On Equivalence of Topological and Quantum 2d Gravity

TL;DR

This paper proves that the continuum limit of the Hermitean 1-matrix model and Kontsevich's model are equivalent, implying that ordinary 2D quantum gravity and topological gravity are essentially the same theory.

Contribution

It establishes the equivalence between Virasoro constraints in matrix models and Ward identities in Kontsevich's model, linking quantum and topological 2D gravity.

Findings

Virasoro constraints match Ward identities in Kontsevich's model

The models describe the same 2D gravity theory

Supports the unification of quantum and topological gravity

Abstract

We demonstrate the equivalence of Virasoro constraints imposed on continuum limit of partition function of Hermitean 1-matrix model and the Ward identities of Kontsevich's model. Since the first model describes ordinary $d = 2$ quantum gravity, while the second one is supposed to coincide with Witten's topological gravity, the result provides a strong implication that the two models are indeed the same.