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arXiv:0912.4789v1 [hep-th] 24 Dec 2009
Dec. 2009
YITP-09-111
ONAGT CONJECTURE FOR PU RE SUPER YA N G -M I L L S
AND W-ALGEBRA
Masato Taki
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
taki@yukawa.kyoto-u.ac.jp
Abstract
Recently Alday, Gaiotto and Tachikawa have proposed relation between 2- and 4-dimensional
conformal field theories. The relation implies that the Nekrasov partition functions of N= 2
superconformal gauge theories are equal to conformal blocks associated with the conformal algebra.
Likewise, a counterpart in pure super Yang-Mills theory exists in conformal field theory. We propose
a simple relation between the Shapovalov matrix of the W3-algebra and the Nekrasov partition
function of N= 2 SU (3) Yang-Mills theory.
1 Introduction
The study of conformal field theory (CFT) in 2-dimensions [1] has been an important subject in
physics and mathematics, since it is available for various applications such as string theory, the
critical phenomena and the representation theory of groups.
N= 2 gauge theories are also important topics of quantum field theory. It is known that these
systems are exactly solvable by using the Seiberg-Witten theory. The microscopic verification of the
theory had been given by Nekrasov [2][3]. He showed that the integral over the ADHM moduli space
of instantons implies the generating function of the Seiberg-Witten prepotential, which is called
the Nekrasov partition function. The localization fomula reduces the integral over the instanton
moduli space to the instanton counting problem. The resulting Nekrasov partition function has a
combinatorial expression, which is summation over Young diagrams.
Recently, remarkable relation between 2- and 4-dimensonal CFT’s was proposed by Alday,
Gaiotto and Tachikawa [4]. The core of their proposal is that the instanton part of the Nekrasov
partition for a SU(2) superconformal gauge theory is exactly equal to a certain conformal block
of 2-dimansional CFT. This means that we can recast the Nekrasov partition function into a CFT
correlator by summing over Young diagrams.
In this paper, we study relation between N= 2 SU (3) pure Yang-Mills theory and conformal
symmetry. In [5] Gaiotto proposed that the Nekrasov partition function of SU (2) pure Yang-
Mills theory can be represented as the norm of a certain state in the Virasoro Verma module.
Moreover Mironov and Morozov showed that the Shapovalov form of the Verma module gives
this state by applying the decoupling limit of flavors to the Alday-Gaiotto-Tachikawa conjecture
(AGT conjecture) [6]. We study the decoupling limit of the Wyllard proposal [7] which implies
that a correlator of AN−1Toda field theory with WN-symmetry [8][9] is related to N= 2 SU(N)
conformal quiver gauge theory. We call it the AGT-W conjecture. In [10][11][12] string theory
derivations were proposed. We focus on the case of W3-symmetry [13]. In [14][15][16] it was shown
that the AGT-W conjecture implies that the 4-point conformal block associated with W3-symmetry
is equal to the Nekrasov partition function of SU(3) superconformal gauge theory. We study the
decoupling limit of SU(3) superconformal gauge theory and the conformal block for W3-symmetry,
and then we relate the Nekrasov partition function SU (3) pure Yang-Mills to the Shapovalov form
for the W3-algebra. The result implies that the Nekrasov partition functions may be deeply linked
to the representation theory of conformal algebra.
This paper is organized as follows. In section 2, we give a brief reviews on the AGT conjecture.
In section 3, we propose a relation between SU(3) pure Yang-Mills and the Shapovalov matrix for
the W3-algebra. We check the proposal for 1-instanton partition function and the level-1 Shapovalov
1
matrix. By applying the decoupling limit of flavors, we show that the AGT-W conjecture implies
our proposal. Conclusions are found in section 4. In Appendix A, we summarize the definition
and properties of the conformal blocks. Our proposal is verified explicitly at 2-instanton level in
Appendix.B.
2 Non-conformal AGT Conjecture and Virasoro Algebra
The main issue we study in this paper is the AGT relation between a Nekrasov partition function
and a conformal blocks of 2-dimensional CFT. As a guide to the latter part, we give a brief review
on the AGT conjecture.
2.1 Alday-Gaiotto-Tachikawa conjecture
The Nekrasov partition function [2] is a generating function of the Seiberg-Witten prepotential
Zinst = exp( 1
ǫ1ǫ2FSW +···). The explicit form of the partition function is found in the next
section. The parameters ǫ1,2correspond to the so-called Ω-background [3].
The Nekrasov partition function for the N= 2 S U (2) gauge theory with 4-flavors is a function
of 8 parameters:
Zinst(a, ~µ, x, ǫ1, ǫ2) =
∞
X
k=0
xkZk(a, ~m, ǫ1, ǫ2).(2.1)
While the 7 parameters a,µf,ǫ1and ǫ2have mass dimension one, the factors Zk(a, ~m, ǫ1, ǫ2) are
dimensionless, which is a reflection of the fact that the gauge theory is conformal. Let us introduce
the scale ~=√−ǫ1ǫ2. Then we can scale the patririon function as follows:
Zinst(a, ~µ, x, ǫ1, ǫ2) =
∞
X
k=0
xkZka
~, ~µ, e1, e2.(2.2)
Here we introduce the dimensionless Ω-background and masses:
eE=ǫE
~, E = 1,2 (2.3)
µf=mf
~, f = 1,··· ,4.(2.4)
Let us introduce the following parametrization [4] of the gauge theory parameters:
c= 1 −6e2
∆ = a2
~2−e2
4(2.5)
Here edenotes e=e1+e2. These new parameters play a role in the AGT relation.
2
A simplest version of the AGT conjecture implies that the above Nekrasov partition function
coincides with the 4-point spherical conformal block of the Virasoro algebra with central charge c
[17]:
Zinst a
~, ~µ, x, e1, e2=B∆h∆1∆2
∆3∆4i(x) (2.6)
Here ∆f=αf(e−αf) is the conformal dimension of the external states. The external momentum
αfare corresponding to the mass parameters of the gauge theory:
µ1=α1+α2−e
2
µ2=−α1+α2+e
2
µ3=α3+α4−e
2
µ4=α3−α4+e
2
See Appendix.A for the definition of the conformal block B.
In this way, the instanton counting of the Nc= 2 Nf= 4 gauge theory keeps the representation
theory of the Virasoro algebra [17] behind. It is very natural to expect that the Nekrasov partition
functions of aN= 2 gauge theory and the representation theory of the symmetry algebra of 2-
dimensional CFT are closely related whether the gauge theory is conformal or not. In the following
we review Gaiotto’s proposal on the SU (2) pure supre Yang-Mills theory.
2.2 Gaiotto conjecture (non-conformal AGT conjecture)
Let us consider the Nekrasov partition function of the SU(2) pure supre Yang-Mills theory:
Zinst(a, Λ, ǫ1, ǫ2) =
∞
X
k=0
Λ4kZk(a, ǫ1, ǫ2)
=
∞
X
k=0
Λ4k
(−ǫ1ǫ2)2kZka
~, e1, e2(2.7)
Notice that the factors Zkhere have mass dimension −4kin order that the full partition function
Zinst = 1 + O(Λ4) is dimensionless.
In [5] Gaiotto proposed that the partition function is realized as the norm of a certain state of
the Virasoro algebra:
Zinst(a~,Λ~, e1~, e2~) = h∆,Λ|∆,Λi.(2.8)
Here we eliminate the overall scale scale ~from the parameters, and the conformal dimension of
the internal state is ∆ = a2−e2/4. He found that the state |∆,Λi=|∆i+··· must satisfy the
3
following conditions:
|∆,Λi=X
n=0
Λ2n|∆, ni
L1|∆, ni=|∆, n −1i,(2.9)
Lk|∆, ni= 0 for k= 2,3,4,··· ,
where |∆iis the primary state with conformal dimension ∆. It is not obvious whether such a state
exists or not. Marshakov and Mironov showed that the Gaiotto state |∆,Λifor pure SU (2) super
Yang-Mills theory is given by the Shapovalov matrix Q∆(Y;Y′) [6][18]:
|∆, ni=X
|Y|=n
Q−1
∆([1n]; Y)L−Y|∆i.(2.10)
Here Y={Y1, Y2,···} = [1m12m2···] is a Young diagram with |Y|=PYi=Pj mjboxes, and
L−Ydenotes L−Yl···L−Y2·L−Y1. The Shapovalov matrix is the following type of Gram matrix:
Q∆(Y;Y′) = h∆|LYL−Y′|∆i.(2.11)
The Gaiotto-Marshakov-Mironov proposal implies that the Nekrasov partition function for pure
SU (2) Yang-Mills is given by
Zinst(a~,Λ~, e1~, e2~) = X
n
Λ4nQ−1
∆([1n]; [1n]).(2.12)
Our interest in this paper has centered on this property of the instanton counting. An important
point is that the Nekrasov partition function of SU (2) pure Yang-Mills corresponds to a basic
quantity in the representation theory of the Virasoro algebra. Now a question arises; what is the
counterpart in the instanton counting of SU (N) pure super Yang-Mills theory? In the next section
we give an explicit answer for the question in the case of SU (3) gauge theory.
3S U (3) Pure Super Yang-Mills and W3-algebra
In this section we propose a relation between the Nekrasov partition function of SU (3) pure super
Yang-Mills theory and the Shapovalov matrix of the W3-algebra. This proposal gives a nontrivial
extension of the Gaiotto conjecture we reviewed in the previous section.
3.1 Nekrasov formula
The Nekrasov partition function is a generating function of the Seiberg-Witten prepotential. It is
given by the localization calculation of the path integral over the instanton moduli space with an
4
appropriate measure. The explicit form of the partition function for SU (N) pure super Yang-Mills
theory is [19][20][21]
Zinst(~a, Λ, ǫ1, ǫ2) = X
~
Y
Λ2Nc|~
Y|
QNc
α,β=1 n~
Y
α,β(~a, ǫ1, ǫ2).(3.1)
~
Y= (Y1,···, YN) is a vector consists of NYoung diagrams and its norm is defined by |~
Y|=
Pn|Yn|. Here the denominator is the eigenvalues of the torus action on the tangent space of the
moduli space [19][20][21], and it is given by the characteristic of the ADHM complex.
n~
Y
α,β(~a, ǫ1, ǫ2) = Y
(i,j)∈Yα
(−lYβ(i, j)ǫ1+ (aYα(i, j) + 1)ǫ2+aα−aβ)
×Y
(i,j)∈Yβ
((lYα(i, j) + 1)ǫ1−aYβ(i, j)ǫ2+aα−aβ).(3.2)
~a = (a1,···, aN) is the eigenvalue vector of the adjoint chiral fields. An arm length and leg length
of a Young diagram are defined by aY(i, j) = Yi−jand lY(i, j) = Ytj−i.
We expand the partition function with respect to the dynamical scale Λ and define the k-
instanton part of the partition function Zk:
Zinst(~a, Λ, ǫ1, ǫ2) = X
k
Λ2NckZk(~a, ǫ1, ǫ2).(3.3)
Let us 1 and 2-instanton partition functions for example.
1-instanton
Terms with |~
Y|= 1 contribute to 1-instanton part of the Nekrasov partition function (3.1). Such
Young diagrams take the form of ~
Y= ( , φ, φ, ···),(φ, , φ, ···),···. Let us compute the factor
(3.2) for ~
Y= ( , φ, φ, ···):
Nc
Y
α,β=1
n(,φ,φ,··· )
α,β (~a, ǫ1, ǫ2) = n~
Y
1,1·n~
Y
1,2·n~
Y
2,1·n~
Y
1,3·n~
Y
3,1···
=ǫ1ǫ2
Nc
Y
α=2
aα,1(a1,α +ǫ).(3.4)
Thus we get a 1-instanton Nekrasov partition function as follows:
Zk=1(~a, ǫ1, ǫ2) = X
β
1
ǫ1ǫ2QNc
α6=βaα,β(aβ,α +ǫ)(3.5)
5
For Nc= 3, the 1-instanton partition function (3.5) becomes
Zk=1(a1, a2, ǫ1, ǫ2)
=6(a12+a22+a1a2−ǫ2)
ǫ1ǫ2(2a1+a2−ǫ)(2a1+a2+ǫ)(a1+ 2a2−ǫ)(a1+ 2a2+ǫ)(a1−a2−ǫ)(a1−a2+ǫ),
(3.6)
where we use a1+a2+a3= 0.
2-instanton
The Young diagrams which satisfy |~
Y|= 2 contribute to the 2-instanton partiton function. There
are three types of such Young diagrams: ~
Y= ( , , φ, ···), ( , φ, φ, ···), ( , φ, φ, ···)···. Let
us compute the contribution from ~
Y= ( , , φ, ···).
Nc
Y
α,β=1
n(, ,φ,··· )
α,β (~a, ǫ1, ǫ2) = n~
Y
1,1·n~
Y
1,2·n~
Y
2,1·n~
Y
2,2
×n~
Y
1,3·n~
Y
3,1·n~
Y
1,4·n~
Y
4,1···n~
Y
2,3·n~
Y
3,2·n~
Y
2,4·n~
Y
4,2···
= (ǫ1ǫ2)2(a1,2+ǫ1)(a1,2−ǫ1)(a1,2+ǫ2)(a1,2−ǫ2)
×Y
β=1,2
Nc
Y
α6=1,2
aα,β(aβ,α +ǫ).(3.7)
The contribution from ~
Y= ( , φ, φ, ···) is given by
Nc
Y
α,β=1
n(,φ,φ,··· )
α,β (~a, ǫ1, ǫ2) = n~
Y
1,1·n~
Y
1,2·n~
Y
2,1·n~
Y
1,3·n~
Y
3,1···
= (2ǫ1ǫ22(ǫ1−ǫ2))
Nc
Y
α6=1
a1,α(a1,α +ǫ)(a1,α +ǫ2)(a1,α +ǫ+ǫ2).(3.8)
Finally, the Young diagram ~
Y= ( , φ, φ, ···)··· gives
Nc
Y
α,β=1
n(,φ,φ,··· )
α,β (~a, ǫ1, ǫ2) = n~
Y
1,1·n~
Y
1,2·n~
Y
2,1·n~
Y
1,3·n~
Y
3,1···
= (2ǫ12ǫ2(ǫ2−ǫ1))
Nc
Y
α6=1
a1,α(a1,α +ǫ)(a1,α +ǫ1)(a1,α +ǫ+ǫ1).(3.9)
Thus we get the 2-instanton partition function:
Zk=2(~a, ǫ1, ǫ2) = X
α<β
1
(ǫ1ǫ2)2(aα,β +ǫ1)(aα,β −ǫ1)(aα,β +ǫ2)(aα,β −ǫ2)Qi=α,β QNc
γ6=α,β aγ,i (ai,γ +ǫ)
+X
α
1
(2ǫ1ǫ22(ǫ1−ǫ2)) QNc
β6=αaα,β(aα,β +ǫ)(aα,β +ǫ2)(aα,β +ǫ+ǫ2)
+X
α
1
(2ǫ12ǫ2(ǫ2−ǫ1)) QNc
β6=αaα,α(aα,α +ǫ)(aα,α +ǫ1)(aα,α +ǫ+ǫ1).(3.10)
6
In Appendix.B, we will use it in order to check our proposal.
Here the Coulomb moduli aαhave the mass dimension one. In the latter part of the articles,
we scale these parameters, and then aαdenotes the dimensionless Coulomb moduli parameter:
aα→~aα.(3.11)
3.2 W3-algebra and SU (3) Nekrasov formula
The basic symmetry of conformal field theories is the Virasoro algebra [1]. In this section we study
the W3-algebra [13], which is an enlarged conformal algebra. This algebra is composed of the spin-2
energy-momentum tensor T(z) and spin-3 current W(z). Their Laurent coefficients Lnand Wn
satisfy the following commutation relations:
[Ln, Lm] = (n−m)Ln+m+c
12(n3−n)δn,−m,(3.12)
[Ln, Wm] = (2n−m)Wn+m,(3.13)
[Wn, Wm] = 9
2hc
3·5!(n2−1)(n2−4)δn,−m+16
22 + 5c(n−m)Λn+m
+ (n−m)(n+m+ 2)(n+m+ 3)
15 −(n+ 2)(m+ 2)
6Ln+mi.(3.14)
Λnis a composite operator
Λn=X
m∈Z
:LmLn−m: +xn
5Ln,(3.15)
where the constants xare defined by
x2l= (1 −l)(1 + l),(3.16)
x2l+1 = (1 −l)(12 + l).(3.17)
We parametrize the central charge as c= 2(1 −12Q2).
The Hilbert space is spanned by the following basis of the descendants of the primary operator
V~α :
L−YLW−YW|∆~α i ∼ L−YLW−YWV~α(z).(3.18)
In this way the descendants are labelled by the pair of the Young diagrams Y={YL, YW}. The
conformal dimension of the operator is given by
∆(~α,Y)= ∆~α +|Y| = ∆~α +|YL|+|YW|.(3.19)
7
The conformal dimensions of the primary with ~α = (α, β) are
∆~α =α2+β2−Q2,
w~α =r4
4−15Q2α(α2−3β2),(3.20)
D(∆) = 4∆
4−15Q2+3Q2
4−15Q2,
where
L0|∆~α i= ∆~α|∆~αi, W0|∆~αi=w~α|∆~α i.(3.21)
Let us consider the Shapovalov matrix of the W3-algebra [14][16]. It is the Gram matrix of the
following type:
Q∆(Y;Y′) = h∆|WYWLYL·L−YLW−YW|∆i.(3.22)
We propose that the Nekrasov partition function for SU(3) pure super Yang-Mills theory coincides
with the following element of the Shapovalov matrix of W3-algebra:
Proposal 3.1
ZSU (3), k (~a, ǫ1, ǫ2) = 27
4ǫ1ǫ2+ 15ǫ2k
Q−1
∆(φ, [1k]; φ, [1k]).(3.23)
In other words, our proposal becomes
ZSU (3), k (~a, e1, e2) = −27
4−15e2k
Q−1
∆(φ, [1k]; φ, [1k]).(3.24)
after the scaling of the Coulomb moduli ~a →~~a. Here ∆ = ∆~α and the identification of the
parameters is [16]
α=√3
2(a1+a2),
β=1
2(−a1+a2),(3.25)
Q=e.
Check
Let us check our conjecture for 1-instanton. The level-1 Shapovalov matrix is [14][16]
Q∆(Y;Y′)||Y|,|Y′|=1 =
(, φ) (φ, )
(, φ) 2∆ 3w
(φ, ) 3w9D∆
2
,(3.26)
8
where the label is Y= (YL, YW). The inverse of the matrix is given by
Q−1
∆(Y;Y′)||Y|,|Y′|=1 =1
9(D∆2−w2)
9D∆
2−3w
−3w2∆
,(3.27)
and the component of our interest is
Q−1
∆(φ, ;φ, ) = 2∆
9(D∆2−w2).(3.28)
Let us evaluate the element (3.28) under (3.25). By using (3.20), we can factorize the determinant
of the Shapovalov matrix [16]:
D∆2−w2=4
4−15 Q2β2−Q2
4(β−Q)2−3α2(β+Q)2−3α2.(3.29)
With (3.25), we can rewrite it as a factor appearing in the Nekrasov partition function:
D∆2−w2=1
4−15 e2(a12 −e)(a12 +e)(a23 −e)(a23 +e)(a31 −e)(a31 +e).(3.30)
Similarly, the conformal dimension becomes
∆ = a2
1+a2
2+a1a2−e2.(3.31)
Thus we can rewrite (3.28) as the 1-instanton partition function (3.6):
ZSU (3), k=1 (~a, e1, e2) = −6
4−15 e2
∆
D∆2−w2
=−27
4−15 e2Q−1
∆(φ, ;φ, ).(3.32)
In this way, we can verify the conjecture explicitly for 1-instanton. We verify it for 2-instanton in
Appendix.B.
3.3 Derivation from AGT conjecture
3.3.1 AGT-W relation for Nc= 3,Nf= 6 gauge theory
AGT-W conjecture [7] gives relation between SU (Nc)Nf= 2Ncgauge theory and the conformal
Toda theory with WN-symmetry. The core of the claim for Nc= 3, Nf= 6 gauge theory is that
the Nekrasov partition function is equal to the 4-point spherical conformal block of the W3-algebra
[16]:
Zinst
SU (3)(a1, a2, ~µ, x, e1, e2) = B∆~α h∆1∆2
∆3∆4i(x).(3.33)
9
Here Bis the 4-point conformal block of the W3-algebra
B∆~α h∆1∆2
∆3∆4i(x) = X
Y,Y′| |Y|=|Y′|
x|Y| ¯ρα1α2;α(Y)Q−1
∆(Y;Y′)ραα3α4(Y′).(3.34)
See Appendix.A for the definition of it. Recall that the theory is conformal and the partition
function does not depend on the overall scale ~:
Zinst
SU (3)(~a~, ~µ~, x, e1~, e2~) = Zinst
SU (3)(~a, ~µ, x, e1, e2).(3.35)
Thus we have to identify these parameters of the gauge theory with the variables of the conformal
block. The identification between the internal momentum ~α and the Coulomb moduli ~a is the same
as the above. The relation between the masses and the external momentum is [16]
µ1=2
√3α2+f(α1, β1), µ4=2
√3α4+f(α3, β3)
µ2=−1
√3α2−β2+f(α1, β1), µ5=−1
√3α4−β4+f(α3, β3) (3.36)
µ3=−1
√3α2+β2+f(α1, β1), µ6=−1
√3α4+β4+f(α3, β3)
There exist 2 ×4 = 8 external momentum in the W-algebra side, on the other hand the gauge
theory has 6 mass parameters. This means 2 parameters of the W-algebra are redundant for AGT
correspondence. In [7] Wyllard therefore proposed that we should make two external states W-null
[22]. Then we have 3×3 = 9 choices of such a state. Now we choose a W-null vector ~α = (α, −Q/2)
for simplicity. Then the factor fis given by [16]
f(α1,−ǫ/2) = −1
√3α1+Q
2, f(α3,−ǫ/2) = 1
√3α3+Q
2.(3.37)
Then we finfd
µ1µ2µ3=2
3√3α2(α22−3β22)−f(~α1)(α22+β22) + f(~α1)3(3.38)
µ4µ5µ6=2
3√3α4(α42−3β42)−f(~α3)(α42+β42) + f(~α3)3.(3.39)
We will use the relations (3.38), (3.39) later.
3.3.2 decoupling limit of massive hypermultiplets
We consider the following decoupling limit of the massive hypermultiplets:
x→0,
µi→ ∞,(3.40)
x
6
Y
i=1
µi= Λ.
10
This limit makes the external momentum infinity:
α1, α2, β2, α3, α4, β4→ ∞.(3.41)
The resulting theory is the SU(3) pure super Yang-Mills theory, and thereby we can ”derive” our
proposal from the AGT-W conjecture by using this scaling limit.
recursion relations and asymptotic behavior
The 4-point conformal block consists of the 3-point spherical conformal blocks and the Shapovalov
matrix (Appendix.A). Then we have to study the asymptotic behavior of the 3-point spherical
conformal blocks in order to evaluate the scaling limit of the 4-point conformal block. For the
purpose, we show the fact that among correlators hVα,Y|V1(1)V2(0)iand hVα,Y(0)V3(1)V4(∞)iwith
fixed |Y| ≡ |YL|+|YW|=n, the 3-point blocks with Y= (φ, [1n]) give the leading contribution in
the limit:
hL−φW[1n]Vα|V1(1)V2(0)i=2w1−w2−3w1
2∆1
(∆1+ ∆2)n
hVα|V1(1)V2(0)i
+ sub-leading terms,(3.42)
h(L−φW[1n]Vα)(0)V3(1)V4(∞)i=w3+w4−3w3
2∆3
(∆3−∆4)n
hVα(0)V1(1)V2(∞)i
+ sub-leading terms.(3.43)
This means that in the limit (3.40) only the following 3-point conformal blocks show the dominant
behavior and can contribute to the 4-point conformal block (3.34):
¯ρα1α2;α(φ, [1n]) = 2w1−w2−3w1
2∆1
(∆1+ ∆2)n
+ sub-leading terms,(3.44)
ραα3α4(φ, [1n]) = w3+w4−3w3
2∆3
(∆3−∆4)n
+ sub-leading terms.(3.45)
Let us prove the above statement. The key is the Mironov-Mironov-Morozov-Morozov recursion
relations [14] for a descendant Va:
hL−nVa|V1(1)V2(0)i= (∆a+n∆1−∆2)hVa|V1(1)V2(0)i,(3.46)
h(L−nVa)(0)V3(1)V4(∞)i= (∆a+n∆3−∆4)hVa(0)V3(1)V4(∞)i,(3.47)
hW−nVa|V1(1)V2(0)i=hW0Va|V1(1)V2(0)i+n(n+ 3)w1
2−w2hVa|V1(1)V2(0)i
+nhVa|(W−1V1)(1)V2(0)i,(3.48)
h(W−nVa)(0)V3(1)V4(∞)i=h(W0Va)(0)V3(1)V4(∞)i+n(3 −n)w3
2+w4hVa(0)V3(1)V4(∞)i
+nhVa(0)(W−1V3)(1)V4(∞)i.(3.49)
11
We now study a pair of partitions Y ≡ {YL, YW}with fixed number of boxes n=|YL|+|YW|.
Let us show the fact that the partition Ywhose correlator hVα,Y|V1(1)V2(0)idominates in the
limit is in the shape of Y ≡ {φ, YW}. We can prove it by inductive argument. First we compare
hL−mVa|V1(1)V2(0)iand hW−mVa|V1(1)V2(0)i. In the right hand sides of the recursion relations
(3.46) and (3.48), L−mcreates a factor ∆i∼αi2and W−mgives wi∼αi3in the limit. Moreover
the first term of (3.48)
hW0Va|V1(1)V2(0)i(3.50)
can not contribute to the dominant behavior, since the action of W0gives
|W0Vα,Yi=X
|Y′|=|Y|
c(α)|Vα,Y′i.(3.51)
As we will show below, the third term of the right hand side of the recursion relation (3.48) is also
negligible in the limit. The correlator hW−mVa|V1(1)V2(0)iis hence the dominant one. By using
this result inductively, the correlator hL−YLW−YWVα|V1(1)V2(0)ibecome dominant for YL=φand
YW= [1n].
Next, let us evaluate the leading correlator hWn
−1Vα|V1(1)V2(0)i. Since we choose the external
state V1as the W-null [7][14], the action of the W−1operator on the primary is W−1V1=3w1
2∆1L−1V1.
Then we obtain
hVa|(W−1V1)(1)V2(0)i=3w1
2∆1hVa|(L−1V1)(1)V2(0)i
=3w1
2∆ (∆a−∆1−∆2)hVa|V1(1)V2(0)i.(3.52)
Here we have used the relation
hVa|(L−1V1)(1)V2(0)i= (∆a−∆1−∆2)hVa|V1(1)V2(0)i.(3.53)
The conformal dimension is
∆a=|YL|+|YW|+ ∆α(3.54)
for Va=L−YLW−YWVα, since L0L−YLW−YWVα= (|YL|+|YW|+ ∆α)Vαholds. This factor is
negligible in the limit αi, βi→ ∞. Therefore the term (3.52) behaves as −∆1−∆2in the decoupling
limit, and the second term of the right hand side of the recursion relation (3.48) is dominant in the
limit. Thus we obtain the leading term of the 3-point function:
hW[1n]Vα|V1(1)V2(0)i=2w1−w2−3w1
2∆1
(∆1+ ∆2)n
hVα|V1(1)V2(0)i+···.(3.55)
Similar argument shows that the following correlator is also leading:
h(W[1n]Vα)(0)V3(1)V4(∞)i=w3+w4−3w3
2∆3
(∆3−∆4)n
hVα(0)V1(1)V2(∞)i+···.(3.56)
12
scaling limit of the conformal block
In this paper we choose a specific W-null state ~α1= (α1,−ǫ
2) for simplicity. The argument does
not change when we choose any other null state. The dimensions of the state with ~α1are
∆1=α12−3
4Q2,
w1=2α1∆1
p4−15Q2,(3.57)
D1=4α12
4−15Q2.
Then the factor appearing in (3.44) becomes
2w1−w2−3w1
2∆1
(∆1+ ∆2)
=−3√3
p4−15Q2 p4−15Q2
3√3w2−−α1
√3(α22+β22) + −α1
√33!−9α1Q2
4p4−15Q2
=−3√3
p4−15Q2µ1µ2µ3−9α1Q2
4p4−15Q2.(3.58)
Here we use (3.38). We can also find the similar relation for (3.45):
w3+w4+3w3
2∆3
(∆3−∆4) = 3√3
p4−15Q2µ4µ5µ6+O(α3).(3.59)
Thus we get the following relation in the limit (3.40):
lim 2w1−w2−3w1
2∆1
(∆1+ ∆2)w3+w4+3w3
2∆3
(∆3−∆4)x=−27
4−15Q2Λ6.(3.60)
Recall that for fixed |Y| =nthe dominant contribution of the sum (3.34) comes from (3.44) and
(3.45). Therefore we get the following result:
lim hx|Y| ¯ρα1α2;α(Y)ραα3α4(Y′)i|Y |=|Y ′|=n=δY,(φ,[1n])δY′,(φ,[1n]) −27
4−15Q2n
Λ6n(3.61)
Therefore only the terms with Y= (φ, [1n]) survive in the limit. This means that the scaling limit
of the conformal block is given by the Shapovalov matrix:
lim B=X
n−27
4−15Q2Λ6n
Q−1
∆(φ, [1n]; φ, [1n]).(3.62)
The gauge theory with 6-flavors becomes SU (3) pure Yang-Mills theory in the decoupling limit. By
assuming the AGT-W conjecture [7][16], we can thus prove our proposal for SU (3) pure Yang-Mills
theory
Zinst
SU (3) =X
n−27
4−15Q2Λ6n
Q−1
∆(φ, [1n]; φ, [1n]).(3.63)
13
4 Conclusion
In this paper we have proposed a relation between representation theory of the W3-symmetry and
the instanton counting of SU(3) pure super Yang-Mills theory. We found that the Nekrasov parti-
tion function for the Yang-Mills theory is equal to the following elements of the inverse Shapovalov
matrix:
Zinst
SU (3)(a, b, Λ, e1, e2) = 1 +
∞
X
n=1 −27
4−15e2Λ6n
Q−1
∆(φ, [1n]; φ, [1n]).(4.1)
We also proved our proposal by assuming the AGT-W conjecture and taking the decoupling limit of
massive hypermultiplets. Then the asymptotic behavior of the 3-point spherical conformal blocks
(3.44)(3.45) played a key role. The proposal (4.1) is a skeleton of the AGT-W conjecture. Thus
the study of our proposal will give an efficient check of the original AGT-W conjecture. In this
paper we verified our proposal explicitly up to 2-instanton.
Our proposal is a simple and nontrivial extension of the Gaiotto-Marshakov-Mironov proposal
[5][6], and it suggests that there exists a direct connection between instanton counting and conformal
symmetry. Then it is very natural to expect that the Nekrasov partition function for SU (N) pure
Yang-Mills theory to be related to the Shapovalov matrix of the WN-symmetry. It would be very
nice to find the explicit relation between them. We also expect that we can recast the theory of
instanton in the language of the representation theory of conformal symmetry and vice versa. It
would deepen our understanding of nonperturbative dynamics of N= 2 theories and 2-dimensional
CFT’s.
In this paper we focused on the pure Yang-Mills theory. It is possible to extend our proposal
for SU (3) gauge theories with Nf= 1,2,···,5 flavors by studying an appropriate decoupling limit
of hypermultiplets. Moreover multi-point conformal blocks for the Virasoro algebra are related to
the Nekrasov partition functions of SU (2) quiver gauge theories [23]. The SU(3) quiver theories
would therefore be related to W3-algebra. Matrix models [10][24][25][26][27] may give an effective
tool to study these extended relations.
In [28], an analog of the AGT conjecture in the 5-dimensional SU (2) gauge theory was found.
The q-deformed Virasoro algebra is the conformal symmetry which plays a role in the extended
AGT correspondence. We would be able to connect the representation theory of the deformed
W-algebra and the 5-dimensional Nekrasov partition functions.
5 Acknowledgement
The auther is supported by JSPS Grant-in-Aid for Creative Scientific Research, No.19GS0219.
14
Appendix
A Conformal Symmetry and Conformal Blocks
A.1 structure of correlators
In this appendix we summarize the definition of the 4-point conformal block. See [17][14][16] for
details.
The basic objects in conformal field theory are the states (vertex operators Va), the norms
(2-point functions) and the structure constants (OPE).
Let us consider the contravariant form (the Shapovalov form) on the Verma module for the
representation of a certain conformal symmetry. The 2-point functions are characterized by the
following type of Gram matrix
Kab =hVa|Vbi.(A.1)
Its matrix element is nonzero if and only if the conformal dimensions are equal ∆a= ∆b. As
we will see, the so-called Shapovalov matrix is the model independent part of the matrix which
depends only on the representation theory of conformal symmetry. The structure constants of the
operator algebra are encoded in the OPE of the vertex operators:
Va(z)Vb(z′) = X
c
Cc
abVc(z′)
(z−z′)∆a+∆b−∆c.(A.2)
Let us consider the following 3 point functions:
¯
Γa1a2;a≡ hVa|Va1(1)Va2(0)i,
=X
b
Cb
a1a2Kab (A.3)
Γba3a4≡ hVb(0)Va3(1)Va4(∞)i.(A.4)
As we will see, they are basic building blocks of the correlators.
In this articles, we are interested in the 4-point spherical correlation functions.
G(4)h∆1∆2
∆3∆4i(x)≡ hVa1(x)Va2(0)Va3(1)Va4(∞)i
=X
b
x−∆1−∆2x∆bCb
a1a2Γba3a4.(A.5)
By using (A.3), we find
G(4)h∆1∆2
∆3∆4i(x) = X
b
x−∆1−∆2x∆b¯
Γa1a2;a(K−1)ab Γba3a4.(A.6)
15
Since we consider the correlation functions of the primaries, let us assume Vaiare primary Vαi.
We introduce the label of the descendants Y. Then Γ and ¯
Γ are
¯
Γα1α2;(α,Y)= ¯ρα1α2;α(Y)Cα1α2;α,(A.7)
Γ(β,Y)α3α4=ρβα3α4(Y)Cβ α3α4.(A.8)
Here ¯ρand ρare 3-point shperical conformal blocks, which are objects of the representation theory
of conformal algebra. Meanwhile, the 3-point functions for primaries
Cα1α2;α≡¯
Γα1α2;(α,φ),(A.9)
Cβα3α4≡Γ(β,φ)α3α4,(A.10)
depend on the choice of a CFT model.
A.2 conformal block of Virasoro algebra
We take the Virasoro symmetry for example. The Virasoro algebra W2is an infinite dimensional
symmetry of 2-dimensional CFT models. It is generated by the energy-momentum tensor T(z) =
Pz−n−2Ln. The symmetry take the following form:
[Ln, Lm] = (n−m)Ln+m+c
12n(n2−1)δn,−m.(A.11)
The action of these generators on a primary operator Vα(z) gives descendants, which are labeled
by the Young diagrams Y={Y1, Y2,···}:
Vα, Y =L−YVα
=L−Ll···L−L2L−Y1Vα.(A.12)
The conformal dimension of the descendant is ∆α, Y = ∆α+|Y|. The Gram matrix for the Verma
module of the Virasoro symmetry is
K(α,Y ),(β,Y ′)=δ|Y|,|Y′|Kαβ Qαβ (Y, Y ′)
=δ|Y|,|Y′|δα,β KαQ∆α(Y, Y ′).(A.13)
Here the norms of primaries characterize the dependence of the norms on a CFT model. The
Shapovalov matrix Q∆α(Y, Y ′) is independent of a choice of a model.
In the case of the Virasoro symmetry, the model-dependent part of the 3-point functions also
possesses a peculiar feature. The Virasoro symmetry implies that the two conformal blocks are
equal [14]:
¯ρα1α2;α(Y) = ραα1α2(Y).(A.14)
16
This property simplify computation of the Virasoro conformal blocks.
Then the spherical 4-point function becomes
G(4)h∆1∆2
∆3∆4i(x) = X
α
x∆α(Cα1α2;α(K−1)αCαα3α4)F∆αh∆1∆2
∆3∆4i(x).(A.15)
Here Fis the model-independent part of the 4-point function. It is given by the conformal block B
F∆αh∆1∆2
∆3∆4i(x) = x−∆1−∆2B∆αh∆1∆2
∆3∆4i(x),(A.16)
and the 4-point sherical conformal block Bis defined by
B∆αh∆1∆2
∆3∆4i(x) = X
|Y|=|Y′|
x|Y|¯ρα1α2;α(Y)Q−1
∆α(Y, Y ′)ραα3α4(Y′)
= 1 +
∞
X
k=1
xkF(k)
∆αh∆1∆2
∆3∆4i.(A.17)
Notice that ¯ρα1α2;α(Y) = ραα1α2(Y) holds for the Virasoro symmetry. The equivalence does not
hold for the extended W-symmetry.
Recall that the conformal block B, which does not depend on a choice of a model, is equal to the
Nekrasov partition function of 2Nc=Nf= 4 gauge theory in the AGT conjecture. This suggests
that we can recast a Nekrasov partition function into an object of the representation theory of
conformal symmetry. It is also conjectured that Cα1α2;α(K−1)αCαα3α4gives the perturbative part
of the Nekrasov partition function for a specific CFT model [4]. Thus the AGT conjecture implies
that a full Nekrasov partition function relate to a physical correlator of a certain CFT model.
Notice that we does not study physical correlators but their holomorphic (chiral) part, since we
concentrate our attention on the instanton part of partition functions.
B Check of the Conjecture for Level-2
In this section, we verify our conjecture at 2-instanton level. Let us recall the identification of
parameters [16]:
c= 2 −24 (e1+e2)2,(B.1)
∆ = a2+b2+ab −(e1+e2)2,(B.2)
w=s27
4−15 (e1+e2)2ab (a+b),(B.3)
D= 4 a2+b2+ab −(e1+e2)2
4−15 (e1+e2)2+ 3 (e1+e2)2
4−15 (e1+e2)2.(B.4)
17
Here e1,2are the dimensionless Ω-background
e1=−1
e2
=e. (B.5)
We also scale the Coulomb moduli in order that aand bbecome dimensionless:
a1=~a, a2=~b. (B.6)
First we compute the level-2 Shapovalov matrix of the W3-symmetry. The labels for this block
matrix must satisfy |YL|+|YW|= 2. There are therefore five choices of such a pair of Young
diagrams:
(YL, YW) = ([2], φ),(12, φ),([1],[1]),(φ, [2]),(φ, 12).(B.7)
The level 2 matrix with these indices is given by [16]
K=
4 ∆ + c
26 ∆ 9 w6w45
2D∆
6 ∆ 4 ∆ (2 ∆ + 1) 6 w(2 ∆ + 1) 12 w27 D∆ + 18 w2
9w6w(2 ∆ + 1) 9 D∆2+ 9 D∆ + 9 w218 D∆27
2Dw (2 ∆ + 3)
6w12 w18 D∆ 9 ∆ (D+ 1) 27
2w(3 D+ 1)
45
2D∆ 27 D∆ + 18 w227
2Dw (2 ∆ + 3) 27
2w(3 D+ 1) 81
4D2∆ (2 ∆ + 1) + 648 D∆ (∆+1)+4 w2
22+5 c
.
An important point is that we can factorize the level-2 Kac determinant as follows [16]:
det K=2438Qi<j (aij 2−e2)2(aij2−(e+e1)2)(aij 2−(e+e2)2)
(4 −15e2)4.(B.8)
Here aij denotes
a12 =a−b, a23 =a+ 2b, a13 = 2a+b. (B.9)
Next we calculate the (φ, 12)-(φ, 12) component of the inverse matrix K−1. For the purpose
we study the following cofactor matrix:
˜
K=
4 ∆ + 1/2c6 ∆ 9 w6w
6 ∆ 4 ∆ (2 ∆ + 1) 6 w(2 ∆ + 1) 12 w
9w6w(2 ∆ + 1) 9 D∆2+ 9 D∆ + 9 w218 D∆
6w12 w18 D∆ 9 ∆ (D+ 1)
.(B.10)
18
The determinant of the matrix is
det ˜
K
=−324 c∆3w2D+ 1458 c∆2w2D+ 486 c∆w2D+ 2592 D2∆6+ 2592 D∆6−2592 ∆4w2
−9072 ∆ w4−648 cw4+ 2592 ∆2w4+ 4860 D2∆4−1620 D∆4+ 1620 ∆2w2−9396 D2∆5
+ 972 D∆5−972 ∆3w2+ 18468 ∆3w2D−10044 ∆2w2D−5184 ∆4w2D+ 324 cD2∆5
+ 324 cD∆5−810 cD2∆4+ 486 cD∆4−324 c∆3w2−486 cD2∆3+ 162 cD∆3−486 c∆2w2
−162 c∆w2+ 5184 w4.(B.11)
By substituting (B1-4), we find the following representation:
det ˜
K=−34
e110 4
4−15e22
×(128 −1444e12−448e12ab −448e12a2−448e12b2+···)Y
i<j
(aij 2−e2).(B.12)
Here (128 −1444e12− ·· ·) is a polynomial of e1,aand b, which is composed of about 150 terms.
Finally we study the 2-instanton Nekrasov partition function. Let us recall the 2-instanton
Nekrasov partition function for N= 2 SU(3) pure Yang-Mills theory (3.10):
ZSU (3), k=2 (a, b, c, e1, e2) = −9(128 −1444e12−448e12ab −448e12a2−448e12b2+···)
e110 Qi<j(aij 2−e2)(aij 2−(e+e1)2)(aij2−(e+e2)2).(B.13)
A remarkable feature is that the polynomial factors (128 −1444e12−·· · ) of (B.12) and (B.13) are
completely equal. Hence the following identity holds for the Nakrasov partition function and the
Shapovalov matrix:
ZSU (3), k=2 (a, b, c, e1, e2) = (4 −15e2)2det ˜
K
2432Qi<j (aij 2−e2)2(aij2−(e+e1)2)(aij 2−(e+e2)2).(B.14)
The point is that the denominator of the right hand side is precisely that of the Kac determinant
(B.8). Then, we can prove that the our proposal also holds for 2-instanton by using (B.8) and by
using (B.14):
ZSU (3), k=2 (a, b, c, e1, e2) = 27
4−15e22det ˜
K
det K
=−27
4−15e22
Q−1
∆(φ, 12;φ, 12).(B.15)
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