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Zamolodchikov periodicity and integrability Pavel Galashin MIT galashin@mit.edu Infinite Analysis 17, Osaka City University, December 7, 2017 Joint work with Pavlo Pylyavskyy 2 2 2 1 1 2 2 3 2 2 2 1 2 Pavel Galashin (MIT)

  1. Zamolodchikov periodicity and integrability Pavel Galashin MIT galashin@mit.edu Infinite Analysis 17, Osaka City University, December 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 Infinite Analysis 17, 12/07/2017 1 / 39

  2. Part 0: Recall

  3. Bipartite recurrent quivers w 1 v w 2 . . . w k w ′ k . . . w ′ 2 u w ′ 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 3 / 39

  4. Bipartite T -system a b c d e f Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 4 / 39

  5. Bipartite T -system b + c b a a b c + bf c − → c d d e f c + f f e Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 4 / 39

  6. Bipartite T -system b + c + c + bf b + c a d b + c a b b a b + c c + f + c + bf c + bf c + bf c a e d d d c c + f c + f + c + bf f c + f e e d e f Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 4 / 39

  7. 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 Infinite Analysis 17, 12/07/2017 5 / 39

  8. Affine ⊠ finite quivers “Affine ⊠ finite quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 6 / 39

  9. Affine ⊠ finite quivers • Bipartite recurrent quiver “Affine ⊠ finite quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 6 / 39

  10. Affine ⊠ finite quivers • Bipartite recurrent quiver • All red components are affine Dynkin diagrams “Affine ⊠ finite quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 6 / 39

  11. Affine ⊠ finite quivers • Bipartite recurrent quiver • All red components are affine Dynkin diagrams • All blue components are finite Dynkin diagrams “Affine ⊠ finite quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 6 / 39

  12. Affine ⊠ finite quivers • Bipartite recurrent quiver • All red components are affine Dynkin diagrams • All blue components are finite Dynkin diagrams “ Affine ⊠ finite quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 6 / 39

  13. Master conjecture Conjecture (G.-Pylyavskyy, 2016) finite ⊠ finite ⇐ ⇒ periodic affine ⊠ finite ⇐ ⇒ linearizable, but not periodic ⇒ grows as exp( t 2 ) affine ⊠ affine ⇐ ⇒ grows as exp(exp( t )) wild ⇐ Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 7 / 39

  14. Results Theorem (G.-Pylyavskyy, 2016) Periodic ⇐ ⇒ finite ⊠ finite Theorem (G.-Pylyavskyy, 2016) Linearizable = ⇒ affine ⊠ finite or finite ⊠ finite Theorem (G.-Pylyavskyy, 2017) Grows slower than exp(exp( t )) = ⇒ affine ⊠ affine, affine ⊠ finite, or finite ⊠ finite What is left: Conjecture (G.-Pylyavskyy, 2017) affine ⊠ finite = ⇒ linearizable ⇒ grows as exp( t 2 ) affine ⊠ affine = Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 8 / 39

  15. Part 1: Type A

  16. Type A m ⊗ ˆ A 2 n − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 10 / 39

  17. Example: A 1 ⊗ ˆ A 1 1 1 89 34 2 1 89 233 2 5 610 233 13 5 . . . 13 34 x n +1 − 3 x n + x n − 1 = 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 11 / 39

  18. Type A m ⊗ ˆ A 2 n − 1 a b Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 12 / 39

  19. Type A m ⊗ ˆ A 2 n − 1 a b b 2 + 1 b a Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 12 / 39

  20. Type A m ⊗ ˆ A 2 n − 1 a b b 2 + 1 b a � 2 b2 + 1 � b 2 +1 + 1 a a b Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 12 / 39

  21. Type A m ⊗ ˆ A 2 n − 1 a b b 2 + 1 b a � 2 b2 + 1 � b 2 +1 + 1 a a b . . . Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 12 / 39

  22. Type A m ⊗ ˆ A 2 n − 1 a b b 2 + 1 b a � 2 b2 + 1 � b 2 +1 + 1 a a b . . . x n +1 − 3 x n + x n − 1 = 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 12 / 39

  23. Type A m ⊗ ˆ A 2 n − 1 a b b 2 + 1 b a � 2 b2 + 1 � b 2 +1 + 1 a a b . . . x n +1 − 3 x n + x n − 1 = 0 � a � b + b a + 1 1 · x n +1 − · x n + 1 · x n − 1 = 0 ab Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 12 / 39

  24. Domino tilings of the cylinder a b Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 13 / 39

  25. Domino tilings of the cylinder a a b a b b a b a b Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 13 / 39

  26. Domino tilings of the cylinder a a b a b b a b a b � a b 1 � 1 · x n +1 − + + · x n + 1 · x n − 1 = 0 b a ab Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 13 / 39

  27. Thurston height Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  28. Thurston height 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  29. Thurston height 0 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  30. Thurston height 0 1 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  31. Thurston height − 1 0 1 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  32. Thurston height 0 − 1 0 1 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  33. Thurston height 0 1 − 1 0 1 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  34. Thurston height 3 2 3 2 0 1 0 1 − 1 − 2 − 1 − 2 0 − 3 0 0 1 1 − 1 − 2 − 1 − 1 2 0 1 0 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 14 / 39

  35. Type A m ⊗ ˆ A 2 n − 1 Theorem (G.-Pylyavskyy, 2016) Recurrence for boundary slice : x t +( m +1) n − H 1 x t + mn + . . . ± H m x t + n ∓ x t = 0 . Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 15 / 39

  36. Type A m ⊗ ˆ A 2 n − 1 Theorem (G.-Pylyavskyy, 2016) Recurrence for boundary slice : x t +( m +1) n − H 1 x t + mn + . . . ± H m x t + n ∓ x t = 0 . � H i = wt ( T ) . T – cylinder tiling of Thurston height i Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 15 / 39

  37. Type A m ⊗ ˆ A 2 n − 1 Theorem (G.-Pylyavskyy, 2016) Recurrence for boundary slice : x t +( m +1) n − H 1 x t + mn + . . . ± H m x t + n ∓ x t = 0 . � H i = wt ( T ) . “Goncharov-Kenyon Hamiltonians” T – cylinder tiling of Thurston height i Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 15 / 39

  38. Type A m ⊗ ˆ A 2 n − 1 Theorem (G.-Pylyavskyy, 2016) Recurrence for boundary slice : x t +( m +1) n − H 1 x t + mn + . . . ± H m x t + n ∓ x t = 0 . � H i = wt ( T ) . “Goncharov-Kenyon Hamiltonians” T – cylinder tiling of Thurston height i Recurrence for r -th slice : express e j [ e r ] in e i ’s and send e i �→ H i . Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 15 / 39

  39. Part 2: Tropical T -system

  40. Tropical T -system 1 4 − 7 2 3 0

  41. Tropical T -system max(4 , − 7) − 1 4 1 4 − 7 − 7 max( − 7 , 4 + 0) − 2 2 3 0 max( − 7 , 0) − 3 0 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 17 / 39

  42. Tropical results Theorem (G.-Pylyavskyy, 2016) Tropical T-system is periodic ⇐ ⇒ finite ⊠ finite Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 18 / 39

  43. Tropical results Theorem (G.-Pylyavskyy, 2016) Tropical T-system is periodic ⇐ ⇒ finite ⊠ finite Theorem (G.-Pylyavskyy, 2016) Tropical T-system grows linearly = ⇒ affine ⊠ finite Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Infinite Analysis 17, 12/07/2017 18 / 39

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