Zamolodchikov periodicity and integrability Pavel Galashin MIT - - PowerPoint PPT Presentation
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
Pavel Galashin
MIT galashin@mit.edu
Infinite Analysis 17, Osaka City University, December 7, 2017 Joint work with Pavlo Pylyavskyy
1 2 1 2 3 2 1 2 2 2 2 2 2
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Bipartite recurrent quivers
w1 v w2 . . . wk w′
k
. . . w′
2
u w′
1
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Bipartite T-system
a b c d e f
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Bipartite T-system
a b c d e f
− →
b+c a
b c
c+bf d c+f e
f
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Bipartite T-system
b+c a b+c a
+ c+bf
d b b+c a c+f e
+ c+bf
d c c+bf d c+f e c+f e
+ c+bf
d f b+c a
b c
c+bf d c+f e
f
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Four classes of quivers
“finite ⊠ finite” “affine ⊠ finite” “affine ⊠ affine” “wild” periodic linearizable grows as exp(t2) grows as exp(exp(t))
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite quivers
“Affine ⊠ finite quiver”
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite quivers
“Affine ⊠ finite quiver”
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite quivers
“Affine ⊠ finite quiver”
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite quivers
“Affine ⊠ finite quiver”
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite quivers
“Affine ⊠ finite quiver”
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Master conjecture
Conjecture (G.-Pylyavskyy, 2016)
finite ⊠ finite ⇐ ⇒ periodic affine ⊠ finite ⇐ ⇒ linearizable, but not periodic affine ⊠ affine ⇐ ⇒ grows as exp(t2) wild ⇐ ⇒ grows as exp(exp(t))
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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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 affine ⊠ affine = ⇒ grows as exp(t2)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−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
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Example: A1 ⊗ ˆ A1
1 1 2 1 2 5 13 5 13 34 89 34 89 233 610 233
xn+1 − 3xn + xn−1 = 0
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
a b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
a b
b2+1 a
b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
a b
b2+1 a
b
b2+1 a
a
2 +1 b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
a b
b2+1 a
b
b2+1 a
a
2 +1 b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
a b
b2+1 a
b
b2+1 a
a
2 +1 b
xn+1 − 3xn + xn−1 = 0
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
a b
b2+1 a
b
b2+1 a
a
2 +1 b
xn+1 − 3xn + xn−1 = 0 1·xn+1 − a b + b a + 1 ab
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Domino tilings of the cylinder
a b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Domino tilings of the cylinder
a b a b a b a b a b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Domino tilings of the cylinder
a b a b a b a b a b
1·xn+1 − a
b
+
b a
+
1 ab
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
1
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
1
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
1 −1
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
1 −1
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
1 −1 1
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Thurston height
1 −1 1 1 −1 −2 −1 2 −3 1 −1 −2 −1 −2 1 1 3 2 3 2
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
Theorem (G.-Pylyavskyy, 2016)
Recurrence for boundary slice: xt+(m+1)n − H1xt+mn + . . . ± Hmxt+n ∓ xt = 0.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
Theorem (G.-Pylyavskyy, 2016)
Recurrence for boundary slice: xt+(m+1)n − H1xt+mn + . . . ± Hmxt+n ∓ xt = 0. Hi =
wt(T).
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
Theorem (G.-Pylyavskyy, 2016)
Recurrence for boundary slice: xt+(m+1)n − H1xt+mn + . . . ± Hmxt+n ∓ xt = 0. Hi =
wt(T). “Goncharov-Kenyon Hamiltonians”
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1
Theorem (G.-Pylyavskyy, 2016)
Recurrence for boundary slice: xt+(m+1)n − H1xt+mn + . . . ± Hmxt+n ∓ xt = 0. Hi =
wt(T). “Goncharov-Kenyon Hamiltonians” Recurrence for r-th slice: express ej[er] in ei’s and send ei → Hi.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system
1 4 −7 2 3
Tropical T-system
1 4 −7 2 3 max(4, −7) − 1 4 −7 max(−7, 4 + 0) − 2 max(−7, 0) − 3
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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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
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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
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Tropical results
Theorem (G.-Pylyavskyy, 2016)
Tropical T-system is periodic ⇐ ⇒ finite ⊠ finite
Theorem (G.-Pylyavskyy, 2016)
Tropical T-system grows linearly = ⇒ affine ⊠ finite
Theorem (G.-Pylyavskyy, 2017)
Tropical T-system grows slower than exp(t) = ⇒ finite ⊠ finite, affine ⊠ finite, or affine ⊠ affine.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical results
Theorem (G.-Pylyavskyy, 2016)
Tropical T-system is periodic ⇐ ⇒ finite ⊠ finite
Theorem (G.-Pylyavskyy, 2016)
Tropical T-system grows linearly = ⇒ affine ⊠ finite
Theorem (G.-Pylyavskyy, 2017)
Tropical T-system grows slower than exp(t) = ⇒ finite ⊠ finite, affine ⊠ finite, or affine ⊠ affine.
Conjecture (G.-Pylyavskyy, 2017)
affine ⊠ finite = ⇒ tropical T-system grows linearly affine ⊠ affine = ⇒ tropical T-system grows quadratically
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system: speed
a b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system: speed
a b max(2b, 0) − a b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system: speed
a b 2b − a b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system: speed
a b b + (b − a) b
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system: speed
a b b + (b − a) b b + (b − a) b + 2(b − a)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system: speed
a b b + (b − a) b b + (b − a) b + 2(b − a) b + 3(b − a) b + 2(b − a)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical T-system: speed
a b b + (b − a) b b + (b − a) b + 2(b − a) b + 3(b − a) b + 2(b − a) b + 3(b − a) b + 4(b − a)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite case
Definition
Blue mutation:
λu >
λw.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite case
Definition
Blue mutation:
λu >
λw.
Proposition
The speed is non-decreasing and increases only during blue mutations.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite case
Definition
Blue mutation:
λu >
λw.
Proposition
The speed is non-decreasing and increases only during blue mutations.
Corollary
Tropical T-system grows linearly ⇔ there are finitely many blue mutations.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Example
spd = 14 spd = −3 spd = 1 spd = 8 1 9 3 2 3 6 −8
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Example
spd = 14 spd = −3 spd = 1 spd = 8 1 9 3 2 3 6 −8 17 9 3 7 6 6 20 spd = 14 spd = 6 spd = 4 spd = 8
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Solitonic behavior
spd = −14 spd = −21 spd = −25 spd = −17 1652 1635 2445 2470 2085 2064 1392 1406 1338 1324 2010 2031 2430 2405 1663 1680 spd = 17 spd = 25 spd = 21 spd = 14
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1: solitonic behavior
Theorem (G.-Pylyavskyy, 2016)
In Type Am ⊗ ˆ A2n−1 we have:
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1: solitonic behavior
Theorem (G.-Pylyavskyy, 2016)
In Type Am ⊗ ˆ A2n−1 we have: (“soliton resolution”) t sufficiently large = ⇒ each affine slice moves independently with constant speed
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Type Am ⊗ ˆ A2n−1: solitonic behavior
Theorem (G.-Pylyavskyy, 2016)
In Type Am ⊗ ˆ A2n−1 we have: (“soliton resolution”) t sufficiently large = ⇒ each affine slice moves independently with constant speed (“speed conservation”) the speed of each slice at t → +∞ equals the speed of its mirror image at t → −∞
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine case
spd = 14 spd = −3 spd = 1 spd = 8 1 9 3 2 3 6 −8 spd = 5 spd = 15
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine case
spd = 14 spd = −3 spd = 1 spd = 8 1 9 3 2 3 6 −8 spd = 5 spd = 15 17 9 3 7 6 6 20 spd = 14 spd = 18 spd = 14 spd = 6 spd = 4 spd = 8
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Acceleration
Definition
Acceleration =
λu −
λw
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Acceleration
Definition
Acceleration =
λu −
λw
Proposition
Tropical T-system grows at most quadratically ⇐ bounded acceleration.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical problems
Problem
Affine ⊠ finite quivers: show that the number of blue mutations is finite.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical problems
Problem
Affine ⊠ finite quivers: show that the number of blue mutations is finite. Case A ⊗ ˆ A: follows from Pylyavskyy (2016);
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical problems
Problem
Affine ⊠ finite quivers: show that the number of blue mutations is finite. Case A ⊗ ˆ A: follows from Pylyavskyy (2016); Case A1 ⊗ ˆ Λ: follows from Keller-Scherotzke (2011)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
26 / 39
Tropical problems
Problem
Affine ⊠ finite quivers: show that the number of blue mutations is finite. Case A ⊗ ˆ A: follows from Pylyavskyy (2016); Case A1 ⊗ ˆ Λ: follows from Keller-Scherotzke (2011)
Problem
Affine ⊠ finite quivers: prove solitonic behavior.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
26 / 39
Tropical problems
Problem
Affine ⊠ finite quivers: show that the number of blue mutations is finite. Case A ⊗ ˆ A: follows from Pylyavskyy (2016); Case A1 ⊗ ˆ Λ: follows from Keller-Scherotzke (2011)
Problem
Affine ⊠ finite quivers: prove solitonic behavior. Case A ⊗ ˆ A: G.-Pylyavskyy (2016)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
26 / 39
Tropical problems
Problem
Affine ⊠ finite quivers: show that the number of blue mutations is finite. Case A ⊗ ˆ A: follows from Pylyavskyy (2016); Case A1 ⊗ ˆ Λ: follows from Keller-Scherotzke (2011)
Problem
Affine ⊠ finite quivers: prove solitonic behavior. Case A ⊗ ˆ A: G.-Pylyavskyy (2016)
Problem
Affine ⊠ affine quivers: show that the acceleration is bounded.
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Tropical problems
Problem
Affine ⊠ finite quivers: show that the number of blue mutations is finite. Case A ⊗ ˆ A: follows from Pylyavskyy (2016); Case A1 ⊗ ˆ Λ: follows from Keller-Scherotzke (2011)
Problem
Affine ⊠ finite quivers: prove solitonic behavior. Case A ⊗ ˆ A: G.-Pylyavskyy (2016)
Problem
Affine ⊠ affine quivers: show that the acceleration is bounded. Case ˆ A ⊗ ˆ A: G.-Pylyavskyy (2017).
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Finite ⊠ finite classification (Stembridge, 2010)
5 infinite families and 11 exceptional quivers
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite classification 15 infinite families and 4 exceptional cases ˆ Dn+1 ∗ ˆ D3n−1 ˆ A3 ∗ ˆ D5 for n = 3
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite classification 15 infinite families and 4 exceptional cases ˆ Dn+1 ∗ ˆ D3n−1 ˆ A3 ∗ ˆ D5 for n = 3
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite classification 15 infinite families and 4 exceptional cases ˆ Dn+1 ∗ ˆ D3n−1 ˆ A3 ∗ ˆ D5 for n = 3
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ finite classification 15 infinite families and 4 exceptional cases ˆ Dn+1 ∗ ˆ D3n−1 ˆ A3 ∗ ˆ D5 for n = 3
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine quivers
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Toric quivers
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine classification: 40 infinite, 13 exceptional
3 2 4 6 5 4 3 2 1 3 2 4 6 5 4 3 2 1 2 1 2 3 4 3 2 1 2 2 4 4 2 2 1 2 1 2 3 2 1 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 1 1 2 1 1 2 2
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine classification: 40 infinite, 13 exceptional
3 2 4 6 5 4 3 2 1 3 2 4 6 5 4 3 2 1 2 1 2 3 4 3 2 1 2 2 4 4 2 2 1 2 1 2 3 2 1 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 1 1 2 1 1 2 2
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine classification: 40 infinite, 13 exceptional
3 2 4 6 5 4 3 2 1 3 2 4 6 5 4 3 2 1 2 1 2 3 4 3 2 1 2 2 4 4 2 2 1 2 1 2 3 2 1 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 1 1 2 1 1 2 2
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine classification: 40 infinite, 13 exceptional
3 2 4 6 5 4 3 2 1 3 2 4 6 5 4 3 2 1 2 1 2 3 4 3 2 1 2 2 4 4 2 2 1 2 1 2 3 2 1 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 1 1 2 1 1 2 2
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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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 affine ⊠ affine = ⇒ grows as exp(t2)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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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 affine ⊠ affine = ⇒ grows as exp(t2)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Affine ⊠ affine classification: 40 infinite, 13 exceptional
3 2 4 6 5 4 3 2 1 3 2 4 6 5 4 3 2 1 2 1 2 3 4 3 2 1 2 2 4 4 2 2 1 2 1 2 3 2 1 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 1 1 2 1 1 2 2
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Stembridge’s classification (2010)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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“Ocneanu graphs” (2001)
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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Bibliography
Slides: http://math.mit.edu/~galashin/slides/japan2.pdf Pavel Galashin and Pavlo Pylyavskyy. The classification of Zamolodchikov periodic quivers.
arXiv:1603.03942 (2016). Pavel Galashin and Pavlo Pylyavskyy Quivers with subadditive labelings: classification and integrability arXiv:1606.04878 (2016). Pavel Galashin and Pavlo Pylyavskyy Quivers with additive labelings: classification and algebraic entropy. arXiv:1704.05024 (2017). Pavel Galashin and Pavlo Pylyavskyy Linear recurrences for cylindrical networks.
arXiv:1704.05160 (2017). Pavel Galashin. Periodicity and integrability for the cube recurrence. arXiv:1704.05570 (2017).
Thank you!
Pavel Galashin (MIT) Zamolodchikov periodicity and integrability
Infinite Analysis 17, 12/07/2017
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3 2 4 6 5 4 3 2 1 3 2 4 6 5 4 3 2 1 2 1 2 3 4 3 2 1 2 2 4 4 2 2 1 2 1 2 3 2 1 2 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 2 1 1 2 1 1 2 2
Bibliography
Slides: http://math.mit.edu/~galashin/slides/japan2.pdf Pavel Galashin and Pavlo Pylyavskyy. The classification of Zamolodchikov periodic quivers.
arXiv:1603.03942 (2016). Pavel Galashin and Pavlo Pylyavskyy Quivers with subadditive labelings: classification and integrability arXiv:1606.04878 (2016). Pavel Galashin and Pavlo Pylyavskyy Quivers with additive labelings: classification and algebraic entropy. arXiv:1704.05024 (2017). Pavel Galashin and Pavlo Pylyavskyy Linear recurrences for cylindrical networks.
arXiv:1704.05160 (2017). Pavel Galashin. Periodicity and integrability for the cube recurrence. arXiv:1704.05570 (2017).