An elliptic vertex of Awata-Feigin-Shiraishi type for M-strings

  • Rui-Dong ZhuEmail author
Open Access
Regular Article - Theoretical Physics


We write down a vertical representation for the elliptic Ding-Iohara-Miki algebra, and construct an elliptic version of the refined topological vertex of Awata, Feigin and Shiraishi. We show explicitly that this vertex reproduces the elliptic genus of M-strings, and that it is an intertwiner of the algebra.


Quantum Groups Topological Strings Conformal and W Symmetry String Duality 


Open Access

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  1. [1]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    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
  3. [3]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A. Iqbal, C. Kozcaz and C. Vafa, The Refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    H. Awata, B. Feigin and J. Shiraishi, Quantum Algebraic Approach to Refined Topological Vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    J.-t. Ding and K. Iohara, Generalization and deformation of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997) 181 [q-alg/9608002] [INSPIRE].
  8. [8]
    K. Miki, A (q, γ) analog of the W 1+∞ algebra, J. Math. Phys. 48 (2007) 3520.ADSCrossRefGoogle Scholar
  9. [9]
    T. Procházka, \( \mathcal{W} \) -symmetry, topological vertex and affine Yangian, JHEP 10 (2016) 077 [arXiv:1512.07178] [INSPIRE].
  10. [10]
    M. Fukuda, S. Nakamura, Y. Matsuo and R.-D. Zhu, SH c realization of minimal model CFT: triality, poset and Burge condition, JHEP 11 (2015) 168 [arXiv:1509.01000] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].ADSGoogle Scholar
  12. [12]
    O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    E. Frenkel and N. Reshetikhin, Quantum affine algebras and deformations of the Virasoro and \( \mathcal{W} \) -algebras, Commun. Math. Phys. 178 (1996) 237 [q-alg/9505025].
  17. [17]
    S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4-D N = 2 gauge theories: 1., Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, (p, q)-webs of DIM representations, 5d N = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
  19. [19]
    D. Gaiotto and M. Rapčák, Vertex Algebras at the Corner, arXiv:1703.00982 [INSPIRE].
  20. [20]
    B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-Strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    F. Nieri, An elliptic Virasoro symmetry in 6d, Lett. Math. Phys. 107 (2017) 2147 [arXiv:1511.00574] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    A. Iqbal, C. Kozcaz and S.-T. Yau, Elliptic Virasoro Conformal Blocks, arXiv:1511.00458 [INSPIRE].
  23. [23]
    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].ADSGoogle Scholar
  25. [25]
    Y. Saito, Elliptic Ding-Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator, arXiv:1301.4912.
  26. [26]
    B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi and S. Yangida, A commutative algebra on degenerate CP 1 and Macdonald polynomials, J. Math. Phys. 50 (2009) 095215 [arXiv:0904.2291].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    J.-E. Bourgine, M. Fukuda, Y. Matsuo and R.-D. Zhu, Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver, JHEP 12 (2017) 015 [arXiv:1709.01954] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    L. Clavelli and J.A. Shapiro, Pomeron factorization in general dual models, Nucl. Phys. B 57 (1973) 490 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    H. Fan, B.-y. Hou, K.-j. Shi and W.-l. Yang, Bosonization of vertex operators for Z n symmetric Belavin model and its correlation functions, J. Phys. A 30 (1997) 5687 [hep-th/9703126] [INSPIRE].ADSzbMATHGoogle Scholar
  30. [30]
    O. Foda and R.-D. Zhu, An elliptic topological vertex, arXiv:1805.12073 [INSPIRE].
  31. [31]
    T. Kimura and V. Pestun, Quiver elliptic W-algebras, Lett. Math. Phys. 108 (2018) 1383 [arXiv:1608.04651] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    T. Kimura, H. Mori and Y. Sugimoto, Refined geometric transition and qq-characters, JHEP 01 (2018) 025 [arXiv:1705.03467] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang and R.-D. Zhu, Coherent states in quantum \( {\mathcal{W}}_{1+\infty } \) algebra and qq-character for 5d Super Yang-Mills, PTEP 2016 (2016) 123B05 [arXiv:1606.08020] [INSPIRE].
  34. [34]
    K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    L. Bhardwaj, Classification of 6d \( \mathcal{N} \) = (1, 0) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
  36. [36]
    M.-x. Huang, K. Sun and X. Wang, Blowup Equations for Refined Topological Strings, arXiv:1711.09884 [INSPIRE].
  37. [37]
    H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP 03 (2018) 192 [arXiv:1712.08016] [INSPIRE].
  38. [38]
    P.H. Ginsparg, Applied conformal field theory, in Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, Les Houches, France, June 28-August 5, 1988, pp. 1-168 (1988) [hep-th/9108028] [INSPIRE].

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhysicsThe University of TokyoTokyoJapan

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