Nodes as Composite Operators in Matrix Models

TL;DR

This paper explores how nodes on Riemann surfaces can be modeled using composite operators in matrix models, specifically in the Kontsevich model, linking graph combinatorics with algebraic geometry and topological gravity.

Contribution

It introduces a simple quadratic composite operator in the Kontsevich model to describe nodal Riemann surfaces and connects this to conjectures about ribbon graphs and Mumford-Morita classes.

Findings

Addition of (tr[X])^2 term models nodes in matrix models

Jenkins-Strebel differentials correspond to identified poles

Provides interpretation of univalent vertices in graph moduli spaces

Abstract

Riemann surfaces with nodes can be described by introducing simple composite operators in matrix models. In the case of the Kontsevich model, it is sufficient to add the quadratic, but ``non-propagating'', term (tr[X])^2 to the Lagrangian. The corresponding Jenkins-Strebel differentials have pairwise identified simple poles. The result is in agreement with a conjecture formulated by Kontsevich and recently investigated by Arbarello and Cornalba that the set ${\cal M}_{m*,s}$ of ribbon graphs with s faces and $m*=(m_0,m_1,\ldots,m_j,\ldots)$ vertices of valencies $(1,3,\ldots,2j+1,\ldots)$ ``can be expressed in terms of Mumford-Morita classes'': one gets an interpretation for univalent vertices. I also address the possible relationship with a recently formulated theory of constrained topological gravity.