Quantum Painlev´e tau-functions

Quantum Painlev ´e tau-functions

Gen Kuroki

Tohoku University

December 11, 2018

Conformal field theory, isomonodromy tau-functions and Painlev ´e equations, 2018

December 10 - 12, 2018

Room B301, Graduate School of Science, Kobe University 2018-12-11 Version 1.0

https://genkuroki.github.io/documents/20181211QuantumPainleveTau.pdf

Summary

For any symmetrizable GCM, we can introduce the proper non-commutativity of quantum τ^-functions by τi = exp(∂/∂α^∨_i ).

Quantum q-Hirota-Miwa equations for A⁽¹⁾

n−1-case.

Quantized Lax and Sato-Wilson forms of the extended affine Weyl group action for A⁽¹⁾

n−1-case.

QuantizedW(e A⁽¹⁾

m−1) ×W(e A⁽¹⁾

n−1)-action for mutually prime mand n.

An appropriate quantization of qP_IV.

General theory of the quantum and q -difference version of

τ ^-functions

generated by the Weyl group action for any symmetrizable GCM

We will consider the birationalaction of the Weyl group (B ¨acklund transformations).

Want to construct quantizations of classical τ^-functions of Painlev ´e systems (differrential and q-difference).

Difficulty. How to find the proper non-commutativity of quantized τ-functions?

My Answer.

parameter variable α^∨_i ↔simple coroot.

classical τi ↔ exp(fundamental weight)

In the situation above, the appropriate definition of quantized τi is

τi = exp(∂/∂α^∨_i ).

Quantum Algebra: Definiton

Consider the associative algebra

(precisely the non-commutative field) generated by dependent variables: f_i

parameter variables: α^∨_i τ-variables: τi

with the relations

q-Serre relations of f_i.

α^∨_i commutes withα^∨_j ^and f_j. τi commutes with τj and f_j.

τiα^∨_jτ⁻_i¹ = α^∨_j +δi j. (τi = exp(∂/∂α^∨_i ))

Quantum Algebra: q -Serre relations

[a_{i j}]_i,j∈I: GCM with d_ia_{i j} = d_ja_ji, d_i ∈ Z>0. q: an inderminate.

q_i := q^di, [x]_q := q^x − q^−x q− q^{−1 ,} [n]_q! := [1]_q[2]_q· · ·[n]_q,

[n k ]

q

:= [n]_q! [k]_q![n− k]_q!^. q-Serre relations: if i, j ∈ I and i , j, then

1∑−ai j

k=0

(−1)^k

[1− a_{i j} k

]

qi

f^{1−ai j−k}

i f_jf^k

i = 0.

Quantum Algebra: Relations

1∑−ai j

k=0

(−1)^k

[1− a_{i j} k

]

qi

f^{1−ai j−k}

i f_jf^k

i = 0 (i , j),

α^∨_i α^∨_j = α^∨_jα^∨_i ^, α^∨_i f_j = f_jα^∨_i ^, τiτj = τjτi, τif_j = f_jτi, τiα^∨_jτ⁻¹ = α^∨_j +δi j

τi’s are the exponentials of the canonical conjugate variables of the parameter variablesα^∨_i ^.

Quantum Algebra: Summary

f_i (i ∈ I)↔ Chevalley generators of U_q(n₋) α^∨_i ↔simple coroot

τi ↔ exp(fundamental weight)

f_i (i ∈ I) satisfy the q-Serre relations.

α^∨_i ^andτi commute with f_i. α^∨_i commutes withα^∨_j^. τi commutes with τj.

τiα^∨_j = (α^∨_j +δi j)τi (τi = exp(∂/∂α^∨_i )).

Weyl group action

Weyl group: W = ⟨s_i | i ∈ I⟩^. s²

i = 1,

a_{i j}a_ji = 0 =⇒ s_is_j = s_js_i, a_{i j}a_ji = 1 =⇒ s_is_js_i = s_js_is_j, a_{i j}a_ji = 2 =⇒ (s_is_j)² = (s_js_i)², a_{i j}a_ji = 3 =⇒ (s_is_j)³ = (s_js_i)³. [A,B]_q := AB − qB A.

(ad_q f_i)(x) := [f_i, x]

q^⟨α

∨ i, β⟩

i

, where β = the weight of x. Then(ad_q f_i)^k+1(f_j) = [f_i,(ad_q f_i)^k(f_j)]_q^2k+ai j.

Weyl group action (B ¨acklund transformations):

s_i(f_i) := f_i,

s_i(f_j) :=

−ai j

∑

k=0

q^{(k+ai j)(α}

∨

i−k)

i

[α^∨_i k

]

qi

(ad_q f_i)^k(f_j)f^−k

i (i , j)

s_i(α^∨_j) := α^∨_j − a_jiα^∨_i ^, s_i(τi) := f_iτi

∏

j∈I

τ⁻_j^{ai j} = f_iτ⁻_i ¹∏

j,i

τ⁻_j^{ai j,} s_i(τj) := τj (i , j).

Remark.

τi = exp(∂/∂α^∨_i ) ↔ the fundamental weightΛi

τi∏

j∈Iτ^−a_j ^{i j} ↔ s_i(Λi) = Λi −αi

(αi = ∑

j∈I a_jiΛj, simple root).

The action of s

is an algebra automorphism.

Proof. We can define the algebra automorphism ˜s_i by

˜

s_i(α^∨_j) = α^∨_j − a_jiα^∨_i ,

˜

s_i(τi) = τi

∏

j∈I

τ^−a_j ^{i j}, s˜_i(τj) = τj (i , j),

˜s_i(f_j) = f_j.

Then we obtain, for x = f_j, α^∨_j, τi,

s_i(x) = f^α

∨ i

i ˜s_i(x)f^−α

∨ i

i .

This is an algebra automorphism. □

Useful formulas

f^γ

i f_jf^−γ

i =

−ai j

∑

k=0

q^{(k+ai j)(γ−k)}

i

[γ k ]

qi

(ad_q f_i)^k(f_j)f^−k

i (i , j).

s_i(f_j) = f^α

∨ i

i f_jf^−α

∨ i

i .

If a_{i j} = −1, then s_i(f_j) = q^−α

∨ i

i f_j +[α^∨_i ]_qi(f_if_j − q⁻¹

i f_jf_i)f⁻¹

i

= [1 −α^∨_i ]_qif_j + [α^∨_i ]_qif_if_jf⁻¹

i .

Therefore

s_i(f_j)f_i = [1−α^∨_i ]_qif_jf_i +[α^∨_i ]_qif_if_j.

Remark: quantum geometric crystal

Since f_i (i ∈ I) satisfy the Verma relations, for example, f^a

i f^a+b

j f^b

i = f^b

j f^a+b

i f^a

j if a_{i j}a_ji = 1, we can consider the actions of f^γ

i , e_i(γ) : x 7→ f^γ

i x f^−γ

i ,

as quantum version of a geometric crystal.

For the definition of classical geometric crystal, see Berenstein-Kazhdan arXiv:math/9912105,

arXiv:math/0601391.

Quantum τ -functions: Definition

Fundamental weights: ⟨α^∨_i ,Λj⟩ = δi j. Weight lattice: P := ⊕

i∈IZΛi, P₊ := ∑

i∈IZ≧0Λi. simple roots: αj := ∑

i∈I a_{i j}Λi

Weyl group action on P: s_i(Λj) = Λj −δi jαi. τ^-monomial: τ^µ := ∏

i∈Iτ^µ_i^{i (}µ = ∑

i∈IµiΛi ∈ P₊) (lattice) quantumτ-functions:

τ(λ) := w(τ^µ) for λ = w(µ) ∈ W P₊.

Quantum τ -functions: Regularity

Regularity Theorem: All quantum τ^-functionsτ(λ) (λ ∈ W P₊) are (non-commutative) polynomials in the

dependent variables f_i. □

Main theorem ofarXiv:1206.3419.

Proof of the regularity theorem

ρ := ∑

i∈IΛi,w ◦ λ := w(λ +ρ) −ρ⁽λ ∈ P, w ∈ W).

Assume λ, µ ∈ P₊ and w ∈ W.

L(µ): highest weight simple module.

M(w ◦λ): Verma module with highest weightw ◦ λ^. M(w ◦λ) ⊂ M(λ).

Translation functor: T^µ

λ(M(λ◦ λ)) ⊂ M(w◦ λ)⊗ L(µ). Sketch of the proof: T^µ

λ(M(w◦ λ)) M(w ◦(λ+µ))

implies the regularity theorem. □

Non-trivial relation between the theory of quantum τ-functions and representation theory!

A

n−1

-case ( n ≧ 3 )

i, j ∈ Z/nZ

a_{i j} =







2 (i = j)

−1 (i− j = ±1) 0 (otherwise) d_i = 1, q_i = q.

We will show that the quantum lattice τ^-functions satisfy the quantum q-Hirota-Miwa equations.

Quantum algebra

Consider the associative algebra generated by dependent variables: f_i

parameter variables: α^∨_i

τ-variables: τi (i ∈ Z/nZ⁾ with the defining relations

f²

i f_i±1 −(q + q⁻¹)f_if_i±1f_i + f_i±1f²

i = 0. f_if_j = f_jf_i (j , i±1).

α^∨_i commutes withα^∨_j ^and f_j. τi = exp(∂/∂α^∨_i ).

Weyl group action

Using the useful formulas above, we can show the following formulas:

s_i(f_i±1) = [1−α^∨_i ]_qf_i±1 +[α^∨_i ]_qf_if_i±1f⁻¹

i ,

s_i(f_j) = f_j (j , i±1).

s_i(α^∨_i ) = −α^∨_i ^, s_i(α^∨_i±1) = α^∨_i±1 +α^∨_i ^, s_i(α^∨_j) = α^∨_j ⁽j , i,i±1).

s_i(τi) = f_iτi−1τi+1

τi

, s_i(τj) = τj (i , j).

Extended coroot and weight lattices

Q^∨ := ⊕_n

i=1Zε^∨_i ⊕Zδ^∨, P := ⊕_n

i=1 Zεi ⊕ZΛ0. Dual bases: ε^∨_i , δ^∨ ←→ εi,Λ^0.

Assume ε^∨_i = ε^∨_i+n +δ^{∨ and} εi+n = εi. α^∨_i := ε^∨_i −ε^∨_i+1^, αi := εi −εi+1. Λi = Λ⁰ +ε¹ +· · · +εi (i ∈ Z≧0).

Then P = ⊕_n−1

i=0 ZΛi ⊕Zε^all, ε^all = ε¹ +· · · +εn

P₊ := ∑n−1

i=0 Z≧0Λi +Zεall.

Assume Λi+n = Λi +ε^{all (}i ∈ Z^).

Extended affine Weyl group

W := W(A⁽¹⁾

n−1) = ⟨s₀,s₁, . . . , s_n−1⟩^, s_i+n = s_i. π(s_i) := s_i+1,

We := W(e A⁽¹⁾

n−1) := ⟨π⟩ ⋉W = ⟨π, s₀, . . . , s_n−1⟩^.

(Do not assumeπⁿ= 1.)

Assume λ ∈ Pand β^∨ ∈ Q^∨.

s_i(λ) := λ − ⟨α^∨_i , λ⟩αi, αi = −Λi−1 +2Λi −Λi+1. s_i(β^∨) := β^∨ − ⟨β^∨, αi⟩α^∨_i

π(Λi) := Λi+1, π(ε^all) := ε^all. π(ε^∨_i ) := ε^∨_i+1^, π(δ^∨) := δ^∨.

Translation part of W e

T_i := s_i−1· · · s₂s₁π s_n−1s_n−2 · · ·s_i ∈ We (i = 1, . . . , n).

Assme ν = ∑n

i=1νiεi ∈ ⊕n

i=1 Zεi. T^ν := ∏_n−1

i=0 T^νi

i . Then

T^ν(ε^∨_i ) = ε^∨_i −νiδ^∨ T^ν(δ^∨) = δ^∨, T^ν(α^∨_i ) = α^∨_i −(νi −νi+1)δ^∨,

T^ν(εi) = εi, T^ν(Λ0) = Λ0 +ν^, T^ν(Λi) = Λi +ν^.

Hirota-Miwa equation (1)

Λi = Λi−1+εi, Λi+1 = Λi−1 +εi+εi+1, s_i(Λi)= Λi−1+εi+1, s_i+1(Λi+1) = Λi−1 +εi+εi+2, s_i+1s_i(Λi)= Λi−1+εi+2, s_is_i+1(Λi+1) = Λi−1 +εi+1+εi+2.

τi = τ(Λi−1+εi), τi+1 = τ(Λi−1 +εi +εi+1), s_i(τi)= τ(Λi−1+εi+1), s_i+1(τi+1) = τ(Λi−1 +εi +εi+2), s_i+1s_i(τi)= τ(Λi−1+εi+2), s_is_i+1(τi+1) = τ(Λi−1 +εi+1 +εi+2).

Lemma:

[α^∨_i+1]_qτi s_is_i+1(τi+1) +[α^∨_i ]_qs_i+1s_i(τi)τi+1

= [α^∨_i +α^∨_i+1]_qs_i(τi)s_i+1(τi+1).

Hirota-Miwa equation (2) Proof of Lemma

Warning: τi does not commute with s_is_i+1(τi+1). τi[α^∨_i ]_q = [α^∨_i +1]_qτi, τi[1 −α^∨_i ]_q = −[α^∨_i ]_qτi. Proof of Lemma:

τi s_is_i+1(τi)

= τi s_i (

f_i+1τiτi+2

τi+1

)

= τi([1−α^∨_i ]_qf_i+1 +[α^∨_i ]_qf_if_i+1f⁻¹

i )f_iτi−1τi+1

τi

τi+2

τi+1,

= τi([1−α^∨_i ]_qf_i+1f_i +[α^∨_i ]_qf_if_i+1)τ⁻_i ¹τi−1τi+2,

= (−[α^∨_i ]_qf_i+1f_i +[α^∨_i +1]_qf_if_i+1)τi−1τi+2.

Thus

τi s_is_i+1(τi) = (−[α^∨_i ]_qf_i+1f_i +[α^∨_i +1]_qf_if_i+1)τi−1τi+2. Similarly we obtain

s_i+1s_i(τi)τi+1 = ([1 −α^∨_i+1]_qf_if_i+1 +[α^∨_i+1]_qf_i+1f_i)τi−1τi+2

s_i(τi)s_i+1(τi+1) = f_if_i+1τi−1τi+2.

q-numbers identity (or addition formula ofsin):

[α^∨_i +1]_q[α^∨_i+1]_q +[α^∨_i ]_q[1− α^∨_i+1]_q = [α^∨_i +α^∨_i+1]_q. Therefore

[α^∨_i+1]_qτi s_is_i+1(τi+1) +[α^∨_i ]_qs_i+1s_i(τi)τi+1

= [α^∨_i +α^∨_i+1]_qs_i(τi)s_i+1(τi+1).

In LHS the f_i+1f_i-terms are canceled. □

Hirota-Miwa equation (3)

Apply T^ν to the formula of Lemma. Then we obtain Theorem: The quantum τ-functions of type A⁽¹⁾

n−1

satisfy the quantum q-Hirota-Miwa equations:

[ε^∨_i (ν)−ε^∨_i+1(ν)]_q τi(ν +εi+2)τi(ν+εi +εi+1) +[ε^∨_i+1(ν) −ε^∨_i+2(ν)]_q τi(ν +εi) τi(ν+εi+1 +εi+2) +[ε^∨_i+2(ν) −ε^∨_i (ν)]_q τi(ν +εi+1)τi(ν+εi+2 +εi) = 0 where

τi(ν) := τ(Λi−1 +ν),

ε^∨_i (ν) := T^ν(ε^∨_i ) = ε^∨_i −νiδ^∨. □

Lax and Sato-Wilson forms of the affine Weyl group action

The relation between

the RLL = LLR formalism of quantum groups and the Lax and Sato-Wilson forms of the Painlev ´e systems is non-trivial.

Assume that mand nare mutually prime.

Lax form: RLL=LLR

A⁽¹⁾

m−1-type R-matrix:

Denote the m× mmatrix units by E_{i j} and R(z) := (q − q⁻¹z)

∑m i=1

E_ii ⊗ E_ii +(1 − z)∑

i,j

E_ii ⊗ E_{j j} +(q− q⁻¹)

∑

i<j

(E_{i j} ⊗ E_ji + zE_ji ⊗ E_{i j}).

Local L-matrices: for k = 1, . . . , n,

L_k(z) =

∑m i=1

a_ikE_ii +

m−1

∑

i=1

b_ikE_i,i+1 + zb_mkE_m1.

L_k(z)¹ := L_k(z) ⊗1, L_k(z)² := 1⊗ L_k(z). Fundamental relations:

R(z/w)L_k(z)¹L_k(w)² = L_k(w)²L_k(z)¹R(z/w), L_k(z)¹L_l(w)² = L_l(w)²L_k(z)¹ (k , l).

Equivalent to the q-commutation relations:

a_ikb_ik = q⁻¹b_ika_ik, a_ikb_i+1,k = qb_i+1,ka_ik, a_ika_jk = a_jka_ik, b_ikb_jk = b_jkb_ik, ^etc.

If k , l, then a_ik and b_ik commute with a_jl and b_jl.

Another form of the bidiagonal matrix L_k(z). a_k := diag(a_1k, . . . , a_mk),

b_k := diag(b_1k, . . . , b_mk), Λ(z) :=

∑m i=1

E_i,i+1 + zE_m1 =









0 1 0 ...

... 1

z 0









(shift matrix).

Then

L_k(z) = a_k + b_kΛ(z) =









a_1k b_1k a_2k ...

... b_m−1,k

zb_mk a_mk







.

Lax form: b L(z)

1. L(z) := L₁(z)· · · L_n(z), the global L-operator.

a˜_i := a_i1· · · a_in.

a˜ := diag( ˜a₁, . . . , a˜_m), the diagonal part of L(z).

2. eL(z) := L(z)a˜ ← doubling the diagonal partof L(z). 3. bL(z) := CeeL(z)Ce⁻¹.

Here Ce = diag( ˜c₁, . . . , c˜_m) is uniquely characterized by

˜

c₁ = 1, bL(z) = ∑_n−1

k=0ℓˆkΛ(z)^k +Λ| {z (z)Λ(rz)· · ·Λ(rⁿ⁻¹z)}

highest term

, whereℓˆ⁰,ℓˆ¹, . . . ,ℓˆn−1 are diagonal matrices.

bL(z) = ℓˆ⁰ +ℓˆ¹Λ(z) +· · · + Λ(z)Λ(rz)· · ·Λ(rⁿ⁻¹z). t = diag(t₁, . . . , t_n) := ˜c˜ac˜⁻¹.

Thenℓˆ0 = t² and t_ibL(z) = bL(z)t_i. Define bˆ_i and fˆ_i by

ℓˆ1 = diag( ˆb_i)_i=ⁿ

1 = diag(

(q⁻¹ − q)t_it_i+1fˆ_i)n i=1 . rbL(z) = bL(z)r.

t_i and r shall be identified with parameter variables.

Assume t_i+n = r⁻¹t_i and fˆ_i+n = rfˆ_i.

Example (qP_IV case): (m,n) = (3,2).

bL(z) =







t²

1 (q⁻¹−q)t₁t₂fˆ₁ 1

rz t²

2 (q⁻¹−q)t₂t₃fˆ₂ rz(q⁻¹−q)t₃t₄fˆ₃ z t²

3





.

The 1,rz, z part is the highest part.

Assume eL(z) = A+ BΛ(z)+CΛ(z)², A,B,C are diagonal, andC = diag(c₁,c₂, c₃). Then

c_i = b_i1b_i+1,2a_i+2,1a_i+2,2,

c˜₁ = 1, c˜₃ = c₁, c˜₂ = c₁c₃, r = c₁c₃c₂. Ce = diag( ˜c₁, c˜₂,c˜₃),bL(z) = eCeL(z)eC⁻¹, rbL(z) = bL(z)r.

Lax form: M( b z)

T_z,r := r^z∂/∂z : f(z) 7→ f(rz), r-difference operator.

4. M(z) :b = bL(z)T_z,rⁿ , matrix coefficient r-difference op.

5. Assume t_i = q^−ε∨i and r = q^−δ∨.

Then[α^∨_i ]_q = (t_i+1/t_i − t_i/t_i+1)/(q− q⁻¹) 6. g_i := (t²

i − t²

i+1)/bˆ_i = [α^∨_i ]_q/fˆ_i.

G_i := g_iE_i+1,i + E (i = 1, . . . , n−1).

(G_n(z) := rz⁻¹g_nE_1n+ E.)

Lax form: Weyl group action

Consider the algebra generated by the matrix elements of bL(z) (precisely of ℓˆ⁰, . . . ,ℓˆn−1).

7. Algebra automorphism Weyl group action:

s_i(M(b z)) := G_iM(b z)G⁻¹

i ,

π(M(b z)) := (Λ(z)T_z,r)M(z)(b Λ(z)T_z,r)⁻¹

= Λ(z)bL(rz)Λ(rⁿz)T_z,rⁿ . Then

s_i(t_i) = t_i+1, s_i(t_i+1) = t_i,

s_i( ˆb_i) = bˆ_i, s_i( ˆb_i±1) = bˆ_i±1 ±(t²

i − t²_i+

1)/bˆ_i.

Sato-Wilson form: z -variables

8. Introduce τ0 and z_i by

τ0 = exp(∂/∂δ^∨): τ0r = q⁻¹rτ0, τ0t_j = t_jτ0. z_i = exp(∂/∂ε^∨_i ): z_ir = rz_i, z_it_j = q^{−δi j}t_jz_i. τ^{0 and} z_i commute withτ^0, z_j, fˆ_j.

9. D_Z := diag(z₁, . . . , z_n), Z(z) := U(z)D_Z, where U(z) = E+∑_∞

k=1 u_kΛ(z)^k,

u₁,u₂, . . .are diagonal matrices, b

M(z) = U(z)t²T_zⁿ_,rU(z)⁻¹. Then

M(b z) = Z(z)(qt)²Tⁿ_z,rZ(z)⁻¹.

Sato-Wilson form: Weyl group action

10. The Weyl group action can be extended by s_i(U(z)) = G_iU(z)S^g

i, s_i(D_Z) = (S^g

i)⁻¹D_ZS_i, s_i(t) = S⁻¹

i tS_i, s_i(Z(z)) = G_i(z)Z(z)S_i, π(A(z)) = (Λ(z)T_z,r)A(z)(Λ(z)T_z,r)⁻¹, (A(z) = U(z), D_Z, t,Z(z))

where g_i = (t²

i − t²

i+1)/bˆ_i = [α^∨_i ]_q/fˆ_i, S^g

i := g⁻¹

i E_i,i+1 − g_iE_i+1,i +∑

j,i,i+1 E_{j j}, S_i := [α^∨_i +1]_qE_i,i+1 −[α^∨_i −1]⁻_q¹E_i+1,i +∑

j,i,i+1 E_{j j}. S^g and S_i are permutation matrices i ↔ i +1.

11. Assume z_j+m = z_j, τj = τj−1z_i,

and s_i(τ0) = τ0 (i = 1,2). Then, for i = 1,2, s_i(z_i) = fˆ_iz_i+1, s_i(z_i+1) = fˆ⁻¹

i z_i, s_i(τi) = fˆ_iτi−1τi+1

τi , π(z_i) = z_i+1, π(τi) = τi+1.

Because g_i = [α^∨_i ]_q/fˆ_i and z_iε^∨_j = (ε^∨_j +δi j)z_i implies

[ 0 g_i

−g⁻¹

i 0

]−1[ z_i 0 0 z_i+1

] [ 0 [α^∨_i +1]_q

−[α^∨_i −1]⁻_q¹ 0 ]

= [g⁻¹

i z_i+1[α^∨_i +1]q 0

0 g_iz_i[α^∨_i −1]⁻_q¹ ]

=

[fˆ_iz_i+1 0 0 fˆ⁻¹

i z_i

] . □

Quantum qP

Both canonically quantized and q-difference.

(m , n) -case: X

(z)

Assume that mand nare mutually prime (gcm = 1).

There exist unique diagonal matricesCe₁, . . . ,eC₁ such that Ce₁ = eC_n+1 = Ceand, for k = 1, . . . , n−1,

X_k(r^k−1z) := Ce_kL_k(z)eC⁻¹

k+1 = x_k + Λ(r^k−1z), X_n(rⁿ⁻¹z) := Ce_nL_n(z) ˜aCe⁻¹

1 = x_n + Λ(rⁿ⁻¹z), x_k = diag(x_1k, . . . ,x_m,k).

Then

bL(z) = X₁(z)X₂(rz)· · ·X_n(rⁿ⁻¹z). Assume x_i+m,k = r⁻¹x_ik and x_i,k+n = x_ik.