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The Liouville side of the vortex

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Abstract

We analyze conformal blocks with multiple (semi-)degenerate field insertions in Liouville/Toda conformal field theories an show that their vector space is fully reproduced by the four-dimensional limit of open topological string amplitudes on the strip with generic boundary conditions associated to a suitable quiver gauge theory. As a by product we identify the non-abelian vortex partition function with a specific fusion channel of degenerate conformal blocks.

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References

  1. M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum Geometry of Refined Topological Strings, arXiv:1105.0630 [SPIRES].

  2. M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  3. 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  MathSciNet  Google Scholar 

  4. 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  ADS  MathSciNet  Google Scholar 

  5. H. Awata et al., Notes on Ding-Iohara algebra and AGT conjecture, arXiv:1106.4088 [SPIRES].

  6. A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  7. A. Belavin, V. Belavin and M. Bershtein, Instantons and 2 d Superconformal field theory, arXiv:1106.4001 [SPIRES].

  8. V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP 07 (2011) 079 [arXiv:1105.5800] [SPIRES].

    Article  ADS  Google Scholar 

  9. G. Bonelli, K. Maruyoshi and A. Tanzini, Instantons on ALE spaces and Super Liouville Conformal Field Theories, arXiv:1106.2505 [SPIRES].

  10. G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin Systems via beta-deformed Matrix Models, arXiv:1104.4016 [SPIRES].

  11. G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [arXiv:0909.4031] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  12. J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, arXiv:1005.2846 [SPIRES].

  13. G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices & Interacting Surface Operators, arXiv:1102.0184 [SPIRES].

  14. T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, arXiv:1006.0977 [SPIRES].

  15. N. Drukker, D. Gaiotto and J. Gomis, The Virtue of Defects in 4D Gauge Theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [SPIRES].

    Article  ADS  Google Scholar 

  16. V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory I, JHEP 11 (2007) 002 [arXiv:0709.3806] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  17. D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES].

  18. S. Gukov and E. Witten, Gauge theory, ramification and the geometric Langlands program, hep-th/0612073 [SPIRES].

  19. A. Iqbal and A.-K. Kashani-Poor, The vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [SPIRES].

    MATH  MathSciNet  Google Scholar 

  20. S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  21. C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  22. A. Marshakov, A. Mironov and A. Morozov, On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [SPIRES].

    Article  ADS  Google Scholar 

  24. M. Taki, Surface Operator, Bubbling Calabi-Yau and AGT Relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [SPIRES].

    Article  ADS  Google Scholar 

  25. D. Gaiotto, Surface Operators in N = 24d Gauge Theories, arXiv:0911.1316 [SPIRES].

  26. H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String, arXiv:008.0574 [SPIRES].

  27. K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  28. U. Bruzzo et al., D-branes, surface operators and ADHM quiver representations, arXiv:1012.1826 [SPIRES].

  29. A. Mironov and A. Morozov, Nekrasov Functions from Exact BS Periods: the Case of SU(N), J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  30. A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Sommerfeld Integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  31. N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].

    MathSciNet  Google Scholar 

  32. T. Nishioka and Y. Tachikawa, Para-Liouville/Toda central charges from M5-branes, Phys. Rev. D 84 (2011) 046009 [arXiv:1106.1172] [SPIRES].

    ADS  Google Scholar 

  33. S. Shadchin, On F-term contribution to effective action, JHEP 08 (2007) 052 [hep-th/0611278] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  34. E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  36. E. Witten, Gauge Theory And Wild Ramification, arXiv: 0710.0631 [SPIRES].

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Authors and Affiliations

  1. International School of Advanced Studies (SISSA), and INFN, Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy

    Giulio Bonelli, Alessandro Tanzini & Jian Zhao

  2. Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014, Trieste, Italy

    Giulio Bonelli

Authors
  1. Giulio Bonelli
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  2. Alessandro Tanzini
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  3. Jian Zhao
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Correspondence to Giulio Bonelli.

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ArXiv ePrint: 1107.2787

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Bonelli, G., Tanzini, A. & Zhao, J. The Liouville side of the vortex. J. High Energ. Phys. 2011, 96 (2011). https://doi.org/10.1007/JHEP09(2011)096

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  • DOI: https://doi.org/10.1007/JHEP09(2011)096

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