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Communications in Mathematical Physics

, Volume 365, Issue 1, pp 93–214 | Cite as

The Kähler Quotient Resolution of \({{\mathbb{C}}^3/ \Gamma}\) Singularities, the McKay Correspondence and \({D = 3\,\,\mathcal{N} = 2}\) Chern–Simons Gauge Theories

  • Ugo BruzzoEmail author
  • Anna Fino
  • Pietro Fré
Article

Abstract

We advocate that the generalized Kronheimer construction of the Kähler quotient crepant resolution \({\mathcal{M}_\zeta \longrightarrow \mathbb{C}^3/ \Gamma}\) of an orbifold singularity where \({\Gamma\subset \mathrm{SU(3)}}\) is a finite subgroup naturally defines the field content and the interaction structure of a superconformal Chern–Simons gauge theory. This latter is supposedly the dual of an M2-brane solution of D = 11 supergravity with \({\mathbb{C}\times\mathcal{M}_\zeta}\) as transverse space. We illustrate and discuss many aspects of this type of constructions emphasizing that the equation p\(\wedge\)p = 0which provides the Kähler analogue of the holomorphic sector in the hyperKähler moment map equations canonically defines the structure of a universal superpotential in the CS theory. Furthermore the kernel \({\mathcal{D}_\Gamma}\) of the above equation can be described as the orbit with respect to a quiver Lie group \({\mathcal{G}_\Gamma}\) of a special locus \({L_\Gamma \subset \mathrm{Hom}_\Gamma(\mathcal{Q}\otimes R,R)}\) that has also a universal definition. We provide an extensive discussion of the relation between the coset manifold \({\mathcal{G}_\Gamma/ \mathcal{F}_\Gamma}\), the gauge group \({\mathcal{F}_\Gamma}\) being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the so named tautological vector bundles that are in one-to-one correspondence with the nontrivial irreps of \({\Gamma}\). These first Chern classes are represented by (1,1)-forms on \({\mathcal{M}_\zeta}\) and provide a basis for the cohomology group \({H^2(\mathcal{M}_\zeta)}\). We also discuss the relation with conjugacy classes of \({\Gamma}\) and we provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of \({\mathbb{C}^2/ \Gamma}\) singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons–Hawking metrics are exhibited.

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Notes

Acknowledgements

We acknowledge important clarifying discussions with our long time collaborators and friends Pietro Antonio Grassi, Dimitri Markushevich, Aleksander Sorin and Mario Trigiante. U.B.’s research is partially supported by PRIN 2015 “Geometria delle varietà algebriche” and INdAM-GNSAGA. This work was completed while U.B. was visiting the Instituto deMatemática e Estatística of the University of São Paulo, Brazil, supported by the FAPESP grant 2017/22091-9. He likes to thank FAPESP for providing support and his hosts, in particular P. Piccione, for their hospitality.

References

  1. 1.
    Fré P., Grassi P.A.: The integral form of D = 3 Chern–Simons theories probing \({{\mathbb{C}}^n/ \Gamma}\) singularities. Fortsch. Phys. 65(10-11), 1700040 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Degeratu, A., Walpuski, T.: Rigid HYM connections on tautological bundles over ALE crepant resolutions in dimension three. In: SIGMA Symmetry Integrability Geometry: Methods and Applications, vol. 12, Paper No. 017, 23 (2016)Google Scholar
  3. 3.
    Fabbri D., Fré P., Gualtieri L., Termonia P.: \({\mathrm{Osp(N|4)}}\) supermultiplets as conformal superfields on \({\partial\mathrm{AdS}_4}\) and the generic form of \({\mathcal{N}=2, D=3}\) gauge theories. Class. Quantum Gravity 17(1), 55 (2000) [arxiv:hep-th/9905134]ADSzbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Billó M., Fabbri D., Fré P., Merlatti P., Zaffaroni A.: Rings of short \({\mathcal{N}=3}\) superfields in three dimensions and M-theory on \({\mathrm{AdS}_4\times \mathrm{N}^{010}}\). Class. Quantum Gravity 18(7), 1269 (2001).  https://doi.org/10.1088/0264-9381/18/7/310 [arxiv:hep-th/0005219]ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Fré P., Gualtieri L., Termonia P.: The Structure of N=3 multiplets in AdS(4) and the complete Osp(3|4) x SU(3) spectrum of M theory on AdS(4) x N0,1,0. Phys. Lett. B 471, 27–38 (1999)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fabbri D., Fré P., Gualtieri L., Termonia P.: M-theory on \({\mathrm{AdS}_4\times\mathrm{M}^{1,1,1}}\): the complete \({\mathrm{Osp(2|4)} \times\mathrm{SU(3)} \times\mathrm{SU(2)}}\) spectrum from harmonic analysis. Nucl. Phys. B 560(1-3), 617–682 (1999) [arxiv:hep-th/9903036]ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Fabbri D., Fré P., Gualtieri L., Reina C., Tomasiello A., Zaffaroni A., Zampa A.: 3D superconformal theories from Sasakian seven-manifolds: new non-trivial evidences for \({\mathrm{AdS}_4 / \mathrm{CFT}_3}\). Nucl. Phys. B 577(3), 547–608 (2000) [arxiv:hep-th/9907219]ADSzbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Billó M., Fabbri D., Fré P., Merlatti P., Zaffaroni A.: Shadow multiplets in \({\mathrm{AdS}_4/ \mathrm{CFT}_3}\) and the super-Higgs mechanism: hints of new shadow supergravities. Nucl. Phys. B. 591(1-2), 139–194 (2000) [arxiv:hep-th/0005220]ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Freund P.G.O., Rubin M.A.: Dynamics of dimensional reduction. Phys. Lett. B 97(2), 233–235 (1980)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Duff, M.J., Pope, C.N.: Kaluza-Klein supergravity and the seven sphere. In: Supersymmetry and supergravity, p. 183 (1983). 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
  11. 11.
    D’Auria R., Fré P.: Spontaneous generation of Osp(4|8) symmetry in the spontaneous compactification of D=11 supergravity. Phys. Lett. B 121(2-3), 141–146 (1983)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Awada M.A., Duff M.J., Pope C.N.: \({\mathcal{N}=8}\) supergravity breaks down to \({\mathcal{N}=1}\). Phys. Rev. Lett. 50(5), 294 (1983)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Biran B., Englert F., de Wit B., Nicolai H.: Gauged N=8 supergravity and its breaking from spontaneous compactification. Phys. Lett. B 124(1-2), 45–50 (1983)ADSCrossRefGoogle Scholar
  14. 14.
    Günaydin M., Warner N.P.: Unitary supermultiplets of Osp(8|4,R) and the spectrum of the S 7 compactification of 11-dimensional supergravity. Nucl. Phys. B 272(1), 99–124 (1986)ADSCrossRefGoogle Scholar
  15. 15.
    Witten E.: Search for a realistic Kaluza–Klein theory. Nucl. Phys. B 186(3), 412–428 (1981)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Castellani L., D’Auria R., Fré P.: \({\mathrm{SU(3)} \otimes \mathrm{SU(2)} \otimes \mathrm{U(1)}}\) from D=11 supergravity. Nucl. Phys. B 239(2), 610–652 (1984)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    D’Auria R., Fré P.: On the spectrum of the \({{\mathcal N}=2\, \mathrm{SU}(3)\otimes\mathrm{SU}(2)\otimes\mathrm{U}(1)}\) gauge theory from D=11 supergravity. Class. Quantum Gravity 1(5), 447 (1984)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Ceresole A., Fré P., Nicolai H.: Multiplet structure and spectra of \({{\mathcal N}=2}\) supersymmetric compactifications. Class. Quantum Gravity 2(2), 133 (1985)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Page D.N., Pope C.N.: Stability analysis of compactifications of D=11 supergravity with \({\mathrm{SU}(3)\times \mathrm{SU}(2)\times\mathrm{U}(1)}\) symmetry. Phys. Lett. B 145(5-6), 337–341 (1984)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    D’Auria R., Fré P., Van Nieuwenhuizen P.: \({{\mathcal N}=2}\) matter coupled supergravity from compactification on a coset G/H possessing an additional killing vector. Phys. Lett. B 136(5-6), 347–353 (1984)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Page D.N., Pope C.N.: Which compactifications of D=11 supergravity are stable?. Phys. Lett. B 144(5-6), 346–350 (1984)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    D’Auria R., Fré P.: Universal Bose–Fermi mass-relations in Kaluza–Klein supergravity and harmonic analysis on coset manifolds with Killing spinors. Ann. Phys. 162(2), 372–412 (1985)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Castellani L., D’Auria R., Fré P., Pilch K., van Nieuwenhuizen P.: The bosonic mass formula for Freund–Rubin solutions of D=11 supergravity on general coset manifolds. Class. Quantum Gravity 1(4), 339 (1984)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Freedman D.Z., Nicolai H.: Multiplet shortening in \({\mathrm{Osp}(N\vert 4)}\). Nucl. Phys. B 237(2), 342–366 (1984)ADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Castellani L., Romans L.J., Warner N.P.: A classification of compactifying solutions for D=11 supergravity. Nucl. Phys. B 241(2), 429–462 (1984)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Fré P.: Supersymmetric M2-branes with Englert fluxes, and the simple group PSL(2, 7). Fortsch. Phys. 64(6-7), 425–462 (2016)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Blichfeldt H.F.: On the order of linear homogeneous groups. IV. Trans. Am. Math. Soc. 12(1), 39–42 (1911)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Blichfeldt H.F.: Blichfeldt’s finite collineation groups. Bull. Am. Math. Soc. 24(10), 484–487 (1918)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Aharony, O., Bergman, O., Jafferis, D.L., Maldacena, J.: N=6 superconformal Chern–Simons-matter theories, M2-branes and their gravity duals. J. High Energy Phys. 2008(10):091 (2008).  https://doi.org/10.1088/1126-6708/2008/10/091 [arXiv:0806.1218 [hep-th]].
  30. 30.
    Kronheimer P.B.: The construction of ALE spaces as hyper-Kähler quotients. J. Differ. Geom. 29(3), 665–683 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Kronheimer P.B.: A Torelli-type theorem for gravitational instantons. J. Differ. Geom. 29(3), 685–697 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Markushevich D.: Resolution of \({C^3/H_{168}}\). Math. Ann. 308(2), 279–289 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Cerchiai, B., Fré, P., Trigiante, M.: Exceptional Field Theory, the Group PSL(2,7) and Englert Equation. Paper in Preparation (2017)Google Scholar
  34. 34.
    Ito, Y.: The McKay correspondence—a bridge from algebra to geometry. In: European Women in Mathematics (Malta, 2001), pp. 127–147. World Scientific Publishing, River Edge (2003)Google Scholar
  35. 35.
    Roan S.-S.: Minimal resolutions of Gorenstein orbifolds in dimension three. Topology 35(2), 489–508 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Ito, Y., Reid, M.: The McKay correspondence for finite subgroups of \({{\rm SL}(3, {\bf C})}\). In: Higher-Dimensional Complex Varieties (Trento, 1994), pp. 221–240. de Gruyter, Berlin (1996)Google Scholar
  37. 37.
    Craw, A.: The McKay Correspondence and Representations of the McKay Quiver. PhD thesis, Warwick University,United Kingdom, (2001)Google Scholar
  38. 38.
    Anselmi D., Billó M., Fré P., Girardello L., Zaffaroni A.: ALE manifolds and conformal field theories. Int. J. Mod. Phys. A 9, 3007–3058 (1994)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Bertolini M., Campos V.L., Ferretti G., Salomonson P., Fré P., Trigiante M.: BPS three-brane solution on smooth ALE manifolds with flux. Fortsch. Phys. 50, 802–807 (2002)ADSzbMATHCrossRefGoogle Scholar
  40. 40.
    Bertolini M., Campos V.L., Ferretti G., Fré P., Salomonson P., Trigiante M.: Supersymmetric three-branes on smooth ALE manifolds with flux. Nucl. Phys. B 617, 3–42 (2001)ADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Arnold V.I.: Normal forms for functions near degenerate critical points, the weyl groups \({a_k,d_k,e_k}\) and Lagrangian singularities. Funct. Anal. Its Appl. 6, 254–272 (1972)CrossRefGoogle Scholar
  42. 42.
    Hitchin N.J., Karlhede A., Lindström U., Roček M.: Hyperkahler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987)ADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Eguchi T., Hanson A.J.: Selfdual solutions to Euclidean gravity. Ann. Phys. 120, 82 (1979)ADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Gibbons G.W., Hawking S.W.: Gravitational multi-instantons. Phys. Lett. 78, 430 (1978)CrossRefGoogle Scholar
  45. 45.
    Gibbons G.W., Hawking S.W.: Classification of gravitational instanton symmetries. Commun. Math. Phys. 66, 291–310 (1979)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Fulton, W.: Introduction to Toric Varieties, vol. 131 of Annals of Mathematics Studies. Princeton University Press, Princeton (1993). The William H. Roever Lectures in Geometry.Google Scholar
  47. 47.
    Dolgachev, I.: Weighted projective varieties. In: Group Actions and Vector Fields (Vancouver, 1981), vol. 956 of Lecture Notes in Mathematics, pp. 34–71. Springer, Berlin (1982)Google Scholar
  48. 48.
    Beltrametti M., Robbiano L.: Introduction to the theory of weighted projective spaces. Expo. Math. 4(2), 111–162 (1986)zbMATHMathSciNetGoogle Scholar
  49. 49.
    Rossi M., Terracini L.: Linear algebra and toric data of weighted projective spaces. Rend. Semin. Mat. Univ. Politec. Torino 70(4), 469–495 (2012)zbMATHMathSciNetGoogle Scholar
  50. 50.
    Hartshorne, R.: Algebraic geometry. Springer, New York (1977). Graduate Texts in Mathematics, No. 52Google Scholar
  51. 51.
    Ross J., Thomas R.: Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics. J. Differ. Geom. 88(1), 109–159 (2011)zbMATHCrossRefGoogle Scholar
  52. 52.
    Craw A.: An explicit construction of the McKay correspondence for A-Hilb \({\mathbb{C}^3}\). J. Algebra 285, 682–705 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Craw A., Ishii A.: Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient. Duke Math. J. 124, 259–307 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Bianchi M., Morales J.F.: Anomalies & tadpoles. JHEP 03, 030 (2000)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Bianchi M., Fucito F., Morales J.F.: Dynamical supersymmetry breaking from unoriented D-brane instantons. JHEP 08, 040 (2009)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    Bianchi M., Fucito F., Morales J.F.: D-brane instantons on the T**6/Z(3) orientifold. JHEP 07, 038 (2007)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    Bianchi, M., Bruzzo, U., Fré, P., Grassi, P.A.: Work in Progress. (to appear) (2018)Google Scholar
  58. 58.
    Chimento, S., Ortin, T.: On 2-dimensional Kaehler metrics with one holomorphic isometry. (2016) arXiv:1610.02078 [hep-th]

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.SISSA, Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly
  2. 2.INFN – Sezione di TriesteTriesteItaly
  3. 3.Dipartimento di Matematica G. PeanoUniversitá di TorinoTurinItaly
  4. 4.Dipartimento di Fisica, INFN – Sezione di TorinoUniversitá di TorinoTurinItaly
  5. 5.National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)MoscowRussia
  6. 6.Arnold-Regge CenterTurinItaly

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