A Combinatorial Study on Quiver Varieties

This is an expository paper which has two parts. In the first part, we study quiver varieties of affine $A$-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating function of Poincar\'e polynomials of quiver varieties in rank 1 cases. Our main tools are cores and quotients of Young diagrams. In the second part, we give a brief survey of instanton counting in physics, where quiver varieties appear as moduli spaces of instantons, focusing on its combinatorial aspects.


Introduction
Quiver varieties, which were introduced by Nakajima [35], stand at the corner of a beautiful interplay between combinatorics, geometry, representation theory, and mathematical physics. This expository paper has two purposes. The first one is to study quiver varieties of type A (1) l−1 from a combinatorial viewpoint (Sections 2, 3 and 4). The second is to give a brief survey of instanton counting in physics, especially results of Maeda et al. [30,31,32], and to discuss combinatorial aspects of them (Section 5).
The goal in the first part is to compute Poincaré polynomials of quiver varieties in rank 1 case via a combinatorial method. Note that Poincaré polynomials of quiver varieties have already been obtained by Nakajima (see Remark 4.11), and our method just confirms his result. However, we hope that our combinatorial viewpoint gives a new insight into the subject. Our strategy is as follows. There is natural torus actions on quiver varieties. Then by localization principle, we can compute topologies of them by studying the fixed point set. We see that they further reduce to enumeration of Young diagrams. More importantly, we compute the Poincaré polynomial of a quiver variety not one by one, but a generating function of them. For that purpose, we use some combinatorial machineries, which are cores and quotient of Young diagrams. They arise from a division algorithm for Young diagrams analogous to that for integers and play important roles in our discussions. They also appear in studies of supersymmetric gauge theories by Maeda et al. [30,31,32] (in slightly different terminologies). The purpose of the second part is to explain it from a mathematical viewpoint. We hope that our exposition is also useful for physics oriented readers.
We introduce quiver varieties of type A (1) l−1 as follows. Let Γ be a cyclic group of order l. Then Γ acts on the framed moduli space M r (n) of torsion free sheaves E on CP 2 with rank r, c 2 = n, where the framing is a trivialization at the line at infinity l ∞ . We consider its Γ-fixed point components, (1.1) where w and v are isomorphism classes of Γ-modules. These M w (v)'s are quiver varieties of type A (1) l−1 . Note that this definition, which is taken from [41], is slightly different from the original one. The original definition was based on the so-called ADHM description of instantons on so-called ALE spaces [25].
The space M r (n) was also introduced by Nakajima [36], as a resolution of the moduli space of instantons on R 4 . Then quiver varieties can be seen as resolutions of the moduli spaces of instantons on C 2 /Γ. It turns out that generating functions of Euler characteristics of quiver varieties can be identified with the partition functions of certain supersymmetric field theory, which is important in mathematical physics. This is one of our motivations to study quiver varieties. We summarize some basic facts of instanton counting in Section 5. Here we should mention a beautiful work by physicists Fucito et al. [10]. They studied instanton moduli spaces on ALE spaces by a combinatorial method. The present work grow out of an attempt at rigorous understanding of their idea mathematically. In this paper, we give a precise mathematical formulation of the idea in [10], and complete proofs to the results announced there.
Let us explain more detail about our analysis in the first part. There is a natural action of (r + 2)-dimensional torus T on a quiver variety M w (v), induced from that on M r (n). We study their fixed point sets. It can be shown that fixed points on a quiver variety M w (v) are parametrized by a certain set P w (v) of Young diagrams, which is determined by the coloring of Young diagrams (see Section 2 for definition). Then localization theorem tells us that we can compute Euler characteristics of quiver varieties via enumeration of P w (v). We can also compute Poincaré polynomials by studying tangent spaces at fixed points. As mentioned above, what we want to compute is the generating function of them with respect to v: The problem is that how to get a closed formula for Z w (t, q, r). In Section 4.2, we give a closed formula for Z w (t, q, r) when w is a class e j of 1-dimensional Γ-module. First, we show a remarkable factorization property of the generating function of the Poincaré polynomials in Theorem 4.6. It decomposes into the product of the contributions of cores and those of quotients of Young diagrams in P e j (v): Z e j (t, q, r) = Z quot (t, q, r)Z core e j (q, r).
Next, we compute each part separately. Computations of the core part Z core e j (q, r) is purely combinatorial. On the other hand, we use geometry of Hilbert scheme of points, to get the quotients part Z quot (t, q, r). Then we obtain a closed formula for Z e j (t, q, r) in Theorem 4.10, where Z e j (t, q, r) is expressed as a ratio of the Riemann theta function and the Dedekind eta function. This result is due to Nakajima, as we mentioned earlier.
The paper is structured as follows. We devote Section 2 to combinatorial preliminaries. The most important notions in Section 2 are cores and quotients of Young diagrams. In Section 3, we introduce quiver varieties and torus actions on them. We study their fixed point sets. In Section 4, we compute two kinds of global invariants on quiver varieties. In Section 4.1, we consider 'equivariant volumes' of quiver varieties, which are defined in Section 4.1.1. In Section 4.2, we study generating functions of Poincaré polynomials and Euler characteristics of quiver varieties, by using combinatorics prepared in Section 2. In Section 5, we summarize developments of instanton counting in physics (up to 2005) and discuss their combinatorial aspects. In Appendix A, we collect some facts about Hilbert scheme of points which are used in this paper.

Note added in 2017
The original version of this paper was submitted to the arXiv in 2005. In this version, we have updated the bibliography and added references [6,14,15,16,53] in order to reflect some recent developments in the subject. We briefly comment on them.
(i) The generating function of Euler characteristics of quiver varieties is re-considered in [15]. The argument in loc. cit. is also combinatorial but different from ours. The method of loc. cit. is used in [14,16] to compute the generating functions of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D.
(ii) The study of instanton counting in gauge theories is still an active field of research in relation to geometric representation theory. For recent advances, see, e.g., [6,53] and references therein. In [53, Section 2], results of this paper are nicely recast into the framework of instanton counting on quotient stacks [C 2 /Z l ]. Their relations to instanton counting on ALE spaces (i.e., minimal resolutions of C 2 /Z l ) are also clarified in loc. cit.

Preliminaries on combinatorics
In this section, we first fix some notations on Young diagrams which are used through the paper. We also introduce the notion of quotients and cores for Young diagrams. Our basic references in this section are [

Young diagrams
A partition is a sequence of non-increasing non-negative integers λ = (λ 1 , . . . , λ m ). The corresponding Young diagram is a collection of rows of square boxes which are left adjusted and with λ i boxes in the ith row for i = 1, . . . , m. We identify partitions with their corresponding Young diagrams.
(i) Let P be the set of Young diagrams. Then we have where P(n) = {Y ∈ P | |Y | = n}.
(ii) Let P l be the set of l-tuples of Young diagrams. Then we have where P l (n) = {Y ∈ P l | |Y | = n}.

Functions on Young diagrams
For a Young diagram Y , we define l Y (s) and a Y (s) as where s = (h, k) ∈ (Z >0 ) 2 , λ h is the length of the h-th row of Y , and λ k is the length of the k-th column of Y .  = l(s) + a(s) + 1.
In the later sections, we need the following generalization of the definition of hook length.
We call h Yα,Y β (s) relative hook length of s with respect to Y β when s ∈ Y α .
Example 2.5. In Fig. 1, the Young diagram drawn by dotted lines is Y α and the Young diagram drawn by solid lines is Y β . The number of diamonds (resp. stars) is l Y β (s) (resp. a Yα (s)). Then we have h Yα,Y β (s) = 7.
By definition, h(s) = #H s (Y ). We call a hook with n boxes a hook of length n.
Definition 2.7. For s = (h, k) ∈ Y , we define the rim of s as Then we have By definition, removing a hook H s (Y ) from Y means that removing all boxes in R s (Y ) from Y . Example 2.8. Let Y be a Young diagram given by a partition (5, 4, 3, 2, 1). We remove H (2,2) (Y ) from Y as follows:

Coloring of Young diagrams
Definition 2.9. Let Y ∈ P. Let us fix an integer l ≥ 2. Assume that we are given a γ ∈ Z/lZ. Then the l-residue of a box s ∈ Y is defined by the following rule: if s is the (i, j)-component of Y , then res(s) = γ + i − j + lZ ∈ Z/lZ.
We call this assignment of an element of Z/lZ to each box in Y as γ-coloring of a Young diagram Y .
We extend this coloring to tuples of Young diagrams. Definition 2.10. Let Y = (Y 1 , . . . , Y r ) ∈ P r . Assume that we are given γ 1 , . . . , γ r ∈ Z/lZ. If a box s ∈ Y is a (i, j)-component of Y k , then we color s with res(s) = γ k + i − j + lZ ∈ Z/lZ.
We call this assignment as a (γ 1 , . . . , γ r )-coloring of an r-tuple of Young diagrams Y .
Example 2.11. Take r = 2, l = 3, and (γ 1 , γ 2 ) = (0, 2) ∈ (Z/3Z) 2 . Let Y = (Y 1 , Y 2 ) ∈ P 2 which is given by a pair of partitions (λ 1 , λ 2 ) = ((4, 3, 1), (3, 2, 1, 1)). Then (γ 1 , γ 2 )-coloring of Y is given as follows:  It is well-known that there is a one-to-one correspondence between the set of Maya diagrams and the set of Young diagrams. We explain how to associate a Young diagram to a Maya diagram. Let us place Young diagrams on the (x, y)-plane by the following way: the bottom-left corner of the Young diagram is at origin (0, 0) and a box in the Young diagram is an unit square. We call an upper-right borderline of {x-axis} ∪ {y-axis} ∪ Y the extended borderline of Y and denote it by ∂Y . The line defined by y = x is called the medium.
Example 2.13. Let Y be the Young diagram corresponding to the partition (3, 2, 1). In Fig. 2, ∂Y is shown by a bold line. The Maya diagram {m Y (ν)} ν∈Z corresponding to a Young diagram Y is defined as follows. We give the direction to the extended borderline ∂Y of Y as in Fig. 2. Then each edge of the extended borderline is numbered by ν ∈ Z, if we set the edge which is located at the next to the medium to be 0. Next we encode a edge ↓ (resp. →) to 0 (resp. 1). By this way, we have a 0/1 sequence {m Y (ν)} ν∈Z , where each m Y (ν) corresponds to a edge of ∂Y . Concretely, {m Y (ν)} ν∈Z is given by Or, equivalently, it is given by Here λ i (resp. λ i ) is the length of the i-th row (resp. i-th column) of Y .
We express this as 0, 1, 0, 1 | 0, 1, 0, 1. Here | represents the position of the medium. This means that the medium's neighbor on the right is the m Y (0).

Remark 2.16.
Since the number of edges → before the medium is same as the number of edges ↓ after the medium, we have Remark 2.17 (removing a hook in terms of the Maya diagrams). Fix (x, y) ∈ Y whose hook length is n. Let Y (x,y) be the Young diagram obtained by removing the hook of length n corresponding to ( else.
where represents locations of exchanges.

Quotients
Then we have an l-tuple . For the latter purpose, let us define the following Definition 2.21. We define k i (Y ) ∈ Z (i = 0, . . . , l − 1) by the following condition: (i) Notice that k i (Y ) = 0 in general, since we take a subsequence. However (i) In this case, (ii) Let us remove the hook of length 5 from Y by means of 5-quotient (Y * 0 , . . . , Y * 4 ) of Y . This is equivalent to exchanging m Y * 2 (−1) = 1 and m Y * 2 (0) = 0, shown as follows:

Cores
Definition 2.24. A Young diagram is called l-core if its l-quotient Y * (l) is empty. We denote C (l) by the subset of P which consists of l-cores.
Definition 2.26. Let Y (l) be the Young diagram obtained by removing as many hooks of length l as possible from Y . 1 It is called the l-core of Y .
(ii) There is a bijection between C (l) and the set  . For any l ≥ 2, a Young diagram Y is uniquely determined by its l-core Y (l) and l-quotient Y * (l) . Thus for given λ ∈ C (l) , there is a bijection between the set and P l (n). Moreover, the weight of Y is given by

Weights of cores
Recall that we place a Young diagram Y on the (x, y)-plane, as shown in Fig. 2.
Hence we obtain, The right hand side of the above equation is equal to By the same argument as above, we have Then, using Remark 2.22(i), it is easy to see that Summing them up, we obtain the claim.

Quotients and cores for P w (v)
Let Γ be a cyclic group of order l. Let w = [W ] be an isomorphism class of r-dimensional Γ-module. Let W = ⊕ r i=1 C ⊗ ρ i be a decomposition of W into 1-dimensional Γ-modules. We regard (ρ 1 , . . . , ρ r ) as an element of (Z/lZ) r , where we fix an order of them: ρ i ≤ ρ i+1 . We give the (ρ 1 , . . . , ρ r )-coloring to P r (n). Let us call it w-coloring. For fixed w, P r (n) has the following decomposition into disjoint sets: where P w (v) is the subset of P r (n) whose number of i-colored boxes is equal to v i for 0 ≤ i ≤ l − 1, if we give the w-coloring to Young diagrams. Here Notation 2.31. Hereafter in this paper, we use the following assumptions and notations on l-dimensional row vector.
(i) We understand that indices of l-dimensional row vectors are in Z/lZ.
(ii) Let e j ∈ Z l (j = 0, . . . , l − 1) be the row vector whose j-th component is 1 and the others are 0.
(iii) Let δ be the row vector defined by δ def.
Let us take Y ∈ P e j (v). Then all boxes on the line y = x − kl − i are colored by the same . Now our claim follows from Proposition 2.27(ii).
By definition, a hook of length nl has n i-colored boxes for all 0 ≤ i ≤ l − 1. We have the following P e j (nδ) = P(nl : ∅).
In particular P e j (nδ) is independent of j.
Proof . In the proof of Proposition 2.32, we obtained the formula

Quiver varieties and torus actions
In this section, we introduce quiver varieties and torus actions on them. We follow [41] and [56] closely.

Quiver varieties
where Q is a 2-dimensional complex vector space. We define an action of GL(V ) on M r (n) by We define a map µ : Note that µ −1 (0) is invariant under the action of GL(V ). (i) M r (n) is a nonsingular complex algebraic variety of dimension 2nr and π : M r (n) → M r (n) 0 is a resolution of singularities.
(ii) M r (n) is isomorphic to the moduli space of the pair (E, φ) where E is a torsion free sheaf over CP 2 of rank E = r, c 2 (E) = n which is locally free in a neighborhood of l ∞ = {[0 : z 1 : z 2 ]} and φ is an isomorphism E| l∞ → O ⊕r CP 2 (framing at infinity). Remark 3.3. When dim(W ) = 1, any torsion free sheaf E of rank 1 with the above condition is a subsheaf of O CP 2 such that Supp(O CP 2 /E) is a 0-dimensional subscheme of C 2 , so we recover the Hilbert schemes of points on C 2 : C 2 [n] ∼ = M 1 (n). Therefore, M r (n) is a higher rank generalization of the Hilbert scheme of points. Note that M r (n) 0 is isomorphic to the n-th symmetric product S n C 2 of C 2 and the above morphism π coincides with the Hilbert-Chow morphism π : C 2 [n] → S n C 2 (see Appendix A.1).

Quiver varieties
Let Γ be a finite cyclic subgroup of SL(2, C) of order l. Let Q be the 2-dimensional Γ-module defined by the inclusion Γ ⊂ SL(2, C). For Γ-modules V and W , let where ( ) Γ means Γ-invariant part. We define GL Γ (V )-action on M W (V ), and the map µ : M W (V ) → End Γ (V ) as above. We also define the stability condition by the same condition as Definition 3.1, where subspace S ⊂ V is replaced by Γ-submodule. Then we define M w (v) and M w (v) 0 exactly the same as above, where v and w are isomorphism classes of V and W as Γ-module respectively. Let {R i } l−1 i=0 be the set of all irreducible representations of Γ, where R 0 is the trivial representation. We can denote Then v and w can be regarded as row vectors (dim V 0 , . . . , dim V l−1 ) and (dim W 0 , . . . , dim W l−1 ) respectively. We call M w (v) and M w (v) 0 quiver varieties. The restriction of π : M r (n) → M r (n) 0 gives a proper morphism π : where C Γ is the affine Cartan matrix of type A l−1 and t ( ) means the transposition.
The statements (i) is due to Nakajima [35]. The statement (ii) is stated in [56, Section 2.3]. Note that there is a natural Γ-action on M r (n) induced by that on CP 2 , if we fix a lift of the Γ-action to O ⊕r l∞ by w. Then the above theorem tells us that Γ-fixed point set M r (n) Γ of M r (n) have the following decomposition:

3.1.3
For the latter purpose, let us introduce the so-called ALE spaces.   This result was first obtained by Kronheimer [24], and rediscovered by Ito-Nakamura [21] and Ginzburg-Kapranov (unpublished) independently.

Torus actions on framed moduli spaces
Let T Q (resp. T W ) be a maximal torus in GL(Q) (resp. GL(W )), and let T = T Q × T W . Note that T Q ∼ = (C * ) 2 and T W ∼ = (C * ) r . Let us consider the following T -action on M r (n): where t = (t 1 , t 2 ) ∈ T Q , e = diag(e 1 , . . . , e r ) ∈ T W , and we denote B = (B 1 , B 2 ) ∈ Q ⊗ Hom(V, V ). This induces a T -action on M r (n) and M r (n) 0 . Note that the morphism π : M r (n) → M r (n) 0 is T -equivariant with respect to this action. This torus action is studied in [44]. It turns out that T -fixed points in M r (n) are parameterized by P r (n) and T -fixed points in M r (n) 0 consists of single point.

Torus actions on quiver varieties
We can regard T Q (resp. T W ) as a maximal torus in GL Γ (Q) (resp. GL Γ (W )). Thus we see that T = T Q × T W also acts on M W (V ) and this induces an action of T on quiver varieties M w (v) and M w (v) 0 . We can study T -fixed points on quiver varieties by using results in [44]. (i) (E, φ) ∈ M r (n) Γ is fixed by the T -action if and only if E has a decomposition E = I 1 ⊕ · · · ⊕ I r satisfying the following conditions for α = 1, . . . , r.
(a) I α is an ideal sheaf of 0-dimensional subscheme Z α whose support is a Γ-orbits in C 2 . Proof . This follows from the same argument as [44, Proposition 2.9].

Parameterization of f ixed points
It is instructive to compare equation (2.2) with equation (1.1).

Proposition 3.8.
There is a one-to-one correspondence between the T -fixed point set of M w (v) and P w (v).
Proof . This follows from Proposition 3.7 and the same argument as in [

3.2.4
We compare Proposition 3.8 with the work of Nakajima [34]. The results are not used later. In [34], an action of a certain 1-parameter subgroup of T on M w (v) is considered. And in the case of v 0 = w 0 = 0, fixed points are parametrized by Young tableaux.
(i) A µ-tableau of shape Y is a Young diagram Y whose boxes are numbered with the figures from 1 to l such that the cardinality of the boxes with figure k is µ k .
(ii) A µ-tableau of shape Y is said to be row-increasing if the entries in each row increase strictly from the left to the right. = v l−1 .
Proof . Note that, in this case, P w (v) consists of tuples of l-cores. Each row in a µ-tableau of shape λ corresponds to a l-core. A correspondence is given as follows. For the i-th row of a µ-tableau of shape λ, we denote C i by the set of contents on it. The data (k 0 , . . . , k l−1 ) of a l-core is given as follows. For 1 ≤ n ≤ λ i , k n−λ i −1 = 0, n ∈ C i , −1 n ∈ C i , and for λ i < n ≤ l, Then it is easy to see that this is a bijection.

Characters at f ixed points
Notation 3.12.
(ii) For a, b ∈ Z, we set Let W = ⊕ r i=1 C ⊗ ρ i be a decomposition of a representative W of w into 1-dimensional Γ-modules as in Section 2.3. Then we have the following Theorem 3.13. Let x be a T -fixed point of M w (v) corresponding to (Y 1 , . . . , Y r ) ∈ P w (v). Then the T -module structure of T x M w (v) is given by Proof . This follows by taking Γ-invariant part of the T -module given in [44,Theorem 2.11].

Enumerative geometry on quiver varieties
This section is devoted to computations of some global invariants of quiver varieties. In Section 4.1, we compute the 'equivariant volume' of M w (v). This is a simple application of Theorem 3.13 and localization theorem. The main part in this section is Section 4.2, where we consider generating functions of Poincaré polynomials and Euler characteristics of quiver varieties. Our computation is based on combinatorial arguments over cores and quotients of Young diagrams in Section 2.

Localization
First, we review some basic facts about equivariant integrals and the localization formula. Let M be an algebraic variety with an action of a torus T and H T * (M ) be the equivariant homology 4 (with Q-coefficient). The ring H T * (M ) is a module over H T * (pt). We consider the T -equivariant integral M 1. This integral takes values in the quotient field of H * T (pt). We refer the reader [43, Section 4.1] for a precise definition of the integral. Assume that M is smooth and T -fixed point set M T is finite. Then the classical Atiyah-Bott localization theorem [1] says that where e x is the T -equivariant Euler class of T x M . (e x = 0 since x is isolated.)

4.1.2
We apply the localization formula to M w (v). In this case, the integral is defined by equivariant pushforward of the fundamental class [M w (v)] to the unique T -fixed point o in M w (v) 0 by π : M w (v) → M w (v) 0 (see Proposition 3.7(ii)). Furthermore, equivariant Euler classes at fixed points are given by Theorem 3.13. Let ( 1 , 2 , a) be a coordinate on the Lie algebra of T given by t 1 = e 1 , t 2 = e 2 , e α = e aα , and a = (a 1 , . . . , a r ). Table 1. The elements of P w (v) for w = (1, 1) and v = (1, 1).
Example 4.2. In rank 1 cases, we have where n = |Y * (l) | for Y ∈ P e j (v). A proof of this identity is given in Proposition A.5.
From the combinatorial discussions in Section 2, we have the following factorization formula for Z e j (t, q, r). Z e j (t, q, r) = Z quot (t, q, r) Z core e j (q, r), Proof . Note that Z quot (resp. Z core e j ) is the contribution from l-quotients (resp. l-cores) of P w (v) corresponding to T -fixed points on M e j (v). By Corollary 2.33, we have Z quot (t, q, r) = Thus Z quot (t, q, r) is independent of e j . Then our claim follows from equation (2.3) and the definition of quotients and cores.
Thus Z quot (t, q reg ) is the generating function of the Poincaré polynomials of M e 0 (nδ). It is known [26,57] that M e 0 (nδ) is diffeomorphic to the Hilbert scheme of n points on the ALE space. By applying the Göttsche's formula (A.1) to the ALE space, we obtain the result.
Notation 4.8. We use the following definition of the Riemann theta function: where u is a complex (l − 1)-dimensional row vector, t ( ) means the transposition, and F is an (l − 1) × (l − 1) matrix. and where C Γ is the Cartan matrix of type A l−1 .
Proof . By Proposition 2.30 and its proof, we have Proof . This follows from Theorem 4.6, Lemmas 4.7 and 4.9.
Remark 4.11. As we mentioned in the introduction, Theorem 4.10 is obtained from the result of Nakajima [41, Section 5.2] (see the comments below the remark). This was informed us by H. Nakajima after the original version of this paper was submitted to e-print archives. He also pointed out that our argument has a close parallel to a geometric Frenkel-Kac construction of the Fock spaces of affine Lie algebras due to Grojnowski [13] (see also [38,Section 9.5]). The authors are grateful to him for these comments.
Let us make some comments on representation theoretical aspects of the result in Theorem 4.10. By Nakajima's pioneering work, it is known that quiver varieties are deeply related to representations of (quantum) affine algebras g (of type A (1) l−1 in our case). It is shown in [37,39] that their (K-)homology groups have structures of representations of affine algebras. Roughly speaking, ⊕ v H * (M w (v)) is a direct sum of certain highest weight representations of g determined by w. When w is the class of trivial 1-dimensional Γ-module, this is the so-called Fock space representation of g. This is explained in [41,Section 5.2]. From this view point, the generating function of Euler characteristics is the character of the representation and the generating function of Poincaré polynomials is the so-called q,t-character [40], which plays a fundamental role in representation theory.

Euler characteristics for higher rank case
Let dim W = r. We consider the following generating function of the Euler characteristics e(M w (v)) of M w (v): Corollary 4.12.
Then by Theorem 4.10, we have the claim.

Relations to instanton counting
In this section, we discuss connections with instanton counting in physics. In Section 5.1, we give a brief explanation of instanton counting, which is one of our motivations of the first part of this paper. In Section 5.2, we explain a part of results of Maeda et al. [30,31,32], where quotients and cores for Young diagrams appeared.

Instanton counting
Mathematically speaking, instanton counting means computations of global invariants on the instanton moduli space, such as the Euler characteristic and the volume of the moduli space.
What kind of invariants one want to compute depends on what kind of physical theory one considers. As we mentioned in introduction, M w (v) is a resolution of the moduli space of instantons on C 2 /Γ, therefore our results in Section 4 can be regarded as an instanton counting on C 2 /Γ.

5.1.1
In four-dimensional (4D) topologically twisted N = 4 supersymmetric (SUSY) Yang-Mills (YM) theory, the partition function is given by the generating function of Euler characteristics of instanton moduli spaces, as we considered in Section 4.2. Such a theory is studied extensively by Vafa and Witten [55]. See also [23,51], which considered issues similar to ours.

5.1.2
Nekrasov [46] introduced the following generating function of equivariant volumes of the instanton moduli spaces, in the sense of Section 4.1, in studies of 4D N = 2 SUSY YM theory: Mr(n)

5.1.3
There is a 5D version Z inst 5D ( 1 , 2 , a; Λ) of the Nekrasov's partition function (5.1), which was also introduced by Nekrasov [46]. Physically speaking, that is N = 1 SUSY YM theory on R 4 × S 1 (in a certain supergravity background). Mathematically speaking, this means that we consider equivariant integrals in (5.1) in the sense of equivariant K-theory [45].

Cores and perturbative gauge theory
In this section, we consider 5D version of the Nekrasov's partition function and discuss a combinatorial aspect of it. We explain works of Maeda et al. [30,31,32], where quotients and cores for Young diagrams appeared in a study of 5D N = 1 SUSY YM theory explained in Section 5.1.3.

5.2.1
So far, we consider only the instanton part of the partition function. There is the so-called perturbative part Z pert 5D ( 1 , 2 , a; Λ) of the partition function, which corresponds to the perturbative part of the Seiberg-Witten prepotential. Although we do not give a definition here, it is defined by an explicit formula (see, for example, [43,Appendix E]). Then the full partition function is define by

Results of Maeda et al.
Recall that T -fixed points on M r (n) are parametrized by P r (n) and an equivariant integral (in the sense of K-theory) over M r (n) can be expressed as a sum over P r (n) by localization theorem, as we discussed in Section 4.1. The authors of loc. cit. found the following identity: where the LHS of (5.2) is the partition function for r = 1 under the condition 5 and summands in the RHS of (5.2) are the partition function for r = N (≥ 2) under the condition (5.3) and specialization a = k , where k = (k 1 , . . . ,k N ) is given bỹ for k = (k 1 , . . . , k N ) ∈ K (N ) . In loc. cit., this identity was proved by the following way. First, Z 5D ( , − , k ; Λ) can be identified with the partition function of a statistical model of plane partitions (= 3D Young diagrams) whose r-core of the 'main diagonal' is Y (r) ( k), which can be described by free fermions. Then, under this identification, the perturbative part Z pert 5D ( , − , k ; Λ) is identified with the contribution of ground states, which correspond to N -cores Y (N ) ( k) ∈ C (N ) determined by k ∈ K (N ) . And the instanton part Z inst 5D ( , − , k ; Λ) is identified with the contribution of excitations, which correspond to N -quotients. Notice that, by Proposition 2.28, we recover the set of all Young diagrams P if we vary all k = (k 1 , . . . , k N ) ∈ K (N ) and Y ∈ P N . Taking these into account, we can see that the identity (5.2) holds, after some Schur function calculus.
It is worth mentioning that we have a similar factorization of the generating function Z e j (t,q, r) of Poincalé polynomials in rank 1 case into 'core-part' Z core and 'quotient-part' Z quot e j in Theorem 4.6. Thus combinatorial structures in our case and the case studied by Maeda et al. are essentially the same.

A Hilbert schemes of points on surfaces
This appendix has two purposes. The one is to summarize some facts about the Hilbert schemes of points 6 (Section A.1). The other is to give another method to compute equivariant volumes of quiver varieties in rank 1 case (Sections A.2 and A.3). We get a closed formula for the generating function of such quantities. By comparing with the result in Section 4.1, we find an interesting combinatorial identity (Proposition A.5).

A.1 Hilbert schemes of points
Let X be a smooth quasi-projective surface. The Hilbert scheme X [n] of n points on X is, by definition, a smooth algebraic variety of dimension 2n, which parametrizes the set of 0dimensional subschemes in X with colength n. Topology of the Hilbert scheme X [n] is wellknown. The generating function of the Poincaré polynomials is given by the following Göttsche's formula [12]: There exists a proper surjective morphism π : X [n] → S n X, where S n X is the n-th symmetric product of X. This morphism, so called the Hilbert-Chow morphism, gives a resolution of singularities of S n X. Note that if X admits a torus action, then there is an induced torus action on X [n] and the Hilbert-Chow morphism π is equivariant.

A.2 Torus actions on Hilbert schemes
We study torus actions on the Hilbert schemes of points on toric surfaces. All the results in this section are due to Ellingsrud-Strømme [8] (see also [27,28,50]).

A.2.1 Toric surfaces
Let N be a 2-dimensional lattice and Σ be a fan in N R . From these datum, we can construct an algebraic surface X = X(N, Σ), equipped with an action of the 2-dimensional algebraic torus T = (C * ) 2 with a dense open orbit. These surfaces are called toric surfaces, and it is known that they are normal and quasi-projective (see, for example, [11,48] for detail).

A.2.2 Torus actions on toric surfaces
Let X be a smooth toric surface associated to (N, Σ). Then the 2-dimensional torus T = N ⊗ Z C * acts on X.
The T -fixed points correspond to vertices of 2-dimensional cones in Σ. In particular, the number of T -fixed points is equal to the Euler number e(X) of X. Let X T = {p 1 , . . . , p e(X) }. Then, by construction, there is an affine chart U i = Spec C[x i , y i ] around each p i which is Tinvariant. We call (x i , y i ) a toric coordinate around p i . We denote w x i (t 1 , t 2 ) (resp. w y i (t 1 , t 2 )) the weight of the T -action on x i (resp. y i ).

A.2.3 Torus actions on Hilbert schemes
As we noticed in Section A.1.2, T acts also on X [n] . First, we identify T -fixed points on X [n] .
Lemma A.1. There is a one-to-one correspondence between T -fixed point set on X [n] and P e(X) (n). n i = n. It is well-known that there is a one-to-one correspondence between T -fixed points in C 2 [n] and partitions of n (see [38,Chapter 5]). This gives the assertion.

Proof
Lemma A.2. The weight decomposition of the cotangent space of X [n] at a T -fixed point corresponding to Y = (Y 1 , . . . , Y e(X) ) ∈ P e(X) (n) is given by Proof . This is obvious from the classical result of Ellingsrud and Strømme [8] on the weight decomposition of the tangent space of C 2 [n] at T -fixed point (see [38,Proposition 5.8]).
For a Young diagram Y , we set n Y (η 1 , η 2 ) def.
Here η 1 , and η 2 are parameters. Then we have the following Proposition A. 3. T -equivariant integral on X [n] is given by

=
Y ∈P e(X) (n) e(X) i=i 1 n Y i log w x i e 1 , e 2 , log w y i (e 1 , e 2 ) .
Here we write t i = e i , where i ∈ Lie(T ).
Let us consider the case X = C 2 equipped with a T -action defined by (x, y) → (t 1 x, t 2 y).

A.3 An example
We study a particular case when X is the ALE space of type A l−1 , i.e., X = M e 0 (δ) and compare the result in Section A.2.4 and the result in Section 4.1.

A.3.1
Recall that the simple singularity of type A l−1 is the quotient space C 2 /Γ, where the action of Γ on C 2 is given by (x, y) → (ζx, ζ −1 y). Here ζ is a primitive l-th root of unity. Note that the above action of Γ on C 2 commutes with the T -action (A.2) on C 2 . It follows that C 2 /Γ is a toric singularity. It is well-known [11,48] that it has the toric minimal resolution. In [21], it is shown that X is isomorphic to the toric minimal resolution, so that X has affine charts U i = Spec C[x i , y i ] (i = 1, . . . , l) defined by where (a, b) is a coordinate of C 2 on which Γ acts by (a, b) → (ζa, ζ −1 b).
Remark A.4. It is easy to see that there is a bijection between the set of these l affine charts and the set P e 0 (δ). This correspondence is pointed out in [20, Corollary 3.10], where P e 0 (δ) is identified with the set of so-called Γ-clusters.
The coefficient of Λ 2n is given by 1 n!l n ( 1 2 ) n , which is nothing but the T -equivariant volume of the orbifold S n C 2 /Γ . This is the desired result, since natural morphism X [n] → S n C 2 /Γ , induced by X → C 2 /Γ, is T -equivariant.