Geodesic
Integrals over moduli spaces, ground ring, and four-point function in
Teoretičeskaâ i matematičeskaâ fizika, Tome 147 (2006) no. 3, pp. 339-371 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Directly evaluating the correlation functions in 2D minimal gravity requires integrating over the moduli space. For degenerate fields, the higher equations of motion of the Liouville field theory allow converting the integrand to a derivative, which reduces the integral to boundary terms and the so-called curvature contribution. The latter is directly related to the vacuum expectation value of the corresponding ground-ring element. The action of this element on the cohomology related to a generic matter primary field is evaluated directly in terms of the operator product expansions of the degenerate fields. This allows constructing the ground-ring algebra and evaluating the curvature term in the four-point function. We also analyze the operator product expansions of the Liouville "logarithmic primaries" and calculate the relevant logarithmic terms. Based on this, we obtain an explicit expression for the four-point correlation number of one degenerate and three generic matter fields. We compare this integral with the numbers obtained from the matrix models of 2D gravity and discuss some related problems and ambiguities.
Keywords: Polyakov string theory, Liouville gravity. 
@article{TMF_2006_147_3_a0,
     author = {A. A. Belavin and Al. B. Zamolodchikov},
     title = {Integrals over moduli spaces, ground ring, and four-point function in},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {339--371},
     year = {2006},
     volume = {147},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2006_147_3_a0/}
}
TY  - JOUR
AU  - A. A. Belavin
AU  - Al. B. Zamolodchikov
TI  - Integrals over moduli spaces, ground ring, and four-point function in
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2006
SP  - 339
EP  - 371
VL  - 147
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2006_147_3_a0/
LA  - ru
ID  - TMF_2006_147_3_a0
ER  - 
%0 Journal Article
%A A. A. Belavin
%A Al. B. Zamolodchikov
%T Integrals over moduli spaces, ground ring, and four-point function in
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2006
%P 339-371
%V 147
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2006_147_3_a0/
%G ru
%F TMF_2006_147_3_a0
A. A. Belavin; Al. B. Zamolodchikov. Integrals over moduli spaces, ground ring, and four-point function in. Teoretičeskaâ i matematičeskaâ fizika, Tome 147 (2006) no. 3, pp. 339-371. http://geodesic.mathdoc.fr/item/TMF_2006_147_3_a0/

[1] A. Polyakov, Phys. Lett. B, 103 (1981), 207 | DOI | MR

[2] H. Dorn, H.-J. Otto, Phys. Lett. B, 291 (1992), 39 ; ; Nucl. Phys. B, 429 (1994), 375 ; ; A. Zamolodchikov, Al. Zamolodchikov, Nucl. Phys. B, 477 (1996), 577 hep-th/9206053hep-th/9403141 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[3] J. Teschner, Class. Quant. Grav., 18 (2001), R153 ; hep-th/0104158 | DOI | MR | Zbl

[4] A. Belavin, A. Polyakov, A. Zamolodchikov, Nucl. Phys. B, 241 (1984), 333 | DOI | MR | Zbl

[5] Al. Zamolodchikov, Int. J. Mod. Phys. A, 19S2 (2004), 510 ; hep-th/0312279 | DOI | MR | Zbl

[6] V. Dotsenko, V. Fateev, Phys. Lett. B, 154 (1985), 291 | DOI | MR

[7] Al. B. Zamolodchikov, TMF, 142:2 (2005), 218 ; hep-th/0505063 | DOI | MR | Zbl

[8] H. Poincaré, J. Math. Pure Appl., 5:4 (1898), 137 ; А. М Поляков, Неопубликованные лекции, 1979 | Zbl

[9] L. Benoit, Y. Saint-Aubin, Phys. Lett. B, 215 (1988), 517 | DOI | MR | Zbl

[10] C. Imbimbo, S. Mahapatra, S. Mukhi, Nucl. Phys. B, 375 (1992), 399 | DOI | MR

[11] B. Feigin, Private communication, 2005

[12] I. Klebanov, A. Polyakov, Mod. Phys. Lett. A, 6 (1991), 3273 ; hep-th/9109032 | DOI | MR | Zbl

[13] E. Witten, Nucl. Phys. B, 373 (1992), 187 ; hep-th/9108004 | DOI | MR

[14] I. K. Kostov, V. B. Petkova, TMF, 146:1 (2006), 132 ; hep-th/0505078 | DOI | MR | Zbl

[15] N. Seiberg, D. Shih, JHEP, 0402 (2004), 021 ; ; M. Douglas, I. Klebanov, D. Kutasov, J. Maldacena, E. Martinec, N. Seiberg, A new hat for the $c=1$ matrix model, hep-th/0312170hep-th/0307195 | DOI | MR | MR

[16] A. A. Belavin, Al. B. Zamolodchikov, Pisma v ZhETF, 82 (2005), 8

[17] D. Kutasov, E. Martinec, N. Seiberg, Phys. Lett. B, 276 (1992), 437 ; hep-th/9111048 | DOI | MR

[18] N. Seiberg, “Notes on quantum Liouville theory and quantum gravity”, Random Surfaces and Quantum Gravity, Proc. NATO Adv. Res. Workshop (Cargese, France, May 27 – June 2, 1990), NATO ASI Ser., Ser. B, 262, eds. O. Alvarez, E. Marinari, P. Windey, Plenum, NY, 1990, 363 | DOI | MR

[19] Al. Zamolodchikov, Commun. Math. Phys., 96 (1984), 419 ; Ал. Б. Замолодчиков, ТМФ, 73:1 (1987), 103 | DOI | MR | MR

[20] I. K. Kostov, Termal flow in the gravitational $O(n)$ model, hep-th/0602075

[21] I. K. Kostov, Mod. Phys. Lett. A, 4 (1989), 217 | DOI | MR

[22] M. Gaudin, I. Kostov, Phys. Lett. B, 220 (1989), 200 | DOI | MR