From topological strings to minimal models

We glue four refined topological vertices to obtain the building block of 5D $U(2)$ quiver instanton partition functions. We take the 4D limit of the result to obtain the building block of 4D instanton partition functions which, using the AGT correspondence, are identified with Virasoro conformal blocks. We show that there is a choice of the parameters of the topological vertices that we start with, as well as the parameters and the intermediate states involved in the gluing procedure, such that we obtain Virasoro minimal model conformal blocks.


We plan to start from the refined topological vertex and obtain Virasoro A-series minimal model conformal blocks times Heisenberg factors.
The refined topological vertex is used to construct refined topological string partition functions Z re f top [1,2,3].The latter partition functions can be identified with 5D quiver gauge theory instanton partition functions, Z 5D instanton which reduce in the limit R → 0, where R is the radius of the M-theory circle, to their 4D counterparts Z 4D instanton [4,5,3,6,7,8].The partition functions Z 4D instanton can be identified via the AGT correspondence with 2D Virasoro generic conformal blocks times Heisenberg factors, B gen, H [9].One can choose the parameters in Z 4D instanton , and restrict their intermediate states to obtain 4D instanton partition functions Z 4D, min  instanton that can be identified with Virasoro A-series minimal conformal blocks times Heisenberg factors B min, H [10,11].
From the above chain of connections, it is expected that one can start from Z re f top , choose the parameters and restrict the intermediate states to obtain 5D instanton partition functions Z 5D, min  instanton that reduce to Z 4D, min instanton , and from these obtain B min, H .In this note, we work out the above chain of connections, which is the same as extending the result of [10,11] by starting from the refined topological vertex, rather than from Z 4D building.block, the normalised contribution of the bifundamental hypermultiplet to Z 4D instanton .Since Z 4D building.blockcan be constructed from refined topological vertices, in the R → 0 limit, it is less elementary than the refined topological vertex, and the construction in this note is in this sense more elementary than that in [10,11].
The Virasoro A-series minimal conformal blocks that we obtain can be computed using other methods, but we view this note as an accessible introduction to one more approach to the minimal conformal blocks that, hopefully, can be extended to minimal blocks beyond what can currently be computed, including the W N minimal blocks that do not satisfy the conditions of [12,13].
Abbreviations and notation.We focus on U(2) quiver gauge theories, and simply say 'the instanton partition functions' .We assume that every conformal block, whether Liouville or minimal, includes a factor from a field theory of a free boson on a line, and omit 'times Heisenberg factors'.The normalised contribution of the bifundamental hypermultiplet is understood as the building block of the instanton partition function, as all other contributions can be obtained from it.We simply say 'the bifundamental partition function' .
Topological string-related notation.C λ µ ν [q, t] is the refined topological vertex.W V W ∆ [q, t, R] is a basic web refined topological string partition function.V and W are two pairs of Young diagrams.∆ is a set of three Kähler parameters.q and t are deformation parameters.R is the radius of the M-theory circle.Z re f top is the topological string partition function.

Gauge theory-related notation. Z 5D
instanton is the 5D U(2) quiver gauge theory instanton partition functions.Z 4D instanton is the 4D U(2) quiver gauge theory instanton partition functions.Z 4D building.blockis the normalised contribution of the bifundamental hypermultiplet.The parameters ǫ 1 and ǫ 2 are deformation parameters.
Conformal field theory-related notation.B gen, H are 2D Virasoro generic conformal blocks times Heisenberg factors.B gen, min are 2D Virasoro A-series minimal model conformal blocks times Heisenberg factors.The parameters p and p ′ are coprime positive integers that label a Virasoro minimal model.The parameters r and s are integers that satisfy 0 < r < p, 0 < s < p ′ and label Virasoro minimal model highest weight representations.

The refined topological vertex
We recall basic definitions related to Young diagrams and Schur functions, followed by the definition of the refined topological vertex.(2,4), in the lower right quadrant of the plane, is in mu, but not in ν, and has A µ = 0, where the sum is over all parts of Y2 .Further, to simplify the equations, we define • • • }, and two variables x and y, we define the 'exponentiated' sequences 1.1.9.A function of the arm-lengths and the leg-lengths.Given a Young diagram λ, we define the function Remark.The expression in the middle of Equation 1.6 corresponds to splitting the cell at the corner of a hook in the partition λ into two halves, then attaching one half to the arm of that hook to form a half-extended arm of length A + ,λ = A ,λ + 1 2 , and attaching the other half to the leg of that hook to form a half-extended leg of length L + ,λ = L ,λ + 1 2 .The expression on the right corresponds to attaching the cell at the corner of a hook to the arm of that hook to form an extended arm of length A ++ ,λ = A ,λ + 1.The length of the leg of that hook remains L ,λ .
where c λ µν are Littlewood-Richardson coefficients defined by (1.9) the refined vertex is not cyclically-symmetric, or at least not manifestly so, and the external legs do not contribute on equal footing.In particular, the contribution of the leg labelled by ν is distinguished from the other two.The original, unrefined topological vertex of Aganagic et al. is recovered by setting q = t.1.3.3.On framing vectors and framing factors.In defining the refined topological vertex, one labels each of the three boundaries of a vertex by a framing vector that indicates a possible twisting of the boundary.On gluing two vertices along a common boundary, there is in general a framing factor that accounts for a possible mismatch in the orientations of the relevant framing vectors.In this note, we glue vertices such that we do not require framing factors.

A 5D U(2) basic web partition function
We glue four copies of the refined topological vertex to obtain a 5D basic web that can be used as a building block of U(2) topological string partition functions.
] is trivalent and depends on two parameters q and t.The segments are labelled by three partitions λ, µ and ν, such that λ is assigned to the vertical segment, µ is assigned to the segment that follows in a clockwise direction, and ν to the segment that follows.
2.1.Geometric engineering.Following [17], we write the normalised U( 2) basic web partition function as where the numerator is where we use and the denominator but with all external partition pairs set to empty.As shown in Figure 2.1, the basic web has two external horizontal legs coming in from the left, two external horizontal legs going out to the right, and a pair of vertical legs, one going up and one down.The horizontal external legs on the left are assigned partitions {V 1 , V 2 }, the horizontal external legs on the right are assigned partitions {W 1 , W 2 }.The internal lines are assigned parameters Q 1 , Q M and Q 2 , and partitions ξ 1 , ξ M and ξ 2 , from top to bottom.The vertical external legs are assigned empty partitions {φ, φ}.
Each trivalent vertex corresponds to a refined topological vertex C λ µ ν .Our convention is such that each vertex has one vertical leg, we associate λ to that vertical leg, regardless of whether it is internal or external, pointing upwards or downwards, then µ and ν to the remaining two legs, encountered sequentially as we start from the vertical leg and move around the vertex clockwise.Using Equation 1.10, we re-write this as 2) basic web diagram.Each external line is labeled by a partition.Each internal line is labeled by a partition and a Kähler parameter.This basic web can be glued to form topological partition functions. (2.4) where we used the fact that for an empty partition φ, the skew partition φ/η exists only for η = φ, a sum over η trivialises, and the skew Schur function s φ/η = s φ = 1.

Two skew Schur function identities.
To evaluate the sums in Equation 2.4 for w num , we need the two identities4 as well as the property that (2.7) which follows from the definition of the skew Schur function.
2.1.2.The basic web in product form.We evaluate the sums over the right hand side of Equation 2.4, using 2.1.3.The sums over ξ 1 and ξ 2 . (2.8) 2.1.4.The sum over ξ M .We re-write this in terms of a sum over a new set of partition τ, (2.10) 2.1.5.The sums over η 1 and η 2 . (2.11) (2.12) 2.1.6.The sum over τ.We finally evaluate the sum over the partitions that were introduced in an intermediate step above, (2.13) Remark.The two products, on the right hand side of the above equation, that involve Q M only as a coefficient are different from the remaining products.
In this notation, the expression for in Equation 2.4, up to replacing each product i, j by a normalised product ′ i, j .
2.1.9.From infinite to finite products.Using Equation 2.15, we have the following identities [19,2] 5 .Firstly, Note that while the product on the left of Equation 2.16 is normalised in the sense of Equation 2.15, the remaining products are not.
2.1.10.The 5D bsic web in product form.Using the identities in Equation 2.16, we can write (2.17) where, using where we have used the second equality in Equation 2. 16 to put the products in the above uniform form.Further, the denominator is where the first four products on the right hand side of Equation 2.19 are due to the product

Remark. One can glue copies of the basic web partition function W 5D
V W ∆ [q, t, R] in several ways.In this note, we restrict our attention to gluing linearly or cyclically, to form linear or cyclic U(2) quiver gauge theories, as described in paragraphs 6.0.1 and 6.0.2.We do not, for example, glue basic webs to form a Hirzbruch surface.

A 4D U(2) basic web partition function
We take the R → 0 limit of the 5D basic web partition function to obtain its 4D analogue.
3.1.Two parameters.We take the relationship between the parameters q and t of the refined vertex and the parameters ǫ 1 and ǫ 2 of the instanton partition function to be, where R, the radius of the M-theory circle, plays the role of a deformation parameter.We write R], then take the limit R → 0. The prefactor on the left hand side of Equation 2.17 tends to 1 in the limit R → 0, and we obtain where, using

The building block of the 4D U(2) quiver instanton partition function
We recall the normalised contribution of the bifundamental hypermultiplet which acts as a building block of the instanton partition function.
In the notation of [11], the normalised bifundamental partition function Z 4D building.blockis where a = {a, −a}, b = {b, −b}, and we use V ′ and W ′ for partition pairs that we will relate shortly to the pairs that appear in the 4D basic web.We refer to [11] for brief explanations of the parameters that appear in Z 4D building.block .Defining (4.2) the numerator z num , as given in [11], is The denominator z den , as given in [11], is , and multiply each factor by −1, which is possible since the number of factors is even by construction, we obtain (5.1) which leads to the identification (5.2) building.block .The denominators Using the identification of parameters obtained in Equation 5.2 in w den [V, W, ∆] and z den as given in Equations 3.4 and 4.4, it is clear that these two functions are not the same.However, what matters is not the denominator of s single factor, but the product of all denominators, as we explain below.
The denominator w den [V, W, ∆] is a natural object, as we can see in the derivation in Section 2. On the other hand, the denominator z den was obtained in [11] by taking the full denominator that appears in expressions for the 4D U(1) linear and cyclic quiver instanton partition functions and factoring that into denominators for the contributions of the bifundamental hypermultiplets.Such a factorisation is not unique and any factorisation is allowed for as long as the product of all factors is equal to the full denominator of the original expression.
In this work, to identify W 4D and Z 4D building.block, we need to work in terms of another denominator z ′ den that is related to z den by a factor F , such that the product of all normalisation factors that appear in a conformal block is equal to 1. Consider the abbreviations In this notation, z den and w den are and and define building.block is constructed such that 1.It has the same numerator as Z 4D building.block, which is the same as that of W norm VW∆ , when we choose the parameters as in Equation 5.2 and 2. It has the same denominator as W norm VW∆ , also when we choose the parameters as in Equation 5.2.Since the denominator of Z ′ building.blockis not manifestly the same as that of Z 4D building.block, we need to show that gluing copies of Z ′ building.blockto build a topological partition function, leads to the same result obtained by gluing copies of the original Z building.block .F can be written in a simpler form as follows, (6.6) F right and F le f t satisfy the obvious properties (6.9) The physical objects that we are interested in are the conformal blocks which are constructed by gluing copies of Z building.block[11].We need to show that gluing copies of Z ′ building.blockleads to the same result, which will be the case if products of the normalisation factors trivialise.This will follow directly from Equations 6.9 and 6.10.There are two cases to consider, the linear conformal block case and the cyclic conformal block case.
building.blockleads to the same conformal blocks as Z building.block .6.0.3.The denominator z ′ den and the Burge conditions.In [10,11], it was shown that for the choice of parameters that leads to Virasoro A-series minimal conformal blocks, the denominator z den will contain non-physical zeros, unless we restrict the partition pairs, that Z 4D, min building.blockdepends on, to obey Burge conditions.These conditions were derived in [11] using z den rather than z ′ den .Using z ′ den leads to the same conditions, since the product of all z ′ den is the same as the product of all z den that show up in the conformal block.In Section 9, we outline the derivation of the Burge conditions from z ′ den .
7. Restricted instanton partition functions for M p, p ′ , H .The parameters We recall the choice of parameters such that Z 4D building.blockreduces to Z 4D, min building.blockwhich is the building block of Virasoro A-series minimal model conformal blocks times Heisenberg factors.

AGT parameterisation of minimal models.
A minimal model M p, p ′ , based on a Virasoro algebra V p, p ′ , characterised by a central charge c p, p ′ < 1, that we parameterise as , where {p, p ′ } are co-prime integers that satisfy 0 < p < p ′ .In the Coulomb gas approach to computing conformal blocks in minimal models with c p, p ′ < 1 [20,21], the screening charges {α + , α − }, and the background charge α back.ground, satisfy The AGT parameterisation of M p, p ′ , H is obtained by choosing 7.2.Two sets of charges in minimal models.We consider two types of charges that, in Coulomb gas terms, are expressed in terms of the screening charges {α + , α − }. 1.The charge a r,s of the highest weight |a r,s of the irreducible highest weight representation H p, p ′ r,s , and 2. The charge µ r,s of the vertex operator O µ that intertwines two highest weight ireducible representations H p, p ′ r 1 ,s 1 and H p, p ′ r 2 ,s 2 .These charges are parameterised in terms of α + and α − as follows

From gauge theory parameters to minimal model parameters
We compare the parameters of W 4D and the parameters of Z 4D building.block .We set We write the denominator w den [V, W, ∆, ǫ 1 , ǫ 2 , g → 0] as w den [V, W, ∆, α + , α − ], and, following [11], we check the conditions required so that that it has no zeros.
Using α + and α − , let us write where we have used Consider the denominator w den [V, W, ∆, α + , α − ] in Equation 9.1, on a product by product basis.We need to check the conditions under which any of these products has a zero, then find the restriction that are necessary and sufficient to remove these zeros.The reasoning that we use to obtain these conditions is the same as that in [11].There are eight products to consider.9.1.The initial four products.In each of the initial four products, the product is over the cells inside a single diagram, thus the arm length A and the leg length L in each of these factors is non-negative.Since α − < 0 < α + , and there is a term α 0 > 0 in each factor, the minimal value of each of these factors is greater than zero.Thus there can be no zeros from these factors.To consider the remaining four factors, we require some preparation.9.2.Two zero-conditions.Following [11], we note that, since α − < 0 < α + , any factor of the type that appears in Equation 9.1 has a zero when an equation of type (9.4) where C + , C − ∈ , is satisfied.Since p and p ′ are coprime, α − and α + are É, the condition in Equation 9.4 is equivalent to the two conditions (9.5) are satisfied, where c is a proportionality constant that needs to be determined.9.3.From two zero-conditions to one zero-condition.Consider the two conditions (9.6) −A ,i = A ′ 0, L , j = L ′ 0 which are satisfied if i j, Y i , and ∈ Y j .If is in row-R and column-C in Y j , then the second condition in (9.6) implies that there is a cell ⊞ ∈ Y 1 , strictly below , with coordinates {R + L ′ , C}, such that there are no cells strictly below ⊞.Since there may, or may not, be cells to the right of ⊞, row-(R + L ′ ) in Y j has length at least C, (9.7) From the definition of A , i , we write the first condition in (9.6) as −A , i = A ′ = C − y i R , that is, C = A ′ + y i R , and using (9.7), we obtain y j R+L ′ A ′ + y i R , which we choose to write as (9.8) The condition in Equation 9.8 is equivalent to the two conditions in Equation 9.6.
vanishes if any factor satisfies (9.12) Given that s and p ′ are non-zero positive integers, the second equation in Equation 9.13 admits a solution only if c = 0, 1, • • • The first equation in Equation 9.13 admits a solution if V 2 .9.5.1.From two zero-conditions to one non-zero-condition.Following paragraphs 9.3 and 9.4, the two zero-conditions in (9.13) are equivalent to one non-zero-condition, The stronger condition.Equation 9.14 is the statement that to eliminate the zeros, we want R+s −r.Thus, we should set c = 0, and obtain (9.15) 9.6.The second product. (9.16) Given that s a and p ′ are non-zero positive integers, the second equation in Equation 9.18 admits a solution only if c = 1, • • • The first equation in Equation 9.18 admits a solution if V 1 .9.6.1.From two zero-conditions to one non-zero-condition.Following paragraphs 9.3 and 9.4, the two zero-conditions in Equation 9.18 are equivalent to one non-zero-condition, 7. The third product.
(9.21) (9.25) which leads to the conditions 9.9.The Burge conditions.Equations 9.15 and 9.20 are conditions on the partition pair V on one side of Z building.block, while Equations 9.24 and 9.28 are conditions on the partition pair W on the other side of Z building.block .When copies of Z building.blockare glued to form conformal blocks, partition pairs on one side are identified with partition pairs on the other side.Thus each partition pair must satisfy all conditions.However, these conditions are not independent as two of them are satisfied when the other two are satisfied.More specifically, following paragraphs 9.3 and 9.4, we can see that it is sufficient to enforce the two conditions in Equations 9.20 and 9.24, which we write in terms of a partition pair {µ, ν} that could be on either side of Z building.block .Partition pairs that satisfy the conditions in Equation 9.29 first appeared in the work of W H Burge on Rogers-Ramanujan-type identities [22].The appeared in earlier studies of Virasoro characters in [23,24,25] and more recently in the context of the AGT correspondence in [10,11].

Comments and open questions
Outline of result.We can generate conformal blocks of Virasoro A-series minimal models, labelled by the co-primes p and p ′ , times a Heisenberg factor, as follows.1. Start from the refined topological vertex of [3] defined in Equation 1.10, 2. Glue four copies of the refined topological vertex to produce a 5D U(2) basic web partition function, then take the R → 0 limit, to obtain its 4D counterpart W 4D VW∆ as in Equations 2.17 The refined topological vertex of Awata and Kanno.We have used the refined topological vertex of Iqbal, Kozcaz and Vafa [3], but could have equally well used that of Awata and Kanno [1,2].The two vertices are equivalent as explained in [26].
Layers.The topic discussed in this note is vast and consists for many layers.We could have started our discussion from M-theory and used the language of M5 branes, but we decided to stay away from this, in this short note.Instead, we started from A-model topological strings, which live in a corner of the M-theory.From the refined topological vertex and topological strings, we obtained the building block of the instanton partition function of a 5D quiver gauge theory.We could have used the K-theoretic version of the AGT correspondence to obtain the minimal model analogues of the q-deformed Liouville conformal blocks discussed in [27,28,29,30,31,32].Instead, we skipped the q-deformed blocks, took the 4D limit, and used the 4D version of AGT to obtain minimal model conformal blocks.The M-theoretic origins of the minimal conformal blocks and their q-deformations should be topics of separate studies.
Interpretation.Missing from this note is an interpretation of the minimal model parameters in topological string or gauge theory terms.A complete interpretation will require working in M-theory terms, which would have taken us far a field and lies outside the scope of this work.

1. 1 . 5 . 1 . 1 . 6 .
Remark.A ,Y and L ,Y can be negative when lies outside Y. Partition pairs.A partition pair Y is a set of two Young diagrams, {Y 1 , Y 2 }, as in Figure 1.2, where |Y| = |Y 1 | + |Y 2 | is the total number of cells in Y.

8. 1 .
The fusion rules.For completeness, let us mention the fusion rules.In the notation(8.4)m i = r i − 1, n i = s i − 1 the fusion rules take the simple form(8.5)m a + m b + m µ = 0 mod 2, n a + n b + n µ = 0 mod 2,where the triple {m a , m b , m µ } satisfies the triangular conditions(8.6)m a + m b m µ , m b + m µ m a , m µ + m a m bwith analogous conditions for the triple {n a , n b , n µ }.9.Restricted instanton partition functions for M p, p ′ , H .The partition pairs − c p, where c = {0, 1, • • • } Since the row-lengths of a partition are by definition weakly decreasing, and c p, where c = {1, 2, • • • }.Since the row-lengths of a partition are by definition weakly decreasing, and c
block , we obtain the same result as using copies of Z building.blockupto a factor(6.11)

One non-zero condition.
Consider a function f Y i ,Y j , of a pair of Young diagrams Y i and Y j , i j, such that f Y i ,Y j = 0, ifand only if (9.8) is satisfied.This implies that f Y i ,Y j 0, if and only if Y i and Y j satisfies the complementary condition ′ 9.4.1.Remark.In the sequel, we refer to Equation 9.6 as 'zero-conditions' , and to Equation 9.10 as 'a non-zero-condition' .Next we consider the latter four products on the right hand side of Equation 9.1.
9.6.2.The stronger condition.Equation (9.19) is the statement that to eliminate the zeros, we want