The Collective Field Theory of a Singular Supersymmetric Matrix Model

TL;DR

This paper develops a supersymmetric collective field theory for a matrix model with a specific potential, revealing a two-band solution, classical configurations, and a spectrum with a massless scalar and Majorana fermion, relevant for two-dimensional string theory in black hole backgrounds.

Contribution

It introduces a supersymmetric collective field framework for a matrix model with a singular potential, deriving classical solutions and analyzing the spectrum in the context of string theory.

Findings

Identifies a two-band solution enforced by supersymmetry.

Constructs eigenfunctions using Chebyshev polynomials.

Connects classical configurations to zeros of Laguerre polynomials.

Abstract

The supersymmetric collective field theory with the potential $v'(x)=\omega x-{\eta\over x}$ is studied, motivated by the matrix model proposed by Jevicki and Yoneya to describe two dimensional string theory in a black hole background. Consistency with supersymmetry enforces a two band solution. A supersymmetric classical configuration is found, and interpreted in terms of the density of zeros of certain Laguerre polynomials. The spectrum of the model is then studied and is seen to correspond to a massless scalar and a majorana fermion. The $x$ space eigenfunctions are constructed and expressed in terms of Chebyshev polynomials. Higher order interactions are also discussed.