The Kontsevich Connection on the Moduli Space of FZZT Liouville Branes

TL;DR

This paper explores the geometric structure of the moduli space of FZZT Liouville branes using the Kontsevich connection, revealing how matrix insertions relate to covariant derivatives and contact term contributions in open string amplitudes.

Contribution

It introduces a covariant derivative framework on the moduli space, linking Kontsevich models with boundary conformal field theories and analyzing the curvature and singularities of the connection.

Findings

Matrix field insertions are covariant derivatives on moduli space.

Kontsevich connection captures contact term contributions.

Curvature has delta-function singularities at specific moduli points.

Abstract

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 (p,1) or the (1,p) 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 (1,p) minimal string models. The curvature of the connection is of type (1,1) 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 c=1 string at self-dual radius and discuss the problems that have to be understood to reconciliate first and second quantized approaches in this case.