Abstract
Motivated by the problem of the moduli space of superconformal theories, we classify all the (normal) homogeneous Kähler spaces which are allowed in the coupling of vector multiplets toN=2 SUGRA. Such homogeneous spaces are in one-to-one correspondence with the homogeneous quaternionic spaces (≠ℍH n) found by Alekseevskii. There are two infinite families of homogeneous non-symmetric spaces, each labelled by two integers. We construct explicitly the corresponding supergravity models. They are described by acubic functionF, as in flat-potential models. They are Kähler-Einstein if and only if they are symmetric. We describe in detail the geometry of the relevant manifolds. They are Siegel (bounded) domains of the first type. We discuss the physical relevance of this class of bounded domains for string theory and the moduli geometry. Finally, we introduce theT-algebraic formalism of Vinberg to describe in an efficient way the geometry of these manifolds. The homogeneous spaces allowed inN=2 SUGRA are associated to rank 3T-algebras in exactly the same way as the symmetric spaces are related to Jordan algebras. We characterize theT-algebras allowed inN=2 supergravity. They are those for which theungraded determinant is a polynomial in the matrix entries. The Kähler potential is simply minus the logarithm of this “naive” determinant.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Sieberg, N.: Observations on the moduli space of superconformal field theories. Nucl. Phys.303B, 286–304 (1988)
Gepner, D.: Exactly solvable string compactifications on manifolds ofSU(N) holonomy. Phys. Lett.199B, 380–388 (1987)
Cecotti, S., Ferrara, S., Girardello, L.: Geometry of type II superstrings and the moduli of superconformal field theories. J. Mod. Phys.A4, 2475 (1989)
Cecotti, S., Ferrara, S., Girardello, L.: A topological formula for the Kähler, potential of 4D N=1, 2 strings, and its implications for the moduli problem. Phys. Lett.213B, 443 (1988)
de Roo, M.: Matter coupling inN=4 supergravity, Nucl. Phys.B255, 515–553 (1985); de Roo, M.: GaugedN=4 matter couplings. Phys. Lett156B, 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)
Todorov, A. N.: Applications of the Kähler-Einstein Calabi-Yau metrics to moduli ofK3 surfaces. Invent. Math.61, 251–265 (1980); Todorov, A. N.: How many, Kähler metrics has aK3 surface, In Arithmetic and geometry. Papers dedicated to I. R. Shafarevitch, vol. 2 p. 451–464. Boton Basel Stuttgart: Birkhäuser (1983); 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. 107–112. Lectures Notes in Mathematics vol.894, Looijenga, E., Sierma, D., Takens, F. (eds.) Berlin, Heildelberg, New York: Springer 1981
Cremmer, E., Van Proeyen, A.: Classification of Kähler manifolds inN=2 vector multiplet-supergravity couplings. Class. Quant. Gravity2, 445–454 (1985)
Bagger, J., Witten, E.: Matter couplings inN=2 supergravity. Nucl. Phys.B222, 1–10 (1983)
de Wit, B., Van Proeyen, A.: Potentials and symmetries of general gaugedN=2 supergravity-Yang-Mills models. Nucl. Phys.B245, 89–117 (1984)
Alekseevskii, D. V.: Riemannian manifolds with exceptional holonomy groups. Funk. Anal. Pril.2, 1–10 (1968); Funct. Anal. Appl.2, 97–105 (1968)
Alvarez-Gaumé, L., Freedman, D.: Geometrical structure and ultraviolet finiteness in the supersymmetricσ-model. Commun. Math. Phys.80, 443–451 (1981)
Gunaydin, M., Sierra, G., Townsend, P. K.: Exceptional supergravity theories and the magic square. Phys. Lett.133B, 72–76 (1983); Gunaydin, M., Sierra, G., Townsend, P. K.: The geometry ofN=2 Maxwell-Einstein supergravity and Jordan algebras. Nucl. Phys.B242, 244–268 (1983); Gunaydin, M., Sierra, G., Townsend, P. K.: Vanishing potentials in gaugedN=2 supergravity: An application of Jordan algebras. Phys. Lett.144B, 41–45 (1983)
Köcher, M.: Positivitätsbereiche imR n. Am. J. Math.79, 575–596 (1957); Köcher, M.: Analysis in reellen Jordan Algebren. Nachr. Akad. Wiss. Göttingen Math. -Phys. KI. IIa 67–74 (1958); Köcher, M.: Die Geodätischen von Positivitätsbereichen. Math. Ann.135, 192–202 (1958)
Vinberg, E. B.: The theory of convex homogeneous cones. In: Transactions of the Moscow Mathematical Society for the year 1963. pp. 340–403. Providence, RI: American Mathematical Society 1965.
Alekseevskii, D. V.: Classification of quaternionic spaces with a transitive solvable group of motions. Izv. Akad. Nauk SSSR Ser. Mat.9, 315–362 (1975); Math. USSR Izvesstija,9, 297–339 (1975)
Wen, X. G., Witten, E.: World-sheet instantons and the Peccei-Quinn symmetry. Phys. Lett.166B, 397–401 (1986)
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)
Freundenthal, H.: Proc. Koninkl. Ned. Akad. WetenshapA62, 447 (1959); Rozenfeld, B. A.: Dokl. Akad. Nauk SSSR106, 600 (1956); Tits, J.: Mem. Acad. Roy, Belg. Sci.29 (1955)
Pjateckii-Sapiro, I. I.: The structure ofj-algebras. Izv. Akad. Nauk SSSR Ser. Mat.26, 453–484 (1962); Am. Math. Soc. Transl.55, 207–241 (1966)
Gindikin, S. G., Pjateckii-Sapiro, I. I., Vinberg, E. B.: Homogeneous Kähler manifolds. In: Geometry of Homogeneous Bounded Domains. pp. 3–87, (Centro Internazionale Matematico Estivo, C.I.M.E., 3° Ciclo, Urbino 1967) Vesentini, E. (ed.). Rome: Edizioni Cremonese 1968
Gindikin, S. G., Pjateckii-Sapiro, I. I., Vinberg, E. B.: Classification and canonical realization of complex bounded homogeneous domains, In: Transactions of the Moscow Mathematical Society for the year 1963. pp. 404–437, Providence, RI: American Mathematical Society 1965
Borel, A.: Kählerian coset spaces of semisimple Lie groups. Proc. Nat. Acad. Sci. USA40, 1147–1151 (1954); See also: Koszul, J. L.: Sur la forme hermitienne canonique des espaces homogenes complexes. Canad. J. Math.7, 562–576 (1955)
Helgason, S.: Differential geometry and symmetric spaces, (Proposition 3.6). New-York, London: Academic Press 1962, second ed. (1978)
Gaillard, M. K., Zumino, B.: Duality rotations for interacting fields. Nucl. Phys.B193, 221–244 (1981)
Smirnov, V. I.: Course of higher mathematics. Vol. IV, Sect. 117, Oxford: Pergamon Press 1964
Atiyah, M. F., Bott, R., Shapiro, A.: Clifford modules. Topology3 [Suppl.] 1, 3–38 (1964)
Hano, J.: On Kählerian homogeneous spaces of unimodular Lie group. Am. J. Math.79, 885–900 (1957)
Calabi, E.: On Kähler manifolds with vanishing canonical class. In: Algebraic geometry and topology. A symposium in honor of S. Lefschets, pp. 78–89. Princeton University Press (1955)
Yau, S.-T.: On Calabi's conjecture and some new results in algebraic geometry. Proc. Natl. Acad. 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 (1878)
Hirzebruch, F.: Topological methods in algebraic geometry. Third ed. Berlin, Heidelberg, New York: Springer 1966
Griffiths, P., Harris, J.: Algebraic geometry. New York: Wiley 1977
Witten, E., Bagger, J.: Quantization of Newton's constant in certain supergravity theories. Phys. Lett.115, 202–206 (1982)
Author information
Authors and Affiliations
Additional information
Communicated by L. Alvarez Gaumé
Rights and permissions
About this article
Cite this article
Cecotti, S. Homogeneous Kähler manifolds andT-algebras inN=2 supergravity and superstrings. Commun.Math. Phys. 124, 23–55 (1989). https://doi.org/10.1007/BF01218467
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01218467