The quantum DELL system

  • Peter KoroteevEmail author
  • Shamil Shakirov


We propose quantum Hamiltonians of the double-elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the classical DELL system which was previously found in the string theory literature. The eigenfunctions for the N-body model are instanton partition functions of 6d SU(N) gauge theory with adjoint matter compactified on a torus with a codimension-two defect. As a by-product, we discover new family of symmetric orthogonal polynomials which provide an elliptic generalization to Macdonald polynomials.


Integrable systems Supersymmetric gauge theories Quantum K-theory Quantum hydrodynamics Geometric representation theory Elliptic cohomology 

Mathematics Subject Classification

14N35 17B37 20G42 76Y05 81R12 81T30 81T60 81R50 



This manuscript took some time to complete primarily due to the technical computational difficulties of expressions involving double-periodic functions. We would like to thank numerous people with whom we discussed these and other matters, as well as various institutions which we visited in the last several years, but our special acknowledgements go to Babak Haghigat, Wenbin Yan, Can Kozcaz and Francesco Benini who participated in earlier stages of the project.


  1. 1.
    Aminov, G., Braden, H.W., Mironov, A., Morozov, A., Zotov, A.: Seiberg–Witten curves and double-elliptic integrable systems. JHEP 01, 033 (2015). arXiv:1410.0698 ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alday, L.F., Gaiotto, D., Gukov, S., Tachikawa, Y., Verlinde, H.: Loop and surface operators in N=2 gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010). arXiv:0909.0945 ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aharony, O.: A brief review of “little string theories”. Class. Quant. Gravity 17, 929–938 (2000)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Aminov, G., Mironov, A., Morozov, A.: New non-linear equations and modular form expansion for double-elliptic SeibergWitten prepotential. Eur. Phys. J. C 76(8), 433 (2016). arXiv:1606.05274 ADSCrossRefGoogle Scholar
  5. 5.
    Aminov, G., Mironov, A., Morozov, A.: Modular properties of 6d (DELL) systems. JHEP 11, 023 (2017). arXiv:1709.04897 ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Aminov, G., Mironov, A., Morozov, A., Zotov, A.: Three-particle integrable systems with elliptic dependence on momenta and theta function identities. Phys. Lett. B 726, 802–808 (2013). arXiv:1307.1465 ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Aganagic, M., Okounkov, A.: Elliptic stable envelope (2016). arXiv:1604.00423
  8. 8.
    Aganagic, M., Shakirov, S.: Knot homology and refined Chern–Simons index. Commun. Math. Phys. 333(1), 187–228 (2015). arXiv:1105.5117 ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Alday, L.F., Tachikawa, Y.: Affine SL(2) conformal blocks from 4d gauge theories. Lett. Math. Phys. 94, 87–114 (2010). arXiv:1005.4469 ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Braden, H., Gorsky, A., Odessky, A., Rubtsov, V.: Double elliptic dynamical systems from generalized Mukai–Sklyanin algebras. Nucl. Phys. B 633, 414–442 (2002). arXiv:hep-th/0111066 ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Braden, H.W., Hollowood, T.J.: The curve of compactified 6-D gauge theories and integrable systems. JHEP 0312, 023 (2003). arXiv:hep-th/0311024 ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bullimore, M., Kim, H.C., Koroteev, P.: Defects and quantum Seiberg–Witten geometry. JHEP 05, 095 (2015). arXiv:1412.6081 ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Braden, H., Marshakov, A., Mironov, A., Morozov, A.: On double elliptic integrable systems. 1. A duality argument for the case of SU(2). Nucl. Phys. 573, 553–572 (1999) ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Donagi, R., Witten, E.: Supersymmetric Yang–Mills theory and integrable systems. Nucl. Phys. B 460, 299 (1996). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Etingof, P., Kirillov Jr. A.: On the affine analogue of Jack’s and Macdonald’s polynomials. arXiv:hep-th/9403168
  16. 16.
    Etingof, P.: Difference equations with elliptic coefficients and quantum affine algebras. arXiv:hep-th/9312057
  17. 17.
    Gaiotto, D., Gukov, S., Seiberg, N.: Surface defects and resolvents. JHEP 1309, 070 (2013). arXiv:1307.2578 ADSCrossRefGoogle Scholar
  18. 18.
    Gaiotto, D., Koroteev, P.: On three dimensional quiver gauge theories and integrability. JHEP 1305, 126 (2013). arXiv:1304.0779 ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg–Witten exact solution. Phys. Lett. B 355, 466–474 (1995). arXiv:hep-th/9505035 ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Hitchin, N.: Stable bundles and integrable systems. Duke Math. J. 54(1), 91 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). arXiv:hep-th/0310272 ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Haghighat, B., Kim, J., Yan, W., Yau, S.T.: D-type fiber-base duality. JHEP 09, 060 (2018). arXiv:1806.10335 ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Haghighat, B., Yan, W.: M-strings in thermodynamic limit: Seiberg–Witten geometry (2016). arXiv:1607.07873
  24. 24.
    Inozemtsev, V.I.: The finite Toda lattices. Commun. Math. Phys. 121(4), 629–638 (1989)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kim, J., Kim, S., Lee, K.: Higgsing towards e-strings (2015). arXiv:1510.03128
  26. 26.
    Koroteev, P.: A-type quiver varieties and ADHM moduli spaces (2018). arXiv:1805.00986
  27. 27.
    Koroteev, P., Pushkar, P.P., Smirnov, A., Zeitlin, A.M.: Quantum K-theory of quiver varieties and many-body systems (2017). arXiv:1705.10419
  28. 28.
    Koroteev, P., Sciarappa, A.: On elliptic algebras and large-n supersymmetric gauge theories. J. Math. Phys. 57, 112302 (2016). arXiv:1601.08238 ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Koroteev, P., Sciarappa, A.: Quantum hydrodynamics from large-n supersymmetric gauge theories. Lett. Math. Phys. 108, 45–95 (2018). arXiv:1510.00972 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Losev, A., Moore, G., Shatashvili, S.L.: M&m’s. Nucl. Phys. B 522, 105–124 (1998)ADSCrossRefGoogle Scholar
  31. 31.
    Mironov, A.: Seiberg-Witten theory and duality in integrable systems. In: 34th Annual Winter School on Nuclear and Particle Physics (PNPI 2000) Gatchina, Russia, February 14–20 (2000)Google Scholar
  32. 32.
    Mironov, A., Morozov, A.: Commuting hamiltonians from Seiberg–Witten theta functions. Phys. Lett. B 475, 71–76 (2000). arXiv:hep-th/9912088 ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Mironov, A., Morozov, A.: Double elliptic systems: problems and perspectives (1999). arXiv:hep-th/0001168
  34. 34.
    Martinec, E.J., Warner, N.P.: Integrable systems and supersymmetric gauge theory. Nucl. Phys. B 459, 97–112 (1996). arXiv:hep-th/9509161 ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Nawata, S.: Givental J-functions, quantum integrable systems, AGT relation with surface operator (2014). arXiv:1408.4132
  36. 36.
    Negut, A.: Laumon spaces and the Calogero-Sutherland integrable system. Invent. Math. 178(2), 299–331 (2009). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). arXiv:hep-th/0206161(To Arkady Vainshtein on his 60th anniversary) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Nekrasov, N.: BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and qq-characters. JHEP 03, 181 (2016). arXiv:1512.05388 ADSCrossRefGoogle Scholar
  39. 39.
    Nekrasov, N.: BPS/CFT correspondence IV: sigma models and defects in gauge theory. Lett. Math. Phys. (2017). arXiv:1711.11011
  40. 40.
    Nekrasov, N.: BPS/CFT correspondence V: BPZ and KZ equations from qq-characters (2017). arXiv:1711.11582
  41. 41.
    Nieri, F.: An elliptic Virasoro symmetry in 6d (2015). arXiv:1511.00574
  42. 42.
    Nekrasov, N., Okounkov, A.: Seiberg–Witten theory and random partitions. In: Etingof, P., Retakh, V.S., Singer, I.M. (eds.) The Unity of Mathematics. Springer, Berlin (2003). arXiv:hep-th/0306238
  43. 43.
    Nieri, F., Pasquetti, S.: Factorisation and holomorphic blocks in 4d. JHEP 11, 155 (2015). arXiv:1507.00261 ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum geometry and quiver gauge theories. Commun. Math. Phys. 357(2), 519–567 (2018). arXiv:1312.6689 ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories (2009). arXiv:0908.4052
  46. 46.
    Seiberg, N.: Matrix description of m-theory on t5t5 and t5/z2t5/z2. Phys. Lett. B 408, 98–104 (1997)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Seiberg, N., Witten, E.: Monopole condensation, and confinement in N=2 supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19–52 (1994). arXiv:hep-th/9407087 ADSCrossRefGoogle Scholar
  48. 48.
    Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994). arXiv:hep-th/9408099 ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Witten, E.: Some comments on string dynamics (1995)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California BerkeleyBerkeleyUSA
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations