Communications in Mathematical Physics

, Volume 336, Issue 3, pp 1285–1328 | Cite as

N =2 Superconformal Nets

  • Sebastiano Carpi
  • Robin Hillier
  • Yasuyuki Kawahigashi
  • Roberto LongoEmail author
  • Feng Xu


We provide an Operator Algebraic approach to N = 2 chiral Conformal Field Theory and set up the Noncommutative Geometric framework. Compared to the N = 1 case, the structure here is much richer. There are naturally associated nets of spectral triples and the JLO cocycles separate the Ramond sectors. We construct the N = 2 superconformal nets of von Neumann algebras in general, classify them in the discrete series c < 3, and we define and study an operator algebraic version of the N = 2 spectral flow. We prove the coset identification for the N = 2 super-Virasoro nets with c < 3, a key result whose equivalent in the vertex algebra context has seemingly not been completely proved so far. Finally, the chiral ring is discussed in terms of net representations.


Central Charge Superconformal Algebra Irreducible Unitary Representation Chiral Ring Spectral Triple 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Blumenhagen, R., Plauschinn, E.: Introduction to conformal field theory. Lect. Notes Phys. 779 (2009)Google Scholar
  2. 2.
    Böckenhauer J.: Localized endomorphisms of the chiral Ising model. Commun. Math. Phys. 177, 265–304 (1996)CrossRefADSzbMATHGoogle Scholar
  3. 3.
    Böckenhauer J.: An algebraic formulation of level one Wess–Zumino–Witten models. Rev. Math. Phys. 8, 925–947 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Böckenhauer J., Evans D.E.: Modular invariants graphs and α-induction for nets of subfactors I. Commun. Math. Phys. 197, 361–386 (1998)CrossRefADSzbMATHGoogle Scholar
  5. 5.
    Boucher W., Friedan D., Kent A.: Determinant formulae and unitarity for the N = 2 superconformal algebras in two dimensions or exact results on string compactification. Phys. Lett. B 172, 316–322 (1986)CrossRefADSzbMATHMathSciNetGoogle Scholar
  6. 6.
    Buchholz D., Mack G., Todorov I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B Proc. Suppl. 5, 20–56 (1988)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Buchholz D., Schulz-Mirbach H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2, 105–125 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states and conformal QFT. II . Commun. Math. Phys. 315, 771–802 (2012)CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    Carpi S.: On the representation theory of Virasoro nets. Commun. Math. Phys. 244, 261–284 (2004)CrossRefADSzbMATHMathSciNetGoogle Scholar
  10. 10.
    Carpi S., Conti R., Hillier R.: Conformal nets and KK-theory. Ann. Funct. Anal. 4(1), 11–17 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Carpi S., Conti R., Hillier R., Weiner M.: Representations of conformal nets, universal C*-algebras, and K-theory. Commun. Math. Phys. 320(1), 275–300 (2013)CrossRefADSzbMATHMathSciNetGoogle Scholar
  12. 12.
    Carpi S., Hillier R., Kawahigashi Y., Longo R.: Spectral triples and the super-Virasoro algebra. Commun. Math. Phys. 295, 71–97 (2010)CrossRefADSzbMATHMathSciNetGoogle Scholar
  13. 13.
    Carpi, S., Hillier, R., Longo, R.: Superconformal nets and noncommutative geometry. arXiv:1304.4062 [math.OA] (to appear in J. Noncommutative Geom.)
  14. 14.
    Carpi S., Kawahigashi Y., Longo R.: Structure and classification of superconformal nets. Ann. Henri Poincaré 9, 1069–1121 (2008)CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. 15.
    Carpi S., Weiner M.: On the uniqueness of diffeomorphism symmetry in conformal field theory. Commun. Math. Phys. 258, 203–221 (2005)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    Carpi, S., Weiner, M.: Local energy bounds and representations of conformal nets (in preparation)Google Scholar
  17. 17.
    Connes A.: Noncommutative Geometry. Academic Press, London (1994)zbMATHGoogle Scholar
  18. 18.
    Di Vecchia P., Petersen J.L., Yu M.J., Zheng H.: Explicit construction of unitary representations of the N = 2 superconformal algebra. Phys. Lett. B 174, 280–284 (1986)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Dobrev V.: Characters of the unitarizable highest weight modules over the N = 2 superconformal algebra. Phys. Lett. B 186, 43–51 (1987)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Dobrev, V.: Characters of the unitarizable highest weight modules over the N = 2 superconformal algebra. arXiv:0708.1719v1 [hep-th]
  21. 21.
    Dörrzapf M.: Analytic expressions for singular vectors of the N = 2 siperconformal algebra. Commun. Math. Phys. 180, 195–231 (1996)CrossRefADSzbMATHGoogle Scholar
  22. 22.
    Dörrzapf M.: The embedding structure of unitary N = 2 minimal models. Nucl. Phys. B 529, 639–655 (1998)CrossRefADSzbMATHGoogle Scholar
  23. 23.
    Driessler W., Fröhlich J.: The reconstruction of local observable algebras from the Euclidean Green’s functions of relativistic quantum field theory. Ann. Henri Poincaré 27, 221–236 (1977)ADSzbMATHGoogle Scholar
  24. 24.
    Eholzer W., Gaberdiel M.R.: Unitarity of rational N = 2 superconformal theories. Commun. Math. Phys. 186, 61–85 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  25. 25.
    Feigin B.L., Semikhatov A.M., Sirota V.A., Tipunin I.Yu.: Resolutions and characters of irreducible representations of the N = 2 superconformal algebra. Nucl. Phys. B 536, 617–656 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  26. 26.
    Feigin B.L., Semikhatov A.M., Tipunin I.Yu.: Equivalence between chain categories of representations of affine sl(2) and N = 2 superconformal algebras. Nucl. Phys. B 536, 617–656 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  27. 27.
    Fewster C.J., Hollands S.: Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17, 577–612 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Fredenhagen, K.: Generalization of the theory of superselection sectors. In: Kastler, D. The Algebraic Theory of Superselection Sectors. Introduction and Recent Results (Proceedings, Palermo, 1990), World Scientific, Singapore (1990)Google Scholar
  29. 29.
    Fredenhagen, K., Rehren, K.H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras II. Geometric aspects and conformal covariance. Rev. Math. Phys. (special issue) 113–157 (1992)Google Scholar
  30. 30.
    Fröhlich J., Gabbiani F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993)CrossRefADSzbMATHGoogle Scholar
  31. 31.
    Fröhlich, J., Gawedzki, K.: Conformal field theory and geometry of strings. In: Mathematical Quantum Theory. I. Field Theory and Many-Body Theory (Vancouver, BC, 1993), pp. 57–97. CRM Proceedings and Lecture Notes, vol. 7. American Mathematical Society (1994)Google Scholar
  32. 32.
    Gannon T.: U(1)m modular invariants, N = 2 minimal models, and the quantum Hall effect. Nucl. Phys. B 491, 659–688 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  33. 33.
    Gato-Rivera, B.: A note concerning subsingular vectors and embedding diagrams of the N = 2 superconformal algebras. arXiv:hep-th/9910121v1
  34. 34.
    Gato-Rivera, B.: No isomorphism between the affine ŝl(2) algebra and the N = 2 superconformal algebras. arXiv:0809.2549v1
  35. 35.
    Getzler E., Szenes A.: On the Chern character of a theta-summable Fredholm module. J. Funct. Anal. 84, 343–357 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Goodman R., Wallach N.R.: Projective unitary positive-energy representations of Diff(S 1). J. Funct. Anal. 63, 299–321 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Gray O.: On the complete classification of unitary N = 2 minimal superconformal models. Commun. Math. Phys. 312, 611–654 (2012)CrossRefADSzbMATHGoogle Scholar
  38. 38.
    Guido D., Longo R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148, 521–551 (1992)CrossRefADSzbMATHMathSciNetGoogle Scholar
  39. 39.
    Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)CrossRefADSzbMATHMathSciNetGoogle Scholar
  40. 40.
    Iohara K.: Unitarizable highest weight modules of the N = 2 super-Virasoro algebras: untwisted sectors. Lett. Math. Phys. 91, 289–305 (2010)CrossRefADSzbMATHMathSciNetGoogle Scholar
  41. 41.
    Jaffe A., Lesniewski A., Osterwalder K.: Quantum K-theory. Commun. Math. Phys. 118, 1–14 (1988)CrossRefADSzbMATHMathSciNetGoogle Scholar
  42. 42.
    Kawahigashi Y., Longo R.: Classification of local conformal nets. Case c < 1. Ann. Math 160, 493–522 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Kawahigashi Y., Longo R.: Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories. Commun. Math. Phys. 244, 63–97 (2004)CrossRefADSzbMATHMathSciNetGoogle Scholar
  44. 44.
    Kawahigashi Y., Longo R.: Noncommutative spectral invariants and black hole entropy. Commun. Math. Phys. 257, 193–225 (2004)CrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)CrossRefADSzbMATHGoogle Scholar
  46. 46.
    Kent A., Riggs H.: Determinant formulae for the N = 4 superconformal algebras. Phys. Lett. B 198, 491–496 (1987)CrossRefADSMathSciNetGoogle Scholar
  47. 47.
    Kiritsis E.: Character formulae and the structure of the N = 1, N = 2 superconformal algebras. Int. J. Mod. Phys. A 3, 1871–1906 (1988)CrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Lerche W., Vafa C., Warner N.P.: Chiral rings in N = 2 superconformal theories. Nucl. Phys. B 324, 427–474 (1989)CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Longo R.: Notes on a quantum index theorem. Commun. Math. Phys. 222, 45–96 (2001)CrossRefADSzbMATHMathSciNetGoogle Scholar
  50. 50.
    Longo R., Rehren K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Matsuo Y.: Character formula for c < 1 unitary representation of N = 2 superconformal algebra. Progr. Theor. Phys. 77, 793–797 (1987)CrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Schwimmer N., Seiberg A.: Comments on the N = 2,3,4 superconformal algebras in two dimensions. Phys. Lett. B 184, 183 (1987)CrossRefADSMathSciNetGoogle Scholar
  53. 53.
    Toledano Laredo V.: Integrating unitary representations of infinite-dimensional Lie groups. J. Funct. Anal. 161, 478–508 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Wassermann A.: Operator algebras and conformal field theory III. Fusion of positive energy representations of LSU (N) using bounded operators. Invent. Math. 133, 467–538 (1998)CrossRefADSzbMATHMathSciNetGoogle Scholar
  55. 55.
    Weiner M.: Conformal covariance and positivity of energy in charged sectors. Commun. Math. Phys. 265, 493–506 (2006)CrossRefADSzbMATHMathSciNetGoogle Scholar
  56. 56.
    Xu F.: On the equivalence of certain coset conformal field theories. Commun. Math. Phys. 228, 257–279 (2002)CrossRefADSzbMATHGoogle Scholar
  57. 57.
    Xu F.: 3-Manifold invariants from cosets. J. Knot Theory Ramif. 14, 21–90 (2005)CrossRefzbMATHGoogle Scholar
  58. 58.
    Xu F.: Strong additivity and conformal nets. Pac. J. Math. 221, 167–199 (2005)CrossRefzbMATHGoogle Scholar
  59. 59.
    Xu F.: Mirror extensions of local nets. Commun. Math. Phys. 270, 835–847 (2007)CrossRefADSzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Sebastiano Carpi
    • 1
  • Robin Hillier
    • 2
  • Yasuyuki Kawahigashi
    • 3
    • 4
  • Roberto Longo
    • 5
    Email author
  • Feng Xu
    • 6
  1. 1.Dipartimento di EconomiaUniversità di Chieti-Pescara “G. d’Annunzio”PescaraItaly
  2. 2.Department of Mathematics and StatisticsLancaster UniversityLancasterUK
  3. 3.Department of Mathematical SciencesThe University of TokyoTokyoJapan
  4. 4.Kavli IPMU (WPI)The University of TokyoKashiwaJapan
  5. 5.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  6. 6.Department of MathematicsUniversity of California at RiversideRiversideUSA

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