Abstract
Kontsevich introduced a hermitian random matrix model to compute the generating function of intersection numbers of the moduli space of (punctured) Riemann surfaces. He showed that this generating function is also a τ-function for the Korteveg-de Vries (KdV) hierarchy of differential equations. This model is fundamentally different from the usual double scaling limit of random matrix models known to yield analogous τ-functions. Our aim is to clarify the notion of “observables” in both pictures, as related to KdV time evolutions. As a result we prove two conjectures by Kontsevich and Witten about the form of these observables, which involve polynomial matrix averages.
Laboratoire de la Direction des Sciences et de la Matière du Commissariat à l’Energie Atomique.
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References
E. Brézin and V. Kazakov, Phys. Lett. B236 (1990) 144.
M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 127.
D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127.
E. Brézin, C. Itzykson, G. Parisi and J.-B. Zuber, Comm. Math. Phys. 69 (1979) 147.
M. Douglas, Phys. Lett. B238 (1990) 176.
P. Di Francesco and D. Kutasov, Nucl. Phys. B342 (1990) 589.
V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819.
F. David, Mod. Phys. Lett. A3 (1988) 1651.
J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509.
P. Di Francesco and D. Kutasov, Phys. Lett. B261 (1991) 385.
P. Di Francesco and D. Kutasov, and Nucl. Phys. B375 (1992) 119.
T. Banks, M. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B238 (1990) 279.
R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B348 (1991) 435.
E. Witten, Comm. Math. Phys. 117 (1988) 353.
E. Witten, Phys. Lett. B206 (1988) 601.
E. Witten, Nucl. Phys. B240 (1990) 281.
E. Witten, Surv. DifF. Geom. 1 (1991) 243.
M. Kontsevich, Intersection theory on the moduli space of curves, Funk. Anal. & Prilozh., 25 (1991) 50.-57
Intersection theory on the moduli space of curves and the matrix Airy function, lecture at the Arbeitstagung, Bonn, June 1991 and Comm. Math. Phys. 147 (1992) 1.
E. Witten, On the Kontsevich model and other models of two dimensional gravity, preprint IASSNS-HEP-91/24
C. Itzykson and J.-B. Zuber, Int. J. Mod. Phys. A7 (1992) 5661.
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Phys. Lett. B275 (1992) 311.
M. Adler and P. van Moerbeke, The W p-gravity version of the Witten-Kontsevich model, Brandeis preprint, September 1991.
E. Witten, Algebraic Geometry associated with matrix models of two dimensional gravity, preprint IASSNS-HEP-91/74.
P. Di Francesco, C. Itzykson and J.-B. Zuber, Comm. Math. Phys. 151 (1993) 193.
Harish-Chandra, Amer. J. Math. 79 (1957) 87.-120
C. Itzykson and J.-B. Zuber, J. Math. Phys. 21 (1980) 411–421.
J. Duistermaat and G. Heckman, Invent. Math. 69 (1982) 259.
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Di Francesco, P. (1993). Observables in the Kontsevich Model. In: Osborn, H. (eds) Low-Dimensional Topology and Quantum Field Theory. NATO ASI Series, vol 315. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1612-9_5
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DOI: https://doi.org/10.1007/978-1-4899-1612-9_5
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