Semiclassical Approach to Finite-N Matrix Models

TL;DR

This paper reformulates the finite-N hermitean one-matrix model as a collective field theory, revealing non-perturbative effects, IR and UV divergences, and providing a two-loop free energy calculation with comparisons to orthogonal polynomial methods.

Contribution

It introduces an exact treatment of the Jacobian in the collective field formulation for finite N and explores the non-trivial N-dependence of the classical background.

Findings

Derived a classical eigenvalue density integral equation showing non-perturbative behavior.

Identified IR singularities and UV divergencies in the large-N expansion.

Computed the free energy at two-loop level and compared with orthogonal polynomial results.

Abstract

We reformulate the zero-dimensional hermitean one-matrix model as a (nonlocal) collective field theory, for finite~$N$. The Jacobian arising by changing variables from matrix eigenvalues to their density distribution is treated {\it exactly\/}. The semiclassical loop expansion turns out {\it not\/} to coincide with the (topological) ${1\over N}$~expansion, because the classical background has a non-trivial $N$-dependence. We derive a simple integral equation for the classical eigenvalue density, which displays strong non-perturbative behavior around $N\!=\!\infty$. This leads to IR singularities in the large-$N$ expansion, but UV divergencies appear as well, despite remarkable cancellations among the Feynman diagrams. We evaluate the free energy at the two-loop level and discuss its regularization. A simple example serves to illustrate the problems and admits explicit comparison with orthogonal polynomial results.