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Zamolodchikov periodicity and integrability Pavel Galashin MIT galashin@mit.edu Hunter College, May 7, 2017 Joint work with Pavlo Pylyavskyy 2 2 2 1 1 2 2 3 2 2 2 1 2 Pavel Galashin (MIT) Zamolodchikov periodicity and

  1. Zamolodchikov periodicity and integrability Pavel Galashin MIT galashin@mit.edu Hunter College, May 7, 2017 Joint work with Pavlo Pylyavskyy 2 2 2 1 1 2 2 3 2 2 2 1 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 1 / 26

  2. General T -systems (Nakanishi, 2011) Q x 1 x 5 1 1 x 1 x 2 x 5 2 5 2 5 x 4 x 2 3 4 3 4 x 3 x 4 x 3 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

  3. General T -systems (Nakanishi, 2011) Q µ 1 ( Q ) x 1 x 5 1 1 x 1 x 2 x 5 2 5 2 5 x 4 x 2 3 4 3 4 x 3 x 4 x 3 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

  4. General T -systems (Nakanishi, 2011) Q µ 1 ( Q ) x 1 x 5 1 1 x 1 x 2 x 5 2 5 2 5 x 4 x 2 3 4 3 4 x 3 x 4 x 3 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

  5. General T -systems (Nakanishi, 2011) Q µ 1 ( Q ) x 3 x 4 + x 2 x 5 x 1 x 1 1 1 x 2 x 5 x 2 x 5 2 5 2 5 3 4 3 4 x 3 x 4 x 3 x 4 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

  6. Bipartite recurrent quivers w 1 v w 2 . . . w k w ′ k . . . w ′ 2 u w ′ 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 3 / 26

  7. Four classes of quivers “finite ⊠ finite” “affine ⊠ finite” “affine ⊠ affine” “wild” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 4 / 26

  8. Example: finite ⊠ finite 1 1 1 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

  9. Example: finite ⊠ finite 1 1 1 2 1 1 1 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

  10. Example: finite ⊠ finite 1 2 1 1 2 4 1 1 1 2 2 4 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

  11. Example: finite ⊠ finite 1 2 1 1 2 4 1 1 1 2 4 2 4 4 4 4 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

  12. Example: finite ⊠ finite 1 2 1 1 2 4 1 1 1 2 4 2 4 4 2 4 2 4 4 4 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

  13. Example: finite ⊠ finite 1 2 1 1 2 4 1 1 1 2 4 2 2 4 4 1 2 4 2 1 2 4 4 4 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

  14. Four classes of quivers “finite ⊠ finite” “affine ⊠ finite” “affine ⊠ affine” “wild” periodic Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 6 / 26

  15. Example: affine ⊠ finite a b Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  16. Example: affine ⊠ finite a b b 2 + 1 b a Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  17. Example: affine ⊠ finite 1 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  18. Example: affine ⊠ finite 1 1 2 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  19. Example: affine ⊠ finite 1 1 1 2 2 5 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  20. Example: affine ⊠ finite 1 1 2 1 2 5 13 5 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  21. Example: affine ⊠ finite 1 1 1 2 2 5 5 13 13 34 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  22. Example: affine ⊠ finite 1 1 34 89 1 2 2 5 5 13 13 34 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  23. Example: affine ⊠ finite 1 1 34 89 1 2 89 233 2 5 5 13 13 34 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  24. Example: affine ⊠ finite 1 1 34 89 1 2 89 233 2 5 233 610 5 13 13 34 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  25. Example: affine ⊠ finite 1 1 34 89 1 2 89 233 2 5 233 610 5 13 . . . 13 34 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  26. Example: affine ⊠ finite 1 1 34 89 1 2 89 233 2 5 610 233 5 13 . . . 13 34 x n +1 = 3 x n − x n − 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

  27. Four classes of quivers “finite ⊠ finite” “affine ⊠ finite” “affine ⊠ affine” “wild” periodic linearizable Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 8 / 26

  28. Example: affine ⊠ affine 1 1 1 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

  29. Example: affine ⊠ affine 1 1 2 1 1 1 1 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

  30. Example: affine ⊠ affine 2 1 2 1 1 2 3 1 1 1 1 1 2 1 2 3 2 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

  31. Example: affine ⊠ affine 2 1 1 2 1 2 3 1 1 2 1 1 1 1 2 1 2 3 2 3 2 6 2 3 2 6 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

  32. Example: affine ⊠ affine 2 1 2 1 1 2 3 1 1 1 1 1 2 1 2 3 2 1 2 6 2 10 2 6 2 3 2 6 2 3 2 10 2 6 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

  33. Example: affine ⊠ affine 2 1 2 1 1 2 3 1 1 1 1 1 2 1 2 3 2 1 2 15 2 10 2 6 2 10 2 6 2 3 2 10 2 6 2 3 2 15 2 10 2 6 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

  34. Example: affine ⊠ affine 2 1 2 1 2 3 1 1 1 1 1 1 2 1 2 3 2 1 2 10 2 6 2 15 2 10 2 6 2 3 2 10 2 6 2 3 2 15 2 10 2 6 2( n 2 ) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

  35. Four classes of quivers “finite ⊠ finite” “affine ⊠ finite” “affine ⊠ affine” “wild” grows as periodic linearizable exp( t 2 ) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 10 / 26

  36. Example: wild a b Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  37. Example: wild a b b 3 + 1 b a Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  38. Example: wild 1 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  39. Example: wild 1 1 1 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  40. Example: wild 1 1 2 1 2 9 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  41. Example: wild 1 1 2 1 2 9 365 9 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  42. Example: wild 1 1 2 1 2 9 9 365 365 5403014 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  43. Example: wild 1 1 2 1 2 9 9 365 365 5403014 432130991537958813 5403014 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  44. Example: wild 1 1 2 1 2 9 365 9 365 5403014 432130991537958813 5403014 the next number is 14935169284101525874491673463268414536523593057 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

  45. Four classes of quivers “finite ⊠ finite” “affine ⊠ finite” “affine ⊠ affine” “wild” grows as grows as periodic linearizable exp( t 2 ) exp(exp( t )) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 12 / 26

  46. ADE Dynkin diagrams Name Finite diagram Affine diagram Name 1 1 ˆ A n A n − 1 1 1 1 1 1 1 ˆ D n D n − 1 2 2 2 1 1 1 ˆ E 6 E 6 2 1 2 3 2 1 2 ˆ E 7 E 7 1 2 3 4 3 2 1 3 ˆ E 8 E 8 2 4 6 5 4 3 2 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 13 / 26

  47. Affine ⊠ finite quivers “Affine ⊠ finite quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 14 / 26

  48. Affine ⊠ finite quivers • Bipartite recurrent quiver “Affine ⊠ finite quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 14 / 26

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