Chiral algebras from Ω-deformation

  • Jihwan Oh
  • Junya YagiEmail author
Open Access
Regular Article - Theoretical Physics


In the presence of an Ω-deformation, local operators generate a chiral algebra in the topological-holomorphic twist of a four-dimensional \( \mathcal{N} \) = 2 supersymmetric field theory. We show that for a unitary \( \mathcal{N} \) = 2 superconformal field theory, the chiral algebra thus defined is isomorphic to the one introduced by Beem et al. Our definition of the chiral algebra covers nonconformal theories with insertions of suitable surface defects.


Conformal and W Symmetry Conformal Field Theory Supersymmetric Gauge Theory 


Open Access

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  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].
  2. [2]
    P. Liendo, I. Ramírez and J. Seo, Stress-tensor OPE in \( \mathcal{N} \) = 2 superconformal theories, JHEP02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
  3. [3]
    M. Lemos and P. Liendo, \( \mathcal{N} \) = 2 central charge bounds from 2d chiral algebras, JHEP04 (2016) 004 [arXiv:1511.07449] [INSPIRE].
  4. [4]
    A. Kapustin, Holomorphic reduction of \( \mathcal{N} \) = 2 gauge theories, Wilson-’t Hooft operators and S-duality, hep-th/0612119 [INSPIRE].
  5. [5]
    E. Witten, Topological quantum field theory, Commun. Math. Phys.117 (1988) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    E. Witten, Topological σ-models, Commun. Math. Phys.118 (1988) 411 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    K. Costello and D. Gaiotto, Vertex Operator Algebras and 3d \( \mathcal{N} \) = 4 gauge theories, JHEP05 (2019) 018 [arXiv:1804.06460] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  9. [9]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math.244 (2006) 525 [hep-th/0306238] [INSPIRE].
  10. [10]
    D. Butson, Omega backgrounds and boundary theories in twisted supersymmetric gauge theories, in preparation.Google Scholar
  11. [11]
    L. Rozansky and E. Witten, HyperKähler geometry and invariants of three manifolds, Selecta Math.3 (1997) 401 [hep-th/9612216] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  12. [12]
    C. Beem, W. Peelaers and L. Rastelli, Deformation quantization and superconformal symmetry in three dimensions, Commun. Math. Phys.354 (2017) 345 [arXiv:1601.05378] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    M. Dedushenko, S.S. Pufu and R. Yacoby, A one-dimensional theory for Higgs branch operators, JHEP03 (2018) 138 [arXiv:1610.00740] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  14. [14]
    C. Vafa, Topological Landau-Ginzburg models, Mod. Phys. Lett.A 6 (1991) 337 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    E. Witten, Mirror manifolds and topological field theory, hep-th/9112056 [INSPIRE].
  16. [16]
    J. Yagi, Ω-deformation and quantization, JHEP08 (2014) 112 [arXiv:1405.6714] [INSPIRE].
  17. [17]
    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
  18. [18]
    N. Nekrasov, Tying up instantons with anti-instantons, arXiv:1802.04202, [INSPIRE].
  19. [19]
    K. Costello and J. Yagi, Unification of integrability in supersymmetric gauge theories, arXiv:1810.01970 [INSPIRE].
  20. [20]
    C. Cordova, D. Gaiotto and S.-H. Shao, Surface defects and chiral algebras, JHEP05 (2017) 140 [arXiv:1704.01955] [INSPIRE].
  21. [21]
    Y. Pan and W. Peelaers, Chiral algebras, localization and surface defects, JHEP02 (2018) 138 [arXiv:1710.04306] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    D. Friedan, E.J. Martinec and S.H. Shenker, Conformal invariance, supersymmetry and string theory, Nucl. Phys.B 271 (1986) 93 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    Y. Pan and W. Peelaers, Schur correlation functions on S 3 × S 1, JHEP07 (2019) 013 [arXiv:1903.03623] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Dedushenko and M. Fluder, Chiral algebra, localization, modularity, surface defects, and all that, arXiv:1904.02704 [INSPIRE].
  25. [25]
    E. Witten, Two-dimensional models with (0, 2) supersymmetry: perturbative aspects, Adv. Theor. Math. Phys.11 (2007) 1 [hep-th/0504078] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    M.-C. Tan, Two-dimensional twisted σ-models and the theory of chiral differential operators, Adv. Theor. Math. Phys.10 (2006) 759 [hep-th/0604179] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  27. [27]
    M.-C. Tan and J. Yagi, Chiral algebras of (0, 2) σ-models: beyond perturbation theory, Lett. Math. Phys.84 (2008) 257 [arXiv:0801.4782] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    M.-C. Tan and J. Yagi, Chiral algebras of (0, 2) σ-models: beyond perturbation theory, Lett. Math. Phys.84 (2008) 257 [arXiv:0801.4782] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    J. Yagi, Chiral algebras of (0, 2) models, Adv. Theor. Math. Phys.16 (2012) 1 [arXiv:1001.0118] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  30. [30]
    M. Del Zotto and G. Lockhart, Universal features of BPS strings in six-dimensional SCFTs, JHEP08 (2018) 173 [arXiv:1804.09694] [INSPIRE].
  31. [31]
    C.G. Callan Jr. and J.A. Harvey, Anomalies and fermion zero modes on strings and domain walls, Nucl. Phys.B 250 (1985) 427 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    A. Kapustin and K. Vyas, A-models in three and four dimensions, arXiv:1002.4241 [INSPIRE].
  33. [33]
    M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb branch of 3d \( \mathcal{N} \)= 4 theories, Commun. Math. Phys.354 (2017) 671 [arXiv:1503.04817] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, mirror symmetry and symplectic duality in 3d \( \mathcal{N} \)= 4 gauge theory, JHEP10 (2016) 108 [arXiv:1603.08382] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn and H.-C. Kim, Vortices and Vermas, Adv. Theor. Math. Phys.22 (2018) 803 [arXiv:1609.04406] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  36. [36]
    K. Costello, M-theory in the Ω-background and 5-dimensional non-commutative gauge theory, arXiv:1610.04144 [INSPIRE].
  37. [37]
    K. Costello, Holography and Koszul duality: the example of the M 2 brane, arXiv:1705.02500 [INSPIRE].
  38. [38]
    M. Dedushenko, Y. Fan, S.S. Pufu and R. Yacoby, Coulomb branch operators and mirror symmetry in three dimensions, JHEP04 (2018) 037 [arXiv:1712.09384] [INSPIRE].
  39. [39]
    C. Beem, D. Ben-Zvi, M. Bullimore, T. Dimofte and A. Neitzke, Secondary products in supersymmetric field theory, arXiv:1809.00009 [INSPIRE].
  40. [40]
    M. Dedushenko, Y. Fan, S.S. Pufu and R. Yacoby, Coulomb branch quantization and abelianized monopole bubbling, arXiv:1812.08788 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of PhysicsUniversity of CaliforniaBerkeleyU.S.A.

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