Geodesic
Multicut solutions of the matrix Kontsevich–Penner model
Teoretičeskaâ i matematičeskaâ fizika, Tome 93 (1992) no. 2, pp. 354-368 Cet article a éte moissonné depuis la source Math-Net.Ru

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Multicut solutions of the Hermitian one-matrix model parametrized by the recently introduced matrix model [1] with external field and Lagrangian having the form $\,\operatorname {tr}{(\Lambda X\Lambda X)} - \alpha N$ $(\log {(1+X)} -X)$ are considered. A brief review of the model, which describes the discretized moduli space of Riemann surfaces, is given. The general structure of multicut solutions is investigated, and it is shown that there arises an additional symmetry and that $s$ parameters remain free for the $(s+1)$-cut solution. A detailed analysis of the one-cut solution is made. Among other results, all solutions of Kazakov type are reproduced. We also discuss the general form for the two-cut solution which arises as generalization of the string equation to the case of two cuts. The entire treatment is given in the approximation of planar diagrams. 
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     title = {Multicut solutions of the matrix {Kontsevich{\textendash}Penner} model},
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K. L. Zarembo; L. O. Chekhov. Multicut solutions of the matrix Kontsevich–Penner model. Teoretičeskaâ i matematičeskaâ fizika, Tome 93 (1992) no. 2, pp. 354-368. http://geodesic.mathdoc.fr/item/TMF_1992_93_2_a11/

[1] Chekhov L., Makeenko Yu., “The multicritical Kontsevich–Penner model”, Mod. Phys. Lett. A, 7 (1992), 1223 | DOI | MR | Zbl

[2] Kontsevich M. L., Funk. Anal. i Prilozh., 25:2 (1991), 50 (Russian) ; Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function, Max-Planck-Institute preprints MPI/91-47 and MPI/91-77 | MR | Zbl | MR

[3] Witten E., Nucl. Phys. B, 340 (1990), 281 | DOI | MR

[4] Witten E., On the Kontsevich model and other models of two-dimensional gravity, preprint IASSNS-HEP-91/24, june 1991 | MR

[5] Marshakov A., Mironov A., Morozov A., On equivalence of topological and quantum $2D$ gravity, FIAN/ITEP preprint, august 1991 | MR | Zbl

[6] Gross D. J., Newman M. J., Unitary and hermitian matrices in an external field. II, preprint PUPT-1282, december 1991 | MR

[7] Kazakov V. A., Rostov I. K., unpublished

[8] Gross D. J., Newman M. J., Phys. Lett. B, 266 (1991), 291 | DOI | MR

[9] Makeenko Yu., Semenoff G., Mod. Phys. Lett. A, 6 (1991), 3455 | DOI | MR | Zbl

[10] Brower R. C., Nauenberg M., Nucl. Phys. B, 180 (1981), 221 ; Brézin E., Gross D., Phys. Lett. B, 97 (1980), 120 ; Brower R. C., Rossi P., Tan C.-I., Phys. Rev. D, 23 (1981), 942 | DOI | MR | DOI | MR | DOI

[11] Itzykson C., Zuber J.-B., Combinatorics of the modular group. II, Saclay preprint SPhT/92-001, january 1992 | MR

[12] Kharchev S., Marshakov A., Mironov A., Morozov A., Zabrodin A., Unification of all string models with $c1$; Towards unified theory of $2d$ gravity, FIAN/ITEP preprints, october 1991 | MR

[13] Ambjørn J., Chekhov L., Makeenko Yu., Higher Genus Correlators and W-infinity from the Hermitian One-Matrix Model, preprint NBI-HE-92-22, february 1992 | MR

[14] Chekhov L., Makeenko Yu. Hint on the External Field Problem for Matrix Models, Phys. Lett. B, 278 (1992), 271 | DOI | MR

[15] Kharchev S., Marshakov A., Mironov A., Morozov A., Generalized Kontsevich Model versus Toda hierarchy, Lebedev Institute preprint FIAN/TD-03/92, february, 1992 | MR

[16] Chekhov L., Matrix Model for Discretized Moduli Space, preprint HEP-TH@9205106, 1992 | MR

[17] Cicuta G. M., Molinari L., Montaldi E., Mod. Phys. Lett. A, 1 (1986), 125 | DOI | MR

[18] Demeterfi K., Deo N., Jain S., Tan C.-I., Multi-band structure and critical behaviour of matrix models, preprint BROWN-HET-764

[19] Crnkovič Č., Moore G., Multicritical Multi-Cut Matrix Models, preprint RU-90-61, 1990 | MR

[20] Chaudhuri S., Dykstra H., Lykken J., Mod. Phys. Lett. A, 6 (1991), 1665 ; Tan C.-I., Mod. Phys. Lett. A, 6 (1991), 1373 ; Generalized Penner models and multicritical behavior, preprint BROWN-HET-810, may 1991 | DOI | MR | Zbl | DOI | MR | Zbl | MR

[21] Korchemsky G. P., Matrix Model Perturbed by Higher Order Curvature Terms, Parma Univ. preprint UPRF-92-334, april 1992 | MR | Zbl

[22] Das S., Dhar A., Sengupta A., Wadia S., Mod. Phys. Lett. A, 5 (1990), 1041 | DOI | MR | Zbl

[23] Harer J., Zagier D., Invent. Math., 85 (1986), 457 ; Harer J., The Cohomology of the Moduli Space of Curves, Gordon, 1985 | DOI | MR | Zbl

[24] Penner R. C., “The Decorated Teichmuller Space of Punctured Surfaces”, Comm. Math. Phys., 113 (1987), 299–339 ; “Perturbative Series and the Moduli Space of Riemann Surfaces”, J. Diff. Geom., 27 (1988), 35–53 | DOI | MR | Zbl | DOI | MR | Zbl

[25] Strebel K., Quadratic Differentials, Springer-Verlag, Heidelberg–New York, 1984 | MR

[26] Mumford D., Towards an Enumerative Geometry of the Moduli Space of Curves, Arithmetic and Geometry, 11, Birkhauser, 1983 | MR

[27] Migdal A. A., Phys. Rep., 102 (1983), 199 | DOI

[28] Ambjørn J., Jurkiewicz J., Makeenko Yu., Phys. Lett. B, 251 (1990), 517 | DOI | MR

[29] Gross D. J., Migdal A. A., Phys. Rev. Lett., 64 (1990), 127 | DOI | MR | Zbl

[30] Kazakov V. A., Mod. Phys. Lett. A, 4 (1989), 2125 | DOI | MR

[31] Brézin E., Kazakov V., Phys. Lett. B, 236 (1990), 144 | DOI | MR

[32] Douglas M., Shenker S., Nucl. Phys. B, 335 (1990), 635 | DOI | MR