Fermionic Matrix Models

TL;DR

This paper reviews fermionic matrix models with anticommuting matrices, analyzing their properties, solutions, and applications to string theory, quantum gravity, and gauge theories, highlighting their potential advantages over bosonic models.

Contribution

It provides a comprehensive analysis of fermionic matrix models, deriving loop equations, exploring critical behavior, and connecting them to string theory and quantum gravity, which is a novel extension of traditional bosonic models.

Findings

Complete solutions of loop equations for fermionic models.

Potential for better convergence in quantum gravity discretizations.

Applications to induced gauge theories and string phases.

Abstract

We review a class of matrix models whose degrees of freedom are matrices with anticommuting elements. We discuss the properties of the adjoint fermion one-, two- and gauge invariant D-dimensional matrix models at large-N and compare them with their bosonic counterparts which are the more familiar Hermitian matrix models. We derive and solve the complete sets of loop equations for the correlators of these models and use these equations to examine critical behaviour. The topological large-N expansions are also constructed and their relation to other aspects of modern string theory such as integrable hierarchies is discussed. We use these connections to discuss the applications of these matrix models to string theory and induced gauge theories. We argue that as such the fermionic matrix models may provide a novel generalization of the discretized random surface representation of quantum gravity in which the genus sum alternates and the sums over genera for correlators have better convergence properties than their Hermitian counterparts. We discuss the use of adjoint fermions instead of adjoint scalars to study induced gauge theories. We also discuss two classes of dimensionally reduced models, a fermionic vector model and a supersymmetric matrix model, and discuss their applications to the branched polymer phase of string theories in target space dimensions D>1 and also to the meander problem.