEs

Understand

Fix constants \chi >0 and \theta \in [0,2\pi), and let h be an instance of the Gaussian free field on a planar domain.

• We study flow lines of the vector field e^{i(h/\chi+\theta)} starting at a fixed boundary point of the domain.
• Considering all \theta \in [0,2\pi), one obtains a family of curves that look locally like SLE_\kappa, with \kappa \in (0,4), where \chi = 2/\kappa^{1/2} - \kappa^{1/2}/2, which we interpret as the rays of a random geometry with purely imaginary curvature.
• We extend the fundamental existence and uniqueness results about these paths to the case that the paths intersect the boundary.