Advertisement

Journal of High Energy Physics

, 2019:171 | Cite as

SCFT/VOA correspondence via Ω-deformation

  • Saebyeok JeongEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We investigate an alternative approach to the correspondence of four­ dimensional \( \mathcal{N} \) = 2 superconformal theories and two-dimensional vertex operator algebras, in the framework of the Ω-deformation of supersymmetric gauge theories. The two­dimensional Ω-deformation of the holomorphic-topological theory on the product four­ manifold is constructed at the level of supersymmetry variations and the action. The supersymmetric localization is performed to achieve a two-dimensional chiral CFT. The desired vertex operator algebra is recovered as the algebra of local operators of the resulting CFT. We also discuss the identification of the Schur index of the \( \mathcal{N} \) = 2 superconformal theory and the vacuum character of the vertex operator algebra at the level of their path integral representations, using our Ω-deformation point of view on the correspondence.

Keywords

Conformal and W Symmetry Conformal Field Theory Extended Supersymmetry 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]
    C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys.336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Kapustin, Holomorphic reduction of N = 2 gauge theories, Wilson-'t Hooft operators and S-duality, hep-th/0612119 [INSPIRE].
  3. [3]
    N. Nekrasov, Four dimensional holomorphic theories, Ph.D. thesis, Princeton University, Princeton, NJ, U.S.A. (1996).Google Scholar
  4. [4]
    L. Baulieu, A. Losev and N. Nekrasov, Chern-Simons and twisted supersymmetry in various dimensions, Nucl. Phys.B 522 (1998) 82 [hep-th/9707174] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  6. [6]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math.244 (2006) 525 [hep-th/0306238] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, hep-th/0302191 [INSPIRE].
  8. [8]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in Proceedings, 16thInternational Congress on Mathematical Physics ( ICMP09 ), Prague, Czech Republic, 3–8 August 2009, World Scientific, Singapore (2010), pg. 265 [arXiv:0908.4052] [INSPIRE].
  9. [9]
    N. Nekrasov and E. Witten, The Ω deformation, branes, integrability and Liouville theory, JHEP09 (2010) 092 [arXiv:1002.0888] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    N. Nekrasov, Tying up instantons with anti-instantons, World Scientific, Singapore (2018), pg. 351 [arXiv:1802.04202] [INSPIRE].
  11. [11]
    C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP05 (2015) 020 [arXiv:1408.6522] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    C. Beem, L. Rastelli and B.C. van Rees, W symmetry in six dimensions, JHEP05 (2015) 017 [arXiv:1404.1079] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in N = 2 superconformal theories, JHEP02 (2016) 019 [arXiv:1509.00033] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    M. Lemos and P. Liendo, N = 2 central charge bounds from 2d chiral algebras, JHEP04 (2016) 004 [arXiv:1511.07449] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  15. [15]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys.319 (2013) 147 [arXiv:1110.3740] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches and modular differential equations, JHEP08 (2018) 114 [arXiv:1707.07679] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    J. Song, Macdonald index and chiral algebra, JHEP08 (2017) 044 [arXiv:1612.08956] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    K. Maruyoshi and J. Song, Enhancement of supersymmetry via renormalization group flow and the superconformal index, Phys. Rev. Lett.118 (2017) 151602 [arXiv:1606.05632] [INSPIRE].
  19. [19]
    K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, JHEP02 (2017) 075 [arXiv:1607.04281] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    P. Agarwal, K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, part II: non-principal deformations, JHEP12 (2016) 103 [Addendum ibid.04 (2017) 113] [arXiv:1610.05311] [INSPIRE].
  21. [21]
    P. Agarwal, A. Sciarappa and J. Song, N = 1 Lagrangians for generalized Argyres-Douglas theories, JHEP10 (2017) 211 [arXiv:1707.04751] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    P. Agarwal, K. Maruyoshi and J. Song, A “Lagrangian” for the E 7superconformal theory, JHEP05 (2018) 193 [arXiv:1802.05268] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    E. Witten, Analytic continuation of Chern-Simons theory, AMS/ IP Stud. Adv. Math.50 (2011) 347 [arXiv:1001.2933] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  24. [24]
    E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
  25. [25]
    E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
  26. [26]
    J. Yagi, Ω-deformation and quantization, JHEP08 (2014) 112 [arXiv:1405.6714] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    Y. Luo, M.-C. Tan, J. Yagi and Q. Zhao, Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence, JHEP02 (2015) 047 [arXiv:1410.1538] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    K. Costello and J. Yagi, Unification of integrability in supersymmetric gauge theories, arXiv:1810.01970 [INSPIRE].
  29. [29]
    J. Oh and J. Yagi, Chiral algebras from Ω-deformation, JHEP08 (2019) 143 [arXiv:1903.11123] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    Y. Pan and W. Peelaers, Schur correlation functions on S 3 × S 1, JHEP07 (2019) 013 [arXiv:1903.03623] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    M. Dedushenko and M. Fluder, Chiral algebra, localization, modularity, surface defects, and all that, arXiv:1904.02704 [INSPIRE].
  32. [32]
    M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb branch of 3d N = 4 theories, Commun. Math. Phys.354 (2017) 671 [arXiv:1503.04817] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    C. Beem, D. Ben-Zvi, M. Bullimore, T. Dimofte and A. Neitzke, Secondary products in supersymmetric field theory, arXiv:1809.00009 [INSPIRE].
  34. [34]
    A. Johansen, Infinite conformal algebras in supersymmetric theories on four manifolds, Nucl. Phys.B 436 (1995) 291 [hep-th/9407109] [INSPIRE].
  35. [35]
    D. Xie, W. Yan and S.-T. Yau, Chiral algebra of Argyres-Douglas theory from M5 brane, arXiv:1604.02155 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.C.N. Yang Institute for Theoretical PhysicsStony Brook UniversityStony BrookU.S.A.

Personalised recommendations