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Conformalities

  • Ilarion V. Melnikov
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 951)

Abstract

In this chapter we give an overview of two-dimensional conformal field theories and properties of the N=2 superconformal algebra and discuss its representations. We also discuss additional global symmetries and constraints from unitarity and compactness. These are probably familiar to many readers, but we introduce them here as a reminder and for later reference; the emphasis is on the results and perspective most relevant for (0,2) exploration. The author’s favorite introduction to the subject is reference Ginsparg (Applied conformal field theory. http://arxiv.org/abs/hep-th/9108028). We also give an elementary discussion of conformal perturbation theory. This notion is at the heart of much of what we discuss in the rest of the book.

References

  1. 56.
    Baume, F., Keren-Zur, B., Rattazzi, R., Vitale, L.: The local Callan-Symanzik equation: structure and applications. J. High Energy Phys. 08, 152 (2014). http://dx.doi.org/10.1007/JHEP08(2014)152; http://arxiv.org/abs/1401.5983
  2. 63.
    Behr, N., Konechny, A.: Renormalization and redundancy in 2d quantum field theories. J. High Energy Phys. 1402, 001 (2014). http://dx.doi.org/10.1007/JHEP02(2014)001; http://arxiv.org/abs/1310.4185
  3. 65.
    Belavin, A., Polyakov, A.M., Zamolodchikov, A.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984). http://dx.doi.org/10.1016/0550-3213(84)90052-X ADSMathSciNetCrossRefGoogle Scholar
  4. 66.
    Benini, F., Bobev, N.: Exact two-dimensional superconformal R-symmetry and c-extremization. Phys. Rev. Lett. 110, 061601 (2013). http://dx.doi.org/10.1103/PhysRevLett.110.061601; http://arxiv.org/abs/1211.4030
  5. 76.
    Berkovits, N.: Explaining the pure spinor formalism for the superstring. J. High Energy Phys. 01, 065 (2008). http://dx.doi.org/10.1088/1126-6708/2008/01/065; http://arxiv.org/abs/0712.0324 MathSciNetCrossRefGoogle Scholar
  6. 83.
    Bertolini, M., Melnikov, I.V., Plesser, M.R.: Accidents in (0,2) Landau-Ginzburg theories. J. High Energy Phys. 12, 157 (2014). http://dx.doi.org/10.1007/JHEP12(2014)157; http://arxiv.org/abs/1405.4266
  7. 95.
    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. B172, 316 (1986). http://dx.doi.org/10.1016/0370-2693(86)90260-1 ADSMathSciNetCrossRefGoogle Scholar
  8. 96.
    Bouwknegt, P., Schoutens, K.: W symmetry in conformal field theory. Phys. Rep. 223, 183–276 (1993). http://dx.doi.org/10.1016/0370-1573(93)90111-P; http://arxiv.org/abs/hep-th/9210010 ADSMathSciNetCrossRefGoogle Scholar
  9. 123.
    Creutzig, T., Ridout, D.: Logarithmic conformal field theory: beyond an introduction. J. Phys. A46, 4006 (2013). http://dx.doi.org/10.1088/1751-8113/46/49/494006; http://arxiv.org/abs/1303.0847
  10. 126.
    de Boer, J., Manschot, J., Papadodimas, K., Verlinde, E.: The Chiral ring of AdS(3)/CFT(2) and the attractor mechanism. J. High Energy Phys. 03, 030 (2009). http://dx.doi.org/10.1088/1126-6708/2009/03/030; http://arxiv.org/abs/0809.0507 CrossRefGoogle Scholar
  11. 130.
    Deser, S., Schwimmer, A.: Geometric classification of conformal anomalies in arbitrary dimensions. Phys. Lett. B309, 279–284 (1993). http://arxiv.org/abs/hep-th/9302047 ADSMathSciNetCrossRefGoogle Scholar
  12. 131.
    Di Francesco, P., Mathieu, P., Senechal, D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York (1997). http://dx.doi.org/10.1007/978-1-4612-2256-9
  13. 132.
    Di Vecchia, P., Petersen, J., Zheng, H.: N=2 extended superconformal theories in two-dimensions. Phys. Lett. B162, 327 (1985). http://dx.doi.org/10.1016/0370-2693(85)90932-3 ADSMathSciNetCrossRefGoogle Scholar
  14. 133.
    Di Vecchia, P., Knizhnik, V.G., Petersen, J.L., Rossi, P.: A supersymmetric Wess-Zumino lagrangian in two-dimensions. Nucl. Phys. B253, 701–726 (1985). http://dx.doi.org/10.1016/0550-3213(85)90554-1 ADSCrossRefGoogle Scholar
  15. 137.
    Dine, M., Seiberg, N.: Are (0,2) models string miracles? Nucl. Phys. B306, 137 (1988)ADSMathSciNetCrossRefGoogle Scholar
  16. 149.
    Dixon, L.J., Kaplunovsky, V., Vafa, C.: On four-dimensional gauge theories from type II superstrings. Nucl. Phys. B294, 43–82 (1987)ADSMathSciNetCrossRefGoogle Scholar
  17. 159.
    Eguchi, T., Taormina, A.: On the unitary representations of N=2 and N=4 superconformal algebras. Phys. Lett. B210, 125 (1988). http://dx.doi.org/10.1016/0370-2693(88)90360-7 ADSMathSciNetCrossRefGoogle Scholar
  18. 160.
    Eguchi, T., Taormina, A.: Extended superconformal algebras and string compactifications. In: Trieste Spring School and Workshop on Superstrings (SUPERSTRINGS ‘88), Trieste, 11–22 Apr 1988. http://alice.cern.ch/format/showfull?sysnb=0102114
  19. 173.
    Friedan, D., Konechny, A.: Gradient formula for the beta-function of 2d quantum field theory. J. Phys. A43, 215401 (2010). http://dx.doi.org/10.1088/1751-8113/43/21/215401; http://arxiv.org/abs/0910.3109 ADSMathSciNetCrossRefGoogle Scholar
  20. 174.
    Friedan, D., Konechny, A.: Curvature formula for the space of 2-d conformal field theories. J. High Energy Phys. 09, 113 (2012). http://dx.doi.org/10.1007/JHEP09(2012)113; http://arxiv.org/abs/1206.1749
  21. 179.
    Gaberdiel, M.R., Konechny, A., Schmidt-Colinet, C.: Conformal perturbation theory beyond the leading order. J. Phys. A42, 105402 (2009). http://dx.doi.org/10.1088/1751-8113/42/10/105402; http://arxiv.org/abs/0811.3149 ADSMathSciNetCrossRefGoogle Scholar
  22. 195.
    Gerchkovitz, E., Gomis, J., Komargodski, Z.: Sphere partition functions and the Zamolodchikov metric. J. High Energy Phys. 11, 001 (2014). http://dx.doi.org/10.1007/JHEP11(2014)001; http://arxiv.org/abs/1405.7271
  23. 197.
    Ginsparg, P.H.: Applied conformal field theory. http://arxiv.org/abs/hep-th/9108028
  24. 198.
    Goddard, P., Kent, A., Olive, D.I.: Virasoro algebras and coset space models. Phys. Lett. B152, 88 (1985). http://dx.doi.org/10.1016/0370-2693(85)91145-1 ADSMathSciNetCrossRefGoogle Scholar
  25. 201.
    Gomis, J., Hsin, P.-S., Komargodski, Z., Schwimmer, A., Seiberg, N., Theisen, S.: Anomalies, conformal manifolds, and spheres. J. High Energy Phys. 03, 022 (2016). http://dx.doi.org/10.1007/JHEP03(2016)022; http://arxiv.org/abs/1509.08511
  26. 208.
    Green, D., Komargodski, Z., Seiberg, N., Tachikawa, Y., Wecht, B.: Exactly marginal deformations and global symmetries. J. High Energy Phys. 1006, 106 (2010). http://dx.doi.org/10.1007/JHEP06(2010)106; http://arxiv.org/abs/1005.3546
  27. 230.
    Hollands, S.: Action principle for OPE. Nucl. Phys. B926, 614–638 (2018). http://dx.doi.org/10.1016/j.nuclphysb.2017.11.013; http://arxiv.org/abs/1710.05601 ADSMathSciNetCrossRefGoogle Scholar
  28. 237.
    Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror Symmetry. Clay Mathematics Monographs, vol. 1. American Mathematical Society, Providence (2003). With a preface by VafaGoogle Scholar
  29. 263.
    Kastor, D.A., Martinec, E.J., Shenker, S.H.: RG Flow in N=1 discrete series. Nucl. Phys. B316, 590–608 (1989)ADSMathSciNetCrossRefGoogle Scholar
  30. 269.
    Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino model in two-dimensions. Nucl. Phys. B247, 83–103 (1984). http://dx.doi.org/10.1016/0550-3213(84)90374-2 ADSMathSciNetCrossRefGoogle Scholar
  31. 279.
    Kutasov, D.: Geometry on the space of conformal field theories and contact terms. Phys. Lett. B220, 153 (1989)ADSMathSciNetCrossRefGoogle Scholar
  32. 283.
    Lerche, W., Vafa, C., Warner, N.P.: Chiral rings in N=2 superconformal theories. Nucl. Phys. B324, 427 (1989)ADSMathSciNetCrossRefGoogle Scholar
  33. 309.
    Moore, G.W., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177 (1989). http://dx.doi.org/10.1007/BF01238857 ADSMathSciNetCrossRefGoogle Scholar
  34. 314.
    Nakayama, Y.: Scale invariance vs conformal invariance. Phys. Rep. 569, 1–93 (2015). http://dx.doi.org/10.1016/j.physrep.2014.12.003; http://arxiv.org/abs/1302.0884 ADSMathSciNetCrossRefGoogle Scholar
  35. 320.
    Osborn, H.: Local renormalization group equations in quantum field theory. In: 2nd JINR Conference on Renormalization Group Dubna, USSR, 3–6 Sept 1991, pp. 128–138Google Scholar
  36. 321.
    Osborn, H.: Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories. Nucl. Phys. B363, 486–526 (1991). http://dx.doi.org/10.1016/0550-3213(91)80030-P ADSMathSciNetCrossRefGoogle Scholar
  37. 322.
    Pappadopulo, D., Rychkov, S., Espin, J., Rattazzi, R.: OPE convergence in conformal field theory. Phys. Rev. D86, 105043 (2012). http://dx.doi.org/10.1103/PhysRevD.86.105043; http://arxiv.org/abs/1208.6449
  38. 325.
    Poland, D., Rychkov, S., Vichi, A.: The conformal bootstrap: theory, numerical techniques, and applications. http://arxiv.org/abs/1805.04405
  39. 326.
    Polchinski, J.: Scale and conformal invariance in quantum field theory. Nucl. Phys. B303, 226 (1988). http://dx.doi.org/10.1016/0550-3213(88)90179-4 ADSMathSciNetCrossRefGoogle Scholar
  40. 327.
    Polchinski, J.: String Theory, Volume 2. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  41. 328.
    Polchinski, J.: String Theory. Volume 1: An Introduction to the Bosonic String. Cambridge University Press, Cambridge (2007)Google Scholar
  42. 331.
    Ramond, P., Schwarz, J.H.: Classification of dual model gauge algebras. Phys. Lett. B64, 75 (1976). http://dx.doi.org/10.1016/0370-2693(76)90361-0 ADSMathSciNetCrossRefGoogle Scholar
  43. 337.
    Samelson, H.: A class of complex-analytic manifolds. Port. Math. 12, 129–132 (1953)MathSciNetzbMATHGoogle Scholar
  44. 339.
    Schottenloher, M.: A Mathematical Introduction to Conformal Field Theory. Lect. Notes Phys. 759, 1–237 (2008). http://dx.doi.org/10.1007/978-3-540-68628-6 MathSciNetzbMATHGoogle Scholar
  45. 340.
    Schwimmer, A., Seiberg, N.: Comments on the N=2, N=3, N=4 superconformal algebras in two-dimensions. Phys. Lett. B184, 191 (1987). http://dx.doi.org/10.1016/0370-2693(87)90566-1 ADSMathSciNetCrossRefGoogle Scholar
  46. 345.
    Sen, K., Tachikawa, Y.: First-order conformal perturbation theory by marginal operators. http://arxiv.org/abs/1711.05947
  47. 346.
    Sevrin, A., Troost, W., Van Proeyen, A., Spindel, P.: Extended supersymmetric sigma models on group manifolds. 2. Current algebras. Nucl. Phys. B311, 465 (1988). http://dx.doi.org/10.1016/0550-3213(88)90070-3
  48. 351.
    Spindel, P., Sevrin, A., Troost, W., Van Proeyen, A.: Extended supersymmetric sigma models on group manifolds. 1. The complex structures. Nucl. Phys. B308, 662 (1988)Google Scholar
  49. 361.
    Teschner, J.: Liouville theory revisited. Class. Quantum Gravity 18, R153–R222 (2001). http://dx.doi.org/10.1088/0264-9381/18/23/201; http://arxiv.org/abs/hep-th/0104158 ADSMathSciNetCrossRefGoogle Scholar
  50. 374.
    Wang, H.-C.: Closed manifolds with homogeneous complex structure. Am. J. Math. 76, 1–32 (1954)MathSciNetCrossRefGoogle Scholar
  51. 375.
    Weinberg, S.: The Quantum Theory of Fields, vol. 2. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  52. 376.
    West, P.: Introduction to Supersymmetry and Supergravity. World Scientific, Singapore (1990)CrossRefGoogle Scholar
  53. 397.
    Zamolodchikov, A.: Conformal symmetry and multicritical points in two-dimensional quantum field theory (in Russian). Sov. J. Nucl. Phys. 44. 529–533 (1986)Google Scholar
  54. 398.
    Zamolodchikov, A.B.: Renormalization group and perturbation theory near fixed points in two-dimensional field theory. Sov. J. Nucl. Phys. 46, 1090 (1987) [Yad. Fiz.46,1819(1987)]Google Scholar

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Authors and Affiliations

  • Ilarion V. Melnikov
    • 1
  1. 1.Department of Physics and AstronomyJames Madison UniversityHarrisonburgUSA

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