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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

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Zamolodchikov periodicity and integrability

Pavel Galashin

MIT galashin@mit.edu

Hunter College, May 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 Hunter College, May 7, 2017 1 / 26

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General T-systems (Nakanishi, 2011)

Q

1 2 3 4 5 x1 x2 x3 x4 x5 2 3 4 5 1 x1 x2 x3 x4 x5

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

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SLIDE 3

General T-systems (Nakanishi, 2011)

Q µ1(Q)

1 2 3 4 5 x1 x2 x3 x4 x5 2 3 4 5 1 x1 x2 x3 x4 x5

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

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SLIDE 4

General T-systems (Nakanishi, 2011)

Q µ1(Q)

1 2 3 4 5 x1 x2 x3 x4 x5 2 3 4 5 1 x1 x2 x3 x4 x5

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

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SLIDE 5

General T-systems (Nakanishi, 2011)

Q µ1(Q)

1 2 3 4 5 x1 x2 x3 x4 x5 2 3 4 5 1

x3x4+x2x5 x1

x2 x3 x4 x5

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 2 / 26

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SLIDE 6

Bipartite recurrent quivers

w1 v w2 . . . wk w′

k

. . . w′

2

u w′

1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 3 / 26

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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

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Example: finite ⊠ finite

1 1 1 1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

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SLIDE 9

Example: finite ⊠ finite

1 1 1 1 2 1 1 2

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

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SLIDE 10

Example: finite ⊠ finite

1 1 1 1 2 1 1 2 2 4 4 2

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

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Example: finite ⊠ finite

1 1 1 1 2 1 1 2 2 4 4 2 4 4 4 4

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

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Example: finite ⊠ finite

1 1 1 1 2 1 1 2 2 4 4 2 4 2 2 4 4 4 4 4

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

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Example: finite ⊠ finite

1 1 1 1 2 1 1 2 2 4 4 2 1 2 2 1 4 2 2 4 4 4 4 4

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 5 / 26

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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

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Example: affine ⊠ finite

a b

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

a b

b2+1 a

b

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1 2 1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1 2 1 2 5

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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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

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Example: affine ⊠ finite

1 1 2 1 2 5 13 5 13 34

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1 2 1 2 5 13 5 13 34 89 34

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1 2 1 2 5 13 5 13 34 89 34 89 233

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1 2 1 2 5 13 5 13 34 89 34 89 233 610 233

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1 2 1 2 5 13 5 13 34 89 34 89 233 610 233

. . .

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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Example: affine ⊠ finite

1 1 2 1 2 5 13 5 13 34 89 34 89 233 610 233

. . .

xn+1 = 3xn − xn−1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 7 / 26

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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

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Example: affine ⊠ affine

1 1 1 1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

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Example: affine ⊠ affine

1 1 1 1 2 1 1 2

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

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Example: affine ⊠ affine

1 1 1 1 21 1 1 21 21 23 23 21

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

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Example: affine ⊠ affine

1 1 1 1 21 1 1 21 21 23 23 21 26 23 23 26

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

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Example: affine ⊠ affine

1 1 1 1 21 1 1 21 21 23 23 21 26 210 210 26 26 23 23 26

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

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Example: affine ⊠ affine

1 1 1 1 21 1 1 21 21 23 23 21 215 210 210 215 26 210 210 26 26 23 23 26

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

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Example: affine ⊠ affine

1 1 1 1 21 1 1 21 21 23 23 21 215 210 210 215 26 210 210 26 26 23 23 26

2(n

2)

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 9 / 26

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Four classes of quivers

“finite ⊠ finite” “affine ⊠ finite” “affine ⊠ affine” “wild” periodic linearizable grows as exp(t2)

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 10 / 26

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Example: wild a b

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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Example: wild a b

b3+1 a

b

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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Example: wild

1 1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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Example: wild

1 1 2 1

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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Example: wild

1 1 2 1 2 9

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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Example: wild

1 1 2 1 2 9 365 9

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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Example: wild

1 1 2 1 2 9 365 9 365 5403014

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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Example: wild

1 1 2 1 2 9 365 9 365 5403014 432130991537958813 5403014

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 11 / 26

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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

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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 Hunter College, May 7, 2017 12 / 26

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ADE Dynkin diagrams

Name Finite diagram Affine diagram Name An

1 1 1 1 1 1

ˆ An−1 Dn

1 1 2 2 2 1 1

ˆ Dn−1 E6

1 2 1 2 3 2 1

ˆ E6 E7

2 1 2 3 4 3 2 1

ˆ E7 E8

3 2 4 6 5 4 3 2 1

ˆ E8

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 13 / 26

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Affine ⊠ finite quivers

“Affine ⊠ finite quiver”

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 14 / 26

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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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Affine ⊠ finite quivers

  • Bipartite recurrent quiver
  • All red components are affine Dynkin diagrams

“Affine ⊠ finite quiver”

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 14 / 26

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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 Hunter College, May 7, 2017 14 / 26

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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 Hunter College, May 7, 2017 14 / 26

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Master conjecture

Conjecture (G.-Pylyavskyy, 2016)

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 15 / 26

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Master conjecture

Conjecture (G.-Pylyavskyy, 2016)

finite ⊠ finite ⇐ ⇒ periodic

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 15 / 26

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Master conjecture

Conjecture (G.-Pylyavskyy, 2016)

finite ⊠ finite ⇐ ⇒ periodic affine ⊠ finite ⇐ ⇒ linearizable

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 15 / 26

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Master conjecture

Conjecture (G.-Pylyavskyy, 2016)

finite ⊠ finite ⇐ ⇒ periodic affine ⊠ finite ⇐ ⇒ linearizable affine ⊠ affine ⇐ ⇒ grows as exp(t2)

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 15 / 26

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Master conjecture

Conjecture (G.-Pylyavskyy, 2016)

finite ⊠ finite ⇐ ⇒ periodic affine ⊠ finite ⇐ ⇒ linearizable affine ⊠ affine ⇐ ⇒ grows as exp(t2) wild ⇐ ⇒ grows as exp(exp(t))

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 15 / 26

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Results

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 16 / 26

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Results

Theorem (G.-Pylyavskyy, 2016)

Periodic ⇐ ⇒ finite ⊠ finite

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 16 / 26

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SLIDE 59

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 Hunter College, May 7, 2017 16 / 26

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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

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 16 / 26

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SLIDE 61

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 Hunter College, May 7, 2017 16 / 26

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Tensor product

D5 ⊗ A3

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 17 / 26

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Tensor product

D5 ⊗ A3

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 17 / 26

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Tensor product

D5 ⊗ A3

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 17 / 26

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Zamolodchikov periodicity

Theorem (B. Keller, 2013)

Tensor product of finite Dynkin diagrams = ⇒ the T-system is periodic.

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 18 / 26

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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 Hunter College, May 7, 2017 19 / 26

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finite ⊠ finite classification (Stembridge, 2010)

5 infinite families and 11 exceptional quivers

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 20 / 26

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finite ⊠ finite classification (Stembridge, 2010)

5 infinite families and 11 exceptional quivers

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 20 / 26

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SLIDE 69

finite ⊠ finite classification (Stembridge, 2010)

5 infinite families and 11 exceptional quivers

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 20 / 26

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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 Hunter College, May 7, 2017 21 / 26

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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 Hunter College, May 7, 2017 21 / 26

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SLIDE 72

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 Hunter College, May 7, 2017 21 / 26

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SLIDE 73

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 Hunter College, May 7, 2017 21 / 26

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SLIDE 74

affine ⊠ affine classification: 41 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 Hunter College, May 7, 2017 22 / 26

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affine ⊠ affine classification: 41 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 Hunter College, May 7, 2017 22 / 26

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SLIDE 76

affine ⊠ affine classification: 41 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 Hunter College, May 7, 2017 22 / 26

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affine ⊠ affine classification: 41 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 Hunter College, May 7, 2017 22 / 26

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SLIDE 78

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 Hunter College, May 7, 2017 23 / 26

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SLIDE 79

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 Hunter College, May 7, 2017 23 / 26

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SLIDE 80

Bibliography

Slides: http://math.mit.edu/~galashin/slides/nyc.pdf Bernhard Keller. The periodicity conjecture for pairs of Dynkin diagrams.

  • Ann. of Math. (2), 177(1):111–170, 2013.

Pavel Galashin and Pavlo Pylyavskyy. The classification of Zamolodchikov periodic quivers.

  • Amer. J. Math., to appear.

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).

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SLIDE 81

Thank you!

Pavel Galashin (MIT) Zamolodchikov periodicity and integrability Hunter College, May 7, 2017 25 / 26

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

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SLIDE 82

Bibliography

Slides: http://math.mit.edu/~galashin/slides/nyc.pdf Bernhard Keller. The periodicity conjecture for pairs of Dynkin diagrams.

  • Ann. of Math. (2), 177(1):111–170, 2013.

Pavel Galashin and Pavlo Pylyavskyy. The classification of Zamolodchikov periodic quivers.

  • Amer. J. Math., to appear.

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).