The Supersymmetric Method in Random Matrix Theory and Applications to QCD

TL;DR

This paper reviews the supersymmetric method in Random Matrix Theory, demonstrating its applications to quantum chromodynamics (QCD), spectral density calculations, and recent advances involving the Toda lattice hierarchy, emphasizing symmetry considerations.

Contribution

It provides an elementary introduction to the supersymmetric method, applies it to Gaussian Unitary Ensemble and QCD Dirac operator spectra, and explores the connection with Toda lattice hierarchy for superintegral calculations.

Findings

Supersymmetric method can be reformulated as integrals over supermanifolds.

Application to Gaussian Unitary Ensemble illustrates symmetry roles.

Relation between supersymmetric partition function and Toda lattice hierarchy enhances superintegral computations.

Abstract

The supersymmetric method is a powerful method for the evaluation of quenched averages in disordered systems. Among others, this method has been applied to the theory of S-matrix fluctuations, the theory of universal conductance fluctuations and the microscopic spectral density of the QCD Dirac operator. We start this series of lectures with a general review of Random Matrix Theory and the statistical theory of spectra. An elementary introduction of the supersymmetric method in Random Matrix Theory is given in the second and third lecture. We will show that a Random Matrix Theory can be rewritten as an integral over a supermanifold. This integral will be worked out for the Gaussian Unitary Ensemble that describes systems with broken time reversal invariance. We especially emphasize the role of symmetries. As a second example of the application of the supersymemtric method we discuss the calculation of the microscopic spectral density of the QCD Dirac operator. which is known to be given by chiral Random Matrix Theory. Also in this case we use symmetry considerations to rewrite the generating function for the resolvent as an integral over a supermanifold. The main topic of the second last lecture is the recent developments on the relation between the supersymmetric partition function and the Toda lattice hierarchy. We will show that this relation is an efficient way to calculate superintegrals. Finally, we will discuss the quenched QCD Dirac spectrum at nonzero chemical potential. Because of the nonhermiticity of the Dirac operator the usual supersymmetric method has not been successful in this case. However, we will show that the supersymmetric partition function can be evaluated by means of the replica limit of the Toda lattice equation.