Published for SISSA by

Springer

Received: May 26, 2014 Accepted: July 20, 2014 Published: September 3, 2014

Yutaka Matsuo,a Chaiho Rimb and Hong Zhangb a

Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033 Japan b Department of Physics and Center for Quantum Spacetime (CQUeST), Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, 121-742 Korea

E-mail: [email protected], [email protected], [email protected] Abstract: We construct Gaiotto states with fundamental multiplets in SU(N ) gauge theories, in terms of the orthonormal basis of spherical degenerate double affine Hecke algebra (SH in short), the representations of which are equivalent to those of Wn algebra with additional U(1) current. The generalized Whittaker conditions are demonstrated under the action of SH, and further rewritten in terms of Wn algebra. Our approach not only consists with the existing literature but also holds for general SU(N ) case. Keywords: Duality in Gauge Field Theories, Conformal and W Symmetry ArXiv ePrint: 1405.3141

c The Authors. Open Access, Article funded by SCOAP3 .

doi:10.1007/JHEP09(2014)028

JHEP09(2014)028

Construction of Gaiotto states with fundamental multiplets through degenerate DAHA

Contents 1 Introduction

1

2 Construction of Gaiotto states

2

3 Brief introduction of SH

4

4 The relation between SHc and W -algebra

5 9 9 11

6 Whittaker conditions in terms of W -algebra

14

7 Conclusion

19

A Derivation of L±2 constraints on the bifundamental multiplets A.1 Modified vertex operator for U(1) factor A.2 Ward identities for J±1 and L±1 A.3 Ward identities for L±2

19 20 20 21

1

Introduction

The instanton Nekrasov partition function [1–3] for 4 dimensional N = 2 supersymmetric SU(2) quiver gauge theory has the remarkable correspondence with 2 dimensional Liouville conformal field theories, so called AGT conjecture [4]. And the correspondence was generalized into SU(N ) quiver gauge theories in [5, 6] . There are various proofs for the AGT conjecture [7–11]. In most cases, the Virasoro and W algebras play the essential role. In contrast, spherical degenerate double affine Hecke algebra (spherical DDAHA or SH) [12–18] turns out to be another useful tool to prove the AGT conjecture. DDAHA is generated by 2N operators, zi and Di (i = 1, · · · , N ) where X X 1 ∂ Di = zi ∇i + σij , ∇i = +β (1 − σij ) , (1.1) ∂zi zi − zj j
j(6=i)

and permutation operators. Here ∇i is the Dunkl operator which plays a fundamental role in Calogero-Sutherland system and σij is the transposition of variables, zi σij = σij zj . The operators zi and Di satisfies the following commutation relations, [zi , zj ] = 0, [Di , Dj ] = 0 ,   −βzi σij    P  P [Di , zj ] = zi + β z σ + z σ ki i ik     −βz σ j ij

–1–

(1.2) ij.

(1.3)

JHEP09(2014)028

5 Whittaker conditions in terms of SHc 5.1 Nf = 0 case 5.2 Nf = k case

2

Construction of Gaiotto states

For the pure super Yang-Mills theory where the fundamental multiplet is absent, Nf = 0, the instanton part of the partition function has the form, Z(~a) =

X

~ ~ ), Λ4|Y | Zvect (~a, Y

~ Y

–2–

(2.1)

JHEP09(2014)028

DDAHA is the algebra freely generated by zi , Di and σ ∈ SN . Spherical DDAHA (SH) is obtained by the restriction to the symmetric part. For the special value of β = 1, SH reduces to W1+∞ algebra which is described by free fermions. Recently it was found that some representations of SH are equivalent to those of Wn algebra with additional U(1) current [16]. It is known that SH has a natural action on the equivariant cohomology class of the instanton moduli space while Wn algebra describes the symmetry of Toda field theory. This correspondence was used to prove the AGT conjecture. For example, in [16] such mechanism was applied to the pure SU(N ) super Yang-Mills theory, and the representative of the cohomology class is mapped to the orthogonal basis in the Hilbert space of Wn algebra. In this way, the Gaiotto state [19] is constructed to arbitrary order through the conditions on the action of the generators of SH. Later in [17, 18], such correspondence was applied to quiver type gauge theories. The action of SH on the basis appears as the recursion relation for the Nekrasov partition function, which is then interpreted as the Ward identities associated with the Wn -algebra. Here we apply the similar trick to construct explicit Gaiotto states with fundamental multiplets in SU(N ) gauge theories. The computation is in parallel with those in [16]. Note that the Gaiotto state appears as an irregular module of Virasoro and Wn algebra. There were already a few attempts to construct the irregular states algebraically in [19–22]. Our construction is not limited to SU(3) but is extended to SU(N ) with Nf < N . It is also noted that the Gaiotto state construction was proposed but in a different manner, which uses the coherent state approach in [23–25]. Some of irregular state was constructed explicitly using random matrix formalism in connection with SU(2) quiver gauge theories [26, 27]. Thus, our construction will be instructive and complementary to understand the Gaiotto state in different approaches. This paper is organizes as follows. In section 2 we define the Gaiotto states with fundamental multiplets in terms of the orthonormal basis of SH. In section 3, we briefly review the algebra SH and the relation with Wn algebra. In section 4, we give the explicit correspondence between SH and Wn generators through the use of free boson fields. In section 5 we show that the states satisfy generalized Whittaker condition in terms of SH. Finally in section 6 we rewrite the conditions in terms of the generators of Wn algebra and confirm the consistency with the existing literature [20–22, 24]. In the appendix, we derive the Ward identities for the Virasoro operator L±2 . Though this is not directly relevant to the main claim of this paper, we include it since the analysis is technically very close and also it completes the analysis of [17, 18].

with ~ ) := f (~a, Y ~ ) := Zvect (~a, Y

Y p,q

1 gYp Yq (ap − aq )

(2.2)

where Λ is the dynamical scale, ~a ∈ Cn is the VEV for an adjoint scalar field in the vector ~ = (Y1 , · · · , YN ) is a set of Young tableaux characterizing fixed points of multiplet and Y localization in the instanton moduli space. And Y Y gY,W (x) = (x + β(Yj0 − i + 1) + Wi − j) (−x + β(Wj0 − i) + Yi − j + 1) , (2.3) (i,j)∈Y

(i,j)∈W

~ Y

~ i is introduced in [17, 18] as an basis of a Hilbert space H~a . The dual basis h~a, Y ~| Here |~a, Y ~ |~a, W ~ i = δ ~ ~ . It is trivial to confirm that it has the desired inner is defined such that h~a, Y Y ,W product due to the orthonormal property of the basis. However, it is nontrivial to confirm that it satisfies the condition for generalized Whittaker condition as given in [16]. One may proceed likewise for Nf = 2. The partition function has extra contributions from the fundamental multiplets with masses mi , X ~ ~ )Zfund (~a, Y ~ , m1 )Zfund (~a, Y ~ , m2 ) Z Nf =2 (~a, m1 , m2 , Λ) = Λ2|Y | Zvect (~a, Y (2.5) ~ Y

where ~ , m) = Zfund (~a, Y

N Y Y

(ap + βi − j − m) .

(2.6)

p=1 (i,j)∈Yp

Noting that Z Nf =2 (~a, m1 , m2 , Λ) = hG, m2 |G, m1 i one may have the Gaiotto state with one additional parameter m X ~ ~ ))1/2 Zfund (~a, Y ~ , m) |~a, Y ~ i. |G, mi = Λ|Y | (Zvect (~a, Y

(2.7)

(2.8)

~ Y

In this way, it is straightforward to generalize it to additional k < N parameters m1 , m2 ,· · ·, mk , namely, |G, m1 , · · · , mk i =

X

~ ~ ))1/2 Λ|Y | (Zvect (~a, Y

k Y A=1

~ Y

–3–

~ , mA ))|~a, Y ~ i. (Zfund (~a, Y

(2.9)

JHEP09(2014)028

where Yi is the length of the ith column of Y , and Y 0 stands for the transposed Young tableaux. β is related to Ω-deformation parameters by β = −1 /2 . According to AGT conjecture, we may put the partition function as the inner product ˜ of two Gaiotto states Z(~a) = hG|Gi. It is a nontrivial issue to realize |Gi in the Hilbert space of W-algebra. On the other hand, in SH, we know the orthonormal basis and the action of generators which will be reviewed in the next section. The Gaiotto state takes the form, X ~ ~ ))1/2 |~a, Y ~ i. |Gi = Λ2|Y | (Zvect (~a, Y (2.4)

One may easily confirm that the inner product of two Gaiotto states with k parameters will give the instanton partition function with Nf = 2k. The nontrivial part is to confirm the Whittaker vector conditions. The case for Nf = 0 was given by [16]. The proof for additional fundamental multiplets is new. Our task is to find the generalized Whittaker conditions using SH generators and rewrite them in terms of Wn generators.

3

Brief introduction of SH

pn := α−n ,

n

∂ := αn , ∂pn

n ∈ Z≥0

(3.1)

which satisfies the standard commutation relation [αn , αm ] = nδn+m,0 . The space of symmetric functions is described by the Fock space F of the free boson. The Hilbert space of Wn -algebra shows up when we take coproduct of n representations of F and make some restriction on the representation (taking the ‘symmetric part’ which is referred as [1n ] representation in [16]). After taking such coproduct it has nontrivial central charges given below. To distinguish the algebra with central extensions from others, we will denote the algebra SHc . It has generators Dr,s with r ∈ Z and s ∈ Z≥0 . The commutation relations for degree ±1, 0 generators are defined by, l ≥ 1,

(3.2)

l ≥ 1,

(3.3)

[D−1,k , D1,l ] = Ek+l

l, k ≥ 0 ,

(3.4)

[D0,l , D0,k ] = 0 ,

k, l ≥ 0 ,

(3.5)

[D0,l , D1,k ] = D1,l+k−1 , [D0,l , D−1,k ] = −D−1,l+k−1 ,

where Ek is a nonlinear combination of D0,k determined in the form of a generating function,     X X X 1 + (1 − β) El sl+1 = exp  (−1)l+1 cl πl (s) exp  D0,l+1 ωl (s) , (3.6) l≥0

l≥0

l≥0

with πl (s) = sl Gl (1 + (1 − β)s) , X ωl (s) = sl (Gl (1 − qs) − Gl (1 + qs)) ,

(3.7) (3.8)

q=1,−β,β−1

G0 (s) = − log(s),

Gl (s) = (s−l − 1)/l

–4–

l ≥ 1.

(3.9)

JHEP09(2014)028

The generators of Spherical DDAHA (SH) are obtained by symmetrizing those of DDAHA P by S = N1 ! σ∈SN σ, SOS where O ∈ DDAHA. Such generators act naturally on the ring of symmetric functions of zi . The independent generators of SH is given by Dnm ∼ P n m S N i=1 (zi ) (Di ) S (n ∈ Z, m ∈ Z≥0 ) in N → ∞ limit. The definition of Dnm is only sketchy here and will be more carefully defined later. For a special value for β = 1, SH reduces to W1+∞ algebra which is described by free fermions. In large N limit, one may introduce free boson description of SH in terms of power P n sum polynomial pn = ∞ i=1 (zi ) . We identify,

The parameters cl (l ≥ 0) are central charges. Other generators are defined recursively by, 1 [D1,1 , Dl,0 ] , l = [D0,l+1 , Dr,0 ]

1 [D−l,0 , D−1,1 ] , l = [D−r,0 , D0,l+1 ] .

Dl+1,0 =

D−l−1,0 =

(3.10)

Dr,l

D−r,l

(3.11)

for l ≥ 0, r > 0 . ~ i, There is an explicit form of the action on the orthonormal basis |~a, Y ~ i = (−1)l D−1,l |~a, Y

fq N X X ~ )|~a, Y ~ (t,−),q i , (aq + Bt (Yq ))l Λ(t,−) (Y q

(3.12)

~ i = (−1)l D1,l |~a, Y

q +1 N fX X ~ )|~a, Y ~ (t,+),q i , (aq + At (Yq ))l Λ(t,+) (Y q

(3.13)

q=1 t=1

~ i = (−1)l D0,l+1 |~a, Y

N X X

~ i. (aq + c(µ))l |~a, Y

(3.14)

q=1 µ∈Yq (t,−)

where c(µ) = βi − j for µ = (i, j). The factor Λq

~ ) is defined by (~a, Y

~)= Λ(k,+) (~a, Y (3.15) p   1/2 fq N Y Y ap −aq +Ak (Yp )−B` (Yq )+ξ Y0fq+1 ap −aq +Ak (Yp )−A` (Yq )−ξ    , `=1 ap −aq +Ak (Yp )−B` (Yq ) ap −aq +Ak (Yp )−A` (Yq ) q=1

`=1

~)= Λ(k,−) (~a, Y (3.16) p   1/2 fq+1 N Y Y ap −aq +Bk (Yp )−A` (Yq )−ξ Y0fq ap −aq +Bk (Yp )−B` (Yq )+ξ    . `=1 ap −aq +Bk (Yp ) − B` (Yq ) ap −aq +Bk (Yp )−A` (Yq ) q=1

`=1

We decompose Y into rectangles Y = (r1 , · · · , rf ; s1 , · · · , sf ) (with 0 < r1 < · · · < rf , s1 > · · · > sf > 0, see figure 1 for the parametrization). We use fp (resp. f¯p ) to represent the number of rectangles of Yp (resp Wp ). The factors Ak (Yp ), B` (Yq ) are Ak (Y ) = βrk−1 − sk − ξ,

(k = 1, · · · , f + 1) ,

(3.17)

Bk (Y ) = βrk − sk ,

(k = 1, · · · , f ) ,

(3.18)

where ξ := 1−β. Ak (Y ) (resp. Bk (Y )) represents the k th location where a box may be added to (resp. deleted from) the Young diagram Y composed with a map from location to C. We denote Y (k,+) (resp. Y (k,−) ) as the Young diagram obtained from Y by adding (resp. deleting) a box at (rk−1 + 1, sk + 1) (resp. (rk , sk )). Similarly we use the notation ~ (k±),p = (Y1 , · · · , Yp(k,±) , · · · , YN ) to represent the variation of one Young diagram in a Y ~ . For more detail of the notation, we refer [17, 18]. set of Young tables Y

4

The relation between SHc and W -algebra

SHc and Wn -algebra look very different but the Hilbert space of both algebras are identical for [1n ] representation of SHc . The content of this section is a brief summary of [16].

–5–

JHEP09(2014)028

q=1 t=1

rk

r1

Y=

rf k sf

1 sk

Figure 1. Decomposition of Young diagram by rectangles

The generators of Wn -algebra are defined through the quantum Miura transformation, −:

n Y

(Q∂z + ~hj · ∂ ϕ ~ ) :=

j=1

n X

W (d) (z)(Q∂z )n−d .

(4.1)

d=0

P where ~hi = ~ei − n1 ni=1 ~ei and ~ei is the i-th orthonormal basis of Rn . ∂ ϕ ~ = (∂ϕ1 , · · · , ∂ϕn ) is n free bosons with the standard OPE, ∂ϕi (z)∂ϕj (0) ∼

δij , z2

∂ϕi (z) =

X

αr(i) z −r−1 .

(4.2)

r∈Z

P We introduce J (z) = ni=1 ∂ϕi (z) to describe the U(1) factor. Expansion of (4.1) gives, W (0) (z) = −1,

(4.3)

W (1) (z) = 0,

(4.4)

W (2) (z) =

1 1 (∂ ϕ ~ )2 − : J 2 (z) : +Q~ ρ · ∂2ϕ ~, 2 2n

(4.5)

n−3 n−1 (2) is the standard form of Virasoro generators with with ρ ~ = (− n−1 2 , − 2 , · · · , 2 ). W the central charge, c = (n − 1)(1 − Q2 n(n + 1)). The higher generators are in general complicated but the part with highest power of ∂ϕ is written in a relatively simple way,

W (d) = −

X

: (~hj1 · ∂ ϕ ~ ) · · · (~hjd · ∂ ϕ ~ ) : + lower terms

j1 <···
n−s n−d

! X

: J (z)d−s ∂ϕj1 (z) · · · ∂ϕjs (z) + lower terms . (4.6)

j1 <···
–6–

JHEP09(2014)028

s1

Meanwhile, SHc is given in free boson representation, obtained from the expression for D±1,0 and D0,2 . For [1n ] representation, they are D±1,0 = −

n X

(i)

α∓1 ,

(4.7)

i=1

D0,2 =

 n √ X β X i

 6

(i)

(: αr(i) αs(i) α−r−s

 n X  X ξX (i) (i) :)+ (r + 1 − 2i)α−r αr(i) +ξ rα−r αr(j) .  2 r>0

r,s∈Z

i0

(4.8)

P l ~ i. The elements Dl,1 are obtained from the where cl = N a, Y p=1 (ap − ξ) when act on |~ P commutation relation, D±r,1 = ±[D0,2 , D±r,0 ] . Here J(z) = √1β ni=1 ∂ϕi (z), and one may evaluate the Virasoro generator as, Ln =

X 1 X X (i) (i) : αn+m α−m : +Q nρi αn(i) . 2 m i

(4.10)

i

This agrees with the Virasoro generator in (4.5) (with the contribution from U(1) factor). It implies that the Hilbert space of the Wn algebra with U(1) factor coincides with the [1n ] representation of SHc . In the following, we derive the explicit form of some generators of SH which are used in the next sections. The relation between higher generators can be similarly obtained using the commutators. The procedure is simplified once we compare the terms with highest generators. For such purpose it is more convenient to introduce a new set of elements Yl,d which are defined inductively starting from Y±1,d = D±1,d . For l ≥ 2 and d ≥ 1, Yl,d =

( [D1,1 , Yl−1,d ] [D1,0 , Yl−1,d+1 ]

if l − 1 6= d if l − 1 = d,

Y−l,d =

( [D−1,1 , Y1−l,d ] [D−1,0 , Y1−l,d+1 ]

if l − 1 6= d if l − 1 = d,

(4.11)

There exists a constant c(l, d) 6= 0 such that Yl,d ≡ c(l, d)

r X

X

(i)

(i)

: αl0 · · · αld : + lower order terms.

i=1 l0 +...+ld =−l

–7–

(4.12)

JHEP09(2014)028

While D±1,0 is diagonal with respect to the sum over i, there exist off-diagonal term in D0,2 which represents the nontrivial twist in the coproduct. D0,2 for n = 1 case is identical to the Hamiltonian of Calogero-Sutherland [28–30]. Generators of Heisenberg (Jl ) and Virasoro algebras (Ll ) are embedded in SHc as [16], p p Jl = (− β)−l D−l,0 , J−l = (− β)−l Dl,0 , J0 = E1 /β, p −l Ll = (− β) D−l,1 /l + (1 − l)c0 ξJl /2 , p L−l = (− β)−l Dl,1 /l + (1 − l)c0 ξJ−l /2 ,   1 ξ2 L0 = [L1 , L−1 ]/2 = D0,1 + c2 + c1 (1 − c0 )ξ + c0 (c0 − 1)(c0 − 2) , (4.9) 2β 6

In particular,

√

d−1

β c(0, d) = , d(d + 1) and

p d c(1, d) = − β /(d + 1),

(4.13)

p d c(−1, d) = − β /(d + 1).

(4.14)

i


Furthermore we use a notation u(z) i = ui when u(z) with conformal dimension d has the P expansion u(z) = i ui z −i−d . With this preparation, we use the power sum polynomial P pl (z) = i (zi )l to represent the first few generators in a compact form, D−1,d

√ d
− β ∼ : pd+1 (z) : 1 , d+1

√

D0,d

d−1


β ∼ : pd+1 (z) : 0 . d(d + 1)

(4.16)

Here ∼ is used to imply that we neglect lower powers of ∂ϕ. The next generator D−2,d has the form: √ d+1
2 β D−2,d ∼ : pd+1 (z) : 2 (4.17) d+1 which will be used in the next sections. Here is an explicit proof of (4.17). We start with D−1,d ∼ c(−1, d)

r X

(i)

X

(i)

: αl0 · · · αld : .

(4.18)

i=1 l0 +...+ld =1

By [αn , αm ] = nδn+m , we obtain [D−1,1 , D−1,d ] = Y−2,d ∼ c(−1, 1) c(−1, d) × 2(d − 1)

r X

X

(i)

(i)

: αl0 · · · αld : . (4.19)

i=1 l0 +...+ld =2

Compare this with (4.12), it follows that c(−2, d) = c(−1, 1) c(−1, d) × 2(d − 1) =

p d+1 β (d − 1)/(d + 1) .

(4.20)

Similarly, [D−1,0 , D−1,d+1 ] ∼ c(−1, 0) c(−1, d + 1) × (d + 2)

r X

X

(i)

(i)

: αl0 · · · αld :

i=1 l0 +...+ld =2

=

p d+1 β

r X

X

(i)

(i)

: αl0 · · · αld : .

i=1 l0 +...+ld =2

–8–

(4.21)

JHEP09(2014)028

The other coefficients are determined recursively. P Here we introduce a notation which is useful later. Let f (z1 , . . . , zn ) = i ai z1i1 · · · zrin is a symmetric polynomial with respect to n variables z1 , · · · , zn . We will also denote the n-powers of bosonic fields with coefficients ai by X : f (z) := ai : (∂ϕ1 (z))i1 · · · (∂ϕn (z))in : . (4.15)

Therefore, we have D−2,d

r X p d+1. = [D−1,0 , D−1,d+1 ]−[D−1,1 , D−1,d ] ∼ 2 β (d + 1)

X

(i)

(i)

: αl0 · · · αld :

i=1 l0 +...+ld =2

√ d+1
2 β = : pd+1 (z) : 2 . d+1

(4.22)

Some of the explicit expressions of W -algebra in terms of SHc are given in the end of section 6.

Whittaker conditions in terms of SHc

5.1

Nf = 0 case

In order to prepare the generalization for Nf 6= 0, we present the Whittaker condition for Nf = 0 using our notation. In the following, we demonstrate, D−1,d |Gi = κd |Gi

0≤d≤N

(5.1)

with   0    κd = (−1)N −1 √1β Λ2     (−1)N √1 PN (ap − ξ)Λ2 p β

d
(5.2)

d=N.

Proof. Set the coefficients in the Gaiotto state as, ~

2|W | ~ ))1/2 . uW (Zvect (~a, W ~ := Λ

(5.3)

Considering the action of SH operator given in (3.12) and (3.13), one has that for the Gaiotto state ˜

fq N X XX ~ )u ~ |~a, W ~ (t,−),q i. D−1,d |Gi = (−1) (aq + Bt (Wq ))l Λ(t,−) (W q W d

(5.4)

~ q=1 t=1 W

If the Gaiotto state satisfies the Whittaker condition in (5.1), the following relation should hold: (−1)d

X

~) (aq + Bt (Wq ))l Λ(t,−) (W q

~ (⊃Y ~) W

uW ~ = κd , uY~

(5.5)

~ is obtained from W ~ by removing one box: Wq(t,−) = Yq , i.e., Wq = Yq(t,+) . We where Y note that

~

Λ2|W | Λ2|Y~ |

= Λ2 . For a Young diagram with one box removed or added (see figure 2),

–9–

JHEP09(2014)028

5

we find At (Y ), Bt (Y ) (defined in (3.17) and (3.18)) in terms of their counterparts of the original Young diagram W :    At (W )     (k,−) At W = Bk (W )     At−1 (W )

1≤t≤k t=k+1

  Bt W (k,−) =

,

k + 2 ≤ t ≤ f˜ + 2

  Bt (W )       Bk (W ) − β

1≤t≤k−1

  Bk (W ) + 1      Bt−1 (W )

t=k+1

t=k , k + 2 ≤ t ≤ f˜ + 1

(5.6)

As W

(k,+)



=

1≤s≤k−1

  Ak (W ) + β      As−1 (W )

s=k+1

s=k , k + 2 ≤ s ≤ f˜ + 2

   Bs (W )     (k,+) Bs W = Ak (W )     Bs−1 (W )

1≤s≤k−1 .

s=k k + 1 ≤ t ≤ f˜ + 1

(5.7) Using the above relations, after some lengthy computation referring to the appendix A.2 of [17, 18], we arrive at uW u ~ (t,+),q ~ = Y (5.8) uY~ uY~  !1/2 Q fp N Y 1 `=1 (aq −ap +At (Yq )−B` (Yp )+ξ)(aq −ap +At (Yq )−B` (Yp ))  = Λ2 . Q 0f p+1 β (aq −ap +At (Yq )−A` (Yp )−ξ)(aq −ap +At (Yq )−A` (Yp )) p=1

`=1

Therefore, ˜

fq N X N X Y d 1 κd = (−1) √ (aq + At (Yq ))d β q=1 t=1 p=1

Q fp

`=1 (aq − ap + At (Yq ) − B` (Yp ) + ξ) Q0fp +1 `=1 (aq − ap + At (Yq ) − A` (Yp ))

! Λ2 .

Setting   xI = {ap + Ak (Yp )}  y = {a + B (Y ) − ξ} p J ` p

1≤I≤

PN

+ 1) = N

1≤J ≤

PN

=M

p=1 (fp p=1 fp

(5.9)

where N − M = N , we have κd in a simplified form QM N X (xI − yJ ) 2 d 1 d κd = Λ (−1) √ (xI ) QNJ=1 . β I=1 J(6=I) (xI − xJ )

(5.10)

According to the formula used in [17, 18]: QM N m+1+M−N X X (xI − yJ ) m (xI ) QNJ=1 = fm−n+1+M−N (−y)bn (x), (5.11) (x − x ) I J J(6 = I) n=0 I=1 P P where fn (x) = I1 <··· N .

– 10 –

JHEP09(2014)028



  As (W )       Ak (W ) − 1

Figure 2. Locations of boxes

Nf = k case

In the following, we demonstrate that for k < N , I:

D−1,d |G, m1 , . . . , mk i = λd |G, m1 , . . . , mk i

II :

λ0d |G, m1 , . . . , mk i

D−2,d |G, m1 , . . . , mk i =

0≤d≤N −k

(5.12)

0 ≤ d ≤ 2N − 2k

(5.13)

with  0      N −k−1 √1 Λ β λd = (−1)     PN Pk  1  N −k √  (−1) p (ap − ξ) − i=1 mi Λ β  0     2 λ0d = Λ    PN Pk   2  −2 p (ap − ξ) − i=1 mi Λ

d
(5.14)

d=N −k

d < 2N − 2k − 1 d = 2N − 2k − 1

(5.15)

d = 2N − 2k

The above expressions still hold for k = 0 case, but with the replacements Λ → Λ2 . Notice that λN −k+1 is not an eigenvalue but an operator which contains derivative of Λ:  λN −k+1

N

∂ 1X 1 1 + (ap − ξ)2 + = (−1)N −k+1 √ βΛ ∂Λ 2 p 2 β  ! N k k X X X + mi mj − mi (ap − ξ) Λ. i
N X

!2 (ap − ξ)

p

p

i=1

We include this expression for later convenience. Proof of I. Our proposal for the Gaiotto state takes the following form, |G, m1 , . . . , mk i =

X

~

Λ|Y | (Zvect )1/2

k Y i=1

~ Y

– 11 –

~ , mi )|~a, Y ~ i. Zfund (~a, Y

(5.16)

JHEP09(2014)028

5.2

Since ~ (t,+),q , m1 ) Zfund (~a, Y = aq + Bt (Wq ) − m1 = aq + At (Yq ) − m1 , ~ , m1 ) Zfund (~a, Y we find the action of D−1,l results to the similar form as the one (5.5) of the Nf = 0 case, and λd is the generalized form of κd in (5.10): Qk QM N 1 X d i=1 (xI − mi ) J=1 (xI − yJ ) √ λd = Λ(−1) (xI ) . QN β I=1 J(6=I) (xI − xJ ) d

(5.17)

Proof of II. To evaluate the action of D−2,l , we use the following commutation relations, D−2,0 = [D−1,0 , D−1,1 ]

(5.18)

D−2,1 = [D−1,0 , D−1,2 ]

(5.19)

D−2,d = [D−1,0 , D−1,d+1 ] − [D−1,1 , D−1,d ] .

(5.20)

Let us write the Gaiotto state as the following, |G, m1 , . . . , mk i =

X

~ i, cW a, W ~ |~

~| |W

cW ~ := Λ

(Zvect )

1/2

k Y

~ , mi ) . Zfund (~a, Y

(5.21)

i=1

~ W

The action of D−2,d on the Gaiotto state is evaluated as (−1)d+1 D−2,d |G, m1 i =

fq N X X


~ )c ~ |~a, W ~ (`,−2H),q i β (aq +B` (Wq ))d +(aq +B` (Wq )−β)d Λ(`,−2H) (W q W

q=1 `=1


) ~ ~ (`,−2V ),q i − (aq +B` (Wq ))d +(aq +B` (Wq )+1)d Λ(`,−2V (W )cW a, W ~ |~ q −

fq N X X

(Bu (Wq )−B` (Wq )){(aq +Bu (Wq ))d +(aq +B` (Wq ))d }




q=1 u<`

~ )Λ(u,−) (W ~ (`,−),q )c ~ |~a, W ~ (`,−;u,−),q i ·Λ(`,−) (W q q W −

fq N X X

B` (Wq )−(Bu (Wq )){(aq +Bu (Wq ))d +(aq +B` (Wq ))d }




q=1 u<`

~ )Λ(`+1,−) ~ (u,−),q )c ~ |~a, W ~ (`,−;u,−),q i ·Λ(u,−) (W (W q q W = λ0d

X

~i cY~ |~a, Y

(5.22)

~ Y

where Y (k,+2H) , Y (k,+2V ) and Y (k,+;u,+) (resp. Y (k,−2H) , Y (k,−2V ) and Y (k,−;u,−) ) stand for the Young diagrams obtained from adding (resp. deleting) two boxes horizontally, vertically and two different places, respectively. (`,−2H) Λq etc. are defined in A.3.

– 12 –

JHEP09(2014)028

Again using (5.11), we find that λd reduces to (5.14).

Again, after lengthy computations, we evaluate the four terms on the right hand side of (5.22) as below: QM QM  N X ~ )1/2 Zvect (W 1 I=1 (xI −yJ ) I=1 (xI −yJ +β) = , QN QN ~ β(1+β) Zvect (Y ) J(6=I) (xI −xJ ) J(6=I) (xI −xJ +β)

~) Λ(`,−2H) (W q

I=1

QM QM  N ~ )1/2 Zvect (W 1 X (`,−2V ) ~ I=1 (xI −yJ ) I=1 (xI −yJ −1) Λq (W ) = , QN QN ~ 1+β Zvect (Y ) J(6=I) (xI −xJ ) J(6=I) (xI −xJ −1) I=1 QM  N QM N ~ )1/2 Zvect (W 1 X J=1 (xI −yJ ) X (u,−) ~ (`,−),q J=1 (xK −yJ ) ~ Λ(`,−) ( W )Λ ( W ) = QN QN q q ~) 2β Zvect (Y (x −x ) I J K6=I J6=I J6=K (xK −xJ ) I=1 (xK −xI )(xK −xI +1−β) × , (xK −xI +1)(xK −xI −β) QM  N QM N ~ )1/2 Zvect (W 1 X J=1 (xI −yJ ) X (u,−) ~ (`+1,−) ~ (u,−),q J=1 (xK −yJ ) Λq (W )Λq (W ) = QN QN ~ 2β Zvect (Y ) J6=I (xI −xJ ) J6=K (xK −xJ ) I=1

K6=I

(xK −xI )(xK −xI −1+β) × , (xK −xI − 1)(xK − xI +β)

(5.24)

where the redefinition of variables as in (5.9) are made. As a result, λ0d has the form, (−1)d+1 λ0d

(5.25)

QM QM N
(xI −yJ +β) Λ2 X I=1 (xI −yJ ) = × xdI +(xI +β)d QN QNI=1 1+β J(6=I) (xI −xJ ) J(6=I) (xI −xJ +β) I=1 k Y
× (xI −mi )(xI +β −mi ) i=1

QM QM N
(xI −yJ −1) Λ2 X I=1 (xI −yJ ) × xdI +(xI −1)d − QN QNI=1 1+β J(6=I) (xI −xJ ) J(6=I) (xI −xJ −1) I=1

k Y
× (xI −mi )(xI −1−mi ) i=1

– 13 –

JHEP09(2014)028

The relations between At (W ), Bt (W ) and their counterparts of the original Young diagram Y are  1≤k ≤l−1   Ak (Y )      Al (Y ) − 1 k=l Ak (W ) = Ak Y (l,+2H) = ,  A (Y ) + 2β k = l + 1  l    Ak−1 (Y ) l + 2 ≤ k ≤ f˜ + 2 (5.23)  1≤k ≤l−1  Bk (Y )    Bk (Y ) = Bk Y (l,+2H) = Al (Y ) + β . k=l   Bk−1 (Y ) l + 1 ≤ k ≤ f˜ + 1

N Q N QM X
(xK −yJ ) (xK −xI )2 (xK −xI +1−β) Λ2 X M J=1 (xI −yJ ) + × × xdK +xdI QN QNJ=1 2β (xK −xI +1)(xK −xI −β) J6=I (xI −xJ ) J6=K (xK −xJ ) I=1

K6=I

k Y
× (xK −mi )(xI −mi ) i=1

−

N Λ2 X

2β

N X J=1 (xI −yJ ) QN J6=I (xI −xJ ) K6=I I=1

QM

QM

QNJ=1

(xK −yJ )

J6=K (xK −xJ )

×


(xK −xI )2 (xK −xI −1+β) × xdK +xdI (xK −xI −1)(xK −xI +β)

i=1

We note that a similar computation appears in the recursion formula with bifundamental multiplet (A.16). After some algebra, it is simplified to (−1)d+1 λ0d

(5.26)

QM Q N
M (xI −yJ +β) Λ2 X d d I=1 (xI −yJ ) = × xI +(xI +β) QNI=1 QN 2(1+β) J(6=I) (xI −xJ ) J(6=I) (xI −xJ +β) I=1 k Y
× (xI −mi )(xI +β −mi ) i=1

QM QM N
(xI −yJ −β) Λ2 X I=1 (xI −yJ ) − × xdI +(xI −β)d QN QNI=1 2(1+β) J(6=I) (xI −xJ ) J(6=I) (xI −xJ −β) I=1 k Y
× (xI −mi )(xI −β −mi ) i=1

QM QM N
(xI −yJ −1) Λ2 X I=1 (xI −yJ ) − × xdI +(xI −1)d QN QNI=1 2(1+β) J(6=I) (xI −xJ ) J(6=I) (xI −xJ −1) I=1 k Y
× (xI −mi )(xI −1−mi ) i=1

QM QM N
(xI −yJ +1) Λ2 X I=1 (xI −yJ ) × xdI +(xI +1)d + QN QNI=1 2(1+β) J(6=I) (xI −xJ ) J(6=I) (xI −xJ +1) I=1 k Y
× (xI − mi )(xI +1 − mi ) , i=1

with N − M = N . In this form, one may use the trick (5.11) to arrive at (5.15).

6

Whittaker conditions in terms of W -algebra

In this section, we rewrite the generalized Whittaker conditions obtained in the previous P (d) section in terms of W -algebra W (d) (z) = i Wi z −i−d . Theorem 2 in the following is the main claim of the paper.

– 14 –

JHEP09(2014)028

k Y
× (xK − mi )(xI − mi ) .

Theorem 1 For Nf = 0 case [16], (d)

(d)

W1 |Gi = λ1 |Gi

0≤d≤N +1

(6.1)

with

(d)

λ1

d
1 N +1

p

(ap − ξ) +

(N −1)N 2 ξ 2(N +1)

 Λ2

,

(6.2)

d=N +1

and (d)

W2 |Gi = 0

0 ≤ d ≤ 2N.

(6.3)

Actually, for SU(N ) case we only have to consider up to W (N ) . From the commutation (d) relations, it is obvious that Wm |Gi = 0 for m ≥ 2 and 0 ≤ d ≤ N . Theorem 2 For the Gaiotto state with k fundamentals, one has (d)

(d)

W1 |G, m1 , . . . , mk i = λ1 |G, m1 , . . . , mk i

0 ≤ d ≤ N − k + 1,

(6.4)

(d) W2 |G, m1 , . . . , mk i

0 ≤ d ≤ 2N − 2k + 1.

(6.5)

(d) λ2 |G, m1 , . . . , mk i

=

When N − k > 1,  0    √ k−N (d) Λ λ1 = ( β) √    ( β)k−N −1

d < N −k d = N −k 1 N−k+1

PN p

(N−k)(N−1)N ξ i=1 mi + 2(N−k+1)

(ap −ξ)−

Pk

(d)

d < 2N − 2k + 2.



, (6.6)

Λ d = N −k+1

and λ2 = 0

(6.7)

When N − k = 1, (1) λ1

(2) λ2

1 = − Λ, β 1 2 = Λ , 2β

(2) λ1

(3) λ2

1 = β

X  N k X (ap − ξ) − mi Λ, p

1 = √ 3 ββ

(6.8)

i=1

X N

(ap − ξ) −

p

Before giving the proof of theorems, we give some comments.

– 15 –

k X i=1

 mi Λ2 .

(6.9)

JHEP09(2014)028

 0    √ −N 2 = ( β) Λ  √    ( β)−N −1

Comments on the other generators. (N −k+2)

1. The action of λ1 becomes an operator involving the derivative of Λ as we show later in (6.29), and we see that   1 ∂ (N −k+2) W1 |G, m1 , · · · , mk i ∼ √ Λ + const. Λ|G, m1 , · · · , mk i. (6.10) β ∂Λ On the other hand, referring to [17, 18] we have

1 = β

−

N X p

ξN (N − 1) (ap − ξ) + 2

! |G, m1 , . . . , mk i,

(6.11)

L0 |G, m1 , . . . , mk i ∂ 1 Λ + ∂Λ 2β

=

N X p

!! N 2 X ξ (ap −ξ)2 +(1−N )ξ (ap −ξ)+ N (N −1)(N −2) 6 p

·|G, m1 , . . . , mk i.

(6.12)

 (N −k+2) Compare to (6.12), we find in the action of W1 − Λ cancels. (d)

√1 ΛL0 β



(r)

(r)

, the derivative of (r)

2. W3 and higher can be generated by commutators of W2 , W1 and W0 , with r ≤ d, more precisely speaking, with the help of (4.11). For example, when the (N −k+2) action of both Ln−1 and (W1 − √1β ΛL0 ) on the Gaiotto state are constant, we (3)

have Wn =

(3) 1 2n−3 [Ln−1 , W1 ]

1 √1 2n−3 [Ln−1 , β ΛL0 ],

∼

Wn(3) |G, m1 , . . . , mk i

so

  1 1 = Ln−1 , √ ΛL0 |G, m1 , . . . , mk i 2n − 3 β =

(n − 1)Λ √ Ln−1 |G, m1 , . . . , mk i. (2n − 3) β

(6.13)

Examples. Here we give some simple cases of our theorem which match with the known results in the literature. • SU(2) case 1 2 Λ |Gi, β  2  1 X L1 |G, mi = (ap − ξ) − m Λ|G, mi, β p L1 |Gi =

L2 |G, mi =

1 2 Λ |G, mi. 2β

All higher Ln have eigenvalue 0.

– 16 –

(6.14) (6.15) (6.16)

JHEP09(2014)028

J0 |G, m1 , . . . , mk i

• SU(3) case 1 Λ|G, mi , β  X  3 1 1 (3) W1 |G, mi = √ (ap − ξ) − m + 2ξ Λ2 |G, mi , ββ 3 p  3  1 X L1 |G, m1 , m2 i = (ap − ξ) − (m1 + m2 ) Λ|G, m1 , m2 i , β p L1 |G, mi =

(6.18)

(6.19)

L2 |G, m1 , m2 i =

(6.20) 3 X

!2 (ap −ξ)

+m1 m2 (6.21)

p

) 3 X 1 +2ξ(m1 + m2 ) − (m1 + m2 ) (ap − ξ) + 3ξ 2 Λ|G, m1 , m2 i , 3 p X  3 1 (3) W2 |G, m1 , m2 i = √ (ap − ξ) − (m1 + m2 ) Λ2 |G, m1 , m2 i , (6.22) 3 ββ p 1 (3) W3 |G, m1 , m2 i = √ Λ3 |G, m1 , m2 i . 3 ββ

(6.23)

P All higher Ln , Wn have eigenvalue 0. Since N p (ap − ξ) can take arbitrary value, after setting it to be zero we find the above equations are in agreement with the known results [19–22], up to overall constant coefficients. In order to compare with P the result of [22], we have to remove the U(1) factor J (z) = ni=1 ∂ϕi (z) =: p1 (z) :.

√ Then we have L01 = L1 − N1 : p1 (z) : 0 : p1 (z) : 1 = L1 + N1 D−1,0 βJ0 , and
2 1 1 L02 = L2 − 2N : p1 (z) : 1 = L2 − 2N (D−1,0 )2 , thus L01 |G, m1 , m2 i

1 = β

L02 |G, m1 , m2 i =

 X  3 2 (ap ) − ξ − (m1 + m2 ) Λ|G, m1 , m2 i , 3 p

1 2 Λ |G, m1 , m2 i , 3β

(6.24) (6.25)

which are consistent with those in [22] by setting

PN p

(ap ) = 0.

Proof of the theorems. Up to terms of order d−1, the generators of W-algebra has the form W (d) (z) ∼ −

d X (−d)s−d : p1 (z)d−s es (z) :

(6.26)

s=0

where el = expansion

P

i1 <···
1 1 en = −(−1)n pn + n 2

· · · zil is the elementary symmetric polynomial. Then using the X

(−1)n

r+s=n,r,s≥1

1 1 pr ps − rs 6

X

(−1)n

r+s+t=n,r,s,t≥1

1 pr ps pt + · · · , rst (6.27)

– 17 –

JHEP09(2014)028

1 2 Λ |G, m1 , m2 i , 2β ( 3 1 ∂ 1X 1 (3) W1 |G, m1 , m2 i = √ βΛ + (ap − ξ)2 + ∂Λ 2 p 6 ββ

(6.17)

it is deduced that, up to terms of order d − 1, p (d) W1 = (−1)d−1 ( β)1−d D−1,d−1 + u

(6.28)

where u is a linear combination of monomials (D0,r1 · · · D0,rs D−1,r ) with r < d − 1, most of which vanish when operate on the Gaiotto states. Take into consideration of (5.14), (5.15), we find explicit correspondence between the generators. In the following “≡” means equivalent up to terms which vanish when operate on the Gaiotto states). (d) Firstly for W1 generators,

p k−N−1 N −k+1 p k−N+1 β) D−1,N−k+1 −(−1)N−k+1 ( β) J0 D−1,N−k N −k+2 (N −k+2)2 −2(N −k+2)−2 p k−N+3 2 + (−1)N−k+1 ( β) J0 D−1,(N−k−1) , (6.29) 2(N −k+2)2 p N − k p 2+k−N (N −k+1) W1 ≡ (−1)N −k ( β)k−N D−1,N −k − (−1)N −k ( β) J0 D−1,N −k−1 , N −k+1 (6.30) p (N −k) W1 ≡ (−1)N −k−1 ( β)1+k−N D−1,N −k−1 . (6.31) (N−k+2)

W1

≡ (−1)N−k+1 (

• For N − k = 1, (3)

W1

(2)

W1

(1)

W1 (d)

Secondly for W2

1 2 1 D−1,2 − J0 D−1,1 + βJ02 D−1,0 , β 3 3 p −1 = L1 ≡ (− β) D−1,1 , p = J1 ≡ (− β)−1 D−1,0 . ≡

(6.32) (6.33) (6.34)

generators are related to SH as, (d)

W2

1 = √ d (−1)d D−2,d−1 + u0 . 2 β

(6.35)

This time u0 is a linear combination of monomials (D0,r1 · · · D0,rs D−1,r D−2,r ) with r < d−1, again most of which vanish when operate on the Gaiotto states. Explicitly, • For N − k > 1, (2N −2k+1) W2

(2N −2k)

W2

√ 1 N −k β ≡ − √ N −k D−2,2N −2k + J D N −k 0 −2,2N −2k−1 2N − 2k + 1 β 2 ββ √ 1 β N −k + √ N−k−1 D−1,N−k−1 D−1,N−k − J0 (D−1,N−k−1 )2 , N−k−1 2N −2k+1 β ββ (6.36) 1 1 ≡ D−2,2N −2k−1 − N −k−1 (D−1,N −k−1 )2 . (6.37) 2β N −k 2β

– 18 –

JHEP09(2014)028

• For N − k > 1,

• For N − k = 1, (3) W2 (2)

W2

√ 1 2 1 β ≡ − √ D−2,2 + √ D−1,0 D−1,1 + √ J0 D−2,1 − J0 (D−1,0 )2 , (6.38) 3 2 ββ 3 β 3 β 1 = L2 ≡ D−2,1 . (6.39) 2β

Combining with (5.2), (5.14)and (5.15), the above equations lead straightforwardly to (6.2), (6.6) and (6.7) in the beginning of this section.

Conclusion

Inspired by AGT conjecture, we construct Gaiotto states with fundamental multiplets in SU(N ) gauge theories by splitting the corresponding Nekrasov partition function in a proper way, and prove that they satisfy the requirements of Whittaker vectors. We make use of a useful algebra SH. Though SH is complicated in form, it has nice properties when acts on the Hilbert space. Also by clarifying its relation with Wn algebra, we are able to obtain the eigenvalues of higher spin Wn generators for general SU(N ) case, extending the current methods limited to SU(3). For the future work we will construct Gaiotto states for linear quiver theory, and compare with another type of Gaiotto state arising from the colliding limit [24, 25]. In this way, it would be interesting to find the explicit connection between this result and the coherent state approach found in [26, 27]. As another application of SH we complete the discussion of Virasoro constraint for Nekrasov partition function’s recursion relation, by calculating the L±2 constraints directly. Combined with the J±1 and L±1 constraints showed in [17, 18], this non-trivial relation gives a strong support for SU(N ) AGT conjecture of linear quiver type. Especially for SU(2) case, Virasoro constraint is enough to serve as a proof of AGT conjecture. An interesting extension to W algebra constraint is now made more accessible since we can easily write down the explicit relation between SH and Wn algebra.

Acknowledgments YM thanks Hiroshi Itoyama, Hiroaki Kanno and Yasuhiko Yamada for the discussion on DAHA and Gaiotto states. YM is supported in part by KAKENHI (#25400246). HZ thanks the former members of the particle physics group in Chuo University for helpful discussions, and owes special thanks to Takeo Inami for his instructions and kind support. This work is partially supported by the National Research Foundation of Korea (NRF) (NRF-2013K1A3A1A39073412) (CR), and (NRF-2014R1A2A2A01004951) (CR and HZ).

A

Derivation of L±2 constraints on the bifundamental multiplets

In this appendix, we derive a proof of Ward identities for L±2 which was not given in [17, 18]. While this is extremely technical, it is important to show the Nekrasov partition function for the bifundamental matter has the invariance with respect to Virasoro generators Ln . This section in general follows the same construction as [17, 18].

– 19 –

JHEP09(2014)028

7

The instanton partition function for linear quiver gauge theories is decomposed into matrix like product with a factor ZY~ ,W ~ which depends on two sets of Young diagrams. ~ = (Y1 , · · · , YN ) represent the fixed points of U(N ) instanton Here the Young diagrams Y moduli space under localization. ZY~ ,W ~ consists of contributions from one bifundamental hypermultiplet and vectormultiplets. We find that the building block ZY~ ,W ~ satisfies an infinite series of recursion relations, δ±m,n ZY~ ,W ~ − U±m,n ZY ~ ,W ~ = 0,

(A.1)

~ ; ~b, W ~ ; µ) = h~a + ν~e, Y ~ |V (1)|~b + (ξ + ν + µ)~e, W ~ i, Z(~a, Y A.1

(A.2)

Modified vertex operator for U(1) factor

The free boson field which describes the U(1) part is given by the operators Jn defined in the previous section. We modify the vertex operator V˜ H for the U(1) factor as, VκH (z) = e φ+

√1 N

(N Q−κ)φ−

−1 √

κφ

e N +, ∞ X αn −n = α0 log z − z , n n=1

(A.3) φ− = q +

∞ X n=1

α−n n z . n

(A.4)

The general commutator [Ln , Vκ (z)] is given in [17, 18], here we write the special cases n = ±2 for the convenience of later calculation. √ √ 3(N Q − κ)2 2 [L2 , Vκ (z)] = z 3 ∂z Vκ (z) + z Vκ (z) + N Qz 2 Vκ (z)α0 + N QzVκ (z)α1 2N √ + N QVκ (z)α2 + 3z 2 ∆W Vκ (z) , (A.5) [L−2 , Vκ (z)] = z −1 ∂z Vκ (z) −

√ κ2 −2 z Vκ (z) − N Qz −1 α−1 Vκ (z) − z −2 ∆W Vκ (z) . (A.6) 2N

−1)) κ where ∆W = κ(κ−Q(N − 2N is the conformal dimension of WN vertex operator VκW 2 with Toda momenta p~ = −κ(~eN − N~e ). 2

A.2

Ward identities for J±1 and L±1

These analysis have already been performed in [17, 18], and we obtained the following: • The Ward identity for J1 is proved since it is identified with the recursion formula δ−1,0 ZY~ ,W ~ − U−1,0 ZY ~ ,W ~ = 0. It shows the equivalence between the recursion formula δ1,0 ZY~ ,W ~ − U1,0 ZY ~ ,W ~ = 0 and the Ward identity for J−1 .

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JHEP09(2014)028

where δ±m,n ZY~ ,W ~ represents a sum of the Nekrasov partition function with instanton number larger or less than ZY~ ,W ~ by m with appropriate coefficients, and U±m,n are polynomials of parameters such as the mass of bifundamental matter or the VEV of gauge multilets. The subscript m takes arbitrary integer values and n takes any non-negative integer values. We observe that AGT conjecture can be proved once we prove the relation

The Ward identity for L1 is reduced to the recursion relation δ−1,1 ZY~ ,W ~ −U−1,1 ZY ~ ,W ~ = 0. In the same way, for L−1 , the recursion formula δ1,1 ZY~ ,W ~ − U1,1 ZY ~ ,W ~ = 0 can be identified with the Ward identity. These consistency conditions are highly nontrivial and strongly suggest that the identify (A.1) are a part of the Ward identities for the extended conformal symmetry. A.3

Ward identities for L±2

The action of the commutator on the basis reads, ~ | 1 [D−1,0 , D−1,2 ] h~a +ν~e, Y β N fp 1 XX ~ (k,+2H),p |β(2ap +2ν +2Ak (Yp )+β)Λ(k,+2H) (Y ~) = h~a +ν~e, Y p β p=1 k=1

~ (k,+2V ),p |(2ap +2ν +2Ak (Yp )−1)Λ(k,+2V ) (Y ~) −h~a +ν~e, Y p N fp +1 −1 X X ~ (k,+;u,+),p |Λ(k,+) ~ )Λ(u,+) ~ (k,+),p ) h~a +ν~e, Y (Y (Y + p p β p=1 u
+

N fp +1 −1 X X ~ (k,+;u,+),p |Λ(u,+) (Y ~ )Λ(k+1,+) (Y ~ (u,+),p ) h~a +ν~e, Y p p β p=1 u
(A.8)

1 ~i [D−1,0 , D−1,2 ]|~b+(ξ +ν +µ)~e, W β =

N fq 1 XX ~ )|~b+(ξ +ν +µ)~e, W ~ (`,−2H),q i β(2bq +2ν +2µ+2B` (Wq )+2ξ −β)Λ(`,−2H) (W q β q=1 `=1

) ~ ~ ~ (`,−2V ),q i −(2bq +2ν +2µ+2B` (Wq )+2ξ +1)Λ(`,−2V (W )|b+(ξ +ν +µ)~e, W q

−

N fq
1 XX (Bu (Wq )−B` (Wq ))(2bq +2ν +2µ+Bu (Wq )+B` (Wq )) β q=1 u<`

~ )Λ(u,−) (W ~ (`,−),q )|~b+(ξ +ν +µ)~e, W ~ (`,−;u,−),q i ·Λ(`,−) (W q q −

N fq
1 XX B` (Wq )−(Bu (Wq ))(2bq +2ν +2µ+Bu (Wq )+B` (Wq )) β q=1 u<`

~ )Λ(`+1,−) ~ (u,−),q )|~b+(ξ +ν +µ)~e, W ~ (`,−;u,−),q i. ·Λ(u,−) (W (W q q

– 21 –

(A.9)

JHEP09(2014)028

Our goal is to show the recursion formula δ±2,1 ZY~ ,W ~ − U±2,1 ZY ~ ,W ~ = 0. From the definition of Ln (4.9), √ (− β)−2 Nξ 1 1 (A.7) L2 = D−2,1 − J2 = [D−1,0 , D−1,2 ] − N ξ[D−1,0 , D−1,1 ]. 2 2 2β 2β

In the two above equations, we have used the relation (5.6) and (5.7), and ~) Λ(`,−2H) (W q ( =

˜

fp+1 N 2 Y Y (bq −bp +Bl (Wq )−Ak (Wp )−ξ)(bq −bp +Bl (Wq )−Ak (Wp )−ξ −β) β +1 (bq −bp +Bl (Wq )−Ak (Wp ))(bq −bp +Bl (Wq )−Ak (Wp )−β) p=1

k=1

Y0f˜p (bq −bp +Bl (Wq )−Bk (Wp )+ξ)(bq −bp +Bl (Wq )−Bk (Wp )+ξ −β) k=1 (bq −bp +Bl (Wq )−Bk (Wp ))(bq −bp +Bl (Wq )−Bk (Wp )−β)

!)1/2 (A.10)

p=1

k=1

Y0f˜p (bq −bp +Bl (Wq )−Bk (Wp )+ξ)(bq −bp +Bl (Wq )−Bk (Wp )+ξ +1) k=1 (bq −bp +Bl (Wq )−Bk (Wp ))(bq −bp +Bl (Wq )−Bk (Wp )+1)

!)1/2 . (A.11)

~) Λ(k,+2H),p (Y ( fq N 2 Y Y (ap −aq +Ak (Yp )−B` (Yq )+ξ)(ap −aq +Ak (Yp )−B` (Yq )+ξ +β) = β +1 (ap −aq +Ak (Yp )−B` (Yq ))(ap −aq +Ak (Yp )−B` (Yq )+β) q=1

`=1

Y0f˜q (ap −aq +Ak (Yp )−A` (Yq )−ξ)(ap −aq +Ak (Yp )−A` (Yq )−ξ +β) `=1 (ap −aq +Ak (Yp )−A` (Yq ))(ap −aq +Ak (Yp )−A` (Yq )+β)

!)1/2 (A.12)

~) Λ(k,+2V ),p (Y ( f˜p+1 N 2β Y Y (ap −aq +Ak (Yp )−B` (Yq )+ξ)(ap −aq +Ak (Yp )−B` (Yq )+ξ −1) = β +1 (ap −aq +Ak (Yp )−B` (Yq ))(ap −aq +Ak (Yp )−B` (Yq )−1) p=1

k=1

Y0f˜p (ap −aq +Ak (Yp )−A` (Yq )−ξ)(ap −aq +Ak (Yp )−A` (Yq )−ξ −1) k=1 (ap −aq +Ak (Yp )−A` (Yq ))(ap −aq +Ak (Yp )−A` (Yq )−1)

!)1/2 .

(A.13)

For u < k, ~ (k,+),p ) Λ(u,+) (Y p ~ )× = Λ(u,+) (Y p

Au (Yp )−Ak (Yp )+ξ Au (Yp )−Ak (Yp )+β Au (Yp )−Ak (Yp )−1 Au (Yp )−Ak (Yp ) × × × Au (Yp )−Ak (Yp ) Au (Yp )−Ak (Yp )+1 Au (Yp )−Ak (Yp )−β Au (Yp )−Ak (Yp )−ξ

(A.14) ~ (u,+),p ) Λ(k+1,+) (Y p ~ )× Ak (Yp )−Au (Yp )+ξ × Ak (Yp )−Au (Yp )+β × Ak (Yp )−Au (Yp ) − 1 × Ak (Yp )−Au (Yp ) . = Λ(k,+) (Y p Ak (Yp )−Au (Yp ) Ak (Yp )−Au (Yp )+1 Ak (Yp )−Au (Yp )−β Ak (Yp )−Au (Yp )−ξ

(A.15)

– 22 –

JHEP09(2014)028

~) Λ(`,−2V ),q (W ( f˜p+1 N 2β Y Y (bq −bp +Bl (Wq )−Ak (Wp )−ξ)(bq −bp +Bl (Wq )−Ak (Wp )−ξ +1) = β +1 (bq −bp +Bl (Wq )−Ak (Wp ))(bq −bp +Bl (Wq )−Ak (Wp )+1)

For convenience, we set (this convention is only used in this appendix, different from (5.9)) ( xI =

1≤I≤N

{bp + ν + µ + Bk (Wp )} (

yI =

{ap + ν + Ak (Yp )}

N +1≤I ≤N +M

{ap + ν + Bk (Yp ) − ξ}

1≤I ≤N −N .

{bp + ν + µ + Ak (Wp ) + ξ}

N −N +1≤I ≤N +M

~ |L2 Vκ (1)|~b+(ξ +ν +µ)~e, W ~ i h~a +ν~e, Y ~ |Vκ (1)L2 |~b+(ξ +ν +µ)~e, W ~i h~a +ν~e, Y − ~ |Vκ (1)|~b+(ξ +ν +µ)~e, W ~i ~ |Vκ (1)|~b+(ξ +ν +µ)~e, W ~i h~a +ν~e, Y h~a +ν~e, Y +

~ |[J2 , Vκ (1)]|~b+(ξ +ν +µ)~e, W ~i N ξ h~a +ν~e, Y ~ |Vκ (1)|~b+(ξ +ν +µ)~e, W ~i 2 h~a +ν~e, Y

−

p ~ |Vκ (1)J1 |~b+(ξ +ν +µ)~e, W ~i p ~ |Vκ (1)J2 |~b+(ξ +ν +µ)~e, W ~i h~a +ν~e, Y h~a +ν~e, Y βQ − βQ ~ ~ ~ ~ ~ ~ h~a +ν~e, Y |Vκ (1)|b+(ξ +ν +µ)~e, W i h~a +ν~e, Y |Vκ (1)|b+(ξ +ν +µ)~e, W i

=−

QN+M N QN+M X 1 J=1 (xI −yJ ) J=1 (xI −yJ −1) ×(2xI −1) QN+ Q N+M 2β(β +1) I=1 J6=M I (xI −xJ ) J6=I (xI −xJ −1)

−

QN+M N+ M QN+M X 1 J=1 (xI −yJ ) J=1 (xI −yJ −β) ×(2xI −β) Q M QN+ M 2β(β +1) I=N+1 N+ (x −x ) I J J6=I J6=I (xI −xJ −β)

+

QN+M N QN+M X 1 J=1 (xI −yJ ) J=1 (xI −yJ +β) ×(2xI +β) QN+ Q M N+ 2β(β +1) I=1 J6=I (xI −xJ ) J6=M I (xI −xJ +β)

+

QN+M N+ M QN+M X 1 J=1 (xI −yJ ) J=1 (xI −yJ +1) ×(2xI +1) Q M QN+ M 2β(β +1) I=N+1 N+ (x −x ) I J J6=I J6=I (xI −xJ +1)

+

N QN+M N QN+M 1 X J=1 (xI −yJ ) X J=1 (xK −yJ ) (xK −xI )2 (xK −xI +1−β) × ×(xK +xI ) Q QN+M M 4β 2 I=1 N+ (xK −xI +1)(xK −xI −β) J6=I (xI −xJ ) J6=K (xK −xJ ) K6=I

−

N QN+M N QN+M 1 X J=1 (xI −yJ ) X J=1 (xK −yJ ) (xK −xI )2 (xK −xI −1+β) × ×(xK +xI ) Q QN+M M 4β 2 I=1 N+ (xK −xI −1)(xK −xI +β) J6=I (xI −xJ ) J6=K (xK −xJ ) K6=I

N+M QN+M (xI −yJ ) 1 X − Q J=1M 4β 2 I=N+1 N+ (x I −xJ ) J6=I

+

−

N+M QN+M (xI −yJ ) 1 X Q J=1M 4β 2 I=N+1 N+ (x I −xJ ) J6=I

N+ M X K=N+1,K6=I N+ M X K=N+1,K6=I

QN+M

J=1 QN+ M J6=K

(xK −yJ ) (xK −xJ )

×

(xK −xI )2 (xK −xI +1−β) ×(xK +xI ) (xK −xI +1)(xK −xI −β)

QN+M

(xK − yJ ) (xK −xI )2 (xK −xI −1+β) × ×(xK +xI ) QJ=1 N+M (xK −xI −1)(xK −xI +β) J6=K (xK −xJ )

N+M QN+M (xI − yJ ) 1−β X . Q J=1M β I=N+1 N+ (x I − xJ ) J6=I

(A.16)

– 23 –

JHEP09(2014)028

Like the L1 case performed in [17, 18], anomalous terms arise both from the the action on the ket basis and the modified vertex operator, and again cancels with each √ other: 2ξ terms exactly cancel with the contribution of N QVκ (z)α2 , and the rest has the following form

Using some tricks like redefining x0I = x1 , x2 , . . . , xI−1 , xI+1 , . . . , xN +M plus xI − 1, xI + β, and yI0 = y1 , y2 , . . . , yN +M plus xI − 1 + β, the above can be evaluated by (5.11), and finally reduces to p

β

~ ; µ) p −1 N ξ δ−2,0 Z(~a, Y ~ ; ~b, W ~ ; µ) ; ~b, W + β . ~ ; ~b, W ~ ; µ) ~ ; ~b, W ~ ; µ) 2 Z(~a, Y Z(~a, Y

~ −1 δ−2,1 Z(~ a, Y

On the other hand, the commutator part becomes

+ =

p

−1 N ξ

β

2

~ ; ~b, W ~ ; µ) U−2,0 Z(~a, Y

p −1 p ~ ; ~b, W ~ ; µ)+ β−1 N ξ U−2,0 Z(~a, Y ~ ; ~b, W ~ ; µ) β U−2,1 Z(~a, Y 2 p p ~ |Vκ (1)J1 |~b+(ξ +ν +µ)~e, W ~ i+ βQh~a +ν~e, Y ~ |Vκ (1)J2 |~b+(ξ +ν +µ)~e, W ~ i. + βQh~a +ν~e, Y

(A.17) Compare the above two equations, the Ward identity for L2 is obtained since it is identified with the recursion formula δ−2,1 ZY~ ,W ~ − U−2,1 ZY ~ ,W ~ = 0. L−2 totally follows the same discussion. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References [1] N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE]. [2] N.A. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE]. [3] H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE]. [4] L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE]. [5] N. Wyllard, AN −1 conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].

– 24 –

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~ |[L2 , Vκ (1)]|~b+(ξ +ν +µ)~e, W ~ i+ N ξ h~a +ν~e, Y ~ |[J2 , Vκ (1)]|~b+(ξ +ν +µ)~e, W ~i h~a +ν~e, Y 2 (  !  ~b+(ν +µ)~e ~a +ν~e N +1 N +1 ~ |−∆ − ~| √ = ∆ − √ −Q~ ρ +Q ~e +|Y −Q~ ρ +Q ~e −|W 2 2 β β ) (N Q−κ)2 κ2 ~ ; ~b, W ~ ; µ) + +κ(κ−Q(N −1))− Z(~a, Y N N p p ~ |Vκ (1)J1 |~b+(ξ +ν +µ)~e, W ~ i+ βQh~a +ν~e, Y ~ |Vκ (1)J2 |~b+(ξ +ν +µ)~e, W ~i + βQh~a +ν~e, Y

[6] A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE]. [7] H. Awata, B. Feigin, A. Hoshino, M. Kanai, J.’i. Shiraishi and S. Yanagida, Notes on Ding-Iohara algebra and AGT conjecture, arXiv:1106.4088 [INSPIRE]. [8] A. Morozov and A. Smirnov, Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials, Lett. Math. Phys. 104 (2014) 585 [arXiv:1307.2576] [INSPIRE].

[10] V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE]. [11] V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE]. [12] I. Cherednik, Double affine Hecke algebras, London Mathematical Society Lecture Note Series (Book 319), Cambridge University Press, Cambridge (2005). [13] D. Bernard, M. Gaudin, F.D.M. Haldane and V. Pasquier, Yang-Baxter equation in long range interacting system, J. Phys. A 26 (1993) 5219 [INSPIRE]. [14] D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE]. [15] A. Smirnov, On the Instanton R-matrix, arXiv:1302.0799 [INSPIRE]. [16] O. Schiffmann and E. Vasserot, Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A2 , arXiv:1202.2756. [17] S. Kanno, Y. Matsuo and H. Zhang, Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function, JHEP 08 (2013) 028 [arXiv:1306.1523] [INSPIRE]. [18] S. Kanno, Y. Matsuo and H. Zhang, Virasoro constraint for Nekrasov instanton partition function, JHEP 10 (2012) 097 [arXiv:1207.5658] [INSPIRE]. [19] D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [INSPIRE]. [20] A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [INSPIRE]. [21] E. Felinska, Z. Jaskolski and M. Kosztolowicz, Whittaker pairs for the Virasoro algebra and the Gaiotto-Bonelli-Maruyoshi-Tanzini states, J. Math. Phys. 53 (2012) 033504 [Erratum ibid. 53 (2012) 129902] [arXiv:1112.4453] [INSPIRE]. [22] H. Kanno and M. Taki, Generalized Whittaker states for instanton counting with fundamental hypermultiplets, JHEP 05 (2012) 052 [arXiv:1203.1427] [INSPIRE]. [23] G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [arXiv:1112.1691] [INSPIRE]. [24] D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].

– 25 –

JHEP09(2014)028

[9] H. Itoyama, T. Oota and R. Yoshioka, 2d–4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space, Nucl. Phys. B 877 (2013) 506 [arXiv:1308.2068] [INSPIRE].

[25] H. Kanno, K. Maruyoshi, S. Shiba and M. Taki, W3 irregular states and isolated N = 2 superconformal field theories, JHEP 03 (2013) 147 [arXiv:1301.0721] [INSPIRE]. [26] T. Nishinaka and C. Rim, Matrix models for irregular conformal blocks and Argyres-Douglas theories, JHEP 10 (2012) 138 [arXiv:1207.4480] [INSPIRE]. [27] S.-K. Choi and C. Rim, Parametric dependence of irregular conformal block, JHEP 04 (2014) 106 [arXiv:1312.5535] [INSPIRE]. [28] H. Awata, Y. Matsuo, S. Odake and J.’i. Shiraishi, Collective field theory, Calogero-Sutherland model and generalized matrix models, Phys. Lett. B 347 (1995) 49 [hep-th/9411053] [INSPIRE].

[30] H. Awata, Y. Matsuo, S. Odake and J.’i. Shiraishi, Excited states of Calogero-Sutherland model and singular vectors of the WN algebra, Nucl. Phys. B 449 (1995) 347 [hep-th/9503043] [INSPIRE].

– 26 –

JHEP09(2014)028

[29] K. Mimachi and Y. Yamada, Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomial, Commun. Math. Phys. 174 (1995) 447.