Intersection theory on the moduli space of curves and the matrix Airy function
type: Item
instance of: scholarly article
title: Intersection theory on the moduli space of curves and the matrix Airy function
zbMATH Open document ID: 0756.35081
DOI: 10.1007/BF02099526
author: Maxim Kontsevich
published in: Communications in Mathematical Physics
publication date: 1993-01-16T00:00:00Z
review text: As it is known, in our days there are at least two natural approaches to 2D gravity which are considered mathematically consistent. The first one, usually called ``enumeration of triangulations'', leads to a partition function which is expressed as a series in an infinite number of variables and coincides with the logarithm of some \(\tau\)-function for KdV-hierarchy. The second approach is based on some specific action and by the use of supersymmetry the integral over the space of all metrics reduces to the integral over the finite dimensional space of conformal structures such that some series in an infinite number of variables arises again. It is also known that \textit{E. Witten} [Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243-310 (1991)] conjectured that the partition functions for both approaches coincide because the gravity has to be unique. The aim of the present paper is to check out explicitly Witten's conjecture using the Feynman diagram techniques and matrix integrals in a new and interesting way. Thus, it is shown in several ways that the coincidence of the two integrals is a nontrivial identity which highly leans on the equivalence of both integrals to KdV equations.
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 83C45
Mathematics Subject Classification ID: 81T70
zbMATH DE Number: 95397
zbMATH Keywords: Witten's conjecture
zbMATH Keywords: Feynman diagram techniques
zbMATH Keywords: matrix integrals
Wikidata QID: Q64356443
reviewed by: Ciprian Dariescu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 95397