AGT, Burge pairs and minimal models

We consider the AGT correspondence in the context of the conformal field theory $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$, where $M^{\, p, p^{\prime}}$ is the minimal model based on the Virasoro algebra $V^{\, p, p^{\prime}}$ labeled by two co-prime integers $\{p, p^{\prime}\}$, $1<p<p^{\prime}$, and $M^{H}$ is the free boson theory based on the Heisenberg algebra $H$. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$ leads to ill-defined or incorrect expressions. Let $B^{\, p, p^{\prime}, H}_n$ be a conformal block in $M^{\, p, p^{\prime}}$ $\otimes$ $M^{H}$, with $n$ consecutive channels $\chi_{i}$, $i = 1, \cdots, n$, and let $\chi_{i}$ carry states from $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ $\otimes$ $F$, where $H^{p, p^{\prime}}_{r_{i}, s_{i}}$ is an irreducible highest-weight $V^{\, p, p^{\prime}}$-representation, labeled by two integers $\{r_{i}, s_{i}\}$, $0<r_{i}<p$, $0<s_{i}<p^{\prime}$, and $F$ is the Fock space of $H$. We show that restricting the states that flow in $\chi_{i}$ to states labeled by a partition pair $\{Y_1^{i}, Y_2^{i}\}$ such that $Y^{i}_{2, {\tt R}} - Y^{i}_{1, {\tt R} + s_{i} - 1} \geq 1 - r_{i}$, and $Y^{i}_{1, {\tt R}} - Y^{i}_{2, {\tt R} + p^{\prime} - s_{i} - 1} \geq 1 - p + r_{i}$, where $Y^{i}_{j, {\tt R}}$ is row-${\tt R}$ of $Y^{i}_j, j \in \{1, 2\}$, we obtain a well-defined expression that we identify with $B^{\, p, p^{\prime}, H}_n$. We check the correctness of this expression for ${\bf 1.}$ Any 1-point $B^{\, p, p^{\prime}, H}_1$ on the torus, when the operator insertion is the identity, and ${\bf 2.}$ The 6-point $B^{\, 3, 4, H}_3$ on the sphere that involves six Ising magnetic operators.

, and H is the Heisenberg algebra. The Virasoro central charge of M H is c H = 1. The AGT correspondence of Alday, Gaiotto and Tachikawa [1] identifies conformal blocks in M gen,H [2] with instanton partition functions in four-dimensional N = 2 supersymmetric quiver gauge theories [3]. Conjectured in [1], AGT was proven for c gen = 1 in [4], and for all c gen in [5] for conformal blocks with non-degenerate external primary fields.
1.2. AGT in minimal models. In this note, we consider AGT in the context of M p,p ′ ,H = M p,p ′ ⊗ M H , based on the algebra V p,p ′ ,H = V p,p ′ ⊕ H, where M p,p ′ is the minimal conformal field theory with a chiral spectrum that spans finitely-many V p,p ′ irreps, and V p,p ′ is the Virasoro algebra labeled by two co-prime integers {p, p ′ }, 0 < p < p ′ , of central charge c p,p ′ , Let B p,p ′ ,H n , B p,p ′ n and B H be conformal blocks with n consecutive channels 3 . We wish to compute any B p,p ′ ,H n of vertex operators O p,p ′ ,H ι (z ι ) = O p,p ′ ι (z ι ) × O H ι (z ι ), ι = 0, · · · , n + 2. Since any B p,p ′ ,H n factorizes to B p,p ′ n × B H n and an explicit expression for the M H -factor B H n is known 4 , computing B p,p ′ ,H n is equivalent to computing its M p,p ′ -factor B p,p ′ n which is typically what we want.
1.3. Zeros in denominators and deformations. Applying AGT to minimal model conformal blocks without modification leads to ill-defined expressions, as will be explained in detail below. In particular, setting the parameters that appear in Nekrasov's partition functions to minimal model values leads to zeros in the denominators of the summands. Following [6], one can make the summands well-defined using suitable deformations of the parameters. Doing that, one finds whenever a denominator is zero in the limit of removing the deformations, the corresponding numerator is also zero in such a way that and that limit is well-defined. This is in agreement with [7], where arguments were given to the effect that, analytically continuing the conformal blocks in the conformal dimensions of the primary states that flow in each channel, the only singularities are poles and the sum of all residues is zero. This is the approach that was followed, albeit without discussion, in an earlier work on AGT in minimal models [8]. 1.4. Zeros in denominators and restrictions. In this note, we follow a different approach from that discussed in subsection 1.3. Our idea is that the zeros in the denominators of Nekrasov's partition functions are due to including null states that should not be included. We avoid this by restricting the summations over Young diagrams that appear in Nekrasov partition functions to avoid these null states. We make the summands well-defined by restricting the partition pairs that label the summed-over states to exclude the summands with poles. To compute B p,p ′ ,H n , the summations that label the factors in Nekrasov's instanton partition functions must be restricted to avoid ill-defined or incorrect expressions for B p,p ′ ,H n , and consequently for its M p,p ′ -factor B p,p ′ n . Our approach allows us to characterise the Young diagrams that label the summands that do contribute to B p,p ′ ,H n . 1.5. Unrestricted partition pairs. The AGT expression for a linear conformal block B gen,H n , that has n consecutive channels χ ι , ι = 1, · · · , n, is an n-fold sum 5 , where the summand is a product of (n + 1) factors q , ι = 1, · · · , n + 1, that will be defined in section 2. Each factor Z ι bb is a rational function that depends on two pairs of 'unrestricted' Young diagrams {Y ι−1 1 , Y ι−1 2 } and {Y ι 1 , Y ι 2 }. In other words, there are no conditions on these Young diagrams and all possible pairs are allowed. The denominator z ι den of Z ι bb is a product of the norms of the states that flow in the preceding channel χ ι−1 and the subsequent channel χ ι . Since Z ι bb is labeled by unrestricted partition pairs, and the sums are over all possible unrestricted pairs, the states that flow in each channel belong to a Verma module of V gen,H .
Applying AGT without modification to M p,p ′ ,H , one includes zero-norm states in the summation, and thereby includes states in a Verma module rather than in an irrep of V p,p ′ ,H . This leads to summands in the instanton partition function with zero 3 Only linear conformal blocks, as in Figure 3, are considered in this work. Our notation is such that an n-channel conformal block B indices n , is the expectation value of (n + 3) vertex operators O same indices ι (zι), ι = 0, · · · , (n + 2), in M same indices on a Riemann surface S, and zι ∈ S. 4 See, for example, equation (1.9) in [5]. 5 The partition pairs Y 0 and Y n+1 are trivial, that is they consist of empty partitions, and no summation is performed on them.
denominators. Further, as show, whenever a denominator in a summand vanishes, the corresponding numerator vanishes as well and one ends with ill-defined expressions 1.6. Restricted partition pairs. In this note, we consider B p,p ′ ,H n as an instanton partition Z N ek that consists of building block partition functions Z ι bb that has a numerator z ι num and a denominator z ι den , ι = 1, · · · , n + 1. Z ι bb connects two channels χ ι−1 and χ ι . The denominator z ι of Z ι bb is a product of two factors [z ι−1 norm ] 1/2 and [z ι norm ] 1/2 that account for the norms of the states that flow in the channels χ ι−1 and χι, respectively. We characterise the zeros in these denominators that lead to ill-defined expressions for B p,p ′ ,H n . If channel χ ι , ι = 1, · · · , n, carries states that belong to an irreducible highest weight Virasoro representation that flows is H p,p ′ rι,sι , we attribute these zeros to the flow of null states that do not belong to H p,p ′ rι,sι , and eliminate these zeros by restricting the partition pairs that appear in Nekrasov's original expressions to partition pairs {Y 1 , Y 2 }, that satisfy the conditions Burge pairs. Partition pairs that satisfy conditions (3) were first studied in [9] and appeared more recently in [10,11]. In this work, we refer to them as Burge pairs, and show that when used to restrict AGT to compute B p,p ′ ,H n , that is when we sum over Burge pairs rather than on all possible partition pairs, where ′ indicates that the sum is restricted to partition pairs that satisfy the Burge conditions (3), we obtain well-defined expressions. We check these expressions in two cases 1. Any 1-point B p,p ′ ,H 1 on the torus, when the operator insertion is the identity, and 2. The 6-point B 3,4,H 3 , when all operator insertions involve Ising magnetic operators. We also give arguments why we expect this identification to be correct.
1.8. Outline of contents. In section 2, we recall basic facts related to Nekrasov's instanton partition functions. In 3, we recall the AGT parametrisation of M gen,H , the choice of parameters that allows us to obtain M p,p ′ ,H , then show how the unrestricted instanton partition functions give the wrong answer in the case of B p,p ′ ,H 1 on the torus. In 4, we use the requirement that the summands remain well-defined to characterise the partition pairs that label them. We identify these partition functions with B p,p ′ ,H n In 5, we study the vanishing of the numerator, and show that whenever the denominator of a summand vanishes, then the numerator also vanishes. In 6, we check the correctness of our expressions in the two cases listed above. In 7, we use results from [5,10,11], Proposition 4.1 in section 4, and Conjecture 7.1 in section 7, to explain why the restriction to Burge pairs produces conformal blocks in M p,p ′ ,H . Because we use Conjecture 7.1, this explanation is not a proof. In 8, we extend of our results to conformal blocks in M gen,H , with degenerate intermediate Virasoro representations, and in 9, we collect a number of remarks that include 1. a conjectural generalization to the W N conformal blocks, and 2. a geometric interpretation of the summation over Burge pairs as a summation over isolated torus fixed points on the instanton moduli space.

The instanton partition function
2.1. Partitions. A partition π of an integer |π| is a set of non-negative integers {π 1 , π 2 , · · · , π p }, where p is the number of parts, π i π i+1 , and p i=1 π i = |π|. π is represented as a Young diagram Y , which is a set of p rows {Y 1 , Y 2 , · · · , Y p }, such that row-i has We use for a cell in a Young diagram Y , which is a square in the south-east quadrant of the plane, with coordinates {R, C}, such that R is the row-number, counted from top to bottom, and C is the column number, counted from left to right. We define A + ,Yi = A ,Yi + 1, where A ,Yi is the arm of in Y i , that is, the number of cells in the same row as, but to the right of in Y i , and L ,Wj to be the leg of with respect its position in W j , that is the number of cells in the same column as, but below in Y i . 2.2. Partition pairs. The AGT representation of B p,p ′ ,H involves a multi-sum over internal states labeled by n + 2 partition pairs Y ι , ι = 0, 1, · · · , n, n + 1, where Y ι is a pair of Young diagrams, {Y ι 1 , Y ι 2 }, and | Y ι | = |Y ι 1 | + |Y ι 2 | is the total number of cells in Y ι . The pairs {Y ι 1 , Y ι 2 }, ι ∈ {1, · · · , n}, are non-empty Young diagrams, while {Y ι 1 , Y ι 2 }, ι ∈ {0, n + 1} are empty 7 , Y (0) = Y (n+1) = ∅, where ∅ is a pair of empty Young diagrams. = −1, and L ,Y ⊺ = −1. 6 We use Y i for row-i as well as for the number of cells in that row. 7 We work in terms of n + 2 linearly-ordered partition pairs. Since we consider conformal blocks of primary fields, the initial and final pairs are always empty, but we prefer to work in terms of n + 2 rather than n non-empty pairs to make the notation in the sequel more uniform.

A decomposition of the instanton partition function.
Consider the fourdimensional N = 2 supersymmetric linear quiver gauge theory with a gauge group n+1 ι=1 U (2) ι , that is (n + 1) copies of U (2) [3]. The instanton partition function of this theory can be written in terms of 'building block' partition functions Z ι bb , ι = 1, · · · , n+1, as follows (5) Z where q ι is an indeterminate. In gauge theory, q ι = e 2πiτι , where τ ι is the complexified coupling constant of U (2) ι . In conformal field theory, it is a rational function of the positions z ι , ι = 0, 1, · · · , n + 2, of the vertex operators O ι , whose expectation value is the conformal block, on the Riemann surface S that the conformal field theory is defined on. Z ι bb is defined in subsection 2.4. The decomposition of the instanton partition function in (5) follows that in [12] and mirrors the decomposition of conformal blocks on a sphere, represented as a comb diagram in Figure 3. Figure 3. The comb diagram of a 4-channel conformal block that corresponds to a linear quiver. It consists of an initial state that corresponds to a vertex operator O 0 on the left, five vertex operator insertions O 1 , · · · , O 5 , and a final state that corresponds to a vertex operator O 6 on the right. O ι is placed at z ι , where z 0 , z n+1 and z n+2 are set 0, 1 and ∞, respectively. In this example, n = 4.

2.4.
The building block of the instanton partition function. Z bb is The parameters that appear in Z bb are as follows.
In gauge theory, a ι is the expectation value of the vector multiplet in the adjoint representation of the gauge group U (2) ι . In conformal field theory, a ι is the charge of the highest weight of the Virasoro irrep that flows in channel χ ι in the conformal block under consideration.

2.4.2.
The partition pairs Y and W . In gauge theory, each partition pair Y ι = {Y ι 1 , Y ι 2 } labels the fixed localization points in the instanton moduli space of U (2) ι . In conformal field theory, they label the states that flow in channel χ ι in the corresponding conformal block. In (6), Y and W are attached to the line segments on the left and the right of a given vertex, respectively.

2.4.3.
The scalar µ ι . In gauge theory, µ ι is the mass parameter of the bi-fundamental matter field that interpolates the gauge groups U (2) ι and U (2) ι+1 . In conformal field theory, µ ι is the charge of the vertex operator that connects channels χ ι and χ ι+1 . In the following, we study the structure of the right hand side of (6).
In gauge theory, z norm is a normalization factor related to the contribution of the vector multiplets that the bi-fundamental couples to. In conformal field theory, it accounts for the norms of the states that propagate into and out of the vertex operator insertion in Z bb .
x is an indeterminate, and {ǫ 1 , ǫ 2 } are Nekrasov's deformation parameters, which are generally complex. In gauge theory, z num is the contribution of a bi-fundamental multiplet in U (2) ι and U (2) ι+1 . In conformal field theory, it is the contribution of the vertex operator insertion that inputs a charge µ into the conformal block into Z bb .
2.4.6. Remark. One can think of z num as the basic object in U (2) AGT theory and in this paper, and all other objects can be written in terms of special cases of it.
2.4.7. Normalisation. Consider the special case where the Virasoro part of the vertex operator in Z bb is the identity, that is {r, s} = {1, 1}, and consequently µ = 0 8 . Z bb is defined combinatorially and does not necessarily vanish when the fusion rules are not satisfied. To ensure that the fusion rules are satisfied, we set a = b. Setting µ = 0 and a = b ensures that the Virasoro part of the vertex operator insertion is the identity operator. However, one can show that, in this case, the Heisenberg part of the vertex operator is an exponential of the creation part of the free boson field [5], which in general contributes to a difference between Y and W , and therefore we do not necessarily have Y = W . Setting Y = W , we pick up the contribution of the trivial part of the exponential, that is the identity, and Z bb reduces to Equation (11) is relevant to computing 1-point conformal blocks of the identity operator on the torus in subsections 3.4 and 4.15.

Unrestricted instanton partition functions for M p,p ′ ,H
3.1. AGT parameterisation. Generic models. A generic model is a conformal field theory characterised by a central charge c gen that we parametrise as In the Coulomb gas approach to computing conformal blocks in generic models, the screening charges {β + , β − }, and the background charge, −2β 0 , satisfy 9 where {p, p ′ } are the minimal model parameters, which are co-prime integers and satisfy 0 < p < p ′ , in our conventions. In the Coulomb gas approach to computing conformal blocks in minimal models with c < 1 [13,14], the screening charges {α + , α − }, and the background charge, −2α 0 , satisfy The AGT parameterisation of M p,p ′ ,H is obtained by choosing We need two distinct objects that, in Coulomb gas terms, are expressed in terms of the screening charges {α + , α − }. 1. The charge µ r,s of the vertex operator O µ that intertwines two irrep's H p,p ′ r1,s1 and H p,p ′ r2,s2 , and 2. The highest weight |a r,s of an irrep H p,p ′ r,s . Following [1,5], we use {r, s} as indices for the charge µ r,s of the vertex operator O µr,s , and {r, s} as indices for the charge a r,s of the highest weight |a r,s . These charges are parameterised in terms of α + and α − as follows Note that the same numerical values of {r, s} indicate different charge contents in µ r,s and in a r,s . In particular, 9 We use β + , β − , −2β 0 for generic model charges and reserve α + , α − and −2α 0 for the corresponding minimal model charges. We use bgen and a p,p ′ for the parameters used to describe the generic and minimal models central charges respectively, since a and b are used for other purposes in the sequel.
Unrestricted instanton partition functions give incorrect 1-point functions on the torus. Consider a conformal block in a 1-point function in M p,p ′ ,H on a torus, Figure 4. µ Following [1], this is given by the instanton partition function of the N = 2 ⋆ U (2) theory, where µ is determined by the operator insertion, and a is determined by the states of the H p,p ′ r,s that flow in the torus and determine the conformal block. When the inserted operator is the identity, that is {r, s}, then 10 µ = 0, and if Y is an unrestricted partition pair as in the original AGT prescription, then where χ p,p ′ r,s is the character of the irrep H p,p ′ r,s that flows in the torus, and a = {a, −a}, This simple example makes it clear that applying the prescription of [1] to M p,p ′ ,H without modification, leads to incorrect answers. In the following section, we find that it leads to zeros in the denominators of the summands.

Restricted instanton partition functions for
The denominator Consider the denominator z den of Z bb in (6). To look for zeros in z den , it is sufficient to look for zeros in z norm [ a, Y ] in (8). Consider B p,p ′ ,H n and focus on a channel that carries states that belong to H p,p ′ r,s .
The proof of Proposition 4.1 is based on checking the products that appear in z norm [ a, Y ] for zeros. 4.1. More notation. We set a 1 = −a 2 = a, and a 1 − a 2 = 2a. If a channel χ ι carries states from H p,p ′ rι,sι ⊗ F , then the label a ι of the corresponding highest weight is Two zero-conditions. In the sequel, we find that an instanton partition function has a zero when an equation of type Equivalently, an instanton partition function has a zero when the two conditions are satisfied, where c is some constant that needs to be determined. Given two conditions, such as (27), we need, for the purposes of comparing with known results, to re-write them as one condition.

4.3.
From two zero-conditions to one zero-condition. Consider the two conditions . If is in row-R and column-C in Y 1 , then the first condition in (28) implies that there is a cell ⊞ ∈ Y 1 , to the right of , with coordinates {R, C + A ′ }, that lies on a vertical boundary. In other words, 1. there are no cells to the right of ⊞, and 2. there may or may not be cells below ⊞. This means that column-  11 We chose the labels of the Young diagrams to be concrete. The same arguments apply under 12 All equations and inequalities in the sequel involve rows of Young diagrams, and never columns.
which is the transpose of Y 1 . The subscript C is there only because the corresponding row is a column in a diagram that we started our arguments with.
which is one condition that is equivalent to the two conditions in (28).

4.4.
One non-zero condition. Consider a function z[Y 1 , Y 2 ], of a pair of Young dia- Remark. Since we use equations such as (28) and (32) frequently in the sequel, refer to the former as 'zero-conditions', and to the latter as 'non-zero-conditions'.

4.5.
Products that appear in the denominator. Two types of products appear in z norm , 1. products in the form ∈Yi E[a i − a j , Y i , Y j , ] that we refer to as {Y i , Y j } den , and 2. products in the form In search of zeros. In the following subsections, 1. we consider the products that appear in z den , one at a time, 2. we search for possible zeros, as in subsection 4.2, 3. we find the conditions that we need to impose on the pair {Y 1 , Y 2 } in order to avoid the zeros, and 4. when there is more than one set of conditions to avoid the zeros, we choose the stronger set. That is, the set that ensures that all zeros are eliminated. We use the fact that r, s, p − a and p ′ − s are non-zero positive integers. 4.7. {Y 1 , Y 1 } den . This product does not vanish, since this requires that there is a factor that satisfies den . These products do not vanish for the same reason that {Y 1 , Y 1 } den in paragraph 4.7 does not vanish. 4.9. {Y 1 , Y 2 } den . This product vanishes if any factor satisfies which lead to the conditions Note that from conditions (35), if the Young diagram Y such that ∈ Y , which in this case is Y 1 , is sufficiently large compared to the Young diagram W such that ∈ W , which in this case is Y 2 , then the product under discussion will have more than one zero. This will be the case in the rest of the factors discussed in this section as well. 1 − s − c p ′ , where c = {0, 1, · · · } Since the row-lengths of a partition are by definition weakly decreasing, and c = {0, 1, · · · }, this is the case if Thus, we should set c = 0, and obtain Transposing the (r−1) rows and (s−1) columns that we removed earlier from Y ⊺ 1 in order to obtain Y ⊺ red,1 , we obtain (r − 1) columns and (s − 1) rows that we can add to the top and to the left of Y 1,red , respectively, to obtain which leads to the conditions (47) which, using the same arguments as in subsections 4.9 and 4.11, are possible for c = {0, 1, · · · }, ∈ Y 1 , ∈ Y 2 , and we should choose c = 0 to obtain which leads to the conditions (50) A ,2 = −r + c p, −L ,1 = 1 − s + c p ′ , which, using the same arguments as in subsections 4.9 and 4.11, are possible for c = {1, 2, · · · }, ∈ Y 2 and ∈ Y 1 , and we should choose c = 1 to obtain Comparing condition (44) with condition (51), we see that the former is stronger than the latter, for the same reasons as in paragraph 4.11.2. Thus this case does not offer new conditions on the partition pair.

4.15.
Restricted instanton partition functions give the correct 1-point function on the torus. From the discussion in paragraphs 4.7-4.14, we conclude that z den has no zeros if the conditions in (37) and (44) are satisfied. As mentioned in section 1, these conditions on partition pairs are known. They were introduced and studied in [9], and were further studied and called Burge pairs in [10]. A full and explicit derivation of the fact that the generating function of the Burge pairs, that satisfy conditions (37) and (44), is the q-series in (22), we refer the reader to Appendix A of [10].

4.15.1.
Remark. The conditions obtained in this note were written differently in [9,10] for three reasons. 1. These papers used the notation {a, b, α, β}, which in terms of the variables {r, s, p, p ′ } used in this work are a = r, b = p − r, α = s, and β = p ′ − s, 2.
The partition rows were labeled such that Y i Y i−1 , while in this note, we assume the opposite (and more conventional) labeling, and 3. The conventions in [9,10] We need to examine the conditions that each of these factors imposes on the partition pairs {Y 1 , Y 2 } and {W 1 , W 2 }.

Notation. We set
and use the subscript a (b) to indicate the parameters that appear in the conditions on the partition pairs Y ( W ) that label the states in the incoming (outgoing) channel that flows towards (away from) the vertex operator insertion. It is useful to note that, in this notation, (55) r a = m a + 1, s a = n a + 1, r b = m a + 1, s b = n a + 1 Further, to simplify the presentation, we use the notation The fusion rules. In the notation of subsection 5.2, the fusion rules are while the second choice is and the two choice are related by While the two representations are the same, for the purposes of the proofs in the sequel, we need to use one or the other, as follows. Scanning a linear conformal block B p,p ′ ,H n from left to right, one considers the building block Z ι bb , ι = 1, 2, · · · , with the Virasoro irrep labeled by {r ι−1 , s ι−1 } flowing in from the left, the vertex operator O ι of the primary field labeled by {r µ , s µ } in the middle, and the Virasoro irrep labeled by {r ι , s ι } flowing out to the right. Suppose that the charge content of the incoming primary field in χ ι−1 is fixed 14 . The charge content of primary state µ of the vertex operator O µ in the middle, and that of the outgoing primary field in χ ι are not fixed yet, and each can be chosen in one of two equivalent ways. We wish to show that we can choose these charge contents in such a way that that the upper bounds in (59) are satisfied. This will simplify our proofs in the sequel.
If {m ι , n ι } and {m µ , n µ } are such that the upper bounds in (59) are satisfied, then use this choice. If {m ι , n ι } and {m µ , n µ } are such that the upper bounds in (59) are not satisfied, we choose the dual representation of the vertex operator in the middle and the outgoing Virasoro irrep 15 . In other words, p − 1 2   m ι−1 + m µ + m ι   − 2, that does not satisfy the upper bound, becomes using the triangular conditions (58), and similarly Now the charge content of the outgoing primary field is fixed and goes on to become the incoming primary field of Z ι+1 bb or the primary state of O n+3 . Thus we can always choose the charge contents such that the upper bounds in equations (63) and (64).
14 Starting from Z 1 bb , we can choose the charge of the highest weight state in O 0 either way, but for the purposes of this proof, it is sufficient to consider an arbitrary Z ι bb , ι = 1, 2, · · · , n + 1, and take the charge of the primary field in χ ι−1 to be fixed. 15 Remember that the charge content of the incoming primary field is given and cannot be changed.
In the following subsections, we consider the conditions that products in the numerator must satisfy to be non-zero 16 . 5.5. {Y 1 , W 1 } num . This product vanishes if any factor satisfies is p ′ − 2, thus the stronger condition corresponds to c = 1, and we obtain two zero-conditions that we can write as one non-zero-condition, This product vanishes if any factor satisfies 5.7. The remaining six products. The analysis of the remaining six products is identical to that in subsections 5.5 and 5.6, and it suffices to list the non-zero-condition in each case.

5.8.
If the denominator is zero, then the numerator is zero. The non-zeroconditions on the {Y i , W j } num and {Y i , W j } ′ num products that appear in the numerator can be combined in pairs to produce non-zero-conditions on {Y i , Y j } and {W i , W j } pairs, i = j pairs also in the numerator, that can be compared to the first and second non-zero-conditions (37) and (44) obtained from the denominator. (67) and (76). We eliminate W ⊺ 1 to obtain a non-zero condition on Y 1 and Y 2 by re-writing (67) as which is a weak version of condition (37).
which is a weak version of condition (37).  (67), which is a weak version of condition (44). (72) and (75), which is condition (44). The stronger condition in each of the above cases is one of the Burge conditions. Thus, when the denominator z den of the building block partition function Z bb is non-zero, then the numerator z num is also non-zero. The reverse is not true.
Note that the above result is similar but different from that in [7], where Zamolodchikov argues that 1. The conformal block B gen n is a meromorphic function in ∆ a , the conformal dimension of the Virasoro irrep that flows in a channel, and that B gen n has only simple poles at ∆ a = ∆ ar,s , where a r,s = − 1 2 [rα + + sα − ], and 2. If the fusion rules are satisfied, then the residue at each pole vanishes.
Our result is that 1. When a summand in Z bb has a zero in the denominator, and the fusion rules are satisfied, then it also has a zero in the numerator. This is independent of Zamolodchikov's statement, since in the latter, the whole sum vanishes rather than just the summand with the zero in the denominator. 2. Zamolodchikov has argued that B gen n has only simple poles, while, as far as we can tell, summands in Z bb can have poles of order greater then 1.

An Ising conformal block
In this section, we set p = 3 and p ′ = 4, so that the minimal model component M p,p ′ of the conformal field theory M p,p ′ ,H under consideration, is the Ising model. In this case, there are three primary fields to form conformal blocks from. They can be labeled as follows. {r, s} = {1, 1} is the identity operator ½, {r, s} = {1, 2} is the spin operator σ and {r, s} = {1, 3} is the thermal operator ψ. Explicit expressions for conformal blocks can be found in [15] and references therein. Consider the 6-point conformal block of σ fields in Figure 6. In this case, α + = 4/3, α − = − 3/4, and α 1,2 = − 1 2 α − = − 1 2 3/4, and following [15], setting the coordinates z 0 = 0, z 1 = q 1 q 2 q 3 , z 2 = q 2 q 3 , z 3 = q 3 , z 4 = 1, z 5 = ∞, The instanton partition function should equal the product of the Ising conformal block and a contribution from the Heisenberg algebra H 17 . Using e.g. [5], equation (1.9), Calculating the expansion of Z up to degree 2 in each variable, we find that result coincides with the sum of non-zero terms in the instanton partition function. Using the notation the q 2 1 q 2 2 q 2 3 -term, as an example, is while all other terms, that satisfy Proposition 4.1 and the condition |Y 1 | + |Y 2 | = |Y 3 | + |Y 4 | = |Y 5 | + |Y 6 | = 2, vanish.
7. An explanation, based on a conjecture, of why we obtain M p,p ′ ,H conformal blocks As mentioned in section 1, there is a proof of AGT in the context of conformal blocks in M gen,H with non-degenerate intermediate Virasoro representations in [5]. In this subsection, we use 1. results from [5], 2. Proposition 4.1 of section 4, 3. that the generating function of Burge pairs is the character of H p,p ′ ,H r,s [10,11], and 4. Conjecture 7.1 below, to explain why restricting the summation to Burge pairs as in (4) leads to conformal blocks in M p,p ′ ,H . Proving 3 would amount to proving that restricting to Burge pairs leads to conformal blocks in M p,p ′ ,H , but this is beyond the scope of this work.
Consider the Verma module H a over V gen ⊕ H generated by highest-weight vector |a , where L k , and a k , k ∈ Z, are generators of V gen and H, respectively, and Conformal blocks are defined in terms of vertex operators O µ (z) : H a → H b , that in turn are defined by the commutation relations AGT was proven in [5] for generic central charge c gen , in the following sense Proposition 7.1. Following [5], there exists an orthogonal basis J Y ∈ H a labeled by pairs of Young diagrams such that the matrix elements of vertex operator O µ satisfy From this proposition, it follows that The vectors J Y can be written in the standard basis of the Verma module, where the summation is over partition pairs {λ, µ} such that |λ| + |µ| = |Y 1 | + |Y 2 |.  [5]. Further motivation is provided by the relation between Jack symmetric functions J α Y [16] and J Y , for α = b 2 gen [5]. Namely, from Macdonald's conjectures, proved by Haiman [17], the coefficients of J α Y , in the standard basis, are polynomial in α, so it is natural to expect that J Y satisfies an analogous property.  (101) where B degen,H n is an n-channel generic model conformal block, such that some of the channels carry degenerate intermediate representations, and ′ indicates that, for channels that carry degenerate representations, the sum is restricted to partition pairs that satisfy Proposition 8.1.
The proof of Proposition 8.2 is based on the same line of arguments as in section 7 but without requiring a conjecture analogous to Conjecture 7.1. Indeed, since the coefficients of J Y are polynomial in a, we can set a = a r,s in (99). The vectors J Y for which z norm [ a, Y ] = 0, belong to the kernel of the Shapovalov form on the Verma module H a . Let H gen r,s denote the irreducible quotient of H a . The vectors J Y , where Y satisfy Proposition 8.1, project to the module H gen r,s . In [11], Feigin et al. proved that the generating function of the Y pairs that satisfy Proposition 8.1 is the character of H gen r,s , therefore the corresponding vectors J Y form a basis in H gen r,s . Using (97) and (98), we obtain the expression (101) for the conformal block for degenerate representations. 9. Comments and remarks 9.1. q-gl ∞ Ding-Iohara. Let E be the algebra called q-deformed gl ∞ in [11], and Ding-Iohara in [18] 19 . Following [18], operators in the rank-2 representations F u1 ⊗ F u2 of E generate the sum of a q-deformed Virasoro algebra and a q-deformed Heisenberg algebra. On the other hand, following [11] [Theorem 3.8], for special values of parameters u 1 and u 2 , as well as q 1 and q 3 of E, this representation has a sub-representation with a basis labeled by Burge pairs. In is natural to expect that in the limit q 1 , q 3 → 1, the basis constructed in [11] reduces to the basis J Y described in section 7.

9.2.
Higher-rank AGT-W. AGT was extended to theories based on the higher rank algebras W N ⊕ H, N > 2, by Wyllard in [19], and by Mironov and Morozov in [20]. In this note, we chose to simplify the presentation by focusing on Virasoro minimal models, but we expect that our analysis extends without essential modification to minimal models based on W N algebras with N > 2. We conjecture that the restricted partitions that are relevant to these extended cases are those that appeared in [11].
9.3. The work of Alkalaev and Belavin. In [21], Alkalaev and Belavin independently suggested the Virasoro result in (4) in the 4-point conformal block case. They proved a proposition equivalent to 4.1, made the same comment on conformal blocks in generic models with degenerate intermediate representations as in section 8, albeit without proving an analogue of Proposition 8.2 and made the same W N conjecture as in subsection 9.2. 9.4. Previous works on AGT in minimal models. There are two previous works on AGT in minimal models that we are aware of. In [8], Santachiara and Tanzini identify Moore-Read wave functions, which are minimal model conformal blocks of {r, s} = {1, 2} and {2, 1} vertex operators, with Nekrasov instanton partition functions, AGT is applied without modification to these conformal blocks and ill-defined expressions are made welldefined using a deformation scheme, as outlined in subsection 1.3. In [22], Estienne, Pasquier, Santachiara and Serban interpret W n ⊕ H minimal model conformal blocks of {r, s} = {1, 2} and {2, 1} vertex operators as wave functions of a trigonometric Calogero-Sutherland models with non-trivial braiding properties, and find that the excited states are characterized by (n+1)-partitions, just as in AGT. While Estienne et al. use different notation from ours, preliminary checks indicate that their partitions can be translated to the Burge pairs used in this note, for n = 2, and {r, s} = {1, 2} or {2, 1}. 9.5. Geometry. Let M(r, N ) be the moduli space of U (r) instantons on R 4 . The instanton partition function for n i=1 U (2) ι gauge theory equals the generating function of equivariant integrals over M(2, N 1 ) × · · · × M(2, N n ), where the equivariant integral is taken with respect to the torus T = (C * ) 2 × (C * ) 2 1 × (C * ) 2 2 × · · · × (C * ) 2 n , where the first (C * ) 2 acts on C 2 , and (C * ) 2 i acts on the i-th instanton moduli space M(2, N i ) by constant gauge transformation. These equvariant integrals are computed using localization and are given by the sum over torus fixed points. These points were labeled by n pairs of Young diagrams Y 1 , · · · , Y n . The parameters ǫ 1 , ǫ 2 , and a i are the coordinates on t = Lie(T). In the M p,p ′ ,H case, ǫ 1 and ǫ 2 are linearly-dependent, and a i j is given by (18). Geometrically, this means that we are considering the one-dimensional subgroup C * ǫ1,ǫ2, a i ⊂ T.
The function z norm [ a, Y ] is the determinant of the vector field with coordinates {ǫ 1 , ǫ 2 , a 1 , a 2 } on the tangent space of the point labeled by Y . The condition z norm [ a, Y ] = 0 is equivalent to the fact that corresponding point is an isolated fixed point of the one dimensional torus C * ǫ1,ǫ2, a i . Therefore, summing over Burge pairs is equivalent to summing over the isolated torus fixed points.