### Abstract

We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.

Original language | English (US) |
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Pages (from-to) | 435-456 |

Number of pages | 22 |

Journal | Nuclear Physics, Section B |

Volume | 348 |

Issue number | 3 |

DOIs | |

State | Published - Jan 21 1991 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

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## Cite this

Dijkgraaf, R., Verlinde, H., & Verlinde, E. (1991). Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity.

*Nuclear Physics, Section B*,*348*(3), 435-456. https://doi.org/10.1016/0550-3213(91)90199-8