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On the crossing relation in the presence of defects

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Abstract

The OPE of local operators in the presence of defect lines is considered both in the rational CFT and the c > 25 Virasoro (Liouville) theory. The duality transformation of the 4-point function with inserted defect operators is explicitly computed. The two channels of the correlator reproduce the expectation values of the Wilson and ’t Hooft operators, recently discussed in Liouville theory in relation to the AGT conjecture.

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References

  1. V.B. Petkova and J.-B. Zuber, Generalised twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  2. V.B. Petkova and J.-B. Zuber, The many faces of Ocneanu cells, Nucl. Phys. B 603 (2001) 449 [hep-th/0101151] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  3. G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. G.W. Moore and N. Seiberg, Lectures on RCFT, physics, geometry and topology, Plenum Press, New York U.S.A. (1990).

    Google Scholar 

  5. L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].

    Article  Google Scholar 

  6. N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [SPIRES].

    Article  Google Scholar 

  7. J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601 [cond-mat/0404051] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [SPIRES].

    Article  ADS  Google Scholar 

  9. I. Runkel, Perturbed defects and T-systems in conformal field theory, J. Phys. A 41 (2008) 105401 [arXiv:0711.0102] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  10. C. Bachas and M. Gaberdiel, Loop operators and the Kondo problem, JHEP 11 (2004) 065 [hep-th/0411067] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  11. A. Alekseev and S. Monnier, Quantization of Wilson loops in Wess-Zumino-Witten models, JHEP 08 (2007) 039 [hep-th/0702174] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. J. Gomis, Loops and defects in 4D gauge theories and 2D CFT’s, talk given at the Workshop on Interfaces and Wall-crossings, November 30–December 4, Munich, Germany (2009).

  13. J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  14. R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B. Zuber, Boundary conditions in rational conformal field theories, Nucl. Phys. B 579 (2000) 707 [hep-th/9908036] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  15. A. Ocneanu, Paths on Coxeter diagrams: from platonic solids and singularities to minimal models and subfactors (Notes recorded by S. Goto), in Lectures on operator theory, eds. B.V. Rajarama Bhat et al., Fields Institute Monographs, AMS Publications (1999), pag. 243.

  16. V.G. Kac, Laplace operators in modules of infinite-dimensional Lie algebras and theta functions, Proc. Natl. Acad. Sci. U.S.A. 81 (1984) 645.

    Article  MATH  ADS  Google Scholar 

  17. B.L. Feigin and D.B. Fuks, Casimir operators in modules over Virasoro algebra, Sov. Math. Dokl. 27 (1983) 465.

    Google Scholar 

  18. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360.

    Article  MathSciNet  ADS  Google Scholar 

  19. A. Zamolodchikov and Al. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0001012 [SPIRES].

  20. V. Fateev, A. Zamolodchikov and Al. Zamolodchikov, Boundary Liouville field theory. I: boundary state and boundary two-point function, hep-th/0001012 [SPIRES].

  21. G. Sarkissian, Defects and permutation branes in the Liouville field theory, Nucl. Phys. B 821 (2009) 607 [arXiv:0903.4422] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  22. L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  23. I. Runkel, Boundary structure constants for the A-series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  25. D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  26. B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [SPIRES].

  27. B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U q (sl(2, R)), Commun. Math. Phys. 224 (2001) 613 [math/0007097].

    Article  MathSciNet  ADS  Google Scholar 

  28. B. Ponsot and J. Teschner, Boundary Liouville field theory: boundary three point function, Nucl. Phys. B 622 (2002) 309 [hep-th/0110244] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  29. P. Furlan, V.B. Petkova and M. Stanishkov, Non-critical string pentagon equations and their solutions, J. Phys. A 42 (2009) 304016 [arXiv:0805.0134] [SPIRES].

    MathSciNet  Google Scholar 

  30. K. Hosomichi, Bulk-boundary propagator in Liouville theory on a disc, JHEP 11 (2001) 044 [hep-th/0108093] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. B. Ponsot, Remarks on the bulk-boundary structure constant in Liouville field theory, unpublished preliminary manuscript (private communication) (2005).

  32. M. Nishizawa and K. Ueno, Integral solutions of q-difference equations of the hypergeometric type with |q| = 1, q-alg/9612014

  33. V. Pasquier, Operator content of the ADE lattice models, J. Phys. A 20 (1987) 5707.

    MathSciNet  ADS  Google Scholar 

  34. V.B. Petkova and J.-B. Zuber, On structure constants of sl(2) theories, Nucl. Phys. B 438 (1995) 347 [hep-th/9410209] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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  1. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee, 1784, Sofia, Bulgaria

    V. B. Petkova

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Petkova, V.B. On the crossing relation in the presence of defects. J. High Energ. Phys. 2010, 61 (2010). https://doi.org/10.1007/JHEP04(2010)061

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