Advertisement

Ω-deformation and quantization

  • Junya Yagi
Open Access
Article

Abstract

We formulate a deformation of Rozansky-Witten theory analogous to the Ω-deformation. It is applicable when the target space X is hyperkähler and the spacetime is of the form ℝ×Σ, with Σ being a Riemann surface. In the case that Σ is a disk, the Ω-deformed Rozansky-Witten theory quantizes a symplectic submanifold of X, thereby providing a new perspective on quantization. As applications, we elucidate two phenomena in four- dimensional gauge theory from this point of view. One is a correspondence between the Ω-deformation and quantization of integrable systems. The other concerns supersymmetric loop operators and quantization of the algebra of holomorphic functions on a hyperkähler manifold.

Keywords

Supersymmetric gauge theory Integrable Field Theories 

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]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics, Saclay (1996), vol. 24 of Adv. Ser. Math. Phys., World Science Publisher, River Edge, NJ (1997), pg. 333. [hep-th/9607163] [INSPIRE].
  2. [2]
    R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in XVIth International Congress on Mathematical Physics, World Science Publisher, Hackensack, NJ (2010), pg. 264 [arXiv:0908.4052] [INSPIRE].Google Scholar
  4. [4]
    D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].CrossRefMATHGoogle Scholar
  5. [5]
    Y. Ito, T. Okuda and M. Taki, Line operators on S 1 × R 3 and quantization of the Hitchin moduli space, JHEP 04 (2012) 010 [arXiv:1111.4221] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    L. Rozansky and E. Witten, HyperKähler geometry and invariants of three manifolds, Selecta Math. 3 (1997) 401 [hep-th/9612216] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    C. Vafa, Topological Landau-Ginzburg models, Mod. Phys. Lett. A 6 (1991) 337 [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    S. Gukov and E. Witten, Branes and Quantization, Adv. Theor. Math. Phys. 13 (2009) 1445 [arXiv:0809.0305] [INSPIRE].CrossRefMATHMathSciNetADSGoogle Scholar
  9. [9]
    N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin Systems via β-deformed Matrix Models, arXiv:1104.4016 [INSPIRE].
  11. [11]
    M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum Geometry of Refined Topological Strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    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].CrossRefADSGoogle Scholar
  13. [13]
    A. Kapustin and N. Saulina, Chern-Simons-Rozansky-Witten topological field theory, Nucl. Phys. B 823 (2009) 403 [arXiv:0904.1447] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    D. Gaiotto and E. Witten, Janus Configurations, Chern-Simons Couplings, And The θ-Angle in \( \mathcal{N} \) = 4 Super Yang-Mills Theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  15. [15]
    T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].CrossRefMATHMathSciNetADSGoogle Scholar
  16. [16]
    Y. Terashima and M. Yamazaki, SL(2, ℝ) Chern-Simons, Liouville and Gauge Theory on Duality Walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].CrossRefADSGoogle Scholar
  17. [17]
    T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].CrossRefMATHMathSciNetADSGoogle Scholar
  19. [19]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].CrossRefMATHGoogle Scholar
  20. [20]
    J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    S. Lee and M. Yamazaki, 3d Chern-Simons Theory from M5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  22. [22]
    C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the Squashed Three-Sphere, arXiv:1305.2891 [INSPIRE].
  23. [23]
    S. Shadchin, On F-term contribution to effective action, JHEP 08 (2007) 052 [hep-th/0611278] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  24. [24]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  25. [25]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, in The unity of mathematics, vol. 244 of Progr. Math., Birkhäuser Boston, Boston, MA (2006) 525 [hep-th/0306238] [INSPIRE].
  26. [26]
    E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].CrossRefMATHMathSciNetADSGoogle Scholar
  27. [27]
    S. Hyun, J. Park and J.-S. Park, N=2 supersymmetric QCD and four manifolds: 1. The Donaldson and Seiberg-Witten invariants, hep-th/9508162 [INSPIRE].
  28. [28]
    J.M.F. Labastida and M. Mariño, Twisted N = 2 supersymmetry with central charge and equivariant cohomology, Commun. Math. Phys. 185 (1997) 37 [hep-th/9603169] [INSPIRE].CrossRefMATHADSGoogle Scholar
  29. [29]
    C. Closset and S. Cremonesi, Comments on = (2, 2) supersymmetry on two-manifolds, JHEP 07 (2014) 075 [arXiv:1404.2636] [INSPIRE].CrossRefADSGoogle Scholar
  30. [30]
    J.M.F. Labastida and P.M. Llatas, Topological matter in two-dimensions, Nucl. Phys. B 379 (1992) 220 [hep-th/9112051] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  31. [31]
    K. Hori and M. Romo, Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary, arXiv:1308.2438 [INSPIRE].
  32. [32]
    G. Thompson, On the generalized Casson invariant, Adv. Theor. Math. Phys. 3 (1999) 249 [hep-th/9811199] [INSPIRE].MATHMathSciNetGoogle Scholar
  33. [33]
    Y. Luo, M.-C. Tan and J. Yagi, \( \mathcal{N} \) = 2 supersymmetric gauge theories and quantum integrable systems, JHEP 03 (2014) 090 [arXiv:1310.0827] [INSPIRE].CrossRefADSGoogle Scholar
  34. [34]
    J. Gomis and S. Lee, Exact Kähler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar
  35. [35]
    A. Kapustin and D. Orlov, Remarks on A branes, mirror symmetry and the Fukaya category, J. Geom. Phys. 48 (2003) 84 [hep-th/0109098] [INSPIRE].CrossRefMATHMathSciNetADSGoogle Scholar
  36. [36]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].CrossRefMATHMathSciNetADSGoogle Scholar
  37. [37]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].CrossRefMathSciNetADSGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.International School for Advanced Studies (SISSA)TriesteItaly
  2. 2.INFN, Sezione di TriesteTriesteItaly

Personalised recommendations