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Local Equivalence of Representations of \({{\rm Diff}^+(S^1)}\) Corresponding to Different Highest Weights

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Abstract

Let c, h and \({c,\tilde{h}}\) be two admissible pairs of central charge and highest weight for \({{\rm Diff}^+(S^1)}\). It is shown here that the positive energy irreducible projective unitary representations \({U_{c,h}}\) and \({U_{c,\tilde{h}}}\) of the group \({{\rm Diff}^+(S^1)}\) are locally equivalent. This means that for any \({I\Subset S^1}\) open proper interval, there exists a unitary operator W I such that \({W_I U_{c,h}(\gamma)W_I^* = U_{c,\tilde{h}}(\gamma)}\) for all \({\gamma \in {\rm Diff}^+(S^1)}\) which act identically on \({I^c\equiv S^1{\setminus} I}\) (i.e., which can “displace” or “move” points only in I). This result extends and completes earlier ones that dealt with only certain regions of the “c, h-plane”, and closes the gap in the full classification of superselection sectors of Virasoro nets.

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References

  1. Buccholz D., D’Antoni C., Longo R.: Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270, 267–293 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Buchholz D., Schulz-Mirbach H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2, 105–125 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in conformal QFT. II. Commun. Math. Phys. 315, 771–802 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Carpi S.: The Virasoro algebra and sectors with infinite statistical dimension. Ann. Henri Poincaré 4, 601–611 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Carpi S.: On the representation theory of Virasoro nets. Commun. Math. Phys. 244, 261–284 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Fewster C.J., Hollands S.: Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17, 577–612 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Furlan P., Sotkov G.M., Todorov I.T.: Two-dimensional conformal quantum field theory. Riv. Nuovo Cimento 12, 1–202 (1989)

    Article  MathSciNet  Google Scholar 

  8. Gabbiani F., Fröhlich J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Goddard P., Kent A., Olive D.: Unitary representations of the Virasoro and super-Virasoro algebra. Commun. Math. Phys. 103, 105–119 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Goodman R., Wallach N.R.: Projective unitary positive-energy representations of \({{\rm Diff}(S^1)}\). J. Funct. Anal. 63, 299–321 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kac V.G., Raina A.K.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. World Scientific, Singapore (1987)

    MATH  Google Scholar 

  12. Kawahigashi Y., Longo R.: Classification of local conformal nets. Case \({c < 1}\). Ann. Math. 160, 493–522 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Loke T.: Operator Algebras and Conformal Field Theory of the Discrete Series Representation of \({{\rm Diff}^+(S^1)}\). Ph.D. Thesis, University of Cambridge (1994)

  14. Milnor, J.: Remarks on infinite-dimensional Lie groups. In: De Witt, B.S., Stora, R. (eds.) Relativity, Groups and Topology II. Les Houches, Session XL, 1983, pp. 1007–1057. Elsevier, Amsterdam (1984)

  15. Rehren K. H.: A new view of the Virasoro algebra. Lett. Math. Phys. 30, 125–130 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Strătilă Ş., Zsidó L.: Lectures on von Neumann Algebras. Editura Academiei and Abacus Press, Kent (1979)

    MATH  Google Scholar 

  17. Weiner M.: Conformal covariance and positivity of energy in charged sectors. Commun. Math. Phys. 265, 493–506 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Weiner M.: Restricting positive energy representations of \({{\rm Diff}^+(S^1)}\) to the stabilizer of n points. Commun. Math. Phys. 277, 555–571 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Xu F.: Strong additivity and conformal nets. Pac. J. Math. 221, 167–199 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Budapest University of Technology and Economics, Budapest, Hungary

    Mihály Weiner

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  1. Mihály Weiner
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Correspondence to Mihály Weiner.

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Communicated by Y. Kawahigashi

Supported in part by the ERC advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models” and by OTKA Grant No. 104206.

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Weiner, M. Local Equivalence of Representations of \({{\rm Diff}^+(S^1)}\) Corresponding to Different Highest Weights. Commun. Math. Phys. 352, 759–772 (2017). https://doi.org/10.1007/s00220-016-2824-3

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