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On the origin of integrability in matrix models

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Abstract

The matrix integrals involved in 2d lattice gravity are studied at finiteN. The integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials. The partition function proves to be a tau function of the Toda lattice hierarchy. The associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function. The cases of one Hermitian matrix, one unitary matrix, and Hermitian matrix chains all fall within the Toda framework.

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Authors and Affiliations

  1. Enrico Fermi Institute and Department of Physics, University of Chicago, 60637, Chicago, IL, USA

    Emil J. Martinec

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  1. Emil J. Martinec
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Communicated by N. Yu. Reshetikhin

Research supported in part by DOE contract DE-FG02-90ER-40560, an NSF Presidential Young Investigator Award, and the Alfred P. Sloan Foundation

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Martinec, E.J. On the origin of integrability in matrix models. Commun.Math. Phys. 138, 437–449 (1991). https://doi.org/10.1007/BF02102036

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