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Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. (English) Zbl 0815.53082
Summary: We develop techniques to compute higher loop string amplitudes for twisted \(N = 2\) theories with \(\widehat{c} = 3\) (i.e. the critical base). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of the \(N = 2\) theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the ‘number’ of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective \(4d\) theory resulting from compactification of standard 10d superstrings on the corresponding \(N = 2\) theory. Relations with \(c = 1\) strings are also pointed out.

MSC:
53Z05 Applications of differential geometry to physics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T50 Anomalies in quantum field theory
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References:
[1] Lerche, W., Vafa, C., Warner, N.P.: Nucl. Phys. B324, 427 (1989) · doi:10.1016/0550-3213(89)90474-4
[2] Cecotti, S., Vafa, C.: Nucl. Phys. B367, 359–461 (1991) · Zbl 1136.81403 · doi:10.1016/0550-3213(91)90021-O
[3] Dixon, L.: In: Proc. of the 1987 ICTP Summer Workshop in High Energy Physics and Cosmology, Trieste
[4] Witten, E.: Commun. Math. Phys.117, 353 (1988); Witten, E.: Commun. Math. Phys.118, 411 (1988); · Zbl 0656.53078 · doi:10.1007/BF01223371
[5] Eguchi, T., Yang, S.-K.: Mod. Phys. Lett. A5, 1693 (1900) · Zbl 1020.81833 · doi:10.1142/S0217732390001943
[6] Dijkgraaf, R., Verlinde, H., Verlinde, E.: Nucl. Phys. B352, 59 (1991) · doi:10.1016/0550-3213(91)90129-L
[7] Alvarez-Gaumé, L., Freedman, D.Z.: Phys. Lett.94B, 171 (1980)
[8] Candelas, P., de la Ossa, Xenia C., Green, Paul S., Parkes, L.: Nucl. Phys. B359, 21 (1991) · Zbl 1098.32506 · doi:10.1016/0550-3213(91)90292-6
[9] Aspinwall, P.S., Morrison, D.R.: Commun. Math. Phys.151, 245–262 (1993) · Zbl 0776.53043 · doi:10.1007/BF02096768
[10] Witten, E.: In: Yau, S.T. (ed.) Essays on Mirror Manifolds. Hong Kong: International Press, 1992 · Zbl 0834.58013
[11] Vafa, C.: In: Yau, S.T. (ed.) Essays on Mirror Manifolds. Hong Kong: International Press, 1992
[12] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley, 1978 · Zbl 0408.14001
[13] Zamolodchikov, A.B.: JETP Lett.43, 730 (1986)
[14] Bryant, R., Griffiths, P.: In: Artin, M., Tate, J. (eds.) Arithmetic and Geometry. (Papers dedicated to I.R. Shafarevitch, vol. 2). Boston: Birkhäuser, 1983, p. 77;
[15] Ferrara, S., Strominger, A.: N=2 spacetime supersymmetry and Calabi-Yau moduli space. (Presented at Texas A & M University, String’ 89 Workshop);
[16] Cecotti, S.: Commun. Math. Phys.131, 517 (1990); · Zbl 0712.53045 · doi:10.1007/BF02098274
[17] Strominger, A.: Commun. Math. Phys.133, 163 (1990); · Zbl 0716.53068 · doi:10.1007/BF02096559
[18] Candelas, P., de la Ossa, X.C.: Moduli Space of Calabi-Yau Manifolds. (University of Texas Report UTTG-07-90); · Zbl 0732.53056
[19] D’Auria, R., Castellani, L., Ferrara, S.: Class. Quant. Grav.1, 1767 (1990)
[20] de Wit, B., van Proeyen, A.: Nucl. Phys. B245, 89 (1984); de Wit, B., Lauwers, P. G., van Proyen, A.: Nucl. Phys. B255, 569 (1985); Cremmer, E., Kounnas, C., van Proeyen, A., Derendinger, J.-P., Ferrara, S., de Wit, B., Girardello, L.: Lucl. Phys. B250, 385 (1985) · doi:10.1016/0550-3213(84)90425-5
[21] Cecotti, S.: Int. J. Mod. Phys. A6, 1749 (1991); · Zbl 0743.57022 · doi:10.1142/S0217751X91000939
[22] Cecotti, S.: Nucl. Phys. B355, 755 (1991) · doi:10.1016/0550-3213(91)90493-H
[23] Dijkgraaf, R., Verlinde, H., Verlinde, E.: Notes on Topological String Theory and 2-D Quantum Gravity. (Proceedings of Trieste Spring School on Strings and Quantum Gravity), 1990, pp. 91–156 · Zbl 0985.81681
[24] Gato-Rivera, B., Semikhatov, A.M.: Phys. Lett. B293, 72–80 (1992) · doi:10.1016/0370-2693(92)91482-O
[25] Bershadsky, M., Lerche, W., Nemeschansky, D., Warner, N.P.: Nucl. Phys. B401, 304 (1993) · Zbl 0941.81591 · doi:10.1016/0550-3213(93)90306-A
[26] Polchinski, J.: Commun. Math. Phys.104, 37 (1986) · Zbl 0606.58014 · doi:10.1007/BF01210791
[27] Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Nucl. Phys. B405, 279 (1993) · Zbl 0908.58074 · doi:10.1016/0550-3213(93)90548-4
[28] Witten, E.: Nucl. Phys. B340, 281 (1990); · doi:10.1016/0550-3213(90)90449-N
[29] Dijkgraaf, R., Witten, E.: Nucl. Phys. B342, 486 (1990) · doi:10.1016/0550-3213(90)90324-7
[30] Verlinde, E., Verlinde, H.: Nucl. Phys. B348, 457 (1991) · doi:10.1016/0550-3213(91)90200-H
[31] Li, K.: Nucl. Phys. B354, 725–739 (1991) · doi:10.1016/0550-3213(91)90374-7
[32] Witten, E.: Chern-Simons Gauge Theory as a String Theory. IASSNS-HEP-92/45, hep-th/9207094
[33] Kutasov, D.: Phys. Lett. B220, 153 (1989) · doi:10.1016/0370-2693(89)90028-2
[34] Dijkgraaf, R., Verlinde, E., Verlinde, H.: Nucl. Phys. B352, 59 (1991) · doi:10.1016/0550-3213(91)90129-L
[35] Schmid, W.: Invent. Math.22, 211 (1973) · Zbl 0278.14003 · doi:10.1007/BF01389674
[36] Witten, E.: Quantum Background Independence in String Theory. IASSNS-HEP-93/29, hep-th/9306122
[37] Kodaira, K., Spencer, D.C.: Annals of Math.67, 328 (1958); Kodaira, K., Niremberg, L., Spencer, D. C.: Annals of Math.68, 450 (1958); Kodaira, K., Spencer, D.C.: Acta Math.100, 281 (1958); Kodaira, K., Spencer, D.C.: Annals of Math.71, 43 (1960) · Zbl 0128.16901 · doi:10.2307/1970009
[38] Tian, G.: In: Yau, S.T. (ed.) Essays on Mirror manifolds. Hong Kong: International Press, 1992 Tian, G.: In: Yau, S.T. (ed.) Mathematical aspects of String theory. Singapore: World Scientific, 1987
[39] Todorov, A.: Geometry of Calabi-Yau. (MPI preprint). Weil-Peterson, July 1986
[40] Tian, G.: Private communication
[41] Todorov, A.N.: Commun. Math. Phys.126, 325 (1989) · Zbl 0688.53030 · doi:10.1007/BF02125128
[42] Kuranishi, M.: Annls of Math.75, 536 (1962) · Zbl 0106.15303 · doi:10.2307/1970211
[43] Zwiebach, B.: Nucl. Phys. B390, 33 (1993) · doi:10.1016/0550-3213(93)90388-6
[44] Elitzur, S., Forge, A., Rabinovici, E.: Nucl. Phys. B388, 131 (1992) · doi:10.1016/0550-3213(92)90548-P
[45] Ooguri, H., Vafa, C.: Nucl. Phys. B361, 469 (1991); Mod. Phys. Lett. A5, 1389 (1990) · doi:10.1016/0550-3213(91)90270-8
[46] Batalin, I.A., Vilkovisky, G.A.: Phys. Rev D28, 2567 (1983) · doi:10.1103/PhysRevD.28.2567
[47] Henneaux, M.: Lectures on the antifield-BRST formalism for gauge theories. (Proc. of XXII GIFT Meeting) · Zbl 0957.81711
[48] Ray, B., Singer, I.M.: Ann. Math.98, 154 (1973) · Zbl 0267.32014 · doi:10.2307/1970909
[49] Bismut, J.-M., Freed, D.S.: Commun. Math. Phys.106, 159 (1986);107, 103 (1986); Bismut, J.-M., Gillet, H., Soulé, C.: Commun. Math. Phys.115, 49, 79, 301 (1988); Bismut, J.-M., Köhler, K.: Higher analytic torsion forms for direct images and anomaly formulas. (Univ. de Paris-sud, preprint 91-58) · Zbl 0657.58037 · doi:10.1007/BF01210930
[50] Belavin, A., Knizhnik, V.: Phys. Lett.168 B, 201 (1986); Alvarez-Gaumé, L., Moore, G., Vafa, C.: Commun. Math. Phys.112, 503 (1987)
[51] Hirzebruch, F.: Topological methods in Algebraic Geometry. Berline-Heidelberg-New York: Springer-Verlag, 1966 · Zbl 0138.42001
[52] Schoen, R., Yau, S.-T.: Invent. Math.16, 161 (1972) · Zbl 0242.32015 · doi:10.1007/BF01391215
[53] Tromba, J.: Man. Math.59, 249 (1987) · Zbl 0635.58004 · doi:10.1007/BF01158050
[54] Witten, E.: Nucl. Phys.371, 191 (1992); Witten, E.: In: Yau, S. T. (ed.) Essays on Mirror Manifolds. Hong Kong: International Press, 1992 · Zbl 0834.58013 · doi:10.1016/0550-3213(92)90235-4
[55] Hamidi, S., Vafa, C.: Nucl. Phys. B279, 465 (1987) · doi:10.1016/0550-3213(87)90006-X
[56] Mumford, D.: In: Artin, M, Tate, J. (eds.) Arithmetic and Geometry. (Papers dedicated to I. R. Shafarevitch). Boston: Birkäuser, 1983
[57] Faber, C.: Ann. Math.132, 331 (1990) · Zbl 0721.14013 · doi:10.2307/1971525
[58] Castellani, L., D’Auria, R., Ferrara, S.: Class. Quant. Grav.7, 1767 (1990) · Zbl 0715.53057 · doi:10.1088/0264-9381/7/10/009
[59] Plesser, B.R., Greene, M.R.: Nucl. Phys. B338, 15 (1990) · doi:10.1016/0550-3213(90)90622-K
[60] Kazakov, V., Migdal, A.: Nucl. Phys. B311, 171 (1989); Polchinski, J.: Nucl. Phys. B346, 253 (1990) · Zbl 1232.81045 · doi:10.1016/0550-3213(88)90146-0
[61] Mukhi, S., Vafa, C.: Nucl. Phys. B407, 667 (1993) · Zbl 1043.81694 · doi:10.1016/0550-3213(93)90094-6
[62] Morrison, D.R.: In: Yau, S.T. (ed.) Essays on Mirror Manifolds. Hong Kong: International Press, 1992
[63] Klemm, A., Theisen, S.: Nucl. Phys. B389, 153 (1993) · doi:10.1016/0550-3213(93)90289-2
[64] Antoniadis, I., Gava, E., Narain, K. S., Taylor, T. R.: Topological Amplitudes in String Theory. hepth/9307158 · Zbl 1007.81522
[65] Seiberg, N.: Nucl. Phys. B303, 206 (1988); Cecotti, S., Ferrara, S., Girardello, L.: J. Mod. Phys. A4, 2475 (1989) · doi:10.1016/0550-3213(88)90183-6
[66] Friedan, D., Martinec, E., Shenker, S.: Phys. Lett.160B, 55 (1985)
[67] Derendinger, J.P., Ibanez, L.E., Nilles, H.P.: Phys. Lett.155B, 65 (1985); Dine, M., Rohm, R., Seiberg, N., Witten, E.: Phys. Lett.156 B, 55 (1985)
[68] This is currently being pursued by Narian et. al.
[69] Kaplunovsky, V.: Nucl. Phys. B307, 36 (1988); Dixon, L. J., Kaplunovsky, V. S., Louis, J.: Nucl. Phys. B355, 649 (1991); Ferrara, S., Kounnas, C., Lüst, D., Zwirner, F: Nucl. Phys. B365, 431 (1991); Antoniadis, I., Gava, E., Narain, K. S.: Nucl. Phys. B383, 93 (1992) and Phys. Lett. B283, 209 (1992); Derendinger, J.-P., Ferrara, S., Kounnas, C., Zwirner, F.: Nucl. Phys. B372, 145 (1992) · doi:10.1016/0550-3213(88)90526-3
[70] Witten, E.: Nucl. Phys. B268, 79 (1986) · doi:10.1016/0550-3213(86)90202-6
[71] Shenker, S.: Proceedings of Cargèse Workshop on Random Surfaces, Quantum Gravity and Strings, 1990.
[72] Axelrod, S., Singer, I.: Chern-Simons Perturbation Theory. MIT preprints, hep-th/9110056, hepth/9304087 · Zbl 0813.53051
[73] Kontsevich, M.: Feynman Diagrams and Low-Dimensional Topology. (Talk given at 1st European Congress of Mathematics, Paris, July 1992) · Zbl 0872.57001
[74] Berkovitz, N.: Nucl. Phys. B395, 77 (1993) · doi:10.1016/0550-3213(93)90209-8
[75] Witten, E.: Nucl. Phys. B373, 187 (1992); Witten, E., Zwiebach, B.: Nucl. Phys. B377, 55 (1992) · doi:10.1016/0550-3213(92)90454-J
[76] Cecotti, S., Vafa, C.: Commun. Math. Phys.157, 139 (1993) · Zbl 0787.58008 · doi:10.1007/BF02098023
[77] Losev, A.: Descendants constructed from matter fields in topological Landau-Ginzburg theories coupled to topological gravity. hepth/9211089; Losev, A., Polyubin, I.: On Connection between Topological Landau-Ginsburg Gravity and Integrable Systems. hep-th/9305079 Eguchi, T., Kanno, H., Yamada, Y., Yang, S.-K.: Phys. Lett. B305, 235 (1993)
[78] Witten, E.: Commun. Math. Phys.117, 117 (1988); Witten, E.: Phys. Lett.206 B, 601 (1988) · Zbl 0656.53078 · doi:10.1007/BF01223371
[79] Witten, E.: Commun. Math. Phys.121, 351 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730
[80] Gross, D.: Nucl. Phys. B400, 161 (1993); Gross, D., Taylor, W.: Nucl. Phys. B400, 181 (1993); Nucl. Phys. B403, 395 (1993) · Zbl 0941.81580 · doi:10.1016/0550-3213(93)90402-B
[81] Dijkgraaf, R., Intriligator, K., Rudd, R.: unpublished Communicated by R.H. Dijgraaf
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