The Kontsevich connection on the moduli space of FZZT Liouville branes Original Research Article

We point out that insertions of matrix fields in (connected amputated) amplitudes of (generalized) Kontsevich models are given by covariant derivatives with respect to the Kontsevich moduli. This implies that correlators are sections of symmetric products of the (holomorphic) tangent bundle on the (complexified) moduli space of FZZT Liouville branes. We discuss the relation of Kontsevich parametrization of moduli space with that provided by either the image or the image boundary conformal field theories. It turns out that the Kontsevich connection captures the contribution of contact terms to open string amplitudes of boundary cosmological constant operators in the image minimal string models. The curvature of the connection is of type image and has delta-function singularities at the points in moduli space where Kontsevich kinetic term vanishes. We also outline the extention of our formalism to the image string at self-dual radius and discuss the problems that have to be understood to reconciliate first and second quantized approaches in this case.