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Cauchy Conformal Fields in Dimensions \({d > 2}\)

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Abstract

Holomorphic fields play an important role in 2d conformal field theory. We generalize them to \({d > 2}\) by introducing the notion of Cauchy conformal fields, which satisfy a first order differential equation such that they are determined everywhere once we know their value on a codimension 1 surface. We classify all the unitary Cauchy fields. By analyzing the mode expansion on the unit sphere, we show that all unitary Cauchy fields are free in the sense that their correlation functions factorize on the 2-point function. We also discuss the possibility of non-unitary Cauchy fields and classify them in d = 3 and 4.

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References

  1. Fubini S., Gordon D., Veneziano G.: A general treatment of factorization in dual resonance models. Phys. Lett. B 29, 679–682 (1969)

    Article  ADS  Google Scholar 

  2. Fubini S., Veneziano G.: Algebraic treatment of subsidiary conditions in dual resonance models. Ann. Phys. 63, 12–27 (1971)

    Article  ADS  Google Scholar 

  3. Del Giudice E., Di Vecchia P., Fubini S.: General properties of the dual resonance model. Ann. Phys. 70, 378–398 (1972)

    Article  ADS  Google Scholar 

  4. Brower, R.C., Goddard, P.: Physical states in the dual resonance model. In: Varenna 1972, Proceedings of Developments in High Energy Physics (1972)

  5. Corrigan E., Goddard P.: Gauge conditions in the dual fermion model. Nuovo Cim. A 18, 339–359 (1973)

    Article  ADS  Google Scholar 

  6. Brink L., Olive D.I., Scherk J.: The gauge properties of the dual model pomeron-reggeon vertex—their derivation and their consequences. Nucl. Phys. B 61, 173–198 (1973)

    Article  ADS  Google Scholar 

  7. Friedan, D.: Introduction to Polyankov’s string theory. In: Proceedings of Summer School of Theoretical Physics: Recent Advances in Field Theory and Statistical Mechanics, Les Houches (1982)

  8. Siegel W.: All free conformal representations in all dimensions. Int. J. Modern Phys. A 4, 2015 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  9. Dymarsky, A., Zhiboedov, A.: Scale-invariant breaking of conformal symmetry. J. Phys. A. Math. Theor. 48(41) (2015)

  10. Weinberg S., Witten E.: Limits on massless particles. Phys. Lett. B 96, 59 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  11. Coleman S.R., Mandula J.: All possible symmetries of the S matrix. Phys. Rev. 159, 1251–1256 (1967)

    Article  ADS  MATH  Google Scholar 

  12. Maldacena, J., Zhiboedov, A.: Constraining conformal field theories with a higher spin symmetry. J. Phys. A 46, 214011 (2013). arXiv:1112.1016 [hep-th]

  13. Alba, V., Diab, K.: Constraining conformal field theories with a higher spin symmetry in d = 4. arXiv:1307.8092 [hep-th]

  14. Stanev, Y.S.: Constraining conformal field theory with higher spin symmetry in four dimensions. Nucl. Phys. B 876, 651–666 (2013). arXiv:1307.5209 [hep-th]

  15. Boulanger, N., Ponomarev, D., Skvortsov, E., Taronna, M.: On the uniqueness of higher-spin symmetries in AdS and CFT. Int. J. Modern Phys. A 28, 1350162 (2013). arXiv:1305.5180 [hep-th]

  16. Minwalla, S.: Restrictions imposed by superconformal invariance on quantum field theories. Adv. Theor. Math. Phys. 2, 781–846 (1998). arXiv:hep-th/9712074 [hep-th]

  17. Evans N.T.: Discrete series for the universal covering group of the 3 +  2 de sitter group. J. Math. Phys. 8(2), 170–184 (1967)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Mack G.: All unitary ray representations of the conformal group SU(2,2) with positive energy. Commun. Math. Phys. 55, 1 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Enright, T., Howe, R., Wallach, N.: A classification of unitary highest weight modules. In: Representation Theory of Reductive Groups (Park City, Utah, 1982), Progr. Math., vol. 40, pp. 97–143. Birkhäuser, Boston (1983)

  20. Defosseux M.: Orbit measures, random matrix theory and interlaced determinantal processes. Ann. Inst. H. Poincar Probab. Stat. 46(1), 209–249 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Enright T.J., Wallach N.R.: Embeddings of unitary highest weight representations and generalized Dirac operators. Math. Ann. 307(4), 627–646 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fulton, W., Harris, J.: Representation Theory, Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)

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Authors and Affiliations

  1. NHETC and Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854-8019, USA

    Daniel Friedan & Christoph A. Keller

  2. The Science Institute, The University of Iceland, Reykjavik, Iceland

    Daniel Friedan

  3. Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland

    Christoph A. Keller

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  1. Daniel Friedan
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  2. Christoph A. Keller
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Correspondence to Christoph A. Keller.

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Communicated by N. A. Nekrasov

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Friedan, D., Keller, C.A. Cauchy Conformal Fields in Dimensions \({d > 2}\) . Commun. Math. Phys. 348, 655–694 (2016). https://doi.org/10.1007/s00220-015-2547-x

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  • DOI: https://doi.org/10.1007/s00220-015-2547-x

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