A Lagrangean formalism for Hermitean matrix models

TL;DR

This paper reformulates Hermitean 1-matrix models as a Lagrangian field theory on a hyperelliptic Riemann surface, providing a new perspective on their intrinsic geometric structure.

Contribution

It introduces a Lagrangian formalism for Hermitean matrix models based on the geometry of hyperelliptic surfaces, extending Eynard's formulation.

Findings

Reformulation of matrix models as scalar field theory on Riemann surfaces

Identification of interactions at branch points of the hyperelliptic surface

Provides a new geometric framework for analyzing matrix models

Abstract

Eynard's formulation of Hermitean 1-matrix models in terms of intrinsic quantities of an associated hyperelliptic Riemann surface is rephrased as a Lagrangean field theory of a scalar particle propagating on the hyperelliptic surface with multiple self-interactions and particle-source interactions. Both types of interaction take place at the branch points of the hyperelliptic surface.