Notes on 3-point functions of A_{N-1} Toda theory and AGT-W relation for SU(N) quiver
aa r X i v : . [ h e p - t h ] D ec KEK-TH-1506
Notes on 3-point functions of A N − Toda theoryand AGT-W relation for SU ( N ) quiver Shotaro Shiba
Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization (KEK),1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan [email protected]
Abstract
We study on the property of 3-point correlation functions of 2-dim A N − Toda fieldtheory, and show the correspondence with the 1-loop part of partition function of 4-dim N = 2 SU ( N ) quiver gauge theory. As a result, we can check successfully the 1-loop partof AGT-W relation for all the cases of SU ( N ) quiver gauge group. Introduction
Recently, based on Gaiotto’s discussion on N = 2 dualities [1], the relation between 4-dim N = 2 gauge theories and the quantum geometry of 2-dim Riemann surface has becomeunderstood more clearly. One of the most remarkable progress must be the proposition ofAGT relation [2], which states that the partition function of 4-dim N = 2 SU (2) linearquiver gauge theory corresponds to the correlation function of 2-dim Liouville field theory.As the natural generalization of this relation, the correspondence between N = 2 SU ( N )linear quiver gauge theory and A N − Toda field theory has been also proposed, which iscalled AGT-W relation [3].The correspondence between the parameters of gauge theory and those of Toda theoryin AGT-W relation has been already proposed for a general case of SU ( N ) quiver [4, 5],but the proof is still incomplete. Up to now, the proof by direct calculations has been donein the following cases: For SU (2) linear quivers, the correspondence has been checked for SU (2) n quiver with n = 1 , , SU (3) linear quivers, it hasbeen checked for SU (3) n quiver with n = 1 , SU (3) × SU (2) quiver up to instantonlevel 3 [5, 7, 8]. For SU ( N ) linear quivers with N >
3, only the 1-loop part for SU ( N ) × SU ( N − × · · · × SU (2) quiver has been discussed [5].On the conformal blocks in Toda theory which correspond to the instanton part of par-tition function in gauge theory, the discussion and calculation have been developed [9–30].For example, by using the newly proposed basis with Young tableau indices [9] which is akind of generalization of Jack polynomials, we are getting to understand the reason why thefactorized form of instanton partition function can be reproduced in Toda theory. Now manyresearches restrict themselves to some limited cases, but they are very useful to deepen ourunderstanding of the mechanism of AGT-W relation.In this paper, on the other hand, we concentrate on the 1-loop part of partition functionin gauge theory. The corresponding part of correlation function in Toda theory is reducedto the product of 3-point correlation functions. In fact, we had the following problem in ourprevious paper [8]: when the two of three fields in a 3-point function are degenerate ones,some factors become zero and then make zeros and poles. At that time, we could not findhow to deal with, so we simply neglected them. In this paper, we reconsider carefully on thisproblem. Then we grasp the mechanism of cancellations of undesirable factors including thezeros, and find the physical interpretation of the poles. As a result, we can check successfullythe 1-loop part of AGT-W relation for all the cases of SU ( N ) quiver gauge group.This paper is organized as follows. In § § A N − Toda field theory. Then in § §
3, we check AGT-W relation by directcalculations: We first summarize our ansatz in § A and A Toda theory in § § § A N − Toda theory in an algorithmic way.1 A N − Toda theory
AGT-W relation is the nontrivial correspondence between the partition function of 4-dim N = 2 SU ( N ) quiver gauge theory and the correlation function of 2-dim A N − Toda theory.In general, we consider the quiver gauge theory with a chain of n SU groups SU ( d ) × SU ( d ) × · · · × SU ( d n − ) × SU ( d n ) . (2.1)Here we require that the theory should be conformal in the massless limit of matter fields byintroducing additional hypermultiplets. Since the number of these hypermultiplets must benon-negative, this requirement means that k a = ( d a − d a +1 ) − ( d a − − d a ) ≥ ∀ a = 1 , · · · , n − , (2.2)then the ranks d a satisfy d ≤ d ≤ · · · ≤ d l − ≤ d l = · · · = d r ≥ d r +1 ≥ · · · ≥ d n − ≥ d n . (2.3)For simplicity, in this paper, we concentrate on only the part of decreasing tail N := d ≥ d ≥ · · · ≥ d n − ≥ d n . (2.4)The generalization to the original case (2.3) is almost straightforward.In order to check AGT-W relation, we must calculate the partition function of gaugetheory with quiver gauge group (2.1), which is summarized in Appendix A, and the corre-sponding ( n + 3)-point correlation function of Toda theory: ~β ∞ ~β n +1 ~β ~β n − ~β n ~β ~α ~α n · · · (2.5)Here we believe that the momenta ~α j , ~β k of Toda vertex operators correspond to the quivergauge group (2.1) by following our ansatz [4]. The details will be reviewed in § i.e. the decomposition into3-point functions and propagators. The 3-point functions are given in terms of Υ-function, aswe review in this section. The propagators are given as the inverse Shapovalov matrices. It isknown that this matrix is infinite size, but is block diagonal with respect to each descendantlevel, and each block is finite size.In this paper, we concentrate on only the propagators with descendant level 0. Then thecorrelation function (2.5) is reduced to the product of ( n + 1) 3-point functions, as we willsee in eq. (3.2). According to the proposition of AGT-W relation, this product of 3-pointfunctions should correspond to the 1-loop part of partition function of gauge theory. Inthis section, before checking this correspondence, we briefly review the derivation of 3-pointfunction and discuss its property. 2 .1 A N − Toda theory
First we summarize our notation for 2-dim A N − Toda field theory. The action is S = Z d σ √ g " π g xy ∂ x ~ϕ · ∂ y ~ϕ + µ N − X k =1 e b~e k · ~ϕ + Q π R~ρ · ~ϕ (2.6)where ~ϕ = ( ϕ , · · · , ϕ N ) is Toda field, satisfying P Np =1 ϕ p = 0. g xy is the metric on 2-dimRiemann surface, and R is its curvature. ~e k is the k -th simple root defined as ~e k = (0 , · · · , , , − , , · · · ,
0) (2.7)where 1 is k -th element. ~ρ is Weyl vector ( i.e. half the sum of all positive roots) of A N − algebra: ~ρ = 12 ( N − , N − , · · · , − N, − N ) . (2.8) µ is a scale parameter called the cosmological constant, b is a real parameter called thecoupling constant, and Q := b + b − . The central charge of this conformal field theory is c = ( N − (cid:0) N ( N + 1) Q (cid:1) .The primary field, or the vertex operator, is defined as V ~α ( z ) := e ~α · ~ϕ ( z ) (2.9)where ~α = ( α , · · · , α N ) is called the momentum, satisfying P Np =1 α p = 0. The 2-pointcorrelation function (propagator) of vertex operators is normalized in the usual manner: h V ~α ( z ) V Q~ρ − ~α ( z ) i = 1 | z | ~α (2.10)where ∆ ~α is the conformal dimension of V ~α , and z := z − z . The 3-point correlationfunction must have standard coordinate dependence due to the conformal invariance: h V ~α ( z ) V ~α ( z ) V ~α ( z ) i = C ( ~α , ~α , ~α ) | z | +∆ − ∆ ) | z | +∆ − ∆ ) | z | +∆ − ∆ ) (2.11)where ∆ i := ∆ ~α i . It is known that the function C ( ~α , ~α , ~α ) becomes perturbatively nonzero,only when the screening condition is satisfied [31]: ~α + ~α + ~α + N − X k =1 bs k ~e k = 2 Q~ρ (2.12)where s k are non-negative integers, satisfying s ≤ s ≤ · · · ≤ s N − . In this case, the function C ( ~α , ~α , ~α ) has simple poles at each of the variables(2 Q~ρ − X i =1 ~α i , ~ω k ) = bs k (2.13)where ~ω k ’s are the dual basis to the simple roots, i.e. ~e i · ~ω j = δ ij . For example, ~ω = 1 N ( N − , − , · · · , − , ~ω N − = 1 N (1 , · · · , , − N ) . (2.14)3 .2 Derivation of 3-point function Now we briefly review the derivation of 3-point function. As far as the author knows, the twoways of derivation have been discussed. One way is to use the recurrent relation for Coulombintegrals [31], and the other way is to use the differential equation for 4-point functions [32].
Derivation 1 : by recurrent relation for Coulomb integral
We note that the main residue of the function C ( ~α , ~α , ~α ) at the poles (2.13) can be writtenin terms of Coulomb integral I as N − O k =1 res (2 Q~ρ − P ~α i ) · ~ω k = bs k C ( ~α , ~α , ~α ) = ( − πµ ) s + ··· + s N − D V ~α ( ∞ ) V ~α (1) V ~α (0) N − Y k =1 Q s k k E =: ( − πµ ) s + ··· + s N − I s ··· s N − ( ~α , ~α , ~α ) (2.15)where Q k = R d z e b~e k · ~ϕ is a screening charge. As we will see in § ~α = κ ~ω or κ ~ω N − ( κ ∈ C ). These two momenta are related by the conjugation,and here we choose ~α = κ ~ω N − . Then the Coulomb integral in (2.15) becomes I s ··· s N − ( ~α , ~α , κ ~ω N − ) (2.16)= Z N − Y k =1 dµ s k ( t k ) D − b s k ( t k ) s k Y j =1 (cid:12)(cid:12) t ( j ) N − (cid:12)(cid:12) − b κ (cid:12)(cid:12) t ( j ) k − (cid:12)(cid:12) − b~α · ~e k N − Y l =1 s l Y i =1 s l +1 Y i ′ =1 (cid:12)(cid:12) t ( i ) l − t ( i ′ ) l +1 (cid:12)(cid:12) b where t ( j ) k is the coordinate of the j -th screening field e b~e k · ~ϕ , and we define dµ s k ( t k ) = 1 π s k s k ! s k Y j =1 d t ( j ) k , D s k ( t k ) = s k Y i 2) = 1. Then by using eq. (2.15), the formula of3-point function can be finally proposed as [31] C ( ~α , ~α , κ ~ω N − ) = h πµγ ( b ) b − b i (2 Q~ρ − ~α − ~α − κ ~ω N − ) · ~ρ/b × Υ( b ) N − Υ( κ ) Q e> Υ (( Q~ρ − ~α ) · ~e ) Υ (( Q~ρ − ~α ) · ~e ) Q j,k Υ (cid:16) κ /N + ( ~α − Q~ρ ) · ~λ j + ( ~α − Q~ρ ) · ~λ k (cid:17) (2.25)where e > s k . Such analyticcontinuation is often taken in the expression for the 3-point functions, as we will see in § Q~ρ − ~α ) · k X l =1 ~e l = Q for ∀ k = 1 , · · · , N − . (2.26)Then in the cases where this condition is violated, the formula (2.25) needs to be justified byanother way of derivation. 5 erivation 2 : by differential equation for 4-point function The other way of derivation is to use the differential equation [32]. It is known that a certainkind of 4-point function of A N − Toda field theory can be written as (cid:10) V − b~ω ( x ) V ~α (0) V ~α ( ∞ ) V κ ~ω N − (1) (cid:11) = | x | b~α · ~λ | − x | b κ /N G ( x, ¯ x ) (2.27)where no conditions are imposed on momenta ~α , ~α and a complex number κ . The function G ( x ) satisfies the generalized Pochhammer hypergeometric equation " x · N Y i =1 (cid:18) x ∂∂x + A i (cid:19) − N Y i =1 (cid:18) x ∂∂x + B i − (cid:19) · x ∂∂x G ( x, ¯ x ) = 0 (2.28)where A k = b κ N − N − N b + b ( ~α − Q~ρ ) · ~λ + b ( ~α − Q~ρ ) · ~λ k B k = 1 + b ( ~α − Q~ρ ) · ( ~λ − ~λ k +1 ) . (2.29)Since G ( x, ¯ x ) should satisfy the same equation (2.28) with the replacement of x → ¯ x (thecomplex conjugation of x ), we can obtain the integral representation as G ( x, ¯ x ) = Z N − Y i =1 d t i | t i | A i − B i ) | t i − t i +1 | B i − A i +1 − | t − x | − A (2.30)up to an overall constant. Here we set t N = 1.In order to obtain the formula of 3-point function which we are interested in, we take adecomposition for the 4-point function (2.27) by using OPE V − b~ω ( x ) V ~α k (0) = N X j =1 C ~α k − b~λ j − b~ω ,~α k (cid:16) | x | kj V ~α k − b~λ j (0) + · · · (cid:17) (2.31)where ∆ kj := ∆ ~α k − b~λ j − ∆ − b~ω − ∆ ~α k , and ‘ · · · ’ includes the contribution of descendant fields.Then we can rewrite eq. (2.27) as h V − b~ω ( x ) V ~α (0) V ~α ( ∞ ) V κ ~ω N − (1) i = | x | b~α · ~λ | − x | b κ /N N X j =1 C ~α − b~λ j − b~ω ,~α C ( ~α − b~λ j , ~α , κ ~ω N − ) | G j ( x ) | (2.32)where C ( ~α , ~α , ~α ) is a 3-point function defined in eq. (2.11), and G j ( x ) can be expressed interms of the generalized hypergeometric function of type ( N, N − 1) as G ( x ) = F A · · · A N B · · · B N − x ! G k +1 ( x ) = x − B k F A − B k , · · · , A N − B k B − B k , · · · , − B k , · · · , B N − − B k x ! (2.33)6or k = 1 , · · · , N − 1, and where F A · · · A N B · · · B N − x ! = 1 + Q Nj =1 A j Q N − k =1 B k x + Q Nj =1 A j ( A j + 1) Q N − k =1 B k ( B k + 1) x · · · . (2.34)Therefore, by comparing eq. (2.32) with eq. (2.27) using the integral representation (2.30), wecan find the relation C ~α − b~λ − b~ω ,~α C ( ~α − b~λ , ~α , κ ~ω N − ) C ~α − b~λ k − b~ω ,~α C ( ~α − b~λ k , ~α , κ ~ω N − ) = N Y j =1 γ ( A j ) γ ( B k − − A j ) γ ( B j ) γ ( B k − − B j ) (2.35)where we set B = B N = 1. The structure constants C ~α − b~λ k − b~ω ,~α can be calculated explicitly bythe free field representation [33] C ~α − b~λ k − b~ω ,~α = (cid:20) − πµγ ( − b ) (cid:21) k − k − Y i =1 γ ( b ( ~α − Q~ρ ) · ( ~λ i − ~λ k )) γ (1 + b + b ( ~α − Q~ρ ) · ( ~λ i − ~λ k )) , (2.36)so we can properly obtain the ratio C ( ~α − b~λ k , ~α , κ ~ω N − ) /C ( ~α − b~λ , ~α , κ ~ω N − ). Byusing this relation, and by a similar discussion of the dual screening charge in § Results The formula of 3-point function with one of the momenta ~α = κ ~ω N − is C ( ~α , ~α ; κ ~ω N − ) = h πµγ ( b ) b − b i (2 Q~ρ − ~α − ~α − κ ~ω N − ) · ~ρ/b × Υ( b ) N − Υ( κ ) Q e> Υ(( Q~ρ − ~α ) · ~e )Υ(( Q~ρ − ~α ) · ~e ) Q i,j Υ (cid:16) κ /N + ( ~α − Q~ρ ) · ~λ i + ( ~α − Q~ρ ) · ~λ j (cid:17) . (2.37)The formula with one of the momenta ~α = κ ~ω can be obtained by taking the conjugation ~λ k → ~λ ∗ k = − ~λ N +1 − k (then ~ω ∗ N − = ~ω ) as C ( ~α , ~α ; κ ~ω ) = h πµγ ( b ) b − b i (2 Q~ρ − ~α − ~α − κ ~ω ) · ~ρ/b × Υ( b ) N − Υ( κ ) Q e> Υ(( Q~ρ − ~α ) · ~e )Υ(( Q~ρ − ~α ) · ~e ) Q i,j Υ (cid:16) κ /N − ( ~α − Q~ρ ) · ~λ i − ( ~α − Q~ρ ) · ~λ j (cid:17) . (2.38)Here e > i, j = 1 , · · · , N .7 .3 Property of 3-point function Before beginning the proof of AGT-W relation, we observe and discuss some properties of3-point function. As we mentioned in § C ( ~α , ~α , ~α ) has simple poles whereeq. (2.12) is satisfied. In fact, the denominator of 3-point function (2.37) can be written as N Y j , j =1 Υ (cid:16) ( ~α − Q~ρ ) · ~λ j + ( ~α − Q~ρ ) · ~λ j + κ ~ω N − · ~λ j (cid:17) (2.39)for ∀ j = 1 , · · · , N − 1, and the Υ-function has zeros atΥ( x ) = 0 ⇔ x = − mb − nb − , ( m + 1) b + ( n + 1) b − ( m, n ∈ Z ≥ ) . (2.40)Then we can find that all the factors with j = j = j =: j in eq. (2.39) becomeΥ (cid:0) b ( s j − − s j ) + b − (˜ s j − − ˜ s j ) (cid:1) = 0 for ∀ j = 1 , · · · , N − , (2.41)when the screening condition (2.12) is satisfied. Here s k , ˜ s k are integers, and we set s =˜ s = 0. Therefore, in this case, the whole denominator (2.39) can be rewritten as N − Y j =1 Υ (cid:0) b ( s j − − s j ) + b − (˜ s j − − ˜ s j ) (cid:1) × Υ( κ ) × Y j >j ′ Υ (cid:16) ( ~α − Q~ρ ) · ( ~λ j − ~λ j ′ ) (cid:17) Y j >j ′ Υ (cid:16) ( ~α − Q~ρ ) · ( ~λ j − ~λ j ′ ) (cid:17) (2.42)where j , j = 1 , · · · , N and j ′ , j ′ = 1 , · · · , N − 1. Now we can easily find that all thefactors in the second line are canceled by the factors Υ(( Q~ρ − ~α i ) · ~e ) in the numerator, since ~λ j ′ − ~λ j = − P j ′ − k = j ~e k for j ′ > j . As a result, when the screening condition (2.12) is satisfied,the 3-point function (2.37) can be simplified as C ( ~α , ~α ; κ ~ω N − ) = h πµγ ( b ) b − b i (2 Q~ρ − ~α − ~α − κ ~ω N − ) · ~ρ/b (cid:20) Υ( b )Υ(0) (cid:21) N − . (2.43)However, the story becomes a little more complicated in AGT-W relation. Let us nowdiscuss it, since it is very important for the proof of AGT-W relation. Discussion for the proof of AGT-W relation In AGT-W relation, we usually take an analytic continuation of the number of screeningcharges s k , ˜ s k to non-integer values. The formula of 3-point function (2.37) is still valid inthis case, as we mentioned. However, the number and the positions of poles must be changed,since the factors (2.41) generally don’t become zero.If there are no zeros in the denominator, the zeros in the numerator mean that the 3-pointfunction vanishes. In fact, as we will see in § 3. such a situation is realized in the case of8 ⊗⊗⊗ ⊗ ⊗⊗⊗ · · ·· · · ✲✻ x , x ✛ HW transition Figure 1: Brane configuration of D4- (horizontal), NS5- (vertical), D6- ( ⊗ ) branes SU ( N ) n quiver ( i.e. without descending nor ascending tails). The vanishing 3-point functionmust be inadequate to compare with the partition function of gauge theory. Therefore, wemust avoid such a setting of parameters by imposing the conditionΥ(( Q~ρ − ~α i ) · ~e ) = 0 for ∀ i = 1 , , ∀ e > . (2.44)For the quivers with descending or ascending tails, on the other hand, we always set newpoles by imposing the following condition on some factors in the denominator (2.39):Υ (cid:0) υ j ◦ ,j ◦ (cid:1) := Υ (cid:16) ( ~α − Q~ρ ) · ~λ j ◦ + ( ~α − Q~ρ ) · ~λ j ◦ + κ ~ω N − · ~λ j ◦ (cid:17) = 0 (2.45)for ∃ j ◦ , j ◦ = 1 , · · · , N and ∀ j ◦ = 1 , · · · , N − 1. Here we don’t require j ◦ = j ◦ = j ◦ , which isdifferent from the discussion before an analytic continuation. In fact, this condition (2.45) isindispensable to match with the partition function of gauge theory.In order to discuss the physical meaning of the condition (2.45), it seems convenient toconsider the brane configuration [34]. For the quivers without tails, we consider the systemof intersecting D4-branes and NS5-branes. For the quivers with tails, on the other hand,D6-branes are introduced from the infinite distance in the D4/NS5 system. The directionswhere each kind of branes is extended are shown in the following table:0 , , , , , , −− −− NS5-branes −− −− D6-branes −− −− The arrangement of branes is shown in figure 1. Here we consider the situation where eachD6-brane in the infinite distance | x | = ∞ is coupled to an infinitely extended D4-brane.Then the D6-branes are moved to | x | = finite, passing through some of the NS5-branes. Atthis time, they cause Hanany-Witten transition [35] on D4-branes, which makes the numberof D4-branes ( i.e. the rank of gauge group) smaller. Moreover, superstrings ending on aD4-brane and a D6-brane (4-6 strings) appear, and they behave as new fundamental (or9ntifundamental) matter fields in 4-dim gauge theory defined in the x , , , spacetime. Themass of these new fields is determined by the position of D6-branes. Here we note that thecondition (2.45) in Toda theory determines the mass of new fields, as we will see in § 3. Also,the condition (2.45) means that some factors in the denominator of Toda correlation functionare factored out, and then the rank of gauge group of corresponding 4-dim theory becomessmaller. Therefore, it is natural to consider that the setting of new poles (2.45) correspondsto the setting of D6-branes. In fact, as we will see in § 3, the order of these poles in the wholeof Toda correlation function is always equal to the number of times which D6-branes passthrough NS5-branes, or equivalently, Hanany-Witten transition occurs.Next, we pay attention to the factors in the denominator other than the zero factors(2.45), i.e. Υ ( υ j ,j ) with ( j , j ) = ( j ◦ , j ◦ ). As in the previous case ( i.e. s k , ˜ s k are integers),the cancellation of these factors and some factors in the numerator may occur. However, itshould be in some different manner, so let us now list the ways of cancellation. Case 1 For j > j ◦ and j = j ◦ , the factors Υ ( υ j ,j ) are canceled by corresponding factorsin the numerator, since υ j ,j ◦ = ( Q~ρ − ~α ) · j − X k = j ◦ ~e k . (2.46) Case 2 For j < j ◦ and j = j ◦ , if ~α · ~e j ◦ = 0, the factors Υ ( υ j ,j ) are canceled bycorresponding factors in the numerator, since υ j ,j ◦ = Q − ( Q~ρ − ~α ) · j ◦ X k = j ~e k . (2.47)Here we note that Υ( x ) = Υ( Q − x ) for ∀ x ∈ C . Case 3 For j < j ◦ and j = j ◦ , if ~α is a momentum of a propagator ( i.e. internal line),the factors Υ ( υ j ,j ) are canceled by corresponding factors in the numerator, since υ j ,j ◦ = (cid:0) Q~ρ − (2 Q~ρ − ~α ) (cid:1) · j ◦ − X k = j ~e k . (2.48)Here we note that a propagator momentum ~α appears in two 3-point functions, like C ( ∗ , ~α ; ∗ ) and C (2 Q~ρ − ~α, ∗ ; ∗ ), as we will see in eq. (3.2). Case 4 For j < j ◦ and j = j ◦ + 1, if ~α · ~e j ◦ = 0, the factors Υ ( υ j ,j ) are canceled bycorresponding factors in the numerator, since υ j ,j ◦ +1 = Q − ( Q~ρ − ~α ) · j ◦ − X k = j ~e k . (2.49)10here are other ways of cancellation, but we will consider only these four cases in thefollowing discussion. We note that all these discussions are still valid, if we interchange theindices 1 ↔ 2, of course. By putting all results together, we finally find that the 3-pointfunction (2.37) appearing in AGT-W relation becomes C ( ~α , ~α ; κ ~ω N − ) = h πµγ ( b ) b − b i (2 Q~ρ − ~α − ~α − κ ~ω N − ) · ~ρ/b (2.50) × Υ( b ) N − Υ( κ ) Q ′ e , > Υ (( Q~ρ − ~α ) · ~e ) Υ (( Q~ρ − ~α ) · ~e ) Q j , = j ◦ , Υ (cid:16) κ /N + ( ~α − Q~ρ ) · ~λ j + ( ~α − Q~ρ ) · ~λ j (cid:17) after the cancellation discussed above. Here, Q j , = j ◦ , in the denominator means the productof only the elements with { j , j | ( j = j ◦ , j ◦ + 1) ∩ ( j = j ◦ , j ◦ + 1) } (2.51)for all the sets { j ◦ , j ◦ } satisfying the condition (2.45). And, Q ′ e , > in the numerator meansthe product of only the elements with ~e = P j ◦ k = j ~e k or P j ◦ − k = j ~e k for j = 1 , · · · , j ◦ − P j − k = j ◦ ~e k for j = j ◦ + 1 , · · · , N . (2.52)For the former case, it depends on the way of cancellation (case 2 – 4). For ~e , we shouldreplace all the indices 1 with 2 in eq. (2.52). Based on the discussions in the previous section, now we begin the check of AGT-W relationby direct calculations. Let us now see the diagram (2.5) of the whole correlation function again: ~β ∞ ~β n +1 ~β ~β n − ~β n ~β ~α ~α n · · · (3.1)In this paper, we consider only the correlation function with descendant level 0: V ∅ ( ~β ∞ , ~β n +1 , ~β , · · · , ~β n , ~β ; ~α , · · · , ~α n ) (3.2)= C ( ~β ∞ , ~α ; ~β n +1 ) C (2 Q~ρ − ~α , ~α ; ~β ) · · · C (2 Q~ρ − ~α n − , ~α n ; ~β n − ) C (2 Q~ρ − ~α n , ~β ; ~β n )where the momenta of vertex operators of 2-dim Toda field theory correspond to the quivergauge group (2.1) with the condition (2.4) of 4-dim N = 2 quiver gauge theory as follows:11 ~β ∞ = Q~ρ + i~β ′∞ : This corresponds to a ‘full’ puncture. • ~β k = (cid:18) Q im k (cid:19) N ~ω N − or (cid:18) Q im k (cid:19) N ~ω (for k = 1 , · · · , n + 1) :They correspond to ‘simple’ punctures [3]. We choose the former in this paper. • ~β : This corresponds to a puncture classified by a Young tableau corresponding to thewhole of quiver gauge group SU ( d ) × · · · × SU ( d n ) [4]. • ~α j (for j = 1 , · · · , n ) : They correspond to propagators, but in the weak coupling limitof SU ( d j ), the diagram (3.1) is decomposed into two diagrams as below [1]. This meansthat ~α j should correspond to a puncture with a Young tableau for a quiver gauge group SU ( d ) × · · · × SU ( d j − ) [5]. ~β ∞ ~β n +1 ~β ~β j − ~β j ~β n − ~β n ~α j ~β Q~ρ − ~α j ~α ~α n · · · · · · N (3.3)Here the parameters ~β ′∞ and m k are real, and the following conditions are satisfied: N X p =1 β ′∞ ,p = N X p =1 β ,p = N X p =1 α j,p = 0 . (3.4)There are no more conditions for ~β ∞ and ~β k , while ~β and ~α j must satisfy some additionalconditions. Then let us now explain how to determine their concrete forms. Our ansatz for ~β According to Gaiotto’s discussion [1], a puncture corresponding ~β can be classified by aYoung tableau [ l , · · · , l s ] ( i.e. the number of boxes in i -th column is l i ). In fact, for thegauge theory with quiver gauge group SU ( d ) × · · · × SU ( d n ) under the condition (2.2), ~β corresponds to the puncture with Young tableau whose number of boxes in j -th line is d j − d j +1 . That is, [ l , · · · , l s ] = [ d − d , · · · , d n − − d n , d n ] T (3.5)where T means the transposition of a Young tableau, and then s = d n . Note that the totalnumber of boxes is always N (= d ). Especially, a puncture with tableau [1 N ] is called a ‘full’puncture, and a puncture with tableau [ N − , 1] is called a ‘simple’ puncture.Our ansatz in [4] gives how to determine the form of ~β by using this Young tableau Y = [ l , · · · , l s ] as follows: First, we divide ~β into its real and imaginary parts as ~β = Q~ρ Y + i~β ′ . (3.6)12or the real part, ~ρ Y is defined as ~ρ Y := ~ρ − ( ~ρ l ⊕ · · · ⊕ ~ρ l i ⊕ · · · ⊕ ~ρ l s ) (3.7)= (cid:0) N − l , · · · , N − l | {z } l , · · · , N − l i − P i − j =1 l j , · · · , N − l i − P i − j =1 l j | {z } l i , · · · , − N + l s , · · · , − N + l s | {z } l s (cid:1) where ~ρ k = 12 ( k − , k − , · · · , − k, − k ) . (3.8)For the imaginary part, ~β ′ is defined as ~β ′ = ( β ′ , , · · · , β ′ , | {z } l , · · · , β ′ ,i , · · · , β ′ ,i | {z } l i , · · · , β ′ ,s , · · · , β ′ ,s | {z } l s ) =: ~β ′ l , ··· ,l i , ··· ,l s ] . (3.9)Therefore, we can find that ~β · ~e k = 0 is satisfied for ∀ k = l + l + · · · + l i for ∀ i = 1 , · · · , s .Note that we don’t fix the order of l , · · · , l s at this moment, but in the following discussion,we will consistently fix it as l ≤ · · · ≤ l s . Ansatz for ~α j by Drukker and Passerini According to Gaiotto’s discussion [1] again, in the weak coupling limit of SU ( d j ), i.e. whenthe diagram is decomposed like eq. (3.3), ~α j should correspond to a puncture for a quivergauge group SU ( d ) × · · · × SU ( d j − ). Therefore, we can use our ansatz (3.6) here again.Then the ansatz for ~α j can be written as ~α j = Q~ρ Y + i h ( ~α ′ j ,~ 0) + ~γ j [ d j ,l dj +1 , ··· ,l s ] i (3.10)where Y = [ l , · · · , l s ] is a Young tableau for a quiver gauge group SU ( d ) × · · · × SU ( d j − ),so s = d j − here. Since the condition (2.2) means d j − − ( d j − − d j − ) ≥ d j , if we put inorder as l ≤ · · · ≤ l s , we find that l = · · · = l d j = 1 is always satisfied. Then we canput a traceless d j -component vector ~α ′ j as in eq. (3.10), which can be regarded as an SU ( d j )propagator. The next ~ N − d j )-component zero vector. The remaining part ~γ j shouldbe determined so that the whole imaginary part (cid:2) ( ~α ′ j ,~ 0) + ~γ j (cid:3) is of the same form as ~β ′ l , ··· ,l s ] in eq. (3.9). In fact, eq. (3.10) satisfies it, since( ~α ′ j ,~ 0) + ~γ j [ d j ,l dj +1 , ··· ,l s ] (3.11)= ( α ′ j, |{z} l =1 , · · · , α ′ j,d j |{z} l dj =1 , , · · · , | {z } N − d j ) + ( γ j,d j , · · · , γ j,d j | {z } l + ··· + l dj = d j , γ j,d j +1 , · · · , γ j,d j +1 | {z } l dj +1 , · · · , γ j,s , · · · , γ j,s | {z } l s ) . This exactly agrees with the ansatz by Drukker and Passerini [5]. Note that ~α j · ~e k = 0 issatisfied for ∀ k = l + l + · · · + l i for ∀ i = 1 , · · · , s , just as for ~β .13p to now, the proof of AGT-W relation by direct calculations has been done in [3, 5, 8],but it seems to be still restrictive. Our calculation in the following is also restrictive in thatwe consider only the correlation functions of A N − Toda theory with descendant level 0 andthe corresponding 1-loop part of partition functions of SU ( N ) quiver gauge theory. However,we consider a general case of SU ( N ) quiver gauge group, which is a new point of this paper.In the remainder of this section, we first consider the simple cases of A and A Toda theoryin § § A N − Toda theory in § A Toda theory We have already discussed this case in our previous paper [8], but here let us make somemodifications. Especially, we consider all factors in the correlation function (3.2), althoughthe unrelated factors to 1-loop partition function have been usually neglected in the previousresearches. SU (3) n quiver In this case, ~β corresponds to a ‘full’ puncture, then we set ~α j = Q~ρ + i~α ′ j , ~β = Q~ρ + i~β ′ , ~β k = (cid:18) Q im k (cid:19) ~ω , (3.12)where j = 1 , · · · , n and k = 1 , · · · , n + 1. Here no conditions other than eq. (3.4) are imposed.As we discussed in § υ ζ ,ζ = Q i (cid:0) a linear combination of α ′ j,p , β ′ ,q , m k (cid:1) for ∀ ζ , ζ = 1 , , p, q = 1 , , 3, and υ ζ ,ζ has been defined in eq. (2.45). Then we require that thereshould be no zeros also in the numerator. This requirement (2.44) means that α ′ j,p = α ′ j,q , β ′ ,p = β ′ ,q for p = q . (3.14)Let us now compare the level-0 correlation function (3.2), denoted by V ∅ , with the 1-looppartition function of gauge theory (A.3), denoted by Z . Then we find that when thecorrespondence of parameters is set as SU (3) adjoint scalar VEV ~ ˆ a j = i~α ′ j (for j = 1 , · · · , n ) SU (3) bifundamental mass ν k = Q im k (for k = 1 , · · · , n − SU (3) fundamental mass µ p = Q im n ± iβ ′ ,p (for p = 1 , , SU (3) antifundamental mass ¯ µ p = Q − im n +1 ∓ iβ ′∞ ,p Nekrasov’s parameters ǫ = b , ǫ = b − , (3.15)14e can show that the correlation function can be written as V ∅ = A n +1 h (2 Q~ρ ) n g ( ~β ∞ ) g ( ~β ) n +1 Y k =1 f ( m k ) n Y j =1 Y p Q~ρ − ~β ) · ~e ) h ( ~α ) := h πµγ ( b ) b − b i − ~α · ~ρ/b (3.17)where Q ′ e> has been already defined in eq. (2.50). In the present case, it is equivalent tothe usual Q e> , since any cancellations discussed in § Q ′ e> following the condition (2.52). We willexplain it clearly in the following cases.We finally note that ~ω in the setting of ~β k (3.12) can be replaced by ~ω . This replacementslightly changes the parameter correspondence (3.15): the upper signs are for ~ω , while thelower signs are for ~ω . The coefficient functions in eq. (3.16) is also changed: for the choiceof ~ω instead of ~ω N − in A N − Toda theory, the factor f N − must be replaced as f N − ( m ) → f ( m ) := h πµγ ( b ) b − b i − ( Q + im ) N~ω · ~ρ/b Υ( N ( Q + im )) . (3.18)Therefore, we can successfully show that the level-0 correlation function of Toda theoryproperly corresponds to the 1-loop partition function of SU (3) n quiver gauge theory. SU (3) n − × SU (2) quiver In this case, ~β becomes a ‘simple’ puncture. In the weak coupling limit of the last SU (2), ~α n becomes a ‘full’ puncture. Then we should change a part of the setting (3.12) as ~α n = Q~ρ + i (cid:2) ( ~α ′ n , − m n ~ω (cid:3) , ~β = (cid:18) Q im (cid:19) ~ω , (3.19)where ~α ′ n is a traceless 2-component vector. The last term in ~α n did not appear in ourprevious paper [8], but we add here by following the ansatz (3.10).As we discussed in § C (2 Q~ρ − ~α n , ~β ; ~β n ), since υ , = i ( m n − m − m n ) . (3.20)15ase denominator numerator − υ , −− υ , ( Q~ρ − ~β ) · ~e υ , ( Q~ρ − ~β ) · ( ~e + ~e )3 υ , ( Q~ρ − ~α n ) · ( ~e + ~e ) υ , ( Q~ρ − ~α n ) · ~e υ , ( Q~ρ − (2 Q~ρ − ~α n )) · ( ~e + ~e ) υ , ( Q~ρ − (2 Q~ρ − ~α n )) · ~e Table 1: Cancellations in the case of SU (3) n − × SU (2) quiverThis means that the factor Υ( υ , ) makes a new pole, when we set m n = m + 2 ˜ m n . Thensome of the other factors in the denominator cancel out some factors in the numerator inthe way of case 1 – 4, or eq. (2.46) – (2.49). These cancellations can be summarized as table 1.After the cancellations, the remaining factors in the denominator areΥ( υ ζ, ) = Υ (cid:18) Q i ( − α ′ n,ζ + 3 m + 3 ˜ m n ) (cid:19) for ζ = 1 , SU (2) fundamental matter field, by comparingwith the 1-loop partition function of gauge theory. Therefore, we find the correspondence ofparameters as follows: SU (2) adjoint scalar VEV ~ ˆ a n = i~α ′ n SU (3) × SU (2) bifundamental mass ν n − = Q im n − SU (3) fundamental mass µ (3) = Q i ( m n − + 2 ˜ m n ) SU (2) fundamental mass µ (2) = Q i ( ± m + 3 ˜ m n ) (3.22)with the condition m n = ± m + 2 ˜ m n . We show here only a different part from the previouscase (3.15). The double signs are related to the choice of ~ω / ~ω in the setting of ~β (3.19):the upper signs are for ~ω , while the lower signs are for ~ω . Note that the newly addedparameter ˜ m n makes apart SU (3) fundamental mass from SU (3) × SU (2) bifundamentalmass.Finally we can show that under the correspondence of parameters (3.22), the correlationfunction can be written as V ∅ = A n +1 Υ(0) h (2 Q~ρ ) n g ( ~β ∞ ) g ( ~β ) n +1 Y k =1 f ( m k ) n Y j =1 Y p Now we incline to rush into the proof of AGT-W relation in a general case of A N − Todatheory, but in this subsection, we check the correspondence in the case of A Toda theory,and pile up more observations and discussions on concrete simple examples. SU (4) n quiver In this case, we can show the correspondence very similarly to the case of SU (3) n quiver.Since ~β corresponds to a ‘full’ ([1 ]) puncture, the setting of Toda momenta is ~α j = Q~ρ + i~α ′ j , ~β = Q~ρ + i~β ′ , ~β k = (cid:18) Q im k (cid:19) ~ω , (3.24)where j = 1 , · · · , n and k = 1 , · · · , n + 1. Then the correspondence of parameters becomes SU (4) adjoint scalar VEV ~ ˆ a j = i~α ′ j (for j = 1 , · · · , n ) SU (4) bifundamental mass ν k = Q im k (for k = 1 , · · · , n − SU (4) fundamental mass µ p = Q im n ± iβ ′ ,p (for p = 1 , · · · , SU (4) antifundamental mass ¯ µ p = Q − im n +1 ∓ iβ ′∞ ,p (3.25)where the lower signs are for the choice of ~ω , instead of ~ω in the setting of ~β k . The finalresult is of the same form as eq. (3.16), so we don’t write it down here. SU (4) n − × SU (3) quiver In this case, ~β corresponds to a [2 , , 1] puncture. In the weak coupling limit of SU (3), ~α n becomes a ‘full’ puncture. Then we should change a part of the setting (3.24) as ~α n = Q~ρ + i (cid:2) ( ~α ′ n , − m n ~ω (cid:3) , ~β = Q~ρ [1 , , + i~β ′ , , , (3.26)where ~α ′ n is a traceless 3-component vector. ~ρ [1 , , and ~β ′ , , are defined in eq. (3.7) and(3.9). Let us here comment on the other choices of Young tableau, i.e. Y = [2 , , , , 1] :The former has no problem, if we redo the discussion in § l , · · · , l s ]with l ≥ · · · ≥ l s and change our ansatz for ~α n . The latter, on the other hand, causes aproblem, since it cannot have sufficient degrees of freedom of fundamental matter fields.The remaining part of discussion is parallel to the case of SU (3) n − × SU (2) quiver. Therecan be the zeros in the denominator of the last 3-point function C (2 Q~ρ − ~α n , ~β ; ~β n ), since υ , = i ( m n + β ′ , − m n ) . (3.27)17ase denominator numerator − υ , −− υ , ( Q~ρ − ~β ) · ~e υ ,ζ ( Q~ρ − ~β ) · ( ~e ζ + · · · + ~e )3 υ ζ , ( Q~ρ − ~α n ) · ( ~e ζ + · · · + ~e )4 υ ζ , ( Q~ρ − (2 Q~ρ − ~α n )) · ( ~e ζ + · · · + ~e )Table 2: Cancellations in the case of SU (4) n − × SU (3) quiver( ζ = 1 , , , ζ = 1 , υ , ) makes a new pole, when we set m n = − β ′ , + 3 ˜ m n . After thecancellations listed in table 2, the remaining factors in the denominator areΥ( υ ζ ,ζ ) = Υ (cid:18) Q i ( − α ′ n,ζ + β ′ ,ζ − β ′ , + 4 ˜ m n ) (cid:19) for ζ = 1 , , , ζ = 1 , SU (3) fundamental matter fields, by comparingwith the 1-loop partition function of gauge theory. Therefore, by using β ′ , + β ′ , + 2 β ′ , = 0,we find the correspondence of parameters as SU (3) adjoint scalar VEV ~ ˆ a n = i~α ′ n SU (4) × SU (3) bifundamental mass ν n − = Q im n − SU (4) fundamental mass µ (4) = Q i ( m n − + 3 ˜ m n ) SU (3) fundamental mass µ (3)1 , = Q + i ( β ′ , + β ′ , + 4 ˜ m n ) Q + i ( β ′ , + β ′ , + 4 ˜ m n ) (3.29)with the condition m n = ( β ′ , + β ′ , ) + ˜ m n . The final result is of the same form as eq. (3.23)with p, q = 1 , · · · , j = 1 , · · · , n − p, q = 1 , , j = n . SU (4) n − × SU (3) × SU (2) quiver In this case, ~β corresponds to a simple ([3 , SU (2), ~α n becomes a [2 , , 1] puncture. In the weak coupling limit of SU (3), ~α n − becomes a ‘full’puncture. Then we should change a part of the setting (3.24) as ~α n = Q~ρ [1 , , + i (cid:2) ( ~α ′ n , , − m n ~ω (cid:3) , ~β = (cid:18) Q im (cid:19) ~ω ,~α n − = Q~ρ + i (cid:2) ( ~α ′ n − , − m n − ~ω (cid:3) , (3.30)where ~α ′ n is a traceless 2-component vector, ~α ′ n − is a traceless 3-component vector, and ~ω = (1 , , − , − / 2. 18ase denominator numerator − υ , −− υ , ( Q~ρ − ~β ) · ~e υ ,ζ ( Q~ρ − ~β ) · ( ~e ζ + · · · + ~e )4 υ ζ , ( Q~ρ − (2 Q~ρ − ~α n )) · ( ~e ζ + · · · + ~e ) − υ , −− υ , ( Q~ρ − ~β ) · ~e υ , ( Q~ρ − ~β ) · ( ~e + ~e )3 υ ζ ′ , ( Q~ρ − ~α n ) · ( ~e ζ ′ + · · · + ~e )4 υ ζ ′ , ( Q~ρ − (2 Q~ρ − ~α n )) · ( ~e ζ ′ + · · · + ~e )Table 3: Cancellations in the case of SU (4) n − × SU (3) × SU (2) quiver( ζ = 1 , , , ζ = 1 , , ζ ′ = 1 , C (2 Q~ρ − ~α n − , ~α n ; ~β n − ), we can reuse the results of the previous case of SU (4) n − × SU (3) quiver only after a reparametrization, e.g. ( m n , ˜ m n , β ′ , ) → ( m n − , ˜ m n − , m n ). Thenwe find the condition m n − = − m n + 3 ˜ m n − . Next we consider the last 3-point function C (2 Q~ρ − ~α n , ~β ; ~β n ). There can be the zeros in the denominator, since υ , = υ , = i ( m n − m − m n ) . (3.31)Then the factors Υ( υ , ) and Υ( υ , ) make new two poles, when we set m n = m + 2 ˜ m n .After the cancellations listed in table 3, the remaining factors in the denominator areΥ( υ ζ, ) = Υ (cid:18) Q i ( − α ′ n,ζ + m n + 3 m + 2 ˜ m n ) (cid:19) for ζ = 1 , SU (2) fundamental matter fields, by comparingwith the 1-loop partition function of gauge theory. Therefore, we find the correspondence ofparameters as SU (3) adjoint scalar VEV ~ ˆ a n − = i~α ′ n − SU (2) adjoint scalar VEV ~ ˆ a n = i~α ′ n SU (4) × SU (3) bifundamental mass ν n − = Q im n − SU (3) × SU (2) bifundamental mass ν n − = Q im n − SU (4) fundamental mass µ (4) = Q i ( m n − + 3 ˜ m n − ) SU (2) fundamental mass µ (2) = Q i (4 m + 4 ˜ m n ) (3.33)19ase denominator numerator − υ , −− υ , ( Q~ρ − ~β ) · ~e υ ,ζ ( Q~ρ − ~β ) · ( ~e ζ + · · · + ~e )3 υ ζ , ( Q~ρ − ~α n ) · ( ~e ζ + · · · + ~e )4 υ ζ , ( Q~ρ − (2 Q~ρ − ~α n )) · ( ~e ζ + · · · + ~e ) − υ , −− υ , ( Q~ρ − ~β ) · ~e υ ζ ′ , ( Q~ρ − ~α n ) · ( ~e ζ ′ + · · · + ~e )4 υ ζ ′ , ( Q~ρ − (2 Q~ρ − ~α n )) · ( ~e ζ ′ + · · · + ~e )Table 4: Cancellations in the case of SU (4) n − × SU (2) quiver( ζ = 1 , , , ζ = 1 , , ζ ′ = 1 , m n − = − m n + 3 ˜ m n − and m n = m + 2 ˜ m n .Finally we can show that under the correspondence of parameters (3.33), the correlationfunction can be written as V ∅ = A n +1 Υ(0) h (2 Q~ρ ) n g ( ~β ∞ ) g ( ~β ) n +1 Y k =1 f ( m k ) n Y j =1 Y p 2] puncture. In the weak coupling limit of SU (2), ~α n becomes a ‘full’ puncture. Then we should change a part of the setting (3.24) as ~α n = Q~ρ + i (cid:2) ( ~α ′ n , , 0) + ~γ n [2 , , (cid:3) , ~β = (cid:18) Q im (cid:19) ~ω , (3.35)where ~α ′ n is a traceless 2-component vector, and ~γ n is of the form ( ~γ n, , ~γ n, , ~γ n, , ~γ n, ).As we discussed repeatedly, there can be the zeros in the denominator of the last 3-pointfunction C (2 Q~ρ − ~α n , ~β ; ~β n ), since υ , = i ( m n − m − γ n, ) , υ , = i ( m n + 2 m − γ n, ) . (3.36)Then the factors Υ( υ , ) and Υ( υ , ) make new poles, when we set m n = 2 m + γ n, = − m + γ n, . After the cancellations listed in table 4, we find that there are no remaining20actors in the denominator. This is consistent with that there are no SU (2) fundamentalmatter fields in the corresponding gauge theory. Then the correspondence of parameters is SU (2) adjoint scalar VEV ~ ˆ a n = i~α ′ n SU (4) × SU (2) bifundamental mass ν n − = Q im n − SU (4) fundamental mass µ (4)1 , = Q + i ( m n − + m n + 2 m ) Q + i ( m n − + m n − m ) (3.37)Therefore, we can show that under this correspondence of parameters, the correlation functioncan be written as V ∅ = A n +1 Υ(0) h (2 Q~ρ ) n g ( ~β ∞ ) g ( ~β ) n +1 Y k =1 f ( m k ) n Y j =1 Y p We finally start the proof of AGT-W relation in a general case of A N − Toda field theory.First, we decompose the original diagram (3.1) into the ( n + 1) 3-point functions: ~β n +1 ~β j ~β n ~β ∞ ~α Q~ρ − ~α j ~α j +1 Q~ρ − ~α n ~β · · · · · · N N N N (3.39)These 3-point functions can be classified into four types, so let us discuss the correspondenceto the 1-loop partition function of gauge theory for each type. Type 1 : the first 3-point function C ( ~β ∞ , ~α ; ~β n +1 )For all the quiver gauge group, the corresponding momenta of Toda vertex operators are ~β ∞ = Q~ρ + i~β ′∞ , ~α = Q~ρ + i~α ′ , ~β n +1 = (cid:18) Q im n +1 (cid:19) N ~ω N − . (3.40)Then, just as in the case of SU (3) n and SU (4) n quiver, there never be any zeros in thedenominator, since υ ζ ,ζ = Q i ( β ′∞ ,ζ + α ′ ,ζ + m n +1 ) for ∀ ζ , ζ = 1 , · · · , N . (3.41)Moreover, no cancellations of the factors in the denominator and those in the numeratoroccur. Therefore, if we set the correspondence of parameters as SU ( N ) adjoint scalar VEV ~ ˆ a = i~α ′ SU ( N ) antifundamental mass ¯ µ p = Q − im n +1 − iβ ′∞ ,p (for p = 1 , · · · , N ) (3.42)21e can show the correspondence of the 3-point function and the 1-loop partition function as C ( ~β ∞ , ~α ; ~β n +1 ) = A g ( ~β ∞ ) f N − ( m n +1 ) h ( ~α ) N Y ¯ p =1 (cid:12)(cid:12) z ( ~ ˆ a , ¯ µ ¯ p ) (cid:12)(cid:12) Y e> Υ( − i~α ′ · ~e ) (3.43)where A , g , f N − and h are defined in eq. (3.17), and z is defined in eq. (A.4). Note thatthe last factor, together with the factor from the next 3-point function C (2 Q~ρ − ~α , ~α ; ~β ),corresponds to the factor z ( ~ ˆ a ) in the 1-loop partition function as Y e> Υ( − i~α ′ · ~e )Υ( i~α ′ · ~e ) = Y e> (cid:12)(cid:12) Γ ( i~α ′ · ~e )Γ ( Q + i~α ′ · ~e ) (cid:12)(cid:12) − = Y e> | i~α ′ · ~e | | Γ ( i~α ′ · ~e + b )Γ ( i~α ′ · ~e + b − ) | = Y p 0) + ~γ j [ d j ,l dj +1 , ··· ,l s ] i , ~β j = (cid:18) Q im j (cid:19) N ~ω N − ,~α j +1 = Q~ρ Y j +1 + i h ( ~α ′ j +1 ,~ 0) + ~γ j +1[ d j +1 ,l ′ dj +1+1 , ··· ,l ′ s ′ ] i , (3.48)22here s = d j − and s ′ = d j . The Young tableaux are Y j := [ d − d , · · · , d j − − d j − , d j − ] T = [1 d j − − d j − , l x , · · · , l i , · · · , l s ] (3.49) Y j +1 := [ d − d , · · · , d j − − d j − , d j − − d j , d j ] T = [1 d j − d j − , d j − − d j − d j − , l x + 1 , · · · , l i + 1 , · · · , l s + 1] (3.50)where x := 2 d j − − d j − + 1.As we saw in § § υ ζ ,ζ for ∃ ζ , ζ becomes zero, and theimaginary part gives the conditions for momenta. Let us here check this for a general case.From the definition (2.45), υ ζ ,ζ can be written as υ ζ ,ζ = Q (cid:20) ( ~ρ − ~ρ Y j ) ζ − ( ~ρ − ~ρ Y j +1 ) ζ + 12 (cid:21) + i h − (Im ~α j ) ζ + (Im ~α j +1 ) ζ + m j i (3.51)where ( ~ρ ∗ ) ζ denotes the ζ -th component of a vector ~ρ ∗ . From the definition (3.7), ~ρ − ~ρ Y j = (cid:0) , · · · · · · , | {z } d j − − d j − , l x − , · · · , − l x | {z } l x , · · · , l i − , · · · , − l i | {z } l i , · · · , l s − , · · · , − l s | {z } l s (cid:1) (3.52) ~ρ − ~ρ Y j +1 = (cid:0) , · · · , | {z } d j − d j − , , − , · · · , , − | {z } d j − − d j − d j − ) , l x , · · · , − l x | {z } l x +1 , · · · , l i , · · · , − l i | {z } l i +1 , · · · , l s , · · · , − l s | {z } l s +1 (cid:1) . Therefore, we can find the condition that the real part of v ζ ,ζ becomes zero:( ζ , ζ ) = ( ζ, ζ − d j − − 1) for ζ = d j + 1 , · · · , d j − − d j − ( ζ, ζ − s − i ) for ζ − (cid:2) (2 d j − − d j − ) + P i − ℓ = x l ℓ (cid:3) = 1 , · · · , l i (3.53)where i = x, · · · , s . Note that this condition can be satisfied for ∀ ζ = d j + 1 , · · · , N . Thenwe have the poles 1 / Υ(0) N − d j in the 3-point function, if we impose the following ( N − d j )conditions on the momenta: m j = (Im ~α j ) ζ − (Im ~α j +1 ) ζ for ( ζ , ζ ) ∈ eq. (3.53) . (3.54)Note that these conditions are not always independent, as we saw in the case of SU (4) n − × SU (3) × SU (2) quiver. This is because (Im ~α j ) ζ = (Im ~α j ) ζ ′ and (Im ~α j +1 ) ζ = (Im ~α j +1 ) ζ ′ for ( ζ , ζ ) = ( ζ ′ , ζ ′ ) can be sometimes satisfied simultaneously.Next we discuss the cancellations of the other factors Υ( υ ζ ,ζ ) by some factors in thenumerator, as we did in § § § ~α j +1 · ~e ζ = 0 is required,which is always satisfied. Then after the cancellations for ∀ ( ζ , ζ ) ∈ eq. (3.53), the remainingfactors in the denominator aredenominator : Υ( υ ζ ,ζ ) with ζ = 1 , · · · , d j , ζ = 1 , · · · , d j − d j − (3.55)23ase denominator numerator − υ ζ ,ζ −− υ ζ ,ζ +1 ( Q~ρ − ~α j +1 ) · ~e ζ υ ζ ,ζ − ( Q~ρ − ~α j +1 ) · ( ~e ζ − + · · · + ~e ζ )3 υ ζ − ,ζ ( Q~ρ − ~α j ) · ( ~e ζ − + · · · + ~e ζ − )4 υ ζ − ,ζ +1 ( Q~ρ − (2 Q~ρ − ~α j )) · ( ~e ζ − + · · · + ~e ζ − )Table 5: Cancellations in a general quiver case( ζ − = 1 , · · · , ζ − , ζ − = 1 , · · · , ζ − ζ , ζ ) and ( ζ − , ζ − δζ ) ∈ eq. (3.53) ⇒ δζ = 1 or 2 . (3.56)If δζ = 1, the case 3 for ( ζ , ζ ) and the case 4 for ( ζ − , ζ − 1) means the cancellationof the same factors, so they are never compatible. Then in this case, we must give up theformer cancellation, as we saw in the case of SU (4) n − × SU (3) × SU (2) quiver. On the otherhand, if δζ = 2 or for min ( ζ , ζ ) ∈ eq. (3.53), the cancellations of all the case 1 – 4 can bedone without any problem, as in the case of SU (4) n − × SU (2) quiver. There we gave up thecase 2 cancellation for ( ζ , ζ ) = (3 , ζ − does not run any value.Therefore, after these cancellations, the remaining factors in the numerator arenumerator : Υ(( Q~ρ − (2 Q~ρ − ~α j )) · P ζ ♯ ζ j = ζ ♭ ~e ζ j with ζ ♭ , ζ ♯ = 1 , · · · , d j − Q~ρ − ~α j +1 ) · P ζ ♯ ζ j +1 = ζ ♭ ~e ζ j +1 with ζ ♭ , ζ ♯ = 1 , · · · , d j +1 − Q~ρ − ~α j +1 ) · ( ~e ζ − + · · · + ~e ζ ) with ζ = d j +1 , · · · , d j − d j − , d j − d j − + 2 n , d j − − d j − − i + P i − ℓ = x l ℓ (3.58)where n = 1 , · · · , d j − − d j − d j − and i = x, · · · , s (= d j − ). However, all these factors areremoved by the case 3 cancellation in the next 3-point function C (2 Q~ρ − ~α j +1 , ~α j +2 ; ~β j +1 ).Then the correspondence of parameters can be set as SU ( d j ) adjoint scalar VEV ~ ˆ a j = i~α ′ j SU ( d j +1 ) adjoint scalar VEV ~ ˆ a j +1 = i~α ′ j +1 SU ( d j ) × SU ( d j +1 ) bifundamental mass ν j = Q im j SU ( d j ) fundamental mass µ ( j ) p = Q im j + iα ′ j +1 ,p (3.59)24or p = d j +1 + 1 , · · · , d j − d j − . Note that the number of SU ( d j ) fundamental matter fieldsis 2 d j − d j − − d j +1 ( ≥ C (2 Q~ρ − ~α j , ~α j +1 ; ~β j ) = A f N − ( m j ) h (2 Q~ρ − ~α j ) h ( ~α j +1 ) × N − d j (cid:12)(cid:12) z ( ~ ˆ a j , ~ ˆ a j +1 , ν j ) (cid:12)(cid:12) d j − d j − Y p = d j +1 +1 (cid:12)(cid:12) z ( ~ ˆ a j , µ ( j ) p ) (cid:12)(cid:12) × Y { ~e ζj } Υ( i~α j · ~e ζ j ) Y { ~e ζj +1 } Υ( − i~α j +1 · ~e ζ j +1 ) (3.60)with the conditions (3.54). { ~e ζ j } and { ~e ζ j +1 } are defined in eq. (3.57). Again, as in eq. (3.44),the last two factors correspond to the factors z ( ~ ˆ a j ) and z ( ~ ˆ a j +1 ), together with the factorsfrom the next 3-point functions. Type 4 : the last 3-point function C (2 Q~ρ − ~α n , ~β ; ~β n )The momenta of Toda vertex operators are set as ~α n = Q~ρ Y n + i h ( ~α ′ n ,~ 0) + ~γ j [ d n ,l dn +1 , ··· ,l s ] i , ~β n = (cid:18) Q im n (cid:19) N ~ω N − ,~β = Q~ρ Y + i~β ′ Y , (3.61)This is just a reparametrization of eq. (3.48), so the discussion is almost parallel to type 3.Then if the correspondence of parameters is set as SU ( d n ) adjoint scalar VEV ~ ˆ a n = i~α ′ n SU ( d n ) fundamental mass µ p = Q im n + iβ ′ ,p (3.62)for p = 1 , · · · , d n − d n − , we can show the correspondence of 3-point function and the 1-looppartition function as C (2 Q~ρ − ~α n , ~β ; ~β n ) = A g ( ~β ) f N − ( m n ) h (2 Q~ρ − ~α n ) × N − d n d n − d n − Y p =1 (cid:12)(cid:12) z ( ~ ˆ a n , µ p ) (cid:12)(cid:12) Y { ~e ζn } Υ( i~α ′ n · ~e ζ n ) (3.63)with the conditions m n = (Im ~α n ) ζ − (Im ~β ) ζ for ( ζ , ζ ) ∈ eq. (3.53) with j = n . (3.64)As we defined in eq. (3.17), when we take the product Q ′ e> in g ( ~β ), the following factorsmust be removed: Υ (cid:0) ( Q~ρ − ~β ) · ( ~e ζ − + · · · + ~e ζ ) (cid:1) for ∀ ζ ∈ { ζ in eq. (3.53) with j = n } .25 ummary By putting all the results together, i.e. eq. (3.43), (3.47), (3.60) and (3.63), we can show thatthe whole of Toda correlation function with descendant level 0 (3.2) exactly corresponds tothe 1-loop partition function of gauge theory with a general quiver gauge group (2.1) as V ∅ = A n +1 h (2 Q~ρ ) n g ( ~β ∞ ) g ( ~β ) n +1 Y k =1 f N − ( m k ) n Y j =1 N − d j Y p In this paper, we show the correspondence between the correlation function of A N − Todatheory with descendant level 0 and the 1-loop part of partition function of N = 2 SU ( N )quiver gauge theory with a general quiver gauge group. All the parameters except gaugecoupling constants appear in this part, so in this sense, we claim that the ansatz for corre-spondence of parameters in AGT-W relation [4, 5] is completely justified.The remaining part of AGT-W relation is the correspondence between the descendantpart of correlation function of Toda theory and the instanton part of partition function ofgauge theory. Now the correspondence of parameters is clearly understood, then the onlyunclear point is so-called ‘ U (1) factor’ in AGT-W relation [2]. At this moment, there seemsto be no consensus among researchers with regard to the way of determining this factor, whilesome researchers propose that this factor is nothing but the free string amplitude [6].On this problem, we have a direction of discussion. In our previous paper [10], we pointedout that W ∞ algebra may exist as a symmetry behind AGT-W relation, by showing thatToda correlation function plus the U (1) factor can be simply written in terms of this algebra.In this algebra, U (1) generator naturally coexists with W N generators of A N − Toda theory.Therefore, we consider that this U (1) generator in W ∞ algebra must be related to the U (1) factor in AGT-W relation. Then from this viewpoint, it may be possible to justify theinterpretation of the U (1) factor as the free string amplitude.Anyway in order to check the remaining part of AGT-W relation, we must calculate thecorrelation function of Toda theory with an arbitrary descendant level. The most basic wayis to calculate the inverse Shapovalov matrix of each level as it was done in [6–8], but itmust be not a realistic way if we want to calculate in an arbitrary high level. One choice26s the calculation by Dotsenko-Fateev method, which has been recently discussed also in thecontext of AGT-W relation [11–13]. Up to now, however, these discussions are restricted tothe case of 4-point correlation function and Q = b + b − = 0. Especially, it must be verydifficult to discuss the case of Q = 0, so all we can do may be to calculate the correlationfunction with arbitrary number of points and descendant level, but Q = 0.Finally, we want to say that AGT-W relation is still a very strange relation. In particular,we discuss the considerably general cases of N = 2 SU ( N ) quiver gauge theory, but thecorresponding correlation function of Toda theory seems in very special cases. Through thefurther various investigations, we hope to understand what it means from the viewpoint of,for example, W ∞ algebra, superconformal theory, and M5-brane dynamics. Acknowledgments We would like to thank Yutaka Matsuo and Shoichi Kanno for useful discussions and com-ments. The author is partially supported by Grant-in-Aid ( A Partition function of N = 2 SU ( N ) quiver gauge theory The full partition function of 4-dim N = 2 SU ( N ) quiver gauge theory can be written as Z = Z class Z Z inst . (A.1)We see each part of function in the following [36, 37]. Classical part The classical part of the partition function is Z class = exp " n X k =1 πiτ k | ~ ˆ a k | (A.2)where τ k := θ k π + πig k is the complex UV coupling constant, and ~ ˆ a k := P d k − i =1 a i ~e i is thediagonal of VEV’s a i of adjoint scalars. ~e i are the simple roots of gauge symmetry algebra,which are usually defined as eq. (2.7) for SU ( N ) algebra. It gives, for example, ~ ˆ a = ( a , − a )for SU (2) and ~ ˆ a = ( a , − a + a , − a ) for SU (3). The 1-loop contribution to the partition function is Z = n Y k =1 z ( ~ ˆ a k ) ! d Y ¯ p =1 z ( ~ ˆ a , ¯ µ ¯ p ) × n − Y k =1 z ( ~ ˆ a k , ~ ˆ a k +1 , m k ) ! d n Y p =1 z ( ~ ˆ a n , µ p ) (A.3)27here µ p , ¯ µ ¯ p , m k are the mass of fundamental, antifundamental and bifundamental fields,respectively. The functions z are defined as z ( ~a ) = Y i The instanton contribution is obtained by Nekrasov’s instanton countingformula with Young tableaux as Z inst = X { ~Y , ··· ,~Y n } n Y k =1 q | ~Y k | k z vec ( ~ ˆ a k , ~Y k ) ! d Y ¯ p =1 z afd ( ~ ˆ a , ~Y , ¯ µ ¯ p ) × n − Y k =1 z bfd ( ~ ˆ a k , ~Y k ; ~ ˆ a k +1 , ~Y k +1 ; m k ) ! d n Y p =1 z fd ( ~ ˆ a n , ~Y n , µ p ) (A.10)28here q k := e πiτ k ( τ k is the coupling constant), and ~Y k = ( Y k, , · · · , Y k,d k ) is a set of Youngtableaux. | ~Y k | is the total sum of number of boxes of Young tableaux Y k,i ( i = 1 , · · · , d k ).Each factor of the instanton part is written as z bfd ( ~ ˆ a, ~Y ; ~ ˆ b, ~W ; m ) = Y i,j Y s ∈ Y i ( E (ˆ a i − ˆ b j , Y i , W j , s ) − m ) × Y t ∈ W j ( ǫ + − E (ˆ b j − ˆ a i , W j , Y i , t ) − m ) ,z vec ( ~ ˆ a, ~Y ) = 1 /z bfd ( ~ ˆ a, ~Y ; ~ ˆ a, ~Y ; 0) ,z fd ( ~ ˆ a, ~Y , µ ) = Y i Y s ∈ Y i ( φ (ˆ a i , s ) − µ + ǫ + ) ,z afd ( ~ ˆ a, ~Y , ¯ µ ) = z fd ( ~ ˆ a, ~Y , ǫ + − ¯ µ ) . 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Phys.7 ′ Υ((
in g ( ~β ) defined in eq. (3.17), the factors of numerator listed in table 1,16 .e. ( Q~ρ − ~β ) · ~e and ( Q~ρ − ~β ) · ( ~e + ~e ), must be removed. We also note that the orderof pole 1 / Υ(0) ( i.e. one, in this case) is equal to the number of times which Hanany-Wittentransition occurs in the D4/NS5/D6-brane system, as we discussed in § A Toda theory
in g ( ~β ) must be taken with the factors of numerator in table 3 removed. The order of pole1 / Υ(0) ( i.e. three, in this case) is equal to the total number of times which Hanany-Wittentransition occurs in the D4/NS5/D6-brane system. SU (4) n − × SU (2) quiver In this case, ~β corresponds to a [2 ,
Υ( i~α ′ j · ~e )Υ( − i~α ′ j +1 · ~e ) . (3.47)Just as we discussed in eq. (3.44), the last two factors correspond to the factors z ( ~ ˆ a j ) and z ( ~ ˆ a j +1 ), together with the factors from the next 3-point functions. Type 3 : 3-point function C (2 Q~ρ − ~α j , ~α j +1 ; ~β j ) with the rank d j > d j +1 We finally discuss the part of descending tail. According to the ansatz (3.10), we set themomenta of Toda vertex operators as ~α j = Q~ρ Y j + i h ( ~α ′ j ,~