Toda and q-Toda equations for Nekrasov
partition functions
Kanehisa Takasaki
SIDE13, Hakata, November 13, 2018
1. Introduction
2. Deriving q-Toda equations from Toda hierarchy
3. Toda-like equations for U (1) Nekrasov functions
4. Dual partition function of 5D SU (N ) theory
5. Conclusion

11. Introduction
Background:
Recent studies on isomonodromic tau functions— Conformal blocks, Nekrasov functions, etc.
— Conjectures on Toda-like bilinear equations
This talk:Previous results 1), 2) and outlook on— Origin of q-Toda equations in 2D Toda hierarchy— Toda-like equation for U (1) Nekrasov functions— Dual partition function of 5D SU (N ) theory
1) K.T., J. Geom. Phys. 59 (2009), 1244–1257, arXiv:0903.2607.
2) T. Maeda, T. Nakatsu, K.T. and T. Tamakoshi, JHEP 03(2005), 056, arxiv:hep-th/0412327.

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2. Deriving q-Toda equations from Toda hierarchyRef) K.T., J. Geom. Phys. 59 (2009), 1244–1257, arXiv:0903.2607.

Tau functions of 2D Toda hierarchyτ (s, t, t̄) =∑Sλ (t)Sλ̄ (t̄)gs,λλ̄ ,λ,λ̄∈Ps ∈ Z, t = (t1 , t2 , . . .), t̄ = (t̄1 , t̄2 , . . .),where

Sλ (t) = det(Sλi −i+j (t))∞
i,j=1 ,(∞)∞∑∑
nSn (t)z = exp
tk z k ,n=0

k=1dgs,λλ̄ = ⟨s, λ|g|s, λ⟩, g ∈ GL(∞)

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2. Deriving q-Toda equations

2D diﬀerence Toda equation
τ (s, t, t̄) satisfies the bilinear equation
τ (s, t − [x], t̄)τ (s, t, t̄ − [y]) − τ (s, t, t̄)τ (s, t − [x], t̄ − [y])+xyτ (s + 1, t, t̄)τ (s − 1, t − [x], t̄ − [y]) = 0,
where()
2kxx[x] = x, , . . . , , . . . .
2kThis is a diﬀerence analogue of the 2D Toda equation.

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2. Deriving q-Toda equations

Key to q-diﬀerence analogue
Define a specialized tau function as(T (s, x, y) = τ (s, [x]q , [y]q̄ ),x2xk)x,,...,,... ,[x]q =2k
1 − q 2(1 − q )k(1 − q )()2
kxxx[y]q̄ =,,...,,... .
2k
1 − q̄ 2(1 − q̄ )k(1 − q̄ )Key relations[qx]q = [x]q − [x],[q̄y]q̄ = [y]q̄ − [y],T (s, qx, y) = τ (s, [x]q − [x], [y]q̄ ),T (s, x, q̄y) = τ (s, [x]q , [y]q̄ − [y]).

52. Deriving q-Toda equations

q-diﬀerence analogue of 2D Toda equation
T (s, x, y) = τ (s, [x]q , [y]q̄ ) satisfies the q-diﬀerence equation
T (s, qx, y)T (s, x, q̄y) − T (s, x, y)T (s, qx, q̄y)+xyT (s + 1, x, y)T (s − 1, qx, q̄y) = 0.
This is a q-diﬀerence analogue (Kajiwara & Satsuma 1991) of the 2DToda equation.

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2. Deriving q-Toda equations

Remarks on Schur functions
1) Sλ (t1 , t2 , . . .) can be converted to the Schur function sλ (x1 , x2 , . . .)in Macdonald’s book by substituting
1∑ ktk =xi ,kk = 1, 2, . . . .

i≥12) Sλ ([x]q ) is related to the principal specialization of sλ (x1 , x2 , . . .):Sλ ([x]q ) = sλ (x, xq, xq 2 , . . .) = x|λ| sλ (1, q, q 2 , . . .).
Thus T (s, x, y) is a double sum of the form∑
T (s, x, y) =sλ (1, q, q 2 , . . .)sλ̄ (1, q̄, q̄ 2 , . . .)x|λ| y |λ̄| gs,λλ̄ .
λ,λ̄∈P7

2. Deriving q-Toda equations

Further specialization of tau function
If g takes the diagonal form
( ∞)∑g = exp
log gn :ψ−n ψn∗ : ←→ diag(gn )∞n=−∞ ,n=−∞the matrix elements of g can be expressed as⟨s, λ|g|s, λ̄⟩ = δλλ̄ gs,λ ,gs,λ =∞∏gλ −i+1+si

i=1g−i+1,and T (s, x, y) becomes a single sum∑
T (s, x, y) =sλ (1, q, q 2 , . . .)sλ (1, q̄, q̄ 2 , . . .)(xy)|λ| gs,λ .
λ∈P82. Deriving q-Toda equations

Further specialization of tau function (cont’d)
In this case,T (s, z) =∑T (s, x, y) = T (s, xy),sλ (1, q, q 2 , . . .)sλ (1, q̄, q̄ 2 , . . .)z |λ| gs,λ .

λ∈PT (s, z) satisfies a 1D reduction of the 2D q-diﬀerence Todaequation:
T (s, qz)T (s, q̄z) − T (s, z)T (s, q q̄z)+zT (s + 1, z)T (s − 1, q q̄z) = 0.

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3. Toda-like equations for U (1) Nekrasov functions
5D U (1) Nekrasov function with external potentialZ(s, Q) =∑sλ (q −ρ )2 Q|λ|+s(s+1)/2 eΦ(s,λ) .

λ∈PQ is a new parameter, q −ρ is the special pointq −ρ = (q 1/2 , q 3/2 , . . . , q i−1/2 , . . .)and Φ(s, λ) is an external potential of the form
Φ(s, λ) =∞∑ξλi −i+s+1 −i=1∞∑i=110ξ−i+1 .

3. Toda-like equations for U (1) Nekrasov functions

Special value of Schur functionsλ (q −ρ ) is a slight modification of the principal specialization:sλ (q −ρ ) = q |λ|/2 sλ (1, q, q 2 , . . .)q −κ(λ)/4,=∏−h(i,j)/2h(i,j)/2−q)(i,j)∈λ (qwhereκ(λ) =∑λi (λi − 2i + 1) = 2i≥1∑(j − i),(i,j)∈λh(i, j) = λi + t λj − i − j + 111(hook length)3. Toda-like equations for U (1) Nekrasov functions

Relation to T (s, z)
T (s, z) =∑sλ (1, q, q 2 , . . .)sλ (1, q̄, q̄ 2 , . . .)z |λ| gs,λλ∈P−−−→Z(s, Q) =q̄ = q, z = q, gn = Qn eξn∑sλ (q −ρ )2 Q|λ|+s(s+1)/2 eΦ(s,λ)λ∈P

In particular,Z(s, qQ) = q s(s+1)/2 T (s, q 2 ),Z(s, q 2 Q) = q s(s+1) T (s, q 3 ).

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3. Toda-like equations for U (1) Nekrasov functions

Toda-like equation for Z(s, Q)
The bilinear equation
T (s, qz)T (s, q̄z) − T (s, z)T (s, q q̄z)+zT (s + 1, z)T (s − 1, q q̄z) = 0.
for T (s, z) turns into a bilinear equation for Z(s, Q):Z(s, Q) satisfies the Toda-like equation
Z(s, qQ)2 − Z(s, q 2 Q)Z(s, Q)+q 2s+1 Z(s + 1, Q)Z(s − 1, q 2 Q) = 0.

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3. Toda-like equations for U (1) Nekrasov functions

4D limit of 5D U (1) Nekrasov function
Z(s, Q) =∑sλ (q −ρ )2 Q|λ|+s(s+1)/2 eΦ(s,λ)λ∈P−−−→Z4D (s, Λ) =q = e−Rℏ , Q = (RΛ)2 , R → 0∑ ( dim λ )2 ( Λ )2|λ|+s(s+1)λ∈P|λ|!

ℏeΦ4D (s,λ)(4D U (1) Nekrasov function)
dim λ is the number of standard tableaux of shape λ, anddim λ
1= Sλ (1, 0, 0, . . .) = ∏.
|λ|!
(i,j)∈λ h(i, j)

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3. Toda-like equations for U (1) Nekrasov functions

4D limit of Toda-like equation
Z(s, q 2 Q)
Z(s, Q)(−Z(s, qQ)
Z(s, Q))22 Q)Z(s + 1, Q)
Z(s−1,q= q 2s+1Z(s, Q)2−−−→q = e−Rℏ , Q = (RΛ)2 , R → 0()2d
Z4D (s − 1, Λ)Z4D (s + 1, Λ)Λlog Z4D (s, Λ) = 4dΛ
Z4D (s, Λ)2
The Toda-like equation of Nekrasov and Okounkov for Z4D (s, Λ)(Nekrasov & Okounkov 2003) can be thus recovered.

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4. Dual partition function of SU (N ) theoryRef) T. Maeda, T. Nakatsu, K.T. & T. Tamakoshi, JHEP 03 (2005),
056, arxiv:hep-th/0412327.

External potential in U (1) Nekrasov function
Z(s, Q) =∑sλ (q −ρ )2 Q|λ|+s(s+1)/2 eΦ(s,λ) ,λ∈P

Φ(s, λ) =∞∑ξλi −i+s+1 −i=1∞∑

ξ−i+1 .

i=1

(The definition of Φ(s, λ) is eﬀectively a finite sum. The subtraction terms is related to normal ordering of a fermion bilinear.)16

4. Dual partition function of SU (N ) theory

Decomposition under external periodic potential
If the external potential is N -periodic, i.e.,
ξn+N = ξn ,the sum over the single partition λ can be decomposed to a sum
over the N -tuple p⃗ = (p1 , . . . , pN ) of integers (charges) and theN -tuple ⃗λ = (λ(1) , . . . , λ(N ) ) of partitions (colored partitions):∑⃗p⃗·ξ(s)Z(s, Q) =eQs(s+1)/2 Z(⃗p, Q),p⃗∈ZN , p1 +···+pN =0⃗ = (ξ1+s , . . . , ξN +s ),ξ(s)Z(⃗p, Q) =∑⃗λ∈P N

17··· .

4. Dual partition function of SU (N ) theory

Decomposition under periodic external potential (cont’d)
The summands of Z(⃗p, Q) are factorized to a ⃗λ-independent factor C(⃗p, Q) (perturbative part) and a sum of ⃗λ-dependent factorsp, Q) (instanton part):Z⃗inst (⃗λZ(⃗p, Q) = C(⃗p, Q)∑Z⃗λinst (⃗p, Q).

⃗λ∈P NZ⃗inst (⃗p, Q) turns out to be the 5D SU (N ) Nekrasov function withλChern-Simons corrections. In summary:Under the N -periodic external potential, Z(s, Q) becomes adual partition function of 5D SU (N ) theory in the sense ofNekrasov and Okounkov.
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4. Dual partition function of SU (N ) theory

N = 2: charges and colored partitions
Write the components of p⃗ and ⃗λ asp⃗ = (p, −p),⃗λ = (µ, µ̄)

and identify the triple (p, µ, µ̄) with the partition λ:λ = (p, µ, µ̄).
1) The triple is obtained by splitting the Maya diagram of λ (as asubset of Z) into the subsets of odd and even integers:∞∞{λi − i + 1}∞={2(µ−i+p)+1}∪{2(µ̄−i−p)+2}
iii=1i=1i=1 .

19The RHS is the exact partition function [2] for the five-dimensional supersymmetric SU(2Yang-Mills with the Chern-Simons term. We notice that, as confirmed [2] for four-dimensiona

N = 2 supersymmetric gauge theories,
the gauge
theorypartitionfunction is realized as th4. Dual partition
functionof SU(N ) theory! → 0 limit of the above partition function.

N = 2: charges andIt coloredpartitions(cont’d)
is clear from(3.30) that thefield theory limit is the thermodynamic limit with a, Λ anR fixed. The identification (3.27) makes p = o(!−1 ). The corresponding ground partition i2) (p, ∅, ∅) represents a staircase Young diagram:(3.30) becomes very large. The partitions λ(r) represent U(1) gauge instantons with c2 = |λ(r
on a
non-commutative R4 . In the model of random partitions, it follows from (3.7) that thes
(2p − 1, 2p − 2, . . . , 1) if p > 0,
instantons become excitations from the ground partition. They provide the deformatio
gauge(p, ∅, ∅) of=the groundpartition at the thermodynamic∅
if plimit= (Figure0, 5). These imply that the grounpartitions
describe the perturbative regime of the Coloumb branch while their deformatio(2|p|,2|p| − 1, . . . , 1)
if p < 0.
becomes non-perturbative in the gauge theory.

3) The staircase Young diagram is the core of λ, and thecolored partition (µ, µ̄) represents the quotient of λ by thecore.

−2p + 10

2p − 1x

Figure 5: Deformation of the SU (2) ground partition for p ≫ 1 by two20partitions (3, 1, 1) and (3, 2).

4) SU (N ) dual partition functions

N = 2: Factorization of weights
Q|λ| sλ (q −ρ )2 = Q|(p,∅,∅)| s(p,∅,∅) (q −ρ )2−ρ )2s(q(p,µ,µ̄)× Q|(p,µ,µ̄)|−|(p,∅,∅)|.
−ρ2s(p,∅,∅) (q )The two parts on the right side give the perturbative part C(p, Q)
inst (p, Q):
and the instanton part Zµµ̄C(p, Q) = Q|(p,∅,∅)| s(p,∅,∅) (q −ρ )2 ,−ρ )2s(q(p,µ,µ̄)
instZµµ̄(p, Q) = Q|(p,µ,µ̄)|−|(p,∅,∅)|.
−ρ2s(p,∅,∅) (q )21

4. Dual partition function of SU (N ) theory

N = 2: Computation of C(p, Q)
2p−1

∏

Qk


1/2 k−1/2 −k 2k(Qq−Qq )FF k=1
C(p, Q) = 1


2|p|
k
∏
Q



−1/2 k
1/2 −k 2k
(Qq−Q)
k=1
FF q

if p > 0,
if p = 0,
if p < 0where QF = q 1−4p .
A final expression should be written in terms of the q-Barnes finction?

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4. Dual partition function of SU (N ) theory

inst (p, Q)
N = 2: Computation of Zµµ̄

An intermediate stage (QF = q 1−4p ):|µ|−|µ̄|

instZµµ̄(p, Q) = Q2|µ|+2|µ̄| QFq −2κ(µ)−2κ(µ̄)∏1×(q −hµ (i,j) − q hµ (i,j) )2(i,j)∈µ×∏(i,j)∈µ̄×1(q −hµ̄ (i,j) − q hµ̄ (i,j) )2∞
1/2−1/2 µi −µ̄j −i+j 2∏(Q q −(µi −µ̄j −i+j) − Qq)
Fi,j=1F
1/2−1/2(QF q i−j − QF q j−i )223.

4. Dual partition function of SU (N ) theory

inst (p, Q) (cont’d)
N = 2: Computation of Zµµ̄|µ|−|µ̄| −2κ(µ)−2κ(µ̄)

instZµµ̄(p, Q) = Q2|µ|+2|µ̄| QF

q× Nµµ (q 2 )Nµ̄µ̄ (q 2 )Nµµ̄ (q 2 , QF )Nµ̄µ (q 2 , Q−1F )where QF = q 1−4p , and N•• ’s are the Nekrasov factors
Nλλ (q) =∏(i,j)∈λNλµ (q, u) =×1,−h(i,j)
h(i,j)λλ(1 − q)(1 − q)∏1(i,j)∈λ

1 − uq −λi − µj +i+j−1∏

1(i,j)∈µt
1 − uq µi + λj −i−j+1t24.

5. Conclusion
1) Toda-like equations for U (1) Nekrasov functions
The 5D U (1) Nekrasov function Z(s, Q) satisfies a Toda-like qdiﬀerence equation. This equation can be derived from the 2D
Toda hierarchy by a few steps of specialization. The 4D version
Z4D (s, Λ) satisfies a Toda-like diﬀerential equation.
2) Dual partition function of SU (N ) theory
If the coeﬃcients ξn of the external potential Φ(s, λ) are N -periodic,Z(s, Q) becomes a dual partition function of 5D SU (N ) theory. Adrawback is that some parameters therein are specialized to discrete values (associated with p⃗ ∈ ZN ). To achieve a full generality,
we have to find a suitable deformation of the weights sλ (q −ρ )2 .

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5. Conclusion

3) Including matter hypermultiplets
This issue is not very serious. Contribution of matter fields can beincluded as corrections of the external potential.
4) Discrepancy with isomonodromic tau functions
Our 5D Nekrasov functions looks slightly diﬀerent from the known
isomonodromic tau functions because of Chern-Simons corrections.

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Thank you
for your attention!

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