Advertisement

Higgs and Coulomb branches from vertex operator algebras

  • Kevin Costello
  • Thomas CreutzigEmail author
  • Davide Gaiotto
Open Access
Regular Article - Theoretical Physics
  • 27 Downloads

Abstract

We formulate a conjectural relation between the category of line defects in topologically twisted 3d \( \mathcal{N} \) = 4 supersymmetric quantum field theories and categories of modules for Vertex Operator Algebras of boundary local operators for the theories. We test the conjecture in several examples and provide some partial proofs for standard classes of gauge theories.

Keywords

Conformal Field Theory Supersymmetric Gauge Theory Topological 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.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].CrossRefzbMATHGoogle Scholar
  2. [2]
    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].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  4. [4]
    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].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 [arXiv:1408.6522] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    F. Bonetti, C. Meneghelli and L. Rastelli, VOAs labelled by complex reflection groups and 4d SCFTs, arXiv:1810.03612 [INSPIRE].
  8. [8]
    D. Gaiotto, Twisted compactifications of 3d \( \mathcal{N} \) = 4 theories and conformal blocks, JHEP 02 (2019) 061 [arXiv:1611.01528] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    D. Gaiotto and M. Rapčák, Vertex Algebras at the Corner, JHEP 01 (2019) 160 [arXiv:1703.00982] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    T. Creutzig and D. Gaiotto, Vertex Algebras for S-duality, arXiv:1708.00875 [INSPIRE].
  11. [11]
    K. Costello and D. Gaiotto, Vertex Operator Algebras and 3d \( \mathcal{N} \) = 4 gauge theories, arXiv:1804.06460 [INSPIRE].
  12. [12]
    N. Seiberg, IR dynamics on branes and space-time geometry, Phys. Lett. B 384 (1996) 81 [hep-th/9606017] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics: A memorial volume for Claude Itzykson. Proceedings, Conference, Saclay, France, June 5–7, 1996, pp. 333–366 (1996) [hep-th/9607163] [INSPIRE].
  14. [14]
    J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, ℤ) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  18. [18]
    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].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, I, Adv. Theor. Math. Phys. 20 (2016) 595 [arXiv:1503.03676] [INSPIRE].CrossRefzbMATHGoogle Scholar
  20. [20]
    A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, II, arXiv:1601.03586 [INSPIRE].
  21. [21]
    J. Yagi, Ω-deformation and quantization, JHEP 08 (2014) 112 [arXiv:1405.6714] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    T. Creutzig and D. Ridout, Logarithmic Conformal Field Theory: Beyond an Introduction, J. Phys. A 46 (2013) 4006 [arXiv:1303.0847] [INSPIRE].zbMATHGoogle Scholar
  23. [23]
    H.G. Kausch, Symplectic fermions, Nucl. Phys. B 583 (2000) 513 [hep-th/0003029] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    D. Ridout, \( \widehat{sl}{(2)}_{-1/2} \) : A Case Study, Nucl. Phys. B 814 (2009) 485 [arXiv:0810.3532] [INSPIRE].
  25. [25]
    B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].CrossRefzbMATHGoogle Scholar
  26. [26]
    T. Creutzig and A.R. Linshaw, Cosets of affine vertex algebras inside larger structures, J. Algebra 517 (2019) 396 [arXiv:1407.8512] [INSPIRE].CrossRefzbMATHGoogle Scholar
  27. [27]
    T. Creutzig, D. Gaiotto and A.R. Linshaw, S-duality for the large N = 4 superconformal algebra, arXiv:1804.09821 [INSPIRE].
  28. [28]
    H.G. Kausch, Extended conformal algebras generated by a multiplet of primary fields, Phys. Lett. B 259 (1991) 448 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    D. Adamovic and A. Milas, Logarithmic intertwining operators and W(2, 2p − 1)-algebras, J. Math. Phys. 48 (2007) 073503 [math/0702081] [INSPIRE].
  30. [30]
    T. Creutzig and P.B. Ronne, The GL(1|1)-symplectic fermion correspondence, Nucl. Phys. B 815 (2009) 95 [arXiv:0812.2835] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    E. Witten, Two-dimensional models with (0, 2) supersymmetry: Perturbative aspects, Adv. Theor. Math. Phys. 11 (2007) 1 [hep-th/0504078] [INSPIRE].CrossRefzbMATHGoogle Scholar
  32. [32]
    V. Gorbounov, O. Gwilliam and B. Williams, Chiral differential operators via Batalin-Vilkovisky quantization, arXiv:1610.09657 [INSPIRE].
  33. [33]
    N. Chriss and V. Ginzburg, Representation theory and complex geometry, Birkhäuser (1997).Google Scholar
  34. [34]
    D. Gaiotto and E. Witten, Janus Configurations, Chern-Simons Couplings, And The theta-Angle in N = 4 Super Yang-Mills Theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    D. Adamovic and A. Milas, On the triplet vertex algebra W(p), Adv. Math. 217 (2008) 2664 [arXiv:0707.1857] [INSPIRE].CrossRefzbMATHGoogle Scholar
  36. [36]
    D. Adamovic and A. Milas, The Structure of Zhus algebras for certain W-algebras, arXiv:1006.5134 [INSPIRE].
  37. [37]
    B.L. Feigin, A.M. Gainutdinov, A.M. Semikhatov and I.Y. Tipunin, Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT, Theor. Math. Phys. 148 (2006) 1210 [math/0512621] [INSPIRE].
  38. [38]
    T. Arakawa, Rationality of admissible affine vertex algebras in the category \( \mathcal{O} \), Duke Math. J. 165 (2016) 67 [arXiv:1207.4857] [INSPIRE].CrossRefzbMATHGoogle Scholar
  39. [39]
    D. Adamović and A. Milas, Vertex operator algebras associated to modular invariant representations of A 1(1), q-alg/9509025.
  40. [40]
    T. Creutzig and D. Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models II, Nucl. Phys. B 875 (2013) 423 [arXiv:1306.4388] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    T. Creutzig and D. Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models I, Nucl. Phys. B 865 (2012) 83 [arXiv:1205.6513] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    B.L. Feigin and I. Yu. Tipunin, Logarithmic CFTs connected with simple Lie algebras, arXiv:1002.5047 [INSPIRE].
  43. [43]
    S.D. Lentner, Quantum groups and Nichols algebras acting on conformal field theories, arXiv:1702.06431 [INSPIRE].
  44. [44]
    T. Creutzig and A. Milas, Higher rank partial and false theta functions and representation theory, Adv. Math. 314 (2017) 203 [arXiv:1607.08563] [INSPIRE].CrossRefzbMATHGoogle Scholar
  45. [45]
    T. Creutzig, Logarithmic W-algebras and Argyres-Douglas theories at higher rank, JHEP 11 (2018) 188 [arXiv:1809.01725] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    T. Creutzig, A.M. Gainutdinov and I. Runkel, A quasi-Hopf algebra for the triplet vertex operator algebra, Commun. Contemp. Math. (2019) [arXiv:1712.07260] [INSPIRE].
  47. [47]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and their Generalized Modules, arXiv:1012.4193 [INSPIRE].
  48. [48]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators, arXiv:1012.4196 [INSPIRE].
  49. [49]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, III: Intertwining maps and tensor product bifunctors, arXiv:1012.4197 [INSPIRE].
  50. [50]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, IV: Constructions of tensor product bifunctors and the compatibility conditions, arXiv:1012.4198 [INSPIRE].
  51. [51]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition, arXiv:1012.4199 [INSPIRE].
  52. [52]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators and the associativity isomorphisms, arXiv:1012.4202 [INSPIRE].
  53. [53]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps, arXiv:1110.1929 [INSPIRE].
  54. [54]
    Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra, arXiv:1110.1931 [INSPIRE].
  55. [55]
    Y.-Z. Huang, A. Kirillov and J. Lepowsky, Braided tensor categories and extensions of vertex operator algebras, Commun. Math. Phys. 337 (2015) 1143 [arXiv:1406.3420] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    T. Creutzig, S. Kanade and R. McRae, Tensor categories for vertex operator superalgebra extensions, arXiv:1705.05017 [INSPIRE].
  57. [57]
    T. Creutzig, K. Shashank and A.R. Linshaw, Simple current extensions beyond semi-simplicity, Commun. Contemp. Math. (2015) [arXiv:1511.08754].
  58. [58]
    T. Creutzig, S. Kanade, A.R. Linshaw and D. Ridout, Schur-Weyl Duality for Heisenberg Cosets, Transform. Groups (2018) [arXiv:1611.00305] [INSPIRE].
  59. [59]
    T. Creutzig, D. Ridout and S. Wood, Coset Constructions of Logarithmic (1, p) Models, Lett. Math. Phys. 104 (2014) 553 [arXiv:1305.2665] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  60. [60]
    T. Creutzig and A. Milas, False Theta Functions and the Verlinde formula, Adv. Math. 262 (2014) 520 [arXiv:1309.6037] [INSPIRE].CrossRefzbMATHGoogle Scholar
  61. [61]
    T. Creutzig, A. Milas and M. Rupert, Logarithmic Link Invariants of \( {\overline{U}}_q^H\left(\mathfrak{s}{\mathfrak{l}}_2\right) \) and Asymptotic Dimensions of Singlet Vertex Algebras, J. Pure Appl. Algebra 222 (2018) 3224 [arXiv:1605.05634] [INSPIRE].CrossRefzbMATHGoogle Scholar
  62. [62]
    J. Auger and M. Rupert, On infinite order simple current extensions of vertex operator algebras, Contemp. Math. 711 (2018) [arXiv:1711.05343].
  63. [63]
    T. Creutzig, W-algebras for Argyres-Douglas theories, Eur. J. Math. 3 (2017) 659 [arXiv:1701.05926] [INSPIRE].CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of MathematicsUniversity of AlbertaEdmontonCanada

Personalised recommendations