# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] [AvM] Adler, M., van Moerbeke, P.: TheW p-gravity version of the Witten-Kontsevich model. Commun. Math. Phys. (to appear) [2] [BIZ] Bessis, D., Itzykson, C., Zuber, J.-B.: Qunatum field theory techniques in graphical enumeration. Adv. Appl. Math.1, 109–157 (1980) · Zbl 0453.05035 · doi:10.1016/0196-8858(80)90008-1 [3] [BK] Brézin, E., Kazakov, V.: LettB236, 144 (1990) [4] [DS] Douglas, M., Shenker, S.: Nucl. Phys.B335, 635 (1990) · doi:10.1016/0550-3213(90)90522-F [5] [GM] Gross, D., Migdal, A.: Phys. Rev. Lett.64, 127 (1990) · Zbl 1050.81610 · doi:10.1103/PhysRevLett.64.127 [6] [H] Harer, J.: The cohomology of the moduli space of curves. Lecture Notes of mathematics, vol. 1337. Berlin, Heidelberg, New York: Springer, pp. 138–221 · Zbl 0533.57003 [7] [HZ] Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Inv. Math.85, 457–485 (1986) · Zbl 0616.14017 · doi:10.1007/BF01390325 [8] [HC] Harish-Chandra: Differential operators on a semisimple Lie algebra. Am. J. Math.79, 87–120 (1957) · Zbl 0072.01901 · doi:10.2307/2372387 [9] [IZ1] Itzykson, C., Zuber, J.-B.: J. Math. Phys.21, 411–421 (1980) · Zbl 0997.81549 · doi:10.1063/1.524438 [10] [IZ2] Itzykson, C., Zuber, J.-B.: Commun. Math. Phys.134, 197–207 (1990) · Zbl 0709.57007 · doi:10.1007/BF02102094 [11] [KS] Kac, V., Schwarz, A.: LettB257, 329 (1991) [12] [KMMMZ] Kharchev, S., Marshakov, A., Mironov, A., Morozov, A., Zabrodin, A.: Unification of all string models withc<1, FIAN/TD-9/91 and ITEP-M-8/91 [13] [K1] Kontsevich, M.: Intersection theory on the moduli space of curves. Funct. Anal. and Appl.25 (2), 123–128 (1991) · Zbl 0742.14021 · doi:10.1007/BF01079591 [14] [K2] Kontsevich, M. Intersection theory on the moduli space of curves and the matrix Airy function. 30. Arbeitstagung Bonn. Preprint MPI/91-47 [15] [KM] Kontsevich, M., Mulase, M.: In preparation [16] [Mh] Mehta, M. L.: Random matrices in nuclear physics and number theory. Contemporary Math.50, 295–309 (1986) [17] [M] Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and geometry, vol. II. Boston: Birkhäuser 1983 · Zbl 0554.14008 [18] [P1] Penner, R. C.: The decorated Teichmüller space of punctured surfaces. Commun. Math. Phys.113, 299–339 (1987) · Zbl 0642.32012 · doi:10.1007/BF01223515 [19] [P2] Penner, R. C.: Perturbative series and the moduli space of Riemann surfaces. J. Diff. Geom.27, 35–53 (1988) · Zbl 0608.30046 [20] [P3] Penner, R. C.: The Poincaré dual of the Weil-Petersson Kähler two-form. Preprint, October 1991 [21] [SW] Segal, G. B., Wilson, G.: Loop groops and equations of KdV type. Publ. Math. I.H.E.S.61, 5–65 (1985) · Zbl 0592.35112 [22] [S] Strebel, K.: Quadratic differentials. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0547.30001 [23] [W1] Witten, E.: Two dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom.1, 243–310 (1991) · Zbl 0757.53049 [24] [W2] Witten, E.: On the Kontsevich model and other models of two dimensional gravity. IAS preprint HEP-91/24 [25] [W3] Witten, E.: TheN matrix model and gauged WZW models. IAS preprint HEP-91/26 [26] [Z] Zwiebach, B.: How covariant closed string theory solves a minimal area problem. Commun. Math. Phys.136, 83–118 · Zbl 0725.30032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.