Abstract
The geometry ofN=2 supergravity is related to the variations of Hodge structure for “formal” Calabi-Yau spaces. All known results in this branch of algebraic geometry are easily recovered from supersymmetry arguments. This identification has a physical meaning for a type IIB superstring compactified on a Calabi-Yau 3-fold. We give exact (non-perturbative) results for the string effective lagrangian. Our geometrical framework suggests a re-formulation of the Gepner conjecture about (2,2) superconformal theories as the solution to theSchottky problem for algebraic complex manifolds having trivial canonical bundle.
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Alvarez-Gaumé, L.: Supersymmetry and mathematics. In: Supersymmetry: A decade of development, West, P. C. (ed.) pp. 456–478. Bristol and Boston: Adam Hilger 1986
Witten, E.: Supersymmetry and morse theory. J. Diff. Geom.17, 661 (1982)
Atiyah, M. F., Singer, J. M.: Index of elliptic operators I. Ann. Math.87, 485 (1968); Atiyah, M. F. Singer, J. M.: Index of elliptic operators III. Ann. Math.87, 546 (1968); Atiyah, M. F., Singer, J. M.: Index of elliptic operators IV. Ann. Math.93, 119 (1971); Atiyah, M. F., Singer, J. M.: Index of elliptic operators V. Ann. Math.93, 139 (1971)
Alvarez-Gaumé, L.: Supersymmetry and the Atiyah-Singer index theorem. Commun. Math. Phys.90, 161–173 (1983); Alvarez-Gaumé, L.: A note on the Atiyah-Singer index theorem. J. Phys.A16, 4177–4182 (1983); Friedan, D., Windey, P.: Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly. Nucl. Phys.B235, 395 (1984); Getzler, E.: Commun. Math. Phys.92, 163 (1983)
Witten, E.: Elliptic genera and quantum field theory. Commun. Math. Phys.109, 525–536 (1987); Schellekens, A. N., Warner, N. P.: Anomalies, characters and strings. Nucl. Phys.B287, 317–316 (1987); Alvarez, O., Killingback, T., Mangano, M., Windey, P.: String theory and loop space index theorems. Commun. Math. Phys.111, 1–10 (1987)
Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys.B202, 253 (1982)
Griffiths, P.: Periods of integrals on algebraic manifolds. I, II. Am. J. Math.90, 568–626, 805–865 (1968)
Calabi, E.: On Kähler manifolds with vanishing canonical class. In: Algebraic geometry and topology. A Symposium in honor of Lefschets, S. pp. 78–89. Princeton, NJ: Princeton University Press 1975; Yau, S.-T.: On Calabi's conjecture and some new results in algebraic geometry. Proc. Nat. Aca.d Sci. USA74, 1798–1799 (1977); Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation, I. Commun. Pure Appl. Math.31, 339–411 (1978)
Griffiths, P.: Variations of Hodge Structure. Curvature properties of the Hodge Bundles. Infinitesimal variation of Hodge structure. Asymptotic behaviour of a variation of Hodge Structure. In: Topics in Transcendental Algebraic Geometry. Griffiths, P. (ed.). (Proceeding of the IAS seminar 1981–82). Annals of Mathematical Studies vol106, Princeton, NJ: Princeton Press 1984. Chapters I, II, III and IV. pp. 3–28; 29–49; 51–61; 63–74; and references therein
Bryant, R. L.: Griffiths, P. A.: Some observations on the infinitesimal period relations for regular threefolds with trivial canonical class. In: Arithmetic and Geometry, papers dedicated to I. R. Shafarevitch, Artin, M., Tate, J. (eds.) Boston Basel Stuttgrat: Birkhauser 1983. Vol.2, pp. 77–102
Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum configurations for superstring. Nucl. Phys.B258, 341 (1985)
Green, M. B.: Schwarz, J. H., Witten, E.: Superstring Theory. Vol.1,2. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press 1987
Kuranishi, M.: On the locally complete families of complex analytic structures. Ann. Math.75, 536–577 (1962)
Tian, G.: Smoothness of the Universal Deformation Space of Compact Calabi-Yau Manifolds and Its Peterson-Weil Matric. In: Mathematical Aspects of String Theory. (Proceeding of the San Diego Conference, 1986). Yau, S.-T. (ed.) World Scientific, Advanced Series in Mathematical Physics, Vol.1, pp. 629–646. Singapore: World Scientific 1987
Todorov, A.: Talk at the conference on: Global methods in analysis. Trieste 1988 (unpublished)
Todorov, A. N.: Applications of the Kähler-Einstein Calabi-Yau metrics to moduli ofK3 surface. Invent. Math.61, 251–265 (1980); Todorov, A. N.: How many Kähler metrics has aK3 surface. In: Arithmetic and geometry. Dedicated to I. R. Shafarevitch. Artin, M., Tate, J. (eds.). Boston Basel Stuttgart: Birkhauser 1983. Vol.2, 251–265; Kobayashi, R., Todorov, A. N.: Polarized period map for generalizedK3 surfaces and the moduli of Einstein metrics. Tohoku Math. J.39, 341 (1987); Siu, Y. T.: A simple proof of the surjectivity of the period map ofK3 surfaces. Manuscripta Math.35, 225–255 (1981); Looijenga, E.: A Torelli theorem for Kähler-EinsteinK3 surfaces. In: Geometry Symposium, Utrecht 1980. pp. 101–112. Lectures Notes in Mathematics Vol.894, Looijenga, E., Sierma, D., Takens, F. Berlin, Heidelberg New York: Springer 1981
Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. A Series of Modern Surveys in Mathematics. Berlin, Heidelberg, New York: Springer 1984
Sieberg, S.: Observations on the moduli space of superconformal field theories. Nucl. Phys.B303, 286–304 (1988)
de Roo, M.: Matter coupling inN=4 supergravity. Nucl. Phys.B255, 515–53 (1985); de Roo, M.: GaugedN=4 matter couplings. Phys. Lett.156B, 331–334 (1985); Bergshoeff, E.: Koh, I. G., Sezgin, E.: Coupling of Yang-Mills toN=4,d=4 supergravity. Phys. Lett.155B, 71–75 (1985); de Roo, M., Wagemans, P.: Gauged matter coupling inN=4 supergravity. Nucl. Phys.B262, 646–660 (1985)
Strominger, A.: Yukawa couplings in Superstring compactification. Phys. Rev. Lett.55, 2547–2250 (1985)
Candelas, P.: Yukawa couplings between (2,1)-forms. In: Mathematical Aspects of String Theory. (Proceedings of the San Diego Conference, 1986). Yau, S.-T. (ed.). World Scientific, Advanced Series in Mathematical Physics, Vol.1, pp. 488–542. Singapore: World Scientific 1987
Zamolodchikov, A. B.: Irreversibility of the flux of the renormalization group in a 2D field theory. J.E.T.P. Lett.43, 730–732 (1986)
Gepner, D.: Exactly solvable string compactifications on manifolds ofSU(N) holonomy. Phys. Lett.199B, 380–388 (1987)
Martinec, E.: Algebraic Geometry and effective Lagrangians. Phys. Lett.217B, 431 (1989)
Vafa, C., Warner, N.: Catastrophes and the Classification of Conformal Theories. Phys. Lett.218B, 431 (1989). Greene, B. R., Vafa, C., Warner, N.: Calabi-Yau Manifolds and Renormalization Group Flows. Nucl. Phys.B324, 427 (1989)
Gaillard, M. K., Zumino, B.: Duality rotations for interacting fields. Nucl. Phys.B193, 221–244 (1981)
Banks, T., Dixon, L., Friedan, D., Martinec, E.: Phenomenology and conformal field theory, or can string theory predict the weak mixing angle? Nucl. Phys.B299, 613–626 (1988)
Kodaira, K.: On compact complex analytic surfaces. I. Ann. Math.71, 111–152 (1960)
Witten, E., Bagger, J.: Quantization of Newton's constant in certain supergravity theories. Phys. Lett.115, 202–206 (1982)
Catanese, F. M. E.: Infinitestimal Torelli theorems and counterexamples to Torelli problems. In: Topics in Transcendental Algebraic Geometry. Griffiths, P. ed.. (Preceeding of the IAS seminar 1981–82). Annals of Mathematical Studies. vol.106, pp. 143–156. Princeton, NJ: Princeton Press 1984, Chap. VIII
Griffiths, P., Schmid, W.: Locally homogeneous complex manifolds. Acta Math.123, 253–302 (1969)
Griffiths, P.: Curvature properties of the Hodge Bundles. In: Topics in Transcendental Algebraic Geometry. Griffiths, P. (ed.) (Proceeding of the IAS seminar 1981–82). Annals of Mathematical Studies, Vol.106, pp. 29–49. Princeton, NJ: Princeton Press 1984, Chap. II
Kodaira, K., Spencer, D. C.: On deformations of complex analytic structures, I, II, III. Ann. Math.67, 328–466 (1958);71, 43–76 (1960); Kodaira, K., Nirenberg, L., Spencer, D. C.: On the existence of deformations of complex analytic structures. Ann. Math.68, 450–459 (1958); Kodaira, K.: Complex manifolds and deformations of complex structures. Grundlehren der mathematischen Wissenschaften, Vol.283. New York, Berlin, Heidelberg: Springer 1986
Grimm, R., Sohnius, M., Wess, J.: Nucl. Phys.B133, 275 (1978)
de Wit, B., Van Proeyen, A.: Potentials and symmetries of general gaugedN=2 Supergravity-Yang-Mills models. Nucl. Phys.B245, 89–117 (1984)
de Wit, B., Lauwers, P. D., Van Proeyen, A.: Nucl. Phys.B255, 569 (1985)
Cremmer, E., Kounnas, C., Van Proeyen, A., Derendinger, J.-P., Ferrara, S., de Wit, B., Girardello, L.: Vector multiplets coupled toN=2 Supergravity: Super-Higgs effect, flat potentials and geometrical structure. Nucl. Phys.B250, 385–426 (1985)
Grisaru, M. T., Van de Ven, A. E. M., Zanon, D.: Four-loop β-function for theN=1 andN=2 supersymmetric non-linear sigma-model in two dimensions. Phys. Lett.B173, 423–428 (1986); Two-dimensional supersymmetric sigma models on Ricci flat Kähler manifolds are not finite. Phys. Lett.B177, (1986); Gross, D., Witten, E.: Superstring modifications of Einstein's equations. Nucl. Phys.B277, 1 (1986); Witten, L., Witten, E.: Large radius expansion of superstring compactification. Nucl. Phys.B281, 109 (1987)
Dixon, L.: Proceeding of the 1987 Trieste Summer School in High Energy Physics; Distler, J., Greene, B.: Some exact results on the superpotential from Calabi-Yau compactifications. Nucl. Phys.B309, 295–316 (1988)
Dine, M., Seiberg, N.: Phys. Rev. Lett.55, 366 (1985); Seiberg, N.: String theory from a macroscopic point of view. In: Superstrings'87 (proceedings of the 1987 Trieste Spring School). Alvarez-Gaumé, L., Green, M. B., Grisaru, M. T., Jengo, R., Sezgin, E. (eds.). Singapore: World Scientific 1987
Dine, M., Seiberg, N., Wen, X. G., Witten, E.: Non-perturbative effects on the string world-sheet I, II. Nucl. Phys.B278, 769–789 (1986); Nucl. Phys. B289, 319–363 (1987)
Gepner, D.: Princeton preprint pupt-1093 (April 1988)
Carlson, J., Griffiths, P.: Infinitesimal variation of Hodge structure and the global Torelli problem. In: Journées de geometric algebrique d'Angers. (Sijthoff and Nordhoff, 1980) pp. 51–76, and reference therein
Cecotti, S., Ferrara, S., Girardello, L.: Geometry of type II supestrings and the moduli of superconformal field theories. J. Mod. Phys.A4, 2475–2529 (1989)
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Communicated by L. Alvarez-Gaumé
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Cecotti, S. N=2 supergravity, type IIB superstrings, and algebraic geometry. Commun.Math. Phys. 131, 517–536 (1990). https://doi.org/10.1007/BF02098274
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DOI: https://doi.org/10.1007/BF02098274