Advertisement

Journal of High Energy Physics

, 2016:58 | Cite as

VEV of Baxter’s Q-operator in N = 2 gauge theory and the BPZ differential equation

  • Gabriel Poghosyan
  • Rubik Poghossian
Open Access
Regular Article - Theoretical Physics

Abstract

In this short note using AGT correspondence we express the simplest fully degenerate primary fields of Toda field theory in terms of an analogue of Baxter’s Q-operator naturally emerging on the \( \mathcal{N} \) = 2 gauge theory side. This quantity can be considered as a generating function of certain chiral operators constructed from the scalars of the \( \mathcal{N} \) = 2 vector multiplets. In the special case of Liouville theory, exploring the second order differential equation satisfied by conformal blocks including a primary field which is degenerate at the second level (BPZ equation) we derive a mixed difference-differential relation for Q-operator. Thus we generalize the T -Q difference equation known in Nekrasov-Shatashvili limit of the Ω-background to the generic case.

Keywords

Conformal and W Symmetry Gauge Symmetry Nonperturbative Effects Supersymmetric gauge theory 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu. S. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [INSPIRE].
  3. [3]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
  9. [9]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
  10. [10]
    A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    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] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  12. [12]
    K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    F. Fucito, J.F. Morales, D.R. Pacifici and R. Poghossian, Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    F. Fucito, J.F. Morales and D. Ricci Pacifici, Deformed Seiberg-Witten Curves for ADE Quivers, JHEP 01 (2013) 091 [arXiv:1210.3580] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
  17. [17]
    N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
  18. [18]
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    R. Baxter, Exactly Solved Models in Statistical Mechanics. Academic Press, London U.K. (1982).MATHGoogle Scholar
  21. [21]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory. 3. The Yang-Baxter relation, Commun. Math. Phys. 200 (1999) 297 [hep-th/9805008] [INSPIRE].
  22. [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] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    N. Wyllard, A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    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] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    M. Piatek, Classical conformal blocks from TBA for the elliptic Calogero-Moser system, JHEP 06 (2011) 050 [arXiv:1102.5403] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    M. Piatek, Classical torus conformal block, N = 2 twisted superpotential and the accessory parameter of Lamé equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    S.K. Ashok, M. Billó, E. Dell’Aquila, M. Frau, R.R. John and A. Lerda, Non-perturbative studies of N = 2 conformal quiver gauge theories, Fortsch. Phys. 63 (2015) 259 [arXiv:1502.05581] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    R. Poghossian, Deformed SW curve and the null vector decoupling equation in Toda field theory, JHEP 04 (2016) 070 [arXiv:1601.05096] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    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 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    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] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the Vortex, JHEP 09 (2011) 096 [arXiv:1107.2787] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    F. Fucito, J.F. Morales, R. Poghossian and D. Ricci Pacifici, Exact results in \( \mathcal{N} \) = 2 gauge theories, JHEP 10 (2013) 178 [arXiv:1307.6612] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Yerevan Physics InstituteYerevanArmenia

Personalised recommendations