(Hyper)Kähler Quotients, ALE-Manifolds and \(\mathbb {C}^n/\varGamma \) Singularities

  • Pietro Giuseppe FréEmail author
Part of the Theoretical and Mathematical Physics book series (TMP)


In this last chapter we turn to the analysis of important developments in complex geometry which took place in the 1980–1990s, directly motivated by supersymmetry and supergravity and completely inconceivable outside such a framework. Notwithstanding their roots in the theoretical physics of the superworld, such developments constitute, by now, the basis of some of the most innovative and alive research directions of contemporary geometry.


  1. 1.
    N.J. Hitchin, A. Karlhede, U. Lindström, M. Roček, Hyperkahler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    P.B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients. J. Differ. Geom. 29, 665–683 (1989)CrossRefzbMATHGoogle Scholar
  3. 3.
    P.B. Kronheimer, A Torelli-type theorem for gravitational instantons. J. Differ. l Geom. 29, 685–697 (1989)Google Scholar
  4. 4.
    J. Maldacena, The large-N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 4, 1113–1133 (1999)., arXiv:hep-th/9711200
  5. 5.
    S. Ferrara, C. Fronsdal, Conformal maxwell theory as a singleton field theory on AdS\(_5\), IIB 3-branes and duality. Class. Quantum Gravity 15, 2153 (1998)., arXiv:hep-th/9712239
  6. 6.
    S. Ferrara, C. Fronsdal, A. Zaffaroni, On \({\cal{N}}=8\) supergravity in AdS\(_5\) and \({\cal{N}} =4\) superconformal Yang-Mills theory. Nucl. Phys. B 532, 153–162 (1998)., arXiv:hep-th/9802203
  7. 7.
    R. Kallosh, A. Van Proeyen, Conformal symmetry of supergravities in AdS spaces. Phys. Rev. D 60, 026001 (1999)., arXiv:hep-th/9804099
  8. 8.
    S. Ferrara, C. Fronsdal, Gauge fields as composite boundary excitations. Phys. Lett. B 433, 19–28 (1998)., arXiv:hep-th/9802126
  9. 9.
    P.G.O. Freund, M.A. Rubin, Dynamics of dimensional reduction. Phys. Lett. B 97, 233–235 (1980)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Fabbri, P. Fré, L. Gualtieri, P. Termonia, M-theory on \({\text{AdS}}_{4}\times {\text{ M }}^{1,1,1}\) : the complete \({\text{ Osp }}(2|4) \times {\text{ SU }}(3) \times {\text{ SU }}(2)\) spectrum from harmonic analysis. Nucl. Phys. B 560, 617–682 (1999). arXiv:hep-th/9903036
  11. 11.
    A. Ceresole, G. Dall’Agata, R. D’Auria, S. Ferrara, Spectrum of type IIB supergravity on \(AdS_5\times T^{11}\): predictions on \(\fancyscript {N}=1\) SCFT’s. Phys. Rev. D 61, 066001 (2000). arXiv:hep-th/9905226
  12. 12.
    D. Fabbri, P. Fré, L. Gualtieri, P. Termonia, \({\text{ Osp }}(N|4)\) supermultiplets as conformal superfields on \(\partial {\text{ AdS }}_{4}\) and the generic form of \({N}=2\), D\(=\)3 gauge theories. Class. Quantum Gravity 17, 55 (2000). arXiv:hep-th/9905134
  13. 13.
    P. Fré, L. Gualtieri, P. Termonia, The structure of \({{\fancyscript {N}}}=3\) multiplets in \({\text{ AdS }}_{4}\) and the complete \({\text{ Osp }}(3|4)\times {\text{ SU }}(3)\) spectrum of M-theory on \({\text{ AdS }}_4\times {\text{ N }}^{0,1,0}\). Phys. Lett. B 471, 27–38 (1999). arXiv:hep-th/9909188
  14. 14.
    D. Fabbri, P. Fré, L. Gualtieri, C. Reina, A. Tomasiello, A. Zaffaroni, A. Zampa, 3D superconformal theories from Sasakian seven-manifolds: new non-trivial evidences for \({\text{ AdS }}_{4} /{\text{ CFT }} _{3}\). Nucl. Phys. B 577, 547–608 (2000). arXiv:hep-th/9907219
  15. 15.
    M. Billó, D. Fabbri, P. Fré, P. Merlatti, A. Zaffaroni, Rings of short \({\fancyscript {N}} =3\) superfields in three dimensions and M-theory on \({\text{ AdS }}_{4}\times {\text{ N }}^{010}\). Class. Quantum Gravity 18, 1269 (2001)., arXiv:hep-th/0005219
  16. 16.
    M. Billó, D. Fabbri, P. Fré, P. Merlatti, A. Zaffaroni, Shadow multiplets in \({\text{ AdS }}_{4}/{\text{ CFT }}_{3}\) and the super-Higgs mechanism: hints of new shadow supergravities. Nucl. Phys. B 591, 139–194 (2000). arXiv:hep-th/0005220
  17. 17.
    M.J. Duff, C.N. Pope, Kaluza-Klein supergravity and the seven sphere. Supersymmetry and supergravity (1983), p. 183; ICTP/82/83-7, Lectures given at September school on supergravity and supersymmetry, Trieste, Italy, Sep 6–18, 1982. Published in Trieste Workshop 1982:0183 (QC178:T7:1982)Google Scholar
  18. 18.
    R. D’Auria, P. Fré, Spontaneous generation of Osp(4|8) symmetry in the spontaneous compactification of D \(=\) 11 supergravity. Phys. Lett. B 121, 141–146 (1983)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    M.A. Awada, M.J. Duff, C.N. Pope, \(\fancyscript {N}=8\) supergravity breaks down to \(\fancyscript {N}=1\). Phys. Rev. Lett. 50, 294 (1983)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    B. Biran, F. Englert, B. de Wit, H. Nicolai, Gauged N \(=\) 8 supergravity and its breaking from spontaneous compactification. Phys. Lett. B 124, 45–50 (1983)ADSCrossRefGoogle Scholar
  21. 21.
    M. Günaydin, N.P. Warner, Unitary supermultiplets of Osp(8|4, R) and the spectrum of the \(S^7\) compactification of 11-dimensional supergravity. Nucl. Phys. B 272, 99–124 (1986)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    E. Witten, Search for a realistic Kaluza-Klein theory. Nucl. Phys. B 186, 412–428 (1981)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    L. Castellani, R. D’Auria, P. Fré, \({\text{ SU }}(3)\otimes {\text{ SU }}(2)\otimes {\text{ U }}(1)\) from D \(=\) 11 supergravity. Nucl. Phys. B 239, 610–652 (1984)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    R. D’Auria, P. Fré, On the spectrum of the \({\fancyscript {N}}=2\) \({\text{ SU }}(3)\otimes {\text{ SU }}(2)\otimes {\text{ U }}(1)\) gauge theory from D \(=\) 11 supergravity. Class. Quantum Gravity 1, 447 (1984)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Ceresole, P. Fré, H. Nicolai, Multiplet structure and spectra of \({\fancyscript {N}}=2\) supersymmetric compactifications. Class. Quantum Gravity 2, 133 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D.N. Page, C.N. Pope, Stability analysis of compactifications of D \(=\) 11 supergravity with \({\text{ SU }}(3)\times {\text{ SU }}(2)\times {\text{ U }}(1)\) symmetry. Phys. Lett. B 145, 337–341 (1984)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    R. D’Auria, P. Fré, P. Van Nieuwenhuizen, \({\fancyscript {N}}=2\) matter coupled supergravity from compactification on a coset G/H possessing an additional killing vector. Phys. Lett. B 136, 347–353 (1984)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    D.N. Page, C.N. Pope, Which compactifications of D \(=\) 11 supergravity are stable? Phys. Lett. B 144, 346–350 (1984)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    R. D’Auria, P. Fré, Universal Bose-Fermi mass-relations in Kaluza-Klein supergravity and harmonic analysis on coset manifolds with Killing spinors. Ann. Phys. 162, 372–412 (1985)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    L. Castellani, R. D’Auria, P. Fré, K. Pilch, P. van Nieuwenhuizen, The bosonic mass formula for Freund-Rubin solutions of D \(=\) 11 supergravity on general coset manifolds. Class. Quantum Gravity 1, 339 (1984)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    D.Z. Freedman, H. Nicolai, Multiplet shortening in \({\text{ Osp }}(N\vert 4)\). Nucl. Phys. B 237, 342–366 (1984)ADSCrossRefzbMATHGoogle Scholar
  32. 32.
    L. Castellani, L.J. Romans, N.P. Warner, A classification of compactifying solutions for D \(=\) 11 supergravity. Nucl. Phys. B 241, 429–462 (1984)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    P.G. Fré, A. Fedotov, Groups and Manifolds: Lectures for Physicists with Examples in Mathematica (De Gruyter, Berlin, 2018)zbMATHGoogle Scholar
  34. 34.
    P. Fré, P.A. Grassi, Constrained supermanifolds for AdS M-theory backgrounds. JHEP 01, 036 (2008)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    D. Gaiotto, X. Yin, Notes on superconformal Chern-Simons-matter theories. J. High Energy Phys. 2007, 056 (2007)., arXiv:0704.3740 [hep-th]
  36. 36.
    O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, N \(=\) 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals. J. High Energy Phys. 2008, 091 (2008)., arXiv:0806.1218 [hep-th]
  37. 37.
    J. Bagger, N. Lambert, Comments on multiple M2-branes. J. High Energy Phys. 2008, 105 (2008)., arXiv:0712.3738 [hep-th]
  38. 38.
    P. Fré, P.A. Grassi, The integral form of D \(=\) 3 Chern-Simons theories probing \({\mathbb{C}}^n/\varGamma \) singularities (2017). arXiV:hep-th1705.00752
  39. 39.
    U. Bruzzo, A. Fino, P. Fr, The Kähler quotient resolution of \({\cal{C}}^3/\varGamma \) singularities, the McKay correspondence and D \(=\) 3 \({\cal{N}} =2\) Chern-Simons gauge theories (2017)Google Scholar
  40. 40.
    D. Anselmi, M. Billó, P. Fré, L. Girardello, A. Zaffaroni, ALE manifolds and conformal field theories. Int. J. Mod. Phys. A 9, 3007–3058 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    H.F. Blichfeldt, On the order of linear homogeneous groups. IV. Trans. Am. Math. Soc. 12, 39–42 (1911)MathSciNetzbMATHGoogle Scholar
  42. 42.
    H.F. Blichfeldt, Blichfeldt’s finite collineation groups. Bull. Am. Math. Soc. 24, 484–487 (1918)MathSciNetCrossRefGoogle Scholar
  43. 43.
    G.A. Miller, H.F. Blichfeldt, L.E. Dickson, Theory and Applications of Finite Groups (Dover Publications Inc., New York, 1961)zbMATHGoogle Scholar
  44. 44.
    S.-S. Roan, Minimal resolutions of Gorenstein orbifolds in dimension three. Topology 35, 489–508 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    D. Markushevich, Resolution of \({ C}^3/H_{168}\). Math. Ann. 308, 279–289 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Y. Ito, M. Reid, The McKay correspondence for finite subgroups of \({\text{ SL }}(3,{\mathbf{C)}}\), in Higher-dimensional Complex Varieties (Trento, (1994) (de Gruyter, Berlin, 1996), pp. 221–240Google Scholar
  47. 47.
    G.W. Gibbons, S.W. Hawking, Classification of gravitational instanton symmetries. Commun. Math. Phys. 66, 291–310 (1979)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    N.J. Hitchin, Polygons and gravitons. Math. Proc. Camb. Philos. Soc. 85, 465–476 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    E. Witten, Phases of N \(=\) 2 theories in two-dimensions. Nucl. Phys. B403, 159–222 (1993) [AMS/IP Stud. Adv. Math. 1, 143 (1996)]Google Scholar
  50. 50.
    E. Calabi, Metriques kähleriennes et fibres holomorphes. Ann. Scie. Ec. Norm. Sup. 12, 269 (1979)CrossRefzbMATHGoogle Scholar
  51. 51.
    A. Craw, The McKay correspondence and representations of the McKay quiver. Ph.D. thesis, Warwick University, United Kingdom, 2001Google Scholar
  52. 52.
    T. Eguchi, A.J. Hanson, Selfdual Solutions to Euclidean Gravity. Ann. Phys. 120, 82 (1979)ADSCrossRefzbMATHGoogle Scholar
  53. 53.
    J. McKay, Graphs, singularities and finite groups. Proc. Symp. Pure Math. 37, 183 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    S. Chimento, T. Ortin, On 2-dimensional Kaehler metrics with one holomorphic isometry (2016). arXiv:hep-th1610.02078
  55. 55.
    A. Degeratu, T. Walpuski, Rigid HYM connections on tautological bundles over ALE crepant resolutions in dimension three. SIGMA Symmetry Integr. Geom. Methods Appl. 12, Paper No. 017, 23 (2016)Google Scholar
  56. 56.
    W. Fulton, Introduction to Toric Varieties. Volume 131 of Annals of Mathematics Studies (Princeton University Press, Princeton, 1993) (The William H. Roever Lectures in Geometry)Google Scholar
  57. 57.
    V. Arnold, S. Gusein-Zade, A. Varchenko, Singularities of Differentiable Maps (Birkhäuser, 1975)Google Scholar
  58. 58.
    E. Brieskorn, Unknown, Actes Congrés Intern. Math. Ann. t, 2 (1970)Google Scholar
  59. 59.
    P. Slodowy, Simple Singularities and Simple Algebraic Groups. Lecture Notes in Mathematics, vol. 815 (1980)Google Scholar
  60. 60.
    P.G. Fré, A Conceptual History of Symmetry from Plato to the Superworld (Springer, Berlin, 2018)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Pietra MarazziItaly

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