Deautonomization of cluster integrable systems
M. Bershtein, P. Gavrylenko, A. Marshakov
arXiv:1711.02063 [math-ph]
arXiv:1804.10145 [math-ph]
HSE & Skoltech, Moscow, Russia

SIDE1313 November 2018

P. Gavrylenko

Deautonomization of cluster integrable systems

November 19, 20181 / 26

General agenda [BGM] (with example)

Newton polygon ∆terClus

retediscNon-autonomous q-difference system
2)+Z
G (qZ )G (q −1 Z ) = GG(Z(Z )+1flow

Topological string partition function= general solution (τ -function) of the equation

P. GavrylenkoDeautonomization of cluster integrable systems

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General agenda [BGM]Given Newton polygon ∆:1

Construct q-difference system (non-linear, non-autonomous, but exactly
solvable: like (q-)Painlevé equations):
Find quiver Q = Q(∆) using Goncharov-Kenyon inverse procedure
Find mapping class group of Q: GQ
Take lattice G∆ ⊂ GQ and write down equations

2

Construct solution of the q-difference system:
Compute topological string partition function on corresponding CY3 . Possibly
express it in terms of Nekrasov functions.
Take discrete Fourier transformation and guess appropriate shifts of variables
to get several τ functions
In the q = 1 case: everything is solved by Θ-functions [Fock]

3Quantize equation and solution
Anything is done only partly, for some concrete examples.

P. Gavrylenko

Deautonomization of cluster integrable systems

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Goncharov-Kenyon integrable system (with example)
In the q=1 limit q-difference system turns into discrete symmetries of GK(cluster) integrable system. Switching on q 6= 1 is called deautonomization.
Autonomous picture:Parof t tition
he d func
ime tionrmodelInve
rsealgorith
mBipartite graph on torus

Family of spectral curves=Newton polygon
P. Gavrylenko

Deautonomization of cluster integrable systems

231

42

31

4

QuiverNovember 19, 2018

4 / 26

Weights of the dimer configurations (=perfect matchings)
The weight of each dimer configuration D is a product of weights of the edges
Yw (D) =w (e).
e∈D

If we pick one dimer configuration D0 then D − D0 is cycle, ∂(D − D0 ) = 0, for
any D. Therefore, the weight w (D0 )−1 w (D) is given by weights of elementarycycles — faces and A, B cycles on torusYYYw (e) = xi ,w (e) = λ,w (e) = µe∈∂Faceie∈A−cyclee∈B−cycle

Note that λ, µ depend on concrete choice of A, B cycles.
QQ
We have i xi = e∈∂T2 W (e) = 1, since ∂T2 = 0.
λx1 x2x41µx1

λ−1P. Gavrylenko

1µ−11

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Dimer partition function

x1−1 x2−1x4

1x1−1

Partition function Z(λ, µ) = W (D0 )P−1µ−1 x4µx1−1

λx1−1 x2−1

λ−1W (D), whereZ(λ, µ) = x1−1 x2−1 λ + λ−1 + µx1−1 + µ−1 x4 + H,
where
H = 1 + x1−1 + x1−1 x2−1 + x4
P. Gavrylenko

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Cluster structure
2314Poisson bracket is{x1 , x2 } = 2x1 x2 , {x2 , x3 } = 2x2 x3 , {x3 , x4 } = 2x3 x4 , {x4 , x1 } = 2x4 x1
Using the (C× )3 action Z(λ, µ) 7→ tZ · Z(tλ · λ, tµ · µ) one can getZ(λ, µ) = λ + zλ−1 + µ + µ−1 + H = 0
After rescaling coefficients of Z become Casimirs and Hamiltonians:
rr√
1x1x2++z
1 = x1 x2 x3 x4 , z = x1 x3 , H = x1 x2 + √x1 x2x2
x1
It’s relativistic Toda lattice.
P. Gavrylenko

Deautonomization of cluster integrable systems

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Mapping class group GQ
We have to find all combinations of mutations, permutations of vertices andsimultaneous inversions of edges, that preserve quiver. This is purelycombinatorial problem. Example:

P. Gavrylenko

x2x3

23

1x1

4x4

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Mapping class group GQ
We have to find all combinations of mutations, permutations of vertices andsimultaneous inversions of edges, that preserve quiver. This is purely
combinatorial problem. Example:

P. Gavrylenko

x2x3

2

3

1x1

4x4x2 (1 + x1−1 )−2 x3
23
11/x1

Deautonomization of cluster integrable systems

4x4 (1 + x1 )2

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Mapping class group GQ
We have to find all combinations of mutations, permutations of vertices andsimultaneous inversions of edges, that preserve quiver. This is purely
combinatorial problem. Example:
x2x3

2

3

1x1

4x4x2 (1 + x1−1 )−2 x3
2
3
11/x1

4x4 (1 + x1 )2

1+x3 2x2 ( 1+x−1 )12
11/x1P. Gavrylenko

Deautonomization of cluster integrable systems

1/x3
3
41+x1 2x4 ( 1+x−1 )3November 19, 20188 / 26

Mapping class group GQ
We have to find all combinations of mutations, permutations of vertices andsimultaneous inversions of edges, that preserve quiver. This is purely
combinatorial problem. Example:
x2x3231x1

4x4x2 (1 + x1−1 )−2 x3
2
3
11/x1

4x4 (1 + x1 )21+x3 2x2 ( 1+x−1 )12
1
1/x1P. Gavrylenko

Deautonomization of cluster integrable systems

1/x3
3
4
1+x1 2x4 ( 1+x−1 )3November 19, 20188 / 26

Mapping class group GQ
We have to find all combinations of mutations, permutations of vertices andsimultaneous inversions of edges, that preserve quiver. This is purelycombinatorial problem. Example:
x2x3

2

3

1x14x4

1/x1
2
11+x3 2x2 ( 1+x−1 )1P. Gavrylenko

x2 (1 + x1−1 )−2 x3
2
3
1
1/x1

1+x1 2x4 ( 1+x−1 )3

1+x3 2x2 ( 1+x−1 )
1

32

4
1/x3

4x4 (1 + x1 )2

1
1/x1

Deautonomization of cluster integrable systems

1/x3
3
4
1+x1 2x4 ( 1+x−1 )3November 19, 20188 / 26

Discrete flow
Take a map coming from quiver automorphism.
Forget about x1 x2 x3 x4 = 1 =⇒ no integrable system. x1 x2 x3 x4 = q
1 + x3 2 −1
1 + x1 2 −1
T : (x1 , x2 , x3 , x4 ) 7→ x2 (),x,x(),x4
1
31 + x1−1
1 + x3−1x1 + z 2 −1) , x1 , qz, q
T : (x1 , x2 , z, q) 7→ x2 (x1 + 1
Casimir z becomes “time”, so introduce xi = xi (z), T : xi (z) 7→ xi (qz).
x1 (z) + z
x1 (z) + 1

2x1 x2 + √x11 x2 +qx1 (qz)x1 (q −1 z) =(1)0This is q-Painlevé III3 equation, or P(A7 ).
Only for q = 1 flow T preserves H =P. Gavrylenko√

Deautonomization of cluster integrable systems

x1x2 + Zqx2
x1 .
November 19, 2018

9 / 26

Quantization (one of the main advantages)
In addition to non-autonomous parameter q one may add quantum
deformation p:x̂i x̂j = p −2ij x̂j x̂i
There are quantum mutationsµj : x̂j 7→ x̂j−1 ,

1/|ij |x̂i

1/|ij |7→ x̂i

sgn ij

1 + p x̂j

sgn ij, i 6= jAll groups GQ are the same.
And so there are quantum deformations of all systems. For example,quantum q-Painlevé III3 : x̂1 (q −1 z)1/2 x̂1 (qz)1/2 = x̂1 (z) + pz ,x̂1 (z) + px̂1 (z)x̂1 (q −1 z) = p 4 x̂1 (q −1 z)x̂1 (qz).
Different approaches to quantization were also considered long before by
K. Hasegawa, G. Kuroki, H. Nagoya, Y. Yamada.
P. Gavrylenko

Deautonomization of cluster integrable systems

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Generic solution [BShch]/[BGM] (quantum q-GIL formula)x̂1 (z) = pz 1/2 T̂1 (z)2 T̂3 (z)−2T̂1 (z) = âXŝ n Z 2,0 (ûq 2n ; q1 q2−1 , q22 |z)n∈Z

XT̂3 (z) = i âŝ n Z 2,0 (ûq 2n ; q1 q2−1 , q22 |z)n∈ 21 +Z

Whereq2 = q 1/2 ,q1 = q2−1 p 2 ,ûŝ = p 4 ŝ ûand also
q22 â = p −2 âq22 = âq1−1 q2 ,q1 q2−1 â = p 2 âq1 q2−1 = âq12

So here we have operator Fourier transformation.
Z 2,0 (ûq 2n ; q1 q2−1 , q22 |z) is a topological string partition function (with prefactor),or 5D Nekrasov function for SU(2) pure gauge theory.
Proofs for the classical case: M. Bershtein, A. Shchechkin (also conjectured this);(1)
M. Jimbo, H. Nagoya, H. Sakai (P(A3 ) ) + Y. Matsuhira, H. Nagoya (limit)
P. Gavrylenko

Deautonomization of cluster integrable systems

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Numerology of cluster integrable systems(# of variables) = 2*Area(∆).
Dimension of the phase space = 2*(# of internal points).
Number of Casimirs (without q) = (# of boundary points) - 3.
(# of discrete flows) = number of Casimirs (without q)
Simplest cases: 1) one discrete flow, 2) one Hamiltonian.

P. Gavrylenko

Deautonomization of cluster integrable systems

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Directions of the generalization
4 boundary points, internal points on one line

One
internalpoint

Non-autonomous
discrete Hirota
equations? General Newton polygons

q-difference
Painlevé
equationsP. Gavrylenko

Deautonomization of cluster integrable systems

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4 boundary points, hyperelliptic curves (Toda family)
Classification: Y N,k polygons with 0 ≤ k ≤ N (left picture) and L1,2N−1,2
polygons (right picture):(0,1)(N-k,1)

(0,0)

(N,0)

(0,-1)

N=6, k=2(N-1,0)(-1,-1) (0,-1)

N=6

Quivers for Y N,k theories can be glued from blocks of three types 0, 1, -1,
respectively. N = N1 + N0 + N−1 , k = N1 − N−1 .

Similar non-cyclic quivers appeared in Di Francesco’s papers.
Graphs also computed by S. Franco, A. Hanany, K. Kennaway, D. Vegh, B. Wecht
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Action of the automorphism group

N=6, k=2

P. Gavrylenko

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Action of the automorphism group

N=6, k=2

P. Gavrylenko

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Equations [BGM?]
Mutable “+”-variables labelled by the points of integer lattice: x(n,m) . They satisfy
periodicity condition and Y-system in order to be compatible with mutations:x(n,m+1) x(n,m−1)(1 + x(n+1,m) )(1 + x(n−1,m) ),=2(1 + x(n,m) )2x(n,m)

x(n,m) = x(n+N,m+k)

One can move from Y-system to T-system (from X-clusters to A-clusters):
1/N (kn−Nm+N)/N 2 τ(n−1,m−1) τ(n+1,m−1)q,2τ(n,m−1)x(n,m) = z0

1/N (kn−Nm)/N 22
τ(n,m+1) τ(n,m−1) = τ(n,m)+ z0q

τ(n,m) = τ(n+N,m+k)τ(n+1,m) τ(n−1,m)

And after some change of labeling:τj (qz) τj q −1 z = τj (z)2 + z 1/N τj+1 q k/N z τj−1 q −k/N z ,P. Gavrylenko

Deautonomization of cluster integrable systemsj ∈ Z/NZNovember 19, 201817 / 26

Solution [BGM]τj (qz) τj q −1 z = τj (z)2 + z 1/N τj+1 q k/N z τj−1 q −k/N z ,j ∈ Z/NZ

Fourier transformation of partition function of 5D SU(N) pure gauge theory withChern-Simons term at level k:
XYj ∈ Z/NZ .
τj (z) =(siΛi ) · Z N,k ({ui q Λi }; q, q −1 |z),~Λ∈QN−1 +ωji

where QN−1 is SL(N) root lattice, and ωi are SL(N) fundamental weights(ω0 = 0).
Compare with K. Takasaki’s talk: ui ≈ qP. Gavrylenko

N+1−2i2N,k=N

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Nekrasov functions
N,kN(~u ; q1 , q2 |z) ,Z N,k (~u ; q1 , q2 |z) = ZclN,k (~u ; q1 , q2 |z) · Z1−loop(~u ; q1 , q2 ) · Zinst

where!
P23(log ui )(log ui )log z+k,−2 log q1 log q2−6 log q1 log q2PZclN,k (~u ; q1 , q2 |z) = exp
NZ1−loop(~u ; q1 , q2 ) =Y

(ui /uj ; q1 , q2 )∞ ,1≤i6=j≤N

X z |~λ| QN (Tλ(i) (ui ; q1 , q2 ))ki=1,QNNλ(i) ,λ(j) (ui /uj ; q1 , q2 )i,j=1~λX
X~λ = (λ(1) , . . . , λ(N) ), |~λ| =|λ(i) |, |λ| =λj ,YY−`(s)−1−aµ (s)−1 `λ (s)a (s)(1 − uq2q1 ) ·(1 − uq2λ q1 µ),
Nλ,µ (u, q1 , q2 ) =N,kZinst(~u ; q1 , q2 |z) =s∈λs∈µ|λ0 |− 12 (kλ0 k) 12 (|λ|−kλk)
Tλ (u; q1 , q2 ) = u −|λ| q1q2=

Yu −1 q11−i q21−j ,(i,j)∈λkλk =X

P. Gavrylenko

λ2j .
Deautonomization of cluster integrable systems

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Differential limit (5D → 4D)
τj (qz) τj q −1 z = τj (z)2 + z 1/N τj+1 q k/N z τj−1 q −k/N z
We take q = exp R, z = R 2N z and send R → 0:(∂log z )2 log τj = z1/N

τj+1 τj−1,
τj2j ∈ Z/NZ

So we see no dependence on k. In the different variables
φj = log τj − log τj−1 ,1r = 2Nz 2N

We haved 2 φn
1 dφn+= e φn+1 −φn − e φn −φn−1
dr 2r drThis is radial Toda equation, for N = 2 — radial sinh-Gordon equation (PIII3 ).

P. Gavrylenko

Deautonomization of cluster integrable systems

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Newton polygons with one internal point
34a4b4c5a5b6a6b6c6d7a7b8a8b8c9

Same area ⇔ same quivers, except for 4a and 4b . 3 is trivial.

P. Gavrylenko

Deautonomization of cluster integrable systems

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Quivers and their mapping class groups [BGM]
(1)

(1)

W̃ ((A1 + A1 )(1) )W (A1 )

Dih4 n W (A1 )S3W̃ ((A1 + A2 )(1) )
12
13

2

12

124343(1)236
7P. Gavrylenko

558375354(1)W̃ (E6 )6544
164W̃ (D5 )
413(1)W̃ (D4 )
21232

17896

Deautonomization of cluster integrable systems

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8a,b,c case — q − PVI [BGM]
2

12

30

4
1

3

7

54
5

8

6

s0 = (1, 2),
s1 = (5, 6),s2 = (1, 5) ◦ µ5 ◦ µ1 ,s3 = (3, 7) ◦ µ3 ◦ µ7 ,s4 = (3, 4),s5 = (7, 8),
π = (1, 7, 5, 3)(2, 8, 6, 4),
σ = (1, 7)(2, 8)(3, 5)(4, 6) ◦ ς,here ς — inversion of all arrows
P. Gavrylenko

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9 case [BGM]6541234635

2

178

97

s1 = (2, 3),s2 = (1, 2),s4 = (4, 5),s5 = (5, 6),
s6 = (7, 8),s0 = (8, 9),s3 = (4, 7) ◦ µ1 ◦ µ4 ◦ µ7 ◦ µ1 ,π = (1, 4, 7)(2, 5, 8)(3, 6, 9),
σ = (1, 7)(2, 8)(3, 9) ◦ ς
P. Gavrylenko

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More complicated examplesmK. Kajiwara, M. Noumi, Y. Yamada: action of the(1)(1)
W (An−1 ) × W (Am−1 ) Weyl group(?) m = 2, n = 2k case: N. Okubo, T. Suzuki (?)
nm(work in progress) M. Semenyakin, A. Marshakov:
deautonomization of classical XXZ spin chain (with?

nm

small GL(m) orbits) = q-Schlesinger. Solutions are
given by Fourier transform of quiver Nekrasov partition(1)(1)
functions with matter. W (An−1 )2 × W (Am−1 )2k(partly work in progress) R. Inoue, T. Ishibashi, T. Lam,
(1)
H. Oya, P. Pylyavskyy: quiver Qm (An−1 )kn
P. Gavrylenko

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Thank you for your attention!

P. Gavrylenko

Deautonomization of cluster integrable systems

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