Fermionic Matrix Models

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.

139 pages Latex (99 pages in landscape, two-column option); Section on Supersymmetric Matrix Models expanded, additional references included

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Cited by in corpus (31)

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• Gravity from Spinors
• Meanders: Exact Asymptotics
• A matrix model for a quantum hall droplet with manifest particle-hole symmetry
• Free Energy and Phase Transition of the Matrix Model on a Plane-Wave
• Notes on Matrix Models
• Meanders: A Direct Enumeration Approach
• A transfer matrix approach to the enumeration of plane meanders
• Geometrically constrained statistical systems on regular and random lattices: From folding to meanders
• Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model
• Solvable Quantum Grassmann Matrices
• Super-quantum curves from super-eigenvalue models
• Grassmann Matrix Quantum Mechanics
• From CFT to Ramond super-quantum curves
• Matrix Model Thermodynamics
• Large N matrices from a nonlocal spin system
• W-representations of the fermionic matrix and Aristotelian tensor models
• Exact Meander Asymptotics: a Numerical Check
• Fermionic Quantum Gravity
• Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model
• The Chiral Supereigenvalue Model
• Polynomial solution of quantum Grassmann matrices
• Notes on Curtright-Zachos Deformations of osp(1,2) and Super Virasoro Algebras
• Integrability of Supersymmetric Quantum Matrix Models in the Large-N Limit
• $W$-representations of two-matrix models with infinite set of variables
• Phase transitions and random matrices
• Noncommutative probability, matrix models, and quantum orbifold geometry
• The AdS/CFT Correspondence: Classical, Quantum, and Thermodynamical Aspects
• Fermionic Vector Model Solitons in the Large N Limit
• Fermionic Matrix Models and Bosonization
• A note on observables for counting trails and paths in graphs