Matrix Models: Supersymmetry, Chaos, Loop Equations

duration: 1993 - 1996
researchers: Adamietz, Akemann, Lechtenfeld, Plefka
funding: Studienstiftung (3 PhD students)
other projects

Abstract:

Random matrices have been studied in physics since the work of Wigner in the 1950s. Having found numerous applications in various fields throughout the years, matrix models experienced a great renaissance in high-energy and mathematical physics, following the discovery of their relevance for the quantization of two-dimensional gravity and string theory in 1985. Matrix models have the appealing property that they are exactly solvable in the limit of infinite matrix size N. Subleading corrections to arbitrary order in 1/N may be computed by iterative means. The analysis of the diagrammatic expansion of the Hermitean matrix model reveals that solving the model is equivalent to performing the sum over random equilateral triangulations of two-dimensional surfaces of fixed genus.

The PhD projects conducted in this field focus on

Supersymmetric generalizations of matrix models (PhD thesis, Jan Plefka, 11/95)
Phase transitions in matrix models and multi-cut eigenvalue distributions (PhD thesis, Gernot Akemann, 02/96)
Complex matrix models (PhD thesis, Peter Adamietz, /96)

Papers:

Peter Adamietz:
Kollektive Feldtheorie und Momentenmethode in Matrixmodellen
ITP-UH-14/97 (PhD Thesis)

G. Akemann, P.H. Damgaard, U. Magnea and S. Nishigaki:
Universality of random matrices in the microscopic limit and the Dirac operator spectrum
hep-th/9609174 (Nucl. Phys. B 487 (1997) 721)

Jan Ambjorn and Gernot Akemann:
New universal spectral correlators
cond-mat/9606129 (J. Phys. A 29 (1996) L555)

Gernot Akemann:
Higher genus correlators for the hermitian matrix model with multiple cuts
hep-th/9606004 (Nucl. Phys. B 482 (1996) 403)

Gernot Akemann:
Phasenübergänge in Matrixmodellen
ITP-UH-02/96 (PhD Thesis)

Jan C. Plefka:
Supersymmetric generalizations of matrix models
hep-th/960104 (PhD Thesis)

Jan C. Plefka:
The supereigenvalue model in the double-scaling limit
hep-th/9504089 (Nucl. Phys. B 448 (1995) 355)

Gernot Akemann:
Loop equations for multi-cut matrix models
ITP-UH-11/95 (unpublished)

Jan C. Plefka:
Iterative solution of the supereigenvalue model
hep-th/9501120 (Nucl. Phys. B 444 (1995) 333)

Olaf Lechtenfeld:
Collective field theory for D=0 matrix models
hep-th/9312124 (p.32, DESY 94-053)
(Talk at 27th Int. Ahrenshoop Symp. on Elem. Particle Theory, Wendisch-Rietz, Germany, 1993)