Painlevé equations from Nakajima-Yoshioka blow-up
relations
Anton Shchechkin
HSE & Skoltech
Moscow, Russia

based on ArXiv 1811.04050 with Mikhail Bershtein

13 November 2018

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 November
with Mikhail
2018
Bershtein
1 / 23

Painlevé III3 equation and its q-deformation
Object of our interest: Painlevé III3 equation, its q-deformation and theirsolutions.
Toda-like form of these equations is a two bilinear equations on twofunctions: τ and τ1 . It is symmetric under τ ↔ τ1 .
Painlevé III321/2 2D[logτ1z] (τ, τ ) = −2z
21/2 2D[logτz] (τ1 , τ1 ) = −2z(1)

df
200
020
where second Hirota differential D[logz] (τ, τ ) = 2τ τ − τ , f = z dz .

q-deformationτ τ = τ 2 − z 1/2 τ12
τ1 τ1 = τ12 − z 1/2 τ 2(2)where τ (z) = τ (qz), τ (z) = τ (q −1 z).

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 Novemberwith Mikhail
2018
Bershtein
2 / 23

Gamayun-Iorgov-Lisovyy tau function

Gamayun-Iorgov-Lisovyy in 2012-2013 proposed power series representation for
the tau function of the (continuous) Painlevé equations. Tau function — Fourier
series of Nekrasov partition functionsXτj (σ, s|z) =s n Z(σ + n|z)(3)
n∈Z+j/2

Z(σ|z) is a certain Nekrasov partition function which depends on thePainlevé equation we take.
s and σ play role of the integration constants of the Painlevé equation.
We take j = 0 for τ and j = 1 for τ1 .

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 Novemberwith Mikhail
2018
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Approach to solve bilinear equations on tau functions

We have some bilinear equation on some tau function, schematicallyhτ, τ i = 0(4)We take the Gamayun-Iorgov-Lisovyy formula as the ansatz with somefunction Z(σ|z).
We collect terms with the power s m . Roughly, the relations with s m isequivalent to the relations with s m−1hτ, τ i = 0

τ=Xs n Z(σ + n|z)

so hτ, τ i|s m = 0 ⇔ hτ, τ i|s 0 = 0n∈Z

(5)s 0 relations is some bilinear relation on Z(σ|z). In our cases Z(σ|z) iscertain Nekrasov function or via the AGT — conformal block.

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 November
with Mikhail
2018
Bershtein
4 / 23Nekrasov instanton partition functions
Nekrasov instanton partition function for pure gauge U(r ) YM is defined as
the equivariant volume of the instanton moduli space−Z(1 , 2 , →a ; z) =+∞XzNZ1(6)M(r ,n)N=0This integral localizes on the fixed points of action of the r + 2-dimensional
torus. Fixed points are labeled by r -tuple of the Young diagrams.
Example for the 4D SU(2) case
Zinst (a1 , a2 ; 1 , 2 |z) =z |λX
Q2
λ(1) ,λ(2)(1)|+|λ(2) |i,j=1 Nλ(i) ,λ(j) (ai − aj ; 1 , 2 ),Nλ,µ (a; 1 , 2 ) =YY=
(a − 2 (aµ (s) + 1) + 1 lλ (s))(a + 2 aλ (s) − 1 (lµ (s) + 1))s∈λ(7)s∈µAnton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 November
with Mikhail
2018
Bershtein
5 / 23

Structure of the Nekrasov function

Structure of the Nekrasov function
Z = Zcl Z1−loop Zinst ,(8)Zinst is given by the Nekrasov formula. Case 5D differs from the case 4D inthe Nekrasov formula by a 7→ 1 − q a .
Classical part Zcl = z ∆(σ) .
Z1−loop is given by the double gamma functions with periods 1,2 .

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
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2018
Bershtein
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Nekrasov functions and solutions of Painlevé equations

Different choice of Z:
4D SU(2) Nekrasov function with 1 + 2 = 0 (or Vir c = 1 conformal block)give the solution of continuous Painlevé equations PVI , V , III 0 s(Gamayun-Iorgov-Lisovyy, 2012-2013) — proved by different methods.
5D SU(2) Nekrasov function with 1 + 2 = 0 — q-deformed Painlevé III3equation (Bershtein-S.,2016) and q-deformed Painlevé VI, V, III1 , III2equations (Jimbo-Nagoya-Sakai, 2017, Matsuhira-Nagoya, 2018)
Quiver gauge theories and SU(N), N > 2 Nekrasov functions: generalization
to the isomonodromic problems of rank > 2, larger number of punctures,
generalization of Toda-like equations to the number of nodes > 2,quantization of the τ function. . . (Bershtein, Gavrylenko, Marshakov, Iorgov,Lisovyy 2015-2018)

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
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2018
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Bilinear relations on 5D SU(2) Nekrasov function

q-deformed Painlevé III3 : conjectural bilinear relation on 5D Nekrasov
functionsXX
Z(σ + n|q −1 z)Z(σ − n|qz) = (1 − z 1/2 )Z(σ + n|z), Z(σ − n|z) (9)2n∈Z2n∈Z

One of the aims of the talk: present the proof of this relation. Before it waschecked up to the power z 12 .
The continuous limit z 7→ R 4 z, q = e R , R → 0 give usXX
D 2 (Z(σ + n|z), Z(σ − n|z)) = −z 1/2
Z(σ + n|z), Z(σ − n|z) (10)2n∈Z2n∈Z

We proved these relations by the representation theory of Super Virasoro
algebra when we proved the Gamayun-Iorgov-Lisovyy conjecture .

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 Novemberwith Mikhail2018
Bershtein
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Nakajima-Yoshioka relations

There are blow-up relations on 4D and 5D partition functions [Nakajima, Yoshioka
c2 = (C2 blowed up in2003,2005]. They express instanton partition function on C2the point) as a bilinear relation on C instanton partition function
XZCc2 (a|1 , 2 |z) =ZC2 (a + 1 n|1 , 2 − 1 |z)ZC2 (a + 2 n|1 − 2 , 2 |z) (11)n∈ZandZCc2 (a|1 , 2 |z) = ZC2 (a|1 , 2 |z)(12)

There are also differential (for 4D) and q-difference (for 5D) Nakajima-Yoshioka
relations.

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 Novemberwith Mikhail
2018
Bershtein
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Nakajima-Yoshioka relations: c = −2 tau function

Take particular case 1 + 2 = 0 in Nakajima-Yoshioka relations . Then in CFT
terms c = 1 partition function is a bilinear combination of c = −2
X−+Zc=1 (σ|z) =Zc=−2(σ + n|z/4)Zc=−2(σ − n|z/4),
(13)
n∈Z

where Z ± correspond to ±1 , ∓21 partition functions.
Then it is natural to compose Fourier seriesτ (σ, s|z) = τ− (σ, s|z)τ+ (σ, s|z)

(14)

where
τ± (σ, s|z) =X±s n/2 Zc=−2(σ + n|z/4)

(15)

n∈Z

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 November
with Mikhail
2018
Bershtein
10 / 23c = −2 tau function: differential relations
Differential Nakajima-Yoshioka relations− +1/41
τ1 ,D[logz] (τ , τ ) = z

2− +D[logz] (τ , τ ) = 0,

3− +
1/4 0
D[log
τ1 ,z] (τ , τ ) = z

4− +D[logz] (τ , τ ) = −2z.τ

(16)

The PIII3 equation follow from these equations. Namely, second equation implies
zpd ±1
τ = (ζ ± ζ 0 )τ± ,dz2(17)where ζ = τ 0 /τ is a Hamiltonian of the PIII3 equation. From the fourth equation(ζ 00 − ζ 0 )2 = 4ζ 02 (ζ − ζ 0 ) − 4zζ 0 ,(18)which is Hamiltonian form of PIII3 equation.

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 Novemberwith Mikhail
2018
Bershtein
11 / 23

Toda-like form from NY relations: q-caseq-difference Nakajima-Yoshioka relations
τ+ τ− − τ+ τ− = −2z 1/4 τ1 ,τ+ τ− + τ+ τ− = 2τ .

(19)

Proposition
Take (19) and τ = τ+ τ− . Then τ and τ1 satisfy Toda-like equationτ τ = τ 2 − z 1/2 τ12 .

(20)

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
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2018
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Proof and consequences

Proof.
Proof is extremely elementary. We substitute τ1 and τ in different ways
τ+ τ− τ+ τ− =1 + −1(τ τ + τ+ τ− )2 − (τ+ τ− − τ+ τ− )244(21)Nakajima-Yoshioka relations are proven, so we obtain the proof of theconjectured bilinear relation (9) in q-case automatically.
There is, of course, differential analogue of this Proposition.

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 Novemberwith Mikhail
2018
Bershtein
13 / 23

Toda-like equations: Chern-Simons generalization

Bershtein, Gavrylenko, Marshakov, 2018: (m, N)-generalizations of Toda-likeequations, N ∈ N, 0 ≤ m ≤ N
τm;j (qz)τm;j (q −1 z) = τm;j (z)2 −z 1/N τm;j+1 (q m/N z)τm;j−1 (q −m/N z), j ∈ Z/NZ.
(22)
Corresponding to the Toda family of spectral curves (see soon in PashaGavrylenko talk).
Solutions: Fourier series of SU(N) Nekrasov instanton partition function
modified by level m Chern-Simons term.
Modification: each summand in Nekrasov formula multiplying onYTmu −1 q11−i q21−j .
λ =(23)(i,j)∈λAnton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 November
with Mikhail
2018
Bershtein
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CS generalization:c = −2 tau function

We obtained by the same method Toda-like equations fromNakajima-Yoshioka relations only for N = 2.
For m = 2 Toda-like equation is equivalent to the case m = 0 and moreover,generally
Z2 (u; q1 , q2 |z) = (z; q1 , q2 )∞ Z0 (u; q1 , q2 |z)
(24)
(1)

For m = 1 Toda-like equation is equivalent to the Painlevé A7 equationτ τ = τ 2 − z 1/2 τ τ ,(25)

so we have proved that Fourier series of CS m = 1 Nekrasov functions give asolution of this equation.

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 November
with Mikhail
2018
Bershtein
15 / 23

q-Painleve VI and q-Painlevé III3 c = −2 tau functionsq-Painlevé VI equation: system of eight first order q-difference bilinear
equations on 8 tau functions τ1 . . . τ8 .
In the case q θ0 = q θz = q θ1 = q θ∞ = i it is equivalent to system of four
second order q-difference bilinear equations (j ∈ Z/2Z)
τj+ =−+
τj+ τj− − z 1/4 τj+1
τj−1

τj−,

τj− =−+
τj+ τj− + z 1/4 τj+1
τj−1

τj+.

(26)

which coincide with NY equations on c = −2 tau functions.
According to the solution of the q-Painlevé VI (Jimbo-Nagoya-Sakai, 2017)(−qz 1/2 ; q, q)2∞ Zinst (i, i, i, iq ±1/2 , u|z 1/2 ) = Zinst (u; q −1 , q 2 |z)(27)

and we checked this statement analytically up to z 5 .

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
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2018
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q-Painleve VI and q-Painlevé III3 : cluster dynamics
c = 1 tau functions of q-Painleve VI and q-Painlevé III3 is A-cluster coordinates.
(1)0(1)

A7A3241783
124356

Figure: Quiver of q-Painlevé VI and q-Painlevé III’3

Elementary q-shift z 7→ qz of q-Painlevé III’3 equation corresponds to the theq-shift t 7→ q 1/2 t for q-Painlevé VI equation — half of the elementary shift. Such(1)generator in Weyl group W (D5 ) exists and for θ0 = θt , θ1 = θ∞ it is a q-shift.
Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 Novemberwith Mikhail
2018
Bershtein
17 / 23c = −2 tau function: ABJq-Painlevé III3 tau function for |q| = 1, s = 1 [Bonelli-Grassi-Tanzini, 2017] :τ (κ, ~, ξ) = ZCS (~, ξ) det(1 + κρP1 ×P1 ),(28)where ρ is inverse operator to the Hamiltonian of the relativistic Toda chain
ρP1 ×P1 = (e p + e −p + e x + mP1 ×P1 e −x )−1 .

(29)

Spectral determinant for z = q M , M ∈ Z is ABJ grand canonical partition
function. It factorizes according to the parity of eigenvalues of ρ: τ = τ+ τ− .
”Quantum Wronskian” relations in ABJ th. [Grassi-Hatsuda-Marino, 2014]−+ −iz 1/4 τ+1 τ1 − τ τ= −(1 + z 1/2 )τ+ τ− ,(30)−+ −iz 1/4 τ+1 τ1 + τ τ= (1 + z 1/2 )τ+ τ−(31)which is equivalent to the q-difference NY relations.

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbasedrelations
on ArXiv 1811.04050
13 Novemberwith Mikhail
2018
Bershtein
18 / 23c = −2 τ function: Painlevé VI generalization[Iorgov, Lisovyy, Teschner, 2014] give naturally relation of τc=1 to the
Riemann-Hilbert problem on CP1 with 4 punctures. Their arguments workfor the case b 2 ∈ Z \ 0, in particular for c = −2. Precise formulation of RH
for this case?
Central charge of symplectic fermions is c = −2. We expect that τc=−2 is a
conformal block for the symplectic fermions.
For special resonance values of parameters τc=1 is expressed by thedeterminant of hypergeometric 2 F1 functions [Morozov, Mironov, 2017].
Similarly τc=−2 is equal to the Pfaffian. Nakajima-Yoshioka relations in thisterms?

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
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2018
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Thank you for the attention!

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
relationson ArXiv 1811.04050
13 November
with Mikhail
2018
Bershtein
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Conformal blocks
We have graded Lie algebra (for example, Virasoro algebra) g = n+ ⊕ h ⊕ n− .
We take highest weight vector |vλ i, s.t. U(n+ ) act on it by zero and U(h)act as on eigenvector.
Verma module U(n)− |vλ iWhittaker vector in the Verma module|W (z)i =+∞Xz N |Ni,deg (|Ni) = N(32)

N=0s.t.
g |W (z)i = βg z deg (g ) |W (z)i

(33)

Z(z) = hW (1)|W (z)i(34)

Conformal block

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbasedrelations
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Representation-theoretic interpretation
Nakajima-Yoshioka blow-up relations on 4D Nekrasov functions haverepresentation-theoretic interpretation (Bershtein, Feigin, Litvinov, 2013).
4D Nekrasov function with 1 + 2 = 0 correspond via AGT correspondence
to the c = 1 4-point Virasoro conformal block.
IntroduceLvertex operator algebra Urod as a sum of Heisenberg algebra Fockmodules k∈Z Fk √2 but with modified stress-energy tensor.
There are Vir ⊕ Vir subalgebra with b12 + b2−2 = −1 in the U(Urod ⊗ Vir),
moreover there is a decomposition of the Verma module
M
U1 ⊗ LP,b =LP1 +kb1 ,b1 ⊗ LP2 +kb−1 ,b2(35)2k∈ZTake Whittaker vector v1/√2 ⊗ |W (z)i in l.h.s. Then it decompose into sumof Vir ⊕ Vir Whittaker vectors in r.h.s. Squaring this relation we obtain
Nakajima-Yoshioka blow-up relation.

Anton Shchechkin HSE & Skoltech Moscow, Russia Painlevé equations from Nakajima-Yoshioka blow-upbased
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Algebraic solutions and log limit of c = −2 τ function

As it is for c = 1 τ function there exist limit σ → 0 and ”algebraic” solution forthe σ = 1/4, s = ±1√

It is known that τ (1/4, ±1|z) = z 1/16 e ∓ z . One can find that
τ+ = e 2iz

1/4√z 1/32 e 2 z ,τ− = e 2iz

1/4√z 1/32 e 2 z(36)Logarithmic limit for c = 1 is s = e 2Ωσ , σ → 0. It also could be applied forthe c = −2 τ function

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