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Affine SL(2) Conformal Blocks from 4d Gauge Theories

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Abstract

We study Nekrasov’s instanton partition function of four-dimensional \({\mathcal{N}=2}\) gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.

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References

  1. Gaiotto, D.: \({{\mathcal{N}} =2}\) Dualities. arXiv:0904.2715 [hep-th]

  2. Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010) arXiv:0906.3219 [hep-th]

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004) arXiv:hep-th/0206161

    MathSciNet  Google Scholar 

  4. Wyllard N.: A N–1 Conformal Toda field theory correlation functions from conformal \({{\mathcal{N}} =2SU(N)}\) quiver gauge theories. JHEP 11, 002 (2009) arXiv:0907.2189 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  5. Mironov A., Morozov A.: On AGT relation in the case of U(3). Nucl. Phys. B 825, 1–37 (2010) arXiv:0908.2569 [hep-th]

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Alday L.F., Gaiotto D., Gukov S., Tachikawa Y., Verlinde H.: Loop and surface operators in \({{\mathcal{N}} =2}\) gauge theory and Liouville modular geometry. JHEP 01, 113 (2010) arXiv:0909.0945 [hep-th]

    Article  ADS  Google Scholar 

  7. Kozcaz, C., Pasquetti, S., Wyllard, N.: A & B model approaches to surface operators and Toda theories. arXiv:1004.2025 [hep-th]

  8. Drukker N., Morrison D.R., Okuda T.: Loop operators and S-duality from curves on Riemann surfaces. JHEP 09, 031 (2009) arXiv:0907.2593 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. Drukker, N., Gaiotto, D., Gomis, J.: The virtue of defects in 4D gauge theories and 2D CFTs. arXiv:1003.1112 [hep-th]

  11. Gukov, S., Witten, E.: Gauge theory, ramification, and the geometric Langlands program. arXiv:hep-th/0612073

  12. Braverman A.: Instanton counting via affine Lie algebras. I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors. In: Hurturbise, J., Markman, E. (eds) Workshop on algebraic structures and Moduli Spaces: CRM Workshop, AMS, Providence (2003) arXiv:math/0401409

    Google Scholar 

  13. Braverman A., Etingof P.: Instanton counting via affine Lie algebras. II. From Whittaker vectors to the Seiberg–Witten prepotential. In: Bernstein, J., Hinich, V., Melnikov, A. (eds) Studies in Lie Theory: dedicated to A. Joseph on his 60th birthday, Birkhäuser, Boston (2006) arXiv:math/0409441

    Google Scholar 

  14. Negut A.: Laumon spaces and the Calogero-Sutherland integrable system. Invent. Math. 178, 299 (2008) arXiv:0811.4454 [math.AG]

    Article  MathSciNet  ADS  Google Scholar 

  15. Feigin, B., Finkelberg, M., Negut, A., Rybnikov, L.: “Yangians and cohomology rings of Laumon spaces,” arXiv:0812.4656 [math.AG]

  16. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052 [hep-th]

  17. Teschner, J.: Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence. arXiv:1005.2846 [hep-th]

  18. Flume R., Poghossian R.: An algorithm for the microscopic evaluation of the coefficients of the Seiberg–Witten prepotential. Int. J. Mod. Phys. A 18, 2541 (2003) arXiv:hep-th/0208176

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Nakajima H., Yoshioka K.: Instanton counting on blowup, I. Invent. Math. 162, 313 (2005) arXiv:math/0306198

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Fucito F., Morales J.F., Poghossian R.: Instantons on quivers and orientifolds. JHEP 10, 037 (2004) arXiv:hep-th/0408090

    Article  MathSciNet  ADS  Google Scholar 

  21. Nakajima H., Yoshioka K.: Lectures on instanton counting. In: Hurturbise, J., Markman, E. (eds) Workshop on Algebraic Structures and Moduli Spaces: CRM Workshop, AMS, Providence (2003) arXiv:math/0311058

    Google Scholar 

  22. Carlsson, E., Okounkov, A.: Ext and vertex operators. arXiv:0801.2565 [math.AG]

  23. Gukov, S.: Surface operators and knot homologies. arXiv:0706.2369 [hep-th]

  24. Tan, M.-C.: Integration over the u-plane in Donaldson theory with surface operators. arXiv:0912.4261 [hep-th]

  25. Gaiotto, D.: Surface operators in \({{\mathcal{N}} =2}\) 4D gauge theories. arXiv:0911.1316 [hep-th]

  26. Kronheimer P.B., Mrowka T.S.: Gauge theory for embedded surfaces, I. Topology 32, 773 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kronheimer P.B., Mrowka T.S.: Gauge theory for embedded surfaces, II. Topology 34, 37 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Biquard O.: Sur les Fibrés Paraboliques sur une Surface Complexe. J. London Math. Soc. 53, 302 (1996)

    MathSciNet  MATH  Google Scholar 

  29. Martinec E.J., Warner N.P.: Integrable Systems and Supersymmetric Gauge Theory. Nucl. Phys. B 459, 97–112 (1996) arXiv:hep-th/9509161

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Donagi R., Witten E.: Supersymmetric Yang-Mills Theory and Integrable Systems. Nucl. Phys. B 460, 299–334 (1996) arXiv:hep-th/9510101

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Itoyama H., Morozov A.: Integrability and Seiberg-Witten Theory: Curves and Periods. Nucl. Phys. B 477, 855–877 (1996) arXiv:hep-th/9511126

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Gorsky, A., Nekrasov, N.: Elliptic Calogero-Moser system from two-dimensional current algebra. arXiv:hep-th/9401021

  33. Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke Eigensheaves. c2000. http://www.math.utexas.edu/users/benzvi/BD/hitchin.pdf

  34. Frenkel E.: Lectures on the Langlands program and conformal field theory. In: Cartier, P.E., Julia, B., Moussa, P., Vanhove, P. (eds) Frontiers in Number Theory, Physics and Geometry II, Springer, Berlin (2007) arXiv:hep-th/0512172

    Google Scholar 

  35. Knizhnik V.G., Zamolodchikov A.B.: Current algebra and Wess–Zumino model in two dimensions. Nucl. Phys. B 247, 83–103 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Bernard D.: On the Wess–Zumino–Witten models on the Torus. Nucl. Phys. B 303, 77 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  37. Etingof P.I., Kirillov A.A. Jr.: Representation of affine Lie algebras, parabolic differential equations and Lamé functions. Duke Math. J. 74, 585 (1994) arXiv:hep-th/9310083

    Article  MathSciNet  MATH  Google Scholar 

  38. Felder G., Weiczerkowski C.: Conformal blocks on elliptic curves and the Knizhnik–Zamolodchikov–Bernard equations. Commun. Math. Phys. 176, 133–162 (1996) arXiv:hep-th/9411004

    Article  ADS  MATH  Google Scholar 

  39. Martinec E.J.: Integrable structures in supersymmetric gauge and string theory. Phys. Lett. B 367, 91–96 (1996) arXiv:hep-th/9510204

    Article  MathSciNet  ADS  Google Scholar 

  40. Bonelli, G., Tanzini, A.: Hitchin systems, \({{\mathcal{N}} =2}\) gauge theories and W-gravity. arXiv:0909.4031 [hep-th]

  41. Ribault S., Teschner J.: \({{\rm H}^+_3}\) WZNW correlators from Liouville theory. JHEP 06, 014 (2005) arXiv:hep-th/0502048

    Article  MathSciNet  ADS  Google Scholar 

  42. Reshetikhin, N., Varchenko, A.: Quasiclassical asymptotics of solutions to the KZ equations. arXiv:hep-th/9402126

  43. Nekrasov, N., Witten, E.: The omega deformation, branes, integrability, and Liouville theory. arXiv:1002.0888 [hep-th]

  44. Bershadsky M., Ooguri H.: Hidden SL(N) symmetry in conformal field theories. Commun. Math. Phys. 126, 49 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Feigin B., Frenkel E.: Quantization of the Drinfeld–Sokolov reduction. Phys. Lett. B 246, 75–81 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  46. Ribault S.: On SL(3) Knizhnik–Zamolodchikov equations and W 3 null-vector equations. JHEP 10, 002 (2009) arXiv:0811.4587 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  47. Giribet, G.: On Triality in \({{\mathcal{N}} =2}\) SCFT with N F  = 4. arXiv:0912.1930 [hep-th]

  48. Hikida Y., Schomerus V.: \({{\rm H}^+_3}\) WZNW Model from Liouville Field Theory. JHEP 10, 064 (2007) arXiv:0706.1030 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  49. Giribet G., Nakayama Y., Nicolás L.: Langlands duality in Liouville-\({{\rm H}_3^+}\) WZNW correspondence. Int. J. Mod. Phys. A 24, 3137–3170 (2009) arXiv:0805.1254 [hep-th]

    Article  ADS  MATH  Google Scholar 

  50. Pestun, V.: Localization of Gauge theory on a four-sphere and supersymmetric Wilson loops. arXiv:0712.2824 [hep-th]

  51. Awata H., Yamada Y.: Fusion rules for the fractional level SL(2) algebra. Mod. Phys. Lett. A 7, 1185–1196 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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  1. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Luis F. Alday & Yuji Tachikawa

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  1. Luis F. Alday
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  2. Yuji Tachikawa
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Correspondence to Yuji Tachikawa.

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Alday, L.F., Tachikawa, Y. Affine SL(2) Conformal Blocks from 4d Gauge Theories. Lett Math Phys 94, 87–114 (2010). https://doi.org/10.1007/s11005-010-0422-4

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