### Abstract

We review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related O(n) matrix models. For d < 1 matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising-like matter, the sum over topologies is reduced to the solution of nonlinear differential equations (the Painlevé equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.

Original language | English (US) |
---|---|

Pages (from-to) | 1-133 |

Number of pages | 133 |

Journal | Physics Reports |

Volume | 254 |

Issue number | 1-2 |

DOIs | |

State | Published - Mar 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physics Reports*,

*254*(1-2), 1-133. https://doi.org/10.1016/0370-1573(94)00084-G

**2D gravity and random matrices.** / Francesco, P. Di; Ginsparg, P.; Zinn-Justin, J.

Research output: Contribution to journal › Review article

*Physics Reports*, vol. 254, no. 1-2, pp. 1-133. https://doi.org/10.1016/0370-1573(94)00084-G

}

TY - JOUR

T1 - 2D gravity and random matrices

AU - Francesco, P. Di

AU - Ginsparg, P.

AU - Zinn-Justin, J.

PY - 1995/3

Y1 - 1995/3

N2 - We review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related O(n) matrix models. For d < 1 matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising-like matter, the sum over topologies is reduced to the solution of nonlinear differential equations (the Painlevé equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.

AB - We review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related O(n) matrix models. For d < 1 matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising-like matter, the sum over topologies is reduced to the solution of nonlinear differential equations (the Painlevé equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.

UR - http://www.scopus.com/inward/record.url?scp=0040393258&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040393258&partnerID=8YFLogxK

U2 - 10.1016/0370-1573(94)00084-G

DO - 10.1016/0370-1573(94)00084-G

M3 - Review article

AN - SCOPUS:0040393258

VL - 254

SP - 1

EP - 133

JO - Physics Reports

JF - Physics Reports

SN - 0370-1573

IS - 1-2

ER -