Journal of High Energy Physics

, 2019:100 | Cite as

Cluster integrable systems and spin chains

  • A. Marshakov
  • M. SemenyakinEmail author
Open Access
Regular Article - Theoretical Physics


We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that \( {\mathfrak{gl}}_N \) XXZ-type spin chain on M sites is isomorphic to a cluster integrable system with N × M rectangular Newton polygon and N × M fundamental domain of a ‘fence net’ bipartite graph. The Casimir functions of the Poisson bracket, labeled by the zig-zag paths on the graph, correspond to the inhomogeneities, on-site Casimirs and twists of the chain, supplemented by total spin. The symmetricity of cluster formulation implies natural spectral duality, relating \( {\mathfrak{gl}}_N \) -chain on M sites with the \( {\mathfrak{gl}}_M \) -chain on N sites. For these systems we construct explicitly a subgroup of the cluster mapping class group \( {\mathcal{G}}_{\mathcal{Q}} \) and show that it acts by permutations of zig-zags and, as a consequence, by permutations of twists and inhomogeneities. Finally, we derive Hirota bilinear equations, describing dynamics of the tau-functions or A-cluster variables under the action of some generators of \( {\mathcal{G}}_{\mathcal{Q}} \).


Quantum Groups Supersymmetric Gauge Theory 


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. Bershtein, P. Gavrylenko and A. Marshakov, Cluster integrable systems, q-Painlevé equations and their quantization, JHEP02 (2018) 077 [arXiv:1711.02063] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    M. Bershtein, P. Gavrylenko and A. Marshakov, Cluster Toda chains and Nekrasov functions, Theor. Math. Phys.198 (2019) 157 [arXiv:1804.10145] [INSPIRE].CrossRefGoogle Scholar
  3. [3]
    G. Bonelli, A. Grassi and A. Tanzini, Quantum curves and q-deformed Painlevé equations, Lett. Math. Phys.109 (2019) 1961 [arXiv:1710.11603] [INSPIRE].
  4. [4]
    L. Bao, E. Pomoni, M. Taki and F. Yagi, M5-Branes, Toric Diagrams and Gauge Theory Duality, JHEP04 (2012) 105 [arXiv:1112.5228] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    V.V. Bazhanov and S.M. Sergeev, Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups, J. Phys.A 39 (2006) 3295 [hep-th/0509181] [INSPIRE].
  6. [6]
    M. Bershtein and A. Shchechkin, q-deformed Painlevé τ function and q-deformed conformal blocks, J. Phys.A 50 (2017) 085202 [arXiv:1608.02566] [INSPIRE].
  7. [7]
    M. Bershtein and A. Shchechkin, Painleve equations from Nakajima-Yoshioka blow-up relations, arXiv:1811.04050 [INSPIRE].
  8. [8]
    J.T. Ding and I.B. Frenkel, Isomorphism of two realizations of quantum affine algebra Uq (\( \mathfrak{gl} \) (n)), Commun. Math. Phys.156 (1993) 277 [].
  9. [9]
    R. Eager, S. Franco and K. Schaeffer, Dimer Models and Integrable Systems, JHEP06 (2012) 106 [arXiv:1107.1244] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    V.V. Fock, Inverse spectral problem for GK integrable system, arXiv:1503.00289.
  11. [11]
    V.V. Fock and A.B. Goncharov, Cluster χ-varieties, amalgamation, and Poisson-Lie groups, in Algebraic Geometry Theory and Number Theory , Progress in Mathematics Series, volume 253, V. Ginzburg ed., Birkhäuser, Boston MA U.S.A. (2006), pp. 27–68 [math.RT/0508408].
  12. [12]
    V.V. Fock and A. Marshakov, A Note on Quantum Groups and Relativistic Toda Theory, Nucl. Phys. Proc. Suppl.B 56 (1997) 208 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    V.V. Fock and A. Marshakov, Loop groups, Clusters, Dimers and Integrable systems, in Geometry and Quantization of Moduli Spaces , Advanced Courses in Mathematics — CRM Barcelona Series, L. Alvarez Consul, J. Andersen and I. Mundet i Riera eds., Birkhäuser, Cham Switzerland (2016), pp. 1–65 [arXiv:1401.1606] [INSPIRE].
  14. [14]
    S. Fomin and A. Zelevinsky, Cluster algebras IV: Coefficients, Compos. Math.143 (2007) 112 [math.RA/0602259].
  15. [15]
    S. Franco, Y. Hatsuda and M. Mariño, Exact quantization conditions for cluster integrable systems, J. Stat. Mech.1606 (2016) 063107 [arXiv:1512.03061] [INSPIRE].
  16. [16]
    L.D. Faddeev, N.Y. Reshetikhin and L.A. Takhtajan, Quantization of Lie Groups and Lie Algebras, Leningrad Math. J.1 (1990) 193 [Alg. Anal.1 (1989) 178] [INSPIRE].
  17. [17]
    A.B. Goncharov and R. Kenyon, Dimers and cluster integrable systems, Ann. Sci. Ec. Norm. Sup.46 (2013) 747 [arXiv:1107.5588] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  18. [18]
    O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP10 (2012) 038 [Erratum JHEP10 (2012) 183] [arXiv:1207.0787] [INSPIRE].
  19. [19]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  20. [20]
    P. Gavrylenko, M. Semenyakin and Y. Zenkevich, Bazhanov-Sergeev solution of tetrahedron equation from cluster algebras, to appear.Google Scholar
  21. [21]
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett.B 355 (1995) 466 [hep-th/9505035] [INSPIRE].
  22. [22]
    A. Hone and R. Inoue, Discrete Painlevé equations from Y -systems, J. Phys.A 47 (2014) 474007 [arXiv:1405.5379].
  23. [23]
    R. Inoue, T. Ishibashi and H. Oya, Cluster realizations of Weyl groups and higher Teichmüller theory, arXiv:1902.02716.
  24. [24]
    M. Jimbo, H. Nagoya and H. Sakai, CFT approach to the q-Painlevé VI equation, J. Integr. Syst.2 (2017) xyx009 [arXiv:1706.01940].
  25. [25]
    S. Kharchev, Notes on quantum groups, unpublished.Google Scholar
  26. [26]
    A. Marshakov, Lie Groups, Cluster Variables and Integrable Systems, J. Geom. Phys.67 (2013) 16 [arXiv:1207.1869] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    Y. Matsuhira and H. Nagoya, Combinatorial expressions for the tau functions of q-Painlevé V and III equations, arXiv:1811.03285 [INSPIRE].
  28. [28]
    A. Marshakov and A. Mironov, 5d and 6d supersymmetric gauge theories: Prepotentials from integrable systems, Nucl. Phys.B 518 (1998) 59 [hep-th/9711156] [INSPIRE].
  29. [29]
    A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral dualities in XXZ spin chains and five dimensional gauge theories, JHEP12 (2013) 034 [arXiv:1307.1502] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys.B 531 (1998) 323 [hep-th/9609219] [INSPIRE].
  31. [31]
    N. Okubo, Bilinear equations and q-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras, J. Phys.A 48 (2015) 355201 [arXiv:1505.03067] [INSPIRE].
  32. [32]
    N. Okubo, Co-primeness preserving higher dimensional extension of q-discrete Painlevé I, II equations, arXiv:1704.05403.
  33. [33]
    N. Okubo and T. Suzuki, Generalized q-Painlevé VI systems of type (A 2n+1 + A 1 + A 1)(1)arising from cluster algebra, arXiv:1810.03252.
  34. [34]
    A. Oskin, S. Pakuliak and A. Silantyev, On the universal weight function for the quantum affine algebra Uq \( \left(\hat{\mathfrak{g}}{\mathfrak{l}}_N\right) \), Lett. Math. Phys.91 (2010) 167 [arXiv:0711.2821].MathSciNetCrossRefGoogle Scholar
  35. [35]
    A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math.244 (2006) 597 [hep-th/0309208] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  36. [36]
    S.N.M. Ruijsenaars, Relativistic Toda systems, Commun. Math. Phys.133 (1990) 212 [].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett.B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
  38. [38]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys.B 426 (1994) 19 [Erratum ibid.B 430 (1994) 485] [hep-th/9407087] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Advanced Studies, SkoltechMoscowRussia
  2. 2.Faculty of Mathematics, NRU HSEMoscowRussia
  3. 3.Institute for Theoretical and Experimental PhysicsMoscowRussia
  4. 4.Theory Department of Lebedev Physics InstituteMoscowRussia

Personalised recommendations