Yang-Baxter and braid dynamics

Transcription

Yang-Baxter and braid dynamics
Alexander Veselov, Loughborough University
GADUDIS, Glasgow, April 1, 2009
Plan
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Dynamics and groups
Plan
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Dynamics and groups
I
Braid group and modular group
Plan
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Dynamics and groups
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Yang-Baxter versus braid relations
Plan
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Dynamics and groups
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Diophantine analysis and Markov equation
Plan
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Dynamics and groups
I
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Markov dynamics
Plan
I
Dynamics and groups
I
I
I
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Markov dynamics
I
Links with Painleve-VI and Frobenius manifolds
Plan
I
Dynamics and groups
I
I
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Markov dynamics
I
Links with Painleve-VI and Frobenius manifolds
Thanks to
Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy
for illuminating discussions and to
Vsevolod Adler
for very useful comments and help with computer simulations.
Groups and dynamics
Dynamical Erlangen programme:
GROUPS ——– DYNAMICS
Usual discrete dynamical system can be considered as nonlinear
representation of Z.
Groups and dynamics
INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT
GROUPS (Gromov’s theorem)
Groups and dynamics
Examples:
Julia, Fatou, Ritt ; AV: commuting maps
Okamoto; Noumi, Yamada: affine Weyl group actions
Groups and dynamics
Examples:
Julia, Fatou, Ritt ; AV: commuting maps
Okamoto; Noumi, Yamada: affine Weyl group actions
I would like to discuss today the actions of the braid group B3 , which has
exponential growth.
Braid group
Braid group Bn can be defined as the fundamental group π1 (Mn ) of the
configuration space Mn of n different (unordered) points in C.
Braid group
It is generated by n − 1 elements σ1 , . . . , σn−1 with the following relations
σi σi+1 σi = σi+1 σi σi+1
σi σj = σj σi ,
|i − j| > 1.
Braid group
It is generated by n − 1 elements σ1 , . . . , σn−1 with the following relations
σi σi+1 σi = σi+1 σi σi+1
σi σj = σj σi ,
|i − j| > 1.
If we add the relations
σi2 = 1
the braid group Bn is reduced to the symmetric group Sn .
Braid group B3
bx
bz
bx
bz
=
bx
bz
bx
bz
bz bx bz
=
bx bz bx
braid group defining relations
(bx bz) 3
Braid group B3 and the modular group
Braid group B3 is generated by σ1 and σ2 with the only relation
σ1 σ2 σ1 = σ2 σ1 σ2 .
It has the centre Z generated by
(σ1 σ2 )3 = (σ1 σ2 σ1 )(σ2 σ1 σ2 ).
σ1 σ2 σ1 = σ2 σ1 σ2 .
(σ1 σ2 )3 = (σ1 σ2 σ1 )(σ2 σ1 σ2 ).
The quotient B3 /Z is isomorphic to the modular group
PSL(2, Z) = SL(2, Z)/{±I },
which is itself the free product Z2 ∗ Z3 .
σ1 σ2 σ1 = σ2 σ1 σ2 .
(σ1 σ2 )3 = (σ1 σ2 σ1 )(σ2 σ1 σ2 ).
The quotient B3 /Z is isomorphic to the modular group
PSL(2, Z) = SL(2, Z)/{±I },
which is itself the free product Z2 ∗ Z3 .
A natural homomorphism
PSL(2, Z) → SL(2, Z2 ) ∼
= S3
maps it to the symmetric group.
Let X be any set and R be a map:
R : X × X → X × X.
Let Rij : X n → X n , X n = X × X × ..... × X be the maps which acts as R on
i-th and j-th factors and identically on the others. If P : X 2 → X 2 is the
permutation: P(x, y ) = (y , x), then R21 = PRP.
R : X × X → X × X.
The map R is called Yang-Baxter map if it satisfies the Yang-Baxter relation
R12 R13 R23 = R23 R13 R12 .
R : X × X → X × X.
R12 R13 R23 = R23 R13 R12 .
The twisted maps
Si = Pii+1 Rii+1
satisfy the braid relations
Si Si+1 Si = Si+1 Si Si+1 ,
S12 S23 S12 = S23 S12 S23 .
R : X × X → X × X.
R12 R13 R23 = R23 R13 R12 .
The twisted maps
Si = Pii+1 Rii+1
satisfy the braid relations
Si Si+1 Si = Si+1 Si Si+1 ,
S12 S23 S12 = S23 S12 S23 .
The reversible Yang-Baxter maps satisfy the relation
R21 R = Id,
corresponding to
Si2 = Id.
Yang-Baxter relations
1
1
2
=
2
3
3
Figure: Yang-Baxter relation
1
1
=
2
2
Figure: Reversibility
Transfer dynamics
Define the transfer maps
(n)
Ti
: X n → X n , i = 1, . . . , n
by
(n)
Ti
= Rii+n−1 Rii+n−2 . . . Rii+1 ,
where the indices are considered modulo n. In particular
(n)
T1 = R1n R1n−1 . . . R12 .
Transfer dynamics
(n)
Ti
: X n → X n , i = 1, . . . , n
by
(n)
Ti
= Rii+n−1 Rii+n−2 . . . Rii+1 ,
(n)
T1 = R1n R1n−1 . . . R12 .
(n)
For any reversible Yang-Baxter map R the transfer maps Ti
each other:
(n) (n)
(n) (n)
T i Tj = T j T i
commute with
and satisfy the property
(n)
(n)
T1 T2 . . . Tn(n) = Id.
(n)
Conversely, if Ti
satisfy these properties then R is a reversible YB map.
Transfer dynamics
(n)
Ti
: X n → X n , i = 1, . . . , n
by
(n)
Ti
= Rii+n−1 Rii+n−2 . . . Rii+1 ,
(n)
T1 = R1n R1n−1 . . . R12 .
(n)
For any reversible Yang-Baxter map R the transfer maps Ti
each other:
(n) (n)
(n) (n)
T i Tj = T j T i
commute with
and satisfy the property
(n)
(n)
T1 T2 . . . Tn(n) = Id.
(n)
Conversely, if Ti
satisfy these properties then R is a reversible YB map.
Transfer dynamics corresponds to the translations for a natural action of the
(1)
extended affine Weyl group Ãn−1 . Note that the action of the corresponding
braid group is reduced simply to the symmetric group Sn .
Commutativity of the transfer maps
i+1
i
i-1
i+1
i
i-1
=
j-1
j-1
j
j
j+1
j+1
Figure: Commutativity of the transfer maps
Classification
Quadrirational case, X = CP 1 : Adler, Bobenko, Suris
Y
X
V
U
Figure: A quadrirational map on a pair of conics
History: golden ratio and Hurwitz theorem
It is well-known that any irrational real number α has a rational approximation
p
(given by continued fraction expansion) such that
q
|α −
p
1
| < 2.
q
q
p
q
|α −
p
1
| < 2.
q
q
Hurwitz: For any α one can find infinitely many
|α −
p
q
such that
p
1
|< √
.
q
5q 2
For the golden ratio
α=φ=
√
1+ 5
= [1, 1, 1, ...]
2
(and their equivalents) this is the best possible approximation.
p
q
|α −
p
1
| < 2.
q
q
Hurwitz: For any α one can find infinitely many
|α −
p
q
such that
p
1
|< √
.
q
5q 2
For the golden ratio
α=φ=
√
1+ 5
= [1, 1, 1, ...]
2
(and their equivalents) this is the best possible approximation.
In other words, φ is the most irrational number. What is the next one ?
Markov spectrum
Define Markov constant as the minimal possible c in |α − pq | <
c
:
q2
µ(α) = lim inf q||αq||,
q→∞
where ||x|| = min |x − n|, n ∈ Z.
The set of all possible values of µ(α) is called Markov spectrum.
Markov spectrum
c
:
q2
q→∞
where ||x|| = min |x − n|, n ∈ Z.
Markov theorem. The Markov spectrum above 1/3 is discrete and consists of
the numbers
m
µ= √
,
9m2 − 4
where m is one of the Markov numbers:
m = 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, ...
Markov spectrum
c
:
q2
q→∞
where ||x|| = min |x − n|, n ∈ Z.
Markov theorem. The Markov spectrum above 1/3 is discrete and consists of
the numbers
m
µ= √
,
9m2 − 4
where m is one of the Markov numbers:
m = 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, ...
For example, the next two most irrational numbers after φ are
√
2 = [1, 2, 2, 2....]
and
α=
9+
√
221
= [2, 2, 1, 1].
10
Markov triples
Markov numbers are parts of the Markov triples, which are the positive integer
solutions of the Markov equation
x 2 + y 2 + z 2 − 3xyz = 0.
Markov triples
x 2 + y 2 + z 2 − 3xyz = 0.
Theorem (Markov) All Markov triples can be obtained from (1, 1, 1) by the
involution
σ : (x, y , z) → (x, y , 3xy − z)
and permutations.
Markov triples
x 2 + y 2 + z 2 − 3xyz = 0.
involution
σ : (x, y , z) → (x, y , 3xy − z)
and permutations.
(1, 1, 1) → (1, 1, 2) → (1, 2, 1) → (1, 2, 5) → (1, 5, 2) → (1, 5, 13) → ...
Markov triples
x 2 + y 2 + z 2 − 3xyz = 0.
involution
σ : (x, y , z) → (x, y , 3xy − z)
and permutations.
(1, 1, 1) → (1, 1, 2) → (1, 2, 1) → (1, 2, 5) → (1, 5, 2) → (1, 5, 13) → ...
Markov dynamics is the action of PSL2 (Z) (and hence of the braid group B3 )
on C3 determined by σ and τ (x, y , z) = (z, x, y ). It has an obvious integral
F = x 2 + y 2 + z 2 − 3xyz.
Markov dynamics: F = 0.
Markov triples in logarithmic scale:
Markov tree
Markov tree
Markov tree
Markov tree
Markov tree
Markov tree
Markov tree
Markov tree
F = 49 : ”integrable chaos”
On the special level F =
4
9
the dynamics can be linearised by substitution
2
2
2
2
x = cos φ, y = cos ψ, z = cos(φ + ψ), z 0 = cos(φ − ψ).
3
3
3
3
The hyperbolic elements of SL(2, Z) act as Anosov maps, so the dynamics is
nevertheless chaotic.
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
4
9
2
2
2
2
3
3
3
3
Generic level of F : chaos ?
What happens at generic level is an interesting question (cf. Hone)
Tropical Markov dynamics
It is not clear how to tropicalize the formula z 0 = 3xy − z, so we can use
another relation
zz 0 = x 2 + y 2 + c
to define the following tropical Markov dynamics:
σ : (X , Y , Z ) → (X , Y , 2 max(X , Y , C ) − Z )
and
τ : (X , Y , Z ) → (Y , Z , X ).
It has the following tropical integral
F = 2 max(X , Y , Z , C ) − (X + Y + Z ),
corresponding to
F =
x2 + y2 + z2 + c
.
xyz
Tropical Markov orbit
Markov orbit and quantum cohomology
A remarkable observation due to Dubrovin is that in the theory of Frobenius
manifolds
Markov orbit corresponds to quantum cohomology of projective plane CP 2
manifolds
The corresponding potential is
F =
1
t1 (t1 t3 + t22 ) + G (t2 , t3 ),
2
where
G (t2 , t3 ) =
∞
X
d=1
Nd
t33d−1
e dt2 ,
(3d − 1)!
Nd is the number of rational curves in CP 2 passing by 3d − 1 points in general
position.
manifolds
The corresponding potential is
F =
1
t1 (t1 t3 + t22 ) + G (t2 , t3 ),
2
where
G (t2 , t3 ) =
∞
X
d=1
Nd
t33d−1
e dt2 ,
(3d − 1)!
Nd is the number of rational curves in CP 2 passing by 3d − 1 points in general
position.
Kontsevich, Manin: WDVV equation leads to recurrent formula for Nd :
!
!
X
3d − 1
3d − 1
Nd =
Nk Nl k 2 l[l
−k
].
3k − 2
3k − 1
k+l=d
Painlevé-VI and braid group orbits
Painlevé VI equation describes monodromy preserving deformations of Fuchsian
systems
„
«
dΦ
A2
A3
A1
=
+
+
Φ,
Φ ∈ SL(2, C).
dλ
λ − u1
λ − u2
λ − u3
Painlevé-VI and braid group orbits
Painlevé VI equation describes monodromy preserving deformations of Fuchsian
systems
„
«
dΦ
A2
A3
A1
=
+
+
Φ,
Φ ∈ SL(2, C).
dλ
λ − u1
λ − u2
λ − u3
Analytic continuation induces an action of the braid group B3 on the triples of
the monodromy matrices M1 , M2 , M3 :
σ1 (M1 , M2 , M3 ) = (M2 , M2 M1 M2−1 , M3 ), σ2 (M1 , M2 , M3 ) = (M1 , M3 , M3 M2 M3−1 ).
In 2 × 2 case this leads to the generalised Markov dynamics with
σ : (x, y , z) → (x, y , xy − z − ω3 )
and
τ : (x, y , z; ω1 , ω2 , ω3 ) → (y , z, x; ω2 , ω3 , ω1 ),
preserving
F = x 2 + y 2 + z 2 − xyz − ω1 x − ω2 y − ω3 z.
Markov dynamics and special solutions of PVI
Markov orbit corresponds to quantum cohomology solution (Dubrovin)
Integrable case F =
4
9
corresponds to the Picard solutions (Fuchs, Hitchin)
Integrable case F =
4
9
Dubrovin: Finite orbits correspond to the algebraic solutions of Painleve-VI
(Dubrovin, Mazzocco)
Lisovyy, Tikhiy: classification of the finite braid orbits
Integrable case F =
4
9
Comparison with Watanabe result suggests that in general Markov dynamics
can not be ”integrated in classical functions” (see, however, Picard solutions).
Integrable case F =
4
9
Comparison with Watanabe result suggests that in general Markov dynamics
can not be ”integrated in classical functions” (see, however, Picard solutions).
Relation with Okamoto: affine Weyl groups are acting on the set of all
solutions while braid group describes the monodromy of a given solution.
Other relations and questions
Teichmüller spaces (Chekhov-Penner, Fock-Goncharov)
Cluster algebras and Laurent phenomenon (Berenstein, Fomin, Zelevinsky;
Hone)
2
2
xn+1 xn = xn+1
+ xn+2
+ c.
Hone)
2
2
xn+1 xn = xn+1
+ xn+2
+ c.
Poisson structures (Goldman, Chekhov-Fock)
Hone)
2
2
xn+1 xn = xn+1
+ xn+2
+ c.
Generalisations to K3 surfaces in (CP 1 )3 (Silverman)
Hone)
2
2
xn+1 xn = xn+1
+ xn+2
+ c.
Tropical and geometrical versions
Hone)
2
2
xn+1 xn = xn+1
+ xn+2
+ c.
Tropical and geometrical versions
Relations with arithmetic (Markov, Mordell, Silverman)

Yang-Baxter and braid dynamics

Transcription

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