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Universality in Chern-Simons theory

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Abstract

We show that the perturbative part of the partition function in the ChernSimons theory on a 3-sphere as well as the central charge and expectation value of the unknotted Wilson loop in the adjoint representation can be expressed in terms of the universal Vogel’s parameters α, β, γ. The derivation is based on certain generalisations of the Freudenthal-de Vries strange formula.

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References

  1. P. Vogel, Algebraic structures on modules of diagrams, J. Pure Appl. Algebra 215 (2011) 1292, preprint (1995) available at: www.math.jussieu.fr/~vogel/diagrams.pdf.

  2. P. Vogel The universal Lie algebra, Preprint (1999) available at www.math.jussieu.fr/~vogel/A299.ps.gz.

  3. G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  4. E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933 [INSPIRE].

  5. R.C. King, The dimensions of irreducible tensor representations of the orthogonal and symplectic groups, Can. J. Math. 23 (1971) 176.

    Article  MATH  Google Scholar 

  6. R.L. Mkrtchyan, The equivalence of Sp(2N) and SO(−2N) gauge theories, Phys. Lett. 105B (1981) 174.

    ADS  Google Scholar 

  7. P. Cvitanovic, Group Theory, Princeton University Press, Princeton NJ (2004), available at http://www.nbi.dk/group.theory.

  8. R.L. Mkrtchyan and A.P. Veselov, On duality and negative dimensions in the theory of Lie groups and symmetric spaces, J. Math. Phys. 52 (2011) 083514 [arXiv:1011.0151] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. M. El Houari, Tensor invariants associated with classical Lie algebras: a new classification of simple Lie algebras, Algebras Groups Geom. 14 (1997) 423.

    MathSciNet  MATH  Google Scholar 

  10. E. Angelopoulos, Classification of simple Lie algebras, Panamer. Math. J. 11 (2001) 65.

    MathSciNet  MATH  Google Scholar 

  11. P. Deligne, La série exceptionnelle des groupes de Lie, C.R. Acad. Sci. Paris, Série I 322 (1996) 321.

    MathSciNet  MATH  Google Scholar 

  12. P. Deligne and R. de Man, La série exceptionnelle des groupes de Lie II, C.R. Acad. Sci. Paris, Série I 323 (1996) 577.

    MATH  Google Scholar 

  13. E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. D. Bar-Natan, Perturbative Chern-Simons Theory, J. Knot Theor. Ram. 4–4 (1995) 503.

    Article  MathSciNet  Google Scholar 

  15. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Reshetikhin and V. Turaev, Invariants of three manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. V.G. Turaev, Quantum invariants of knots and 3-manifolds, Walter de Gruyter and Co., Berlin (2010).

    Book  MATH  Google Scholar 

  18. J.M. Landsberg and L. Manivel, A universal dimension formula for complex simple Lie algebras, Adv. Math. 201 (2006) 379.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math. 171 (2002) 59.

    Article  MathSciNet  MATH  Google Scholar 

  20. R.L. Mkrtchyan, A.N. Sergeev and A.P. Veselov, Casimir eigenvalues for universal Lie algebra, arXiv:1105.0115, to appear in JMP.

  21. H. Freudenthal and H. de Vries, Linear Lie Groups, Academic Press (1969).

  22. P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997).

    Book  MATH  Google Scholar 

  23. H. Ooguri and C. Vafa, World sheet derivation of a large-N duality, Nucl. Phys. B 641 (2002) 3 [hep-th/0205297] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. V. Kac, Infinite dimensional Lie algebras, Third Edition, Cambridge University Press, Cambridge U.K. (1995).

    Google Scholar 

  25. I.G. Macdonald, The volume of a compact Lie group, Invent. Math. 56 (1980) 93.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. M. Marinov, Invariant volumes of compact groups, J. Phys. A 13 (1980) 3357 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  27. Y. Hashimoto, On Macdonald’s formula for the volume of a compact Lie group, Comment. Math. Helv. 72 (1997) 660.

    Article  MathSciNet  MATH  Google Scholar 

  28. N. Bourbaki, Groupes et Algèbres de Lie, Chap. VI, Masson (1981).

    MATH  Google Scholar 

  29. G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. sci. de l’ École Normale Supérieure, Sér. IV 4 (1971) 409.

    MathSciNet  MATH  Google Scholar 

  30. S. Sinha and C. Vafa, SO and Sp Chern-Simons at large-N, hep-th/0012136 [INSPIRE].

  31. S.G. Naculich and H.J. Schnitzer, Level-rank duality of the U(N) WZW model, Chern-Simons theory and 2D qYM theory, JHEP 06 (2007) 023 [hep-th/0703089] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. V. Periwal, Topological closed string interpretation of Chern-Simons theory, Phys. Rev. Lett. 71 (1993) 1295 [hep-th/9305115] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  34. S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, arXiv:1103.5628.

  35. J. Lieberum, On Vassiliev invariants not coming from semisimple Lie algebras, J. Knot Theor. Ram. 8 (1999) 659.

    Article  MathSciNet  MATH  Google Scholar 

  36. V. Hinich and A. Vaintrob, Cyclic operads and algebra of chord diagrams, Selecta Math. (N.S.) 8 (2002) 237.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Yerevan Physics Institute, 2 Alikhanian Brothers St., Yerevan, 0036, Armenia

    R. L. Mkrtchyan

  2. Department of Mathematics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.

    A. P. Veselov

  3. Moscow State University, Moscow, Russia

    A. P. Veselov

Authors
  1. R. L. Mkrtchyan
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  2. A. P. Veselov
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Correspondence to A. P. Veselov.

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ArXiv ePrint: 1203.0766

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Mkrtchyan, R.L., Veselov, A.P. Universality in Chern-Simons theory. J. High Energ. Phys. 2012, 153 (2012). https://doi.org/10.1007/JHEP08(2012)153

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