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Zamolodchikov periodicity and integrability Pavel Galashin MIT galashin@mit.edu UCLA, October 26, 2018 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

  1. (affine, finite) quivers “(affine, finite) quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 17 / 40

  2. (affine, finite) quivers • Bipartite recurrent quiver “(affine, finite) quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 17 / 40

  3. (affine, finite) quivers • Bipartite recurrent quiver • All red components are affine Dynkin diagrams “(affine, finite) quiver” Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 17 / 40

  4. (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 UCLA, 10/26/2018 17 / 40

  5. (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 UCLA, 10/26/2018 17 / 40

  6. 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 UCLA, 10/26/2018 18 / 40

  7. Master conjecture Conjecture (G.-Pylyavskyy, 2016) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 19 / 40

  8. Master conjecture Conjecture (G.-Pylyavskyy, 2016) (finite, finite) ⇐ ⇒ periodic Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 19 / 40

  9. Master conjecture Conjecture (G.-Pylyavskyy, 2016) (finite, finite) ⇐ ⇒ periodic (affine, finite) ⇐ ⇒ linearizable, but not periodic Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 19 / 40

  10. Master conjecture Conjecture (G.-Pylyavskyy, 2016) (finite, finite) ⇐ ⇒ periodic (affine, finite) ⇐ ⇒ linearizable, but not periodic ⇒ grows as exp( t 2 ) (affine, affine) ⇐ Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 19 / 40

  11. Master conjecture Conjecture (G.-Pylyavskyy, 2016) (finite, finite) ⇐ ⇒ periodic (affine, finite) ⇐ ⇒ linearizable, but not periodic ⇒ grows as exp( t 2 ) (affine, affine) ⇐ wild ⇐ ⇒ grows as exp(exp( t )) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 19 / 40

  12. Results Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 20 / 40

  13. Results Theorem (G.-Pylyavskyy, 2016) Periodic ⇐ ⇒ (finite, finite) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 20 / 40

  14. Results Theorem (G.-Pylyavskyy, 2016) Periodic ⇐ ⇒ (finite, finite) Theorem (G.-Pylyavskyy, 2016) Linearizable = ⇒ (affine, finite) or (finite, finite) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 20 / 40

  15. 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) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 20 / 40

  16. 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 UCLA, 10/26/2018 20 / 40

  17. Tensor product D 5 ⊗ A 3 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 21 / 40

  18. Tensor product D 5 ⊗ A 3 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 21 / 40

  19. Tensor product D 5 ⊗ A 3 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 21 / 40

  20. Zamolodchikov periodicity Theorem (B. Keller, 2013) Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic. Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 22 / 40

  21. Zamolodchikov periodicity Theorem (B. Keller, 2013) Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic. Zamolodchikov (1991): Λ ⊗ A 1 (conjectured); Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 22 / 40

  22. Zamolodchikov periodicity Theorem (B. Keller, 2013) Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic. Zamolodchikov (1991): Λ ⊗ A 1 (conjectured); Ravanini-Tateo-Valleriani (1993): Λ ⊗ Λ ′ (conjectured); Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 22 / 40

  23. Zamolodchikov periodicity Theorem (B. Keller, 2013) Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic. Zamolodchikov (1991): Λ ⊗ A 1 (conjectured); Ravanini-Tateo-Valleriani (1993): Λ ⊗ Λ ′ (conjectured); Frenkel-Szenes (1995): A n ⊗ A 1 ; Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 22 / 40

  24. Zamolodchikov periodicity Theorem (B. Keller, 2013) Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic. Zamolodchikov (1991): Λ ⊗ A 1 (conjectured); Ravanini-Tateo-Valleriani (1993): Λ ⊗ Λ ′ (conjectured); Frenkel-Szenes (1995): A n ⊗ A 1 ; Fomin-Zelevinsky (2003): Λ ⊗ A 1 ; Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 22 / 40

  25. Zamolodchikov periodicity Theorem (B. Keller, 2013) Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic. Zamolodchikov (1991): Λ ⊗ A 1 (conjectured); Ravanini-Tateo-Valleriani (1993): Λ ⊗ Λ ′ (conjectured); Frenkel-Szenes (1995): A n ⊗ A 1 ; Fomin-Zelevinsky (2003): Λ ⊗ A 1 ; Volkov (2005): A n ⊗ A m ; Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 22 / 40

  26. Zamolodchikov periodicity Theorem (B. Keller, 2013) Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic. Zamolodchikov (1991): Λ ⊗ A 1 (conjectured); Ravanini-Tateo-Valleriani (1993): Λ ⊗ Λ ′ (conjectured); Frenkel-Szenes (1995): A n ⊗ A 1 ; Fomin-Zelevinsky (2003): Λ ⊗ A 1 ; Volkov (2005): A n ⊗ A m ; Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 22 / 40

  27. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  28. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  29. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  30. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Total positivity Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  31. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Total positivity Zamolodchikov periodicity Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  32. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Total positivity Zamolodchikov periodicity Definition A cluster algebra is of finite type if it has finitely many cluster variables. Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  33. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Total positivity Zamolodchikov periodicity Definition A cluster algebra is of finite type if it has finitely many cluster variables. Theorem (Fomin–Zelevinsky (2003)) → finite Dynkin diagrams ← Cluster algebras of finite type Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  34. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Total positivity Zamolodchikov periodicity Definition A cluster algebra is of finite type if it has finitely many cluster variables. Theorem (Fomin–Zelevinsky (2003)) → finite Dynkin diagrams ← Cluster algebras of finite type → finite root systems Φ ← Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  35. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Total positivity Zamolodchikov periodicity Definition A cluster algebra is of finite type if it has finitely many cluster variables. Theorem (Fomin–Zelevinsky (2003)) → finite Dynkin diagrams ← Cluster algebras of finite type → finite root systems Φ ← Cluster variables ↔ almost positive roots Φ ≥− 1 := Φ + ⊔ ( − Π) Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  36. History of cluster algebras Cluster algebras were invented by Fomin–Zelevinsky in 2000. Motivation: Lusztig’s canonical bases Total positivity Zamolodchikov periodicity Definition A cluster algebra is of finite type if it has finitely many cluster variables. Theorem (Fomin–Zelevinsky (2003)) → finite Dynkin diagrams ← Cluster algebras of finite type → finite root systems Φ ← Cluster variables ↔ almost positive roots Φ ≥− 1 := Φ + ⊔ ( − Π) Explicitly, the above bijection sends α ∈ Φ ≥− 1 to the unique cluster variable with denominator x α . Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  37. History of cluster algebras Theorem (Fomin–Zelevinsky (2003)) → finite Dynkin diagrams ← Cluster algebras of finite type → finite root systems Φ ← Cluster variables ↔ almost positive roots Φ ≥− 1 := Φ + ⊔ ( − Π) Explicitly, the above bijection sends α ∈ Φ ≥− 1 to the unique cluster variable with denominator x α . Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  38. History of cluster algebras Theorem (Fomin–Zelevinsky (2003)) → finite Dynkin diagrams ← Cluster algebras of finite type → finite root systems Φ ← Cluster variables ↔ almost positive roots Φ ≥− 1 := Φ + ⊔ ( − Π) Explicitly, the above bijection sends α ∈ Φ ≥− 1 to the unique cluster variable with denominator x α . Theorem (Fomin–Zelevinsky (2003)) Zamolodchikov periodicity conjecture holds for Λ ⊗ A 1 . Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  39. History of cluster algebras Theorem (Fomin–Zelevinsky (2003)) → finite Dynkin diagrams ← Cluster algebras of finite type → finite root systems Φ ← Cluster variables ↔ almost positive roots Φ ≥− 1 := Φ + ⊔ ( − Π) Explicitly, the above bijection sends α ∈ Φ ≥− 1 to the unique cluster variable with denominator x α . Theorem (Fomin–Zelevinsky (2003)) Zamolodchikov periodicity conjecture holds for Λ ⊗ A 1 . Proof. Use the above bijection and then prove periodicity for the tropical dynamics on Φ ≥− 1 . Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 23 / 40

  40. Example: A 2 x 1 x 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

  41. Example: A 2 x 1 x 2 x 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

  42. Example: A 2 x 1 x 2 x 1 x 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

  43. Example: A 2 x 1 x 2 x 1 x 2 x 2 + 1 x 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

  44. Example: A 2 x 1 x 2 x 1 x 2 x 2 + 1 x 1 x 1 + x 2 + 1 x 1 x 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

  45. Example: A 2 x 1 x 2 x 1 x 2 x 2 + 1 x 1 x 1 + x 2 + 1 x 1 x 2 x 1 + 1 x 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

  46. Example: A 2 x 1 x 2 x 1 x 2 x 2 + 1 x 1 x 1 + x 2 + 1 x 1 x 2 x 1 + 1 x 2 x 1 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

  47. Example: A 2 x 1 x 2 x 1 x 2 x 2 + 1 x 1 x 1 + x 2 + 1 x 1 x 2 x 1 + 1 x 2 x 1 x 2 Pavel Galashin (MIT) Zamolodchikov periodicity and integrability UCLA, 10/26/2018 24 / 40

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