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Intersection theory on the moduli space of curves and the matrix Airy function. (English) Zbl 0756.35081
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 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.
Reviewer: C.Dariescu (Iaşi)

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
53D50 Geometric quantization
83C45 Quantization of the gravitational field
81T70 Quantization in field theory; cohomological methods
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