Journal of High Energy Physics

, 2019:131 | Cite as

Gluing two affine Yangians of 𝔤𝔩1

  • Wei LiEmail author
  • Pietro Longhi
Open Access
Regular Article - Theoretical Physics


We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of 𝔤𝔩1. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the \( \mathcal{N} \) = 2 affine Yangian, which is isomorphic to the universal enveloping algebra of 𝔲 (1)⊕ 𝒲\( {}_{\infty}^{\mathcal{N}=2}\left[\lambda \right] \). All algebras that we construct have natural representations in terms of “twin plane partitions”, a pair of plane partitions appropriately joined along one common leg. We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds.


Conformal and W Symmetry Quantum Groups Topological Strings 


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


  1. [1]
    M.R. Gaberdiel and R. Gopakumar, An AdS3 Dual for Minimal Model CFTs, Phys. Rev.D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].ADSGoogle Scholar
  2. [2]
    M.R. Gaberdiel and R. Gopakumar, Minimal Model Holography, J. Phys.A 46 (2013) 214002 [arXiv:1207.6697] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  3. [3]
    M.A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions and two-dimensions, Int. J. Mod. Phys.D 5 (1996) 763 [hep-th/9611024] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    M.A. Vasiliev, Higher spin gauge theories: Star product and AdS space, hep-th/9910096 [INSPIRE].
  5. [5]
    F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Coset Construction for Extended Virasoro Algebras, Nucl. Phys.B 304 (1988) 371 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    C. Beem, L. Rastelli and B.C. van Rees, W symmetry in six dimensions, JHEP05 (2015) 017 [arXiv:1404.1079] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    D. Gaiotto and M. Rapčák, Vertex Algebras at the Corner, JHEP01 (2019) 160 [arXiv:1703.00982] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    O. Schiffmann and E. Vasserot, Cherednik algebras, 𝒲 -algebras and the equivariant cohomology of the moduli space of instantons on 𝔸2, Publications mathématiques de l’IHÉS118 (2013) 213.Google Scholar
  9. [9]
    D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE].
  10. [10]
    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].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    T. Procházka, 𝒲 -symmetry, topological vertex and affine Yangian, JHEP10 (2016) 077 [arXiv:1512.07178] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    M.R. Gaberdiel, R. Gopakumar, W. Li and C. Peng, Higher Spins and Yangian Symmetries, JHEP04 (2017) 152 [arXiv:1702.05100] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    K. Miki, A (q, γ) analog of the 𝒲1+algebra, J. Math. Phys.48 (2007) 123520.ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous 𝔤𝔩∞ Semi-infinite construction of representations, Kyoto J. Math.51 (2011) 337 [arXiv:1002.3100].MathSciNetCrossRefGoogle Scholar
  15. [15]
    B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous glTensor products of Fock modules and Wn characters, arXiv:1002.3113 [INSPIRE].
  16. [16]
    S. Datta, M.R. Gaberdiel, W. Li and C. Peng, Twisted sectors from plane partitions, JHEP09 (2016) 138 [arXiv:1606.07070] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal 𝔤𝔩1algebra: Plane partitions, Kyoto J. Math.52 (2012) 621.MathSciNetCrossRefGoogle Scholar
  18. [18]
    A. Tsymbaliuk, The affine Yangian of 𝔤𝔩1revisited, Adv. Math.304 (2017) 583 [arXiv:1404.5240] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  19. [19]
    M.R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP07 (2012) 127 [arXiv:1205.2472] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    T. Procházka and M. Rapčák, Webs of W-algebras, JHEP11 (2018) 109 [arXiv:1711.06888] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    T. Procházka and M. Rapčák, W -algebra modules, free fields and Gukov-Witten defects, JHEP05 (2019) 159 [arXiv:1808.08837] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    M.R. Gaberdiel, W. Li, C. Peng and H. Zhang, The supersymmetric affine Yangian, JHEP05 (2018) 200 [arXiv:1711.07449] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    M.R. Gaberdiel, W. Li and C. Peng, Twin-plane-partitions and \( \mathcal{N} \) = 2 affine Yangian, JHEP11 (2018) 192 [arXiv:1807.11304] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Num. Theor. Phys.5 (2011) 231 [arXiv:1006.2706] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    M. Rapčák, Y. Soibelman, Y. Yang and G. Zhao, Cohomological Hall algebras, vertex algebras and instantons, arXiv:1810.10402 [INSPIRE].
  26. [26]
    T. Procházka, Instanton R-matrix and W-symmetry, arXiv:1903.10372 [INSPIRE].
  27. [27]
    A. Neguƫ, The q-AGT-W relations via shuffle algebras, Commun. Math. Phys.358 (2018) 101 [arXiv:1608.08613] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    C. Candu and M.R. Gaberdiel, Duality in N = 2 Minimal Model Holography, JHEP02 (2013) 070 [arXiv:1207.6646] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    M.R. Gaberdiel, K. Jin and W. Li, Perturbations of W∞ CFTs, JHEP10 (2013) 162 [arXiv:1307.4087] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys.2 (1998) 91 [hep-th/9711013] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  31. [31]
    B. Szendroi, Non-commutative Donaldson-Thomas theory and the conifold, Geom. Topol.12 (2008) 1171 [arXiv:0705.3419] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  32. [32]
    D.L. Jafferis and G.W. Moore, Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909 [INSPIRE].
  33. [33]
    M. Aganagic, H. Ooguri, C. Vafa and M. Yamazaki, Wall Crossing and M-theory, Publ. Res. Inst. Math. Sci. Kyoto47 (2011) 569 [arXiv:0908.1194] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  34. [34]
    B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Representations of quantum toroidal 𝔤𝔩n, arXiv:1204.5378.
  35. [35]
    B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Branching rules for quantum toroidal gln , Adv. Math.300 (2016) 229 [arXiv:1309.2147] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  36. [36]
    M. Bershtein and A. Tsymbaliuk, Homomorphisms between different quantum toroidal and affine Yangian algebras arXiv:1512.09109.
  37. [37]
    B. Feigin and S. Gukov, VOA[M4 ], arXiv:1806.02470 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute of Theoretical PhysicsChinese Academy of SciencesBeijingP.R. China
  2. 2.Institut für Theoretische PhysikETH ZurichZürichSwitzerland

Personalised recommendations