Geodesic
Witten Solution for the Gelfand--Dikii Hierarchy
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 1, pp. 25-37.

Voir la notice de l'article provenant de la source Math-Net.Ru

We derive formulas making it possible to calculate the Taylor expansion coefficients of the string solution for the Gelfand–Dikii hierarchy. According to the Witten conjecture, these coefficients coincide with the Mumford–Morita–Miller intersection numbers (correlators) of stable cohomology classes for the moduli space of $n$-spin bundles on Riemann surfaces with punctures.
Keywords: Gelfand–Dikii hierarchy, KP hierarchy
Mots-clés : moduli space, Witten conjecture. 
@article{FAA_2003_37_1_a2,
     author = {S. M. Natanzon},
     title = {Witten {Solution} for the {Gelfand--Dikii} {Hierarchy}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {25--37},
     publisher = {mathdoc},
     volume = {37},
     number = {1},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2003_37_1_a2/}
}
TY  - JOUR
AU  - S. M. Natanzon
TI  - Witten Solution for the Gelfand--Dikii Hierarchy
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2003
SP  - 25
EP  - 37
VL  - 37
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2003_37_1_a2/
LA  - ru
ID  - FAA_2003_37_1_a2
ER  - 
%0 Journal Article
%A S. M. Natanzon
%T Witten Solution for the Gelfand--Dikii Hierarchy
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2003
%P 25-37
%V 37
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2003_37_1_a2/
%G ru
%F FAA_2003_37_1_a2
S. M. Natanzon. Witten Solution for the Gelfand--Dikii Hierarchy. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 1, pp. 25-37. http://geodesic.mathdoc.fr/item/FAA_2003_37_1_a2/

[1] Adler M., van Moerbeke P., “A matrix integral solution to two-dimensional $W_p$-gravity”, Commun. Math. Phys., 147 (1992), 25–56 | DOI | MR | Zbl

[2] Date E., Kashiwara M., Jimbo M., Miwa T., “Transformation groups for soliton equation”, Nonlinear integrable systems – classical theory and quantum theory, Proc. of RIMS Symposium (Kyoto 1981), World Science Publ., Singapore, 1983, 39–119 | MR

[3] Dijkgraaf R., “Intersection theory, integrable hierarchies and topological field theory”, New symmetry principles in quantum field theory (Cargese, 1991), NATO Adv. Sci. Inst. Ser. B Phys., 295, Plenum, New York, 1992, 95–158 | MR

[4] Dubrovin B. A., Natanzon S. M., “Veschestvennye teta-funktsionalnye resheniya uravneniya Kadomtseva–Petviashvili”, Izv. AN SSSR, ser. matem., 52:2 (1988), 267–286 | MR | Zbl

[5] Gelfand I. M., Dikii L. A., “Rezolventa i gamiltonovy sistemy”, Funkts. analiz i ego pril., 11:2 (1977), 11–27 | MR

[6] Kontsevich M., “Intersection theory on the moduli space of curves and the matrix Airy function”, Commun. Math. Phys., 147 (1992), 1–23 | DOI | MR | Zbl

[7] Krichever I. M., “Algebro-geometricheskoe postroenie uravnenii Zakharova–Shabata i ikh periodicheskikh reshenii”, Dokl. AN SSSR, 227:2 (1976), 291–294 | MR | Zbl

[8] Natanzon S. M., “Real nonsingular finite zone solutions of solution equations”, Amer. Math. Soc. Transl. (2), 170 (1995), 153–183 | MR | Zbl

[9] Natanzon S. M., “Formulas for $A_n$ and $B_n$-solutions of WDVV equations”, J. Geometry and Physics, 39 (2001), 323–336 | DOI | MR | Zbl

[10] Okounkov A., Pandharipande R., Hurwitz numbers, and matrix models, I, arXiv: /math.AG/0101147 | MR

[11] Sato M., Soliton equations and universal Grassmann manifold, Math. Lect. Notes Ser., 18, Sophia University, Tokyo, 1984 | Zbl

[12] Witten E., “Algebraic geometry associated with matrix models of two-dimensional gravity”, Topological models in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 235–269 | MR | Zbl