The moduli space of instantons on an ALE space from 3d $\mathcal{N}=4$ field theories

The moduli space of instantons on an ALE space is studied using the moduli space of $\mathcal{N}=4$ field theories in three dimensions. For instantons in a simple gauge group $G$ on $\mathbb{C}^2/\mathbb{Z}_n$, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the affine Dynkin diagram of $G$ with flavour nodes of unitary groups attached to various nodes of the Dynkin diagram. We provide a simple prescription to determine the ranks and the positions of these flavour nodes from the order of the orbifold $n$ and from the residual subgroup of $G$ that is left unbroken by the monodromy of the gauge field at infinity. For $G$ a simply laced group of type $A$, $D$ or $E$, the Higgs branch of such a quiver describes the moduli space of instantons in projective unitary group $PU(n) \cong U(n)/U(1)$ on orbifold $\mathbb{C}^2/\hat{G}$, where $\hat{G}$ is the discrete group that is in McKay correspondence to $G$. Moreover, we present the quiver whose Coulomb branch describes the moduli space of $SO(2N)$ instantons on a smooth ALE space of type $A_{2n-1}$ and whose Higgs branch describes the moduli space of $PU(2n)$ instantons on a smooth ALE space of type $D_{N}$.


Introduction
The realisation of the moduli space of instantons using string theory and supersymmetric field theory has long been studied in the literature. It was pointed out in [1,2] that the ADHM construction [3], an algebraic prescription to construct instanton solutions for classical gauge groups on flat space, can be realised from a system of Dp-branes inside D(p + 4)-branes (possibly with the presence of an orientifold plane). In this system, the former dissolve into instantons for the worldvolume gauge fields of the latter. For the worldvolume gauge theory on the Dp-branes, which has 8 supercharges, the Higgs branch of the moduli space can then be identified with the moduli space of instantons in the theory on the worldvolume on the D(p + 4) branes. Indeed, the hyperKähler quotient of such a Higgs branch can be identified with that of the moduli space of instantons [4].
A similar method is also available for instantons on an asymptotically locally Euclidean (ALE) space. The analogue of the ADHM construction for instantons in a unitary gauge group on an ALE space was proposed by Kronheimer and Nakajima (KN) [5]. Such a result was later generalised to instantons in a general classical gauge group on an ALE space of type A or D by several authors, e.g. [6][7][8][9][10][11], using string theoretic constructions.
Both ADHM and KN constructions exist only for instantons in classical gauge groups. Over the past few years, there has been a lot of progress in the study of instantons in exceptional gauge groups on flat space. For example, the instanton partition functions for exceptional gauge groups E 6,7,8 can be derived [12] by computing of superconformal indices [13][14][15] of theories of class S, arising from compactifying M5-branes on a Riemann sphere with appropriate punctures [16]; by using the blowup equations of [17,18] as in [19]; or by computing and extrapolating the generating function of holomorphic functions on the Higgs branch, known as the Higgs branch Hilbert series, of certain supersymmetric field theories as in [20][21][22].
Another method that has been proven to be fruitful for the study of instanton moduli spaces is to make use of the Coulomb branch of certain three dimensional field theories with 8 supercharges. For one instanton in a simply laced gauge group (i.e. of type A, D or E) on flat space, such 3d gauge theories were proposed in [23]. Such results were later extended to higher instanton numbers and to instantons on ALE spaces by [7,8,11]. Due to the method discovered in [24], it became possible to compute the generating function of holomorphic functions on the Coulomb branch, known as the Coulomb branch Hilbert series, of a large class of 3d N = 4 gauge theories (see also [25] for a recent mathematical development). It was indeed shown that this method can be applied to study the moduli space of instantons in the ADE gauge groups on flat space [24,26]. The generalisation to the non-simply laced gauge groups (i.e. of types B, C, F and G) was developed recently in [27]. In this reference, it was proposed that the moduli space of instantons in any simple gauge group G, including non-simply laced ones, on flat space can be realised from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. For G a simply laced group, such a quiver represents a 3d N = 4 gauge theory with a known Lagrangian description, and this is indeed the main reason why the method of [24] can be successfully applied to study such a moduli space. On the other hand, the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, due to the multiple laces that are present in these diagrams. Nevertheless, it is still possible to compute the Coulomb Hilbert series such quivers using a simple prescription presented in [27].
The Hilbert series is, mathematically speaking, a character of the global symmetry group of the ring of holomorphic functions on the moduli space of the supersymmetric gauge theory. It provides useful information about the moduli space, namely the group theoretic properties of the generators of the moduli space and of the relations between them. Important properties of the theory, such as the global symmetry enhancement, can be studied using this approach. For moduli spaces of k pure Yang-Mills instantons, the Hilbert series is also the five-dimensional (K-theoretic) k instanton partition function of [17,18,21,25,28,29]. Moreover, it was recently realised that the Coulomb branch Hilbert series of 3d N = 4 "good" or "ugly" theories [30] with a Lagrangian description [24] can be obtained as a limit of the superconformal index of the theory 1 [31]. Indeed, some of the theories of our interest in this paper, in particular those involving the simply-laced Dynkin diagrams, have a Lagrangian description and the standard formula [32][33][34][35] for the superconformal index can be written down. On the other hand, the theories involving non-simply laced Dynkin diagrams have no known Lagrangian description and the standard formula for the superconformal index is not available. The prescription for dealing with multiple laces proposed in [27] is a natural generalisation of the result obtained in [24] for Lagrangian theories. Such an approach allows us to study in a uniform way the moduli spaces of instantons of all simple Lie groups. It is currently unclear how to extend this to the full superconformal index.
The main goal of is paper is to study 3d N = 4 field theories whose Higgs and/or Coulomb branches can be identified with the moduli space of instantons on an ALE space. We divide our study into two parts, namely for an orbifold singularity and for a smooth ALE space. Let us first discuss our findings on the case of orbifold singularities. We propose, along the line of [27], a simple field theory description of the moduli space of G instantons on orbifold C 2 /Z n , for any simple group G. Such a description involves the Coulomb branch of the quiver given by the affine Dynkin diagram of G with flavour nodes of unitary groups attached to various nodes of the Dynkin diagram. We provide a simple prescription to determine the ranks and the positions of these flavour nodes from the order of the orbifold n and from the residual subgroup of G that is left unbroken by the monodromy at infinity. For G a simply laced group of type A, D or E, the Higgs branch of such a quiver describes the moduli space of instantons in projective unitary group 2 P U (n) on C 2 / G, where G is the discrete group that is in McKay correspondence to G. The exchange of the instanton gauge group and the orbifold group in the Higgs and the Coulomb branches is in accordance with [7,8,11,36]. We use the Hilbert series as a tool to study such moduli spaces of instantons and find several interesting new features. In the second part of the paper, we discuss the moduli space of instantons on a smooth ALE space. We focus on a 3d N = 4 gauge theory whose Coulomb branch describes the moduli space of SO(2N ) instantons on C 2 /Z 2n with a particular monodromy at infinity. In several special cases, we compare the results with those studied in [37] and find an agreement. As proven in [5], the Higgs branch of such a gauge theory describes the moduli space of P U (2n) instantons on smooth ALE space C 2 / D N .
The paper is organised as follows. In section 2, we focus on instantons on a singular orbifold. As a warm-up exercise, we review the Higgs and the Coulomb branches of the 3d N = 4 gauge theory that describes the moduli space of P U (N ) instantons on C 2 /Z n in section 2.1. Extending the previous result, we then propose the field theories whose Coulomb branch is the moduli space of instantons in a general simple gauge group on C 2 /Z n in section 2.2. As a by-product, the moduli spaces of P U (n) instantons on C 2 / D N and C 2 / E 6,7,8  2 Instantons in gauge group G on orbifold C 2 /Γ In this section, we study the moduli space of G instantons on orbifold C 2 /Z n , for any simple group G. As a warm-up exercise, we discuss the case of G = P U (N ) and Γ = Z n in section 2.1. Many results on this topic have been studied in several papers, e.g. [5-8, 25, 38-43], in the past. The purpose of this section is to introduce necessary concepts and set-up the notation. In section 2.2, we then extend this result to the moduli space of G instantons on C 2 /Z n for a general simple gauge group G, and to the moduli spaces of P U (n) instantons on C 2 / D N and C 2 / E 6,7,8 .
Notation. In the quiver diagrams in this paper, we use a circular node to denote a gauge group, a rectangular node to denote a flavour symmetry, and a line connecting two nodes to denote a bi-fundamental hypermultiplet between the two groups. Unless stated otherwise, the node labelled by number r denotes the unitary group U (r).

G = P U (N ) and Γ = Z n
We start by discussing moduli space of P U (N ) instantons on C 2 /Z n . The instanton configuration is specified by two pieces of information, namely the monodromy of the gauge field at infinity and the monodromy at the origin of C 2 /Z n [41]. With these data specified, the moduli space of such instantons are isomorphic to the Higgs branch of a 3d N = 4 gauge theory whose quiver description is given by the flavoured affine A n−1 quiver diagram, known as the Kronheimer-Nakajima (KN) quiver [5]: This theory can be realised as the worldvolume of the D3-branes in the following configuration [44]: where each blue line denotes the D3-branes, each red line denotes an NS5-branes, and each black dot with the label N i denotes the N i D5-branes. The branes span the following directions: 0 1 2 3 4 5 6 7 8 9 D3 X X X X NS5 X X X X X X D5 X X X X X X (2.4) where the 6-th direction corresponds to the circular direction.
From quiver (2.1), we can read off the information about the gauge field at infinity U ∞ and the gauge field at the origin U 0 as follows [41]. The number of eigenvalues of U ∞ that equal to e 2πi /n (for = 1, . . . , n) is N . Here N also has an interpretation as the number of D5-branes with linking number . The number of eigenvalues of U 0 that equal to e 2πi /n is Note that β is the difference between the linking numbers of the ( + 1)-th and the -th NS5-branes. From now on and in the main text, we refer to the paritition (N 1 , . . . , N n ) of N as the framing of the P U (N ) instantons on C 2 /Z n . Indeed the framing is cyclic. For simplicity of the discussion, throughout section 2 we take and use the terminology that k is the instanton number. In this case, the monodromies U 0 and U ∞ have the same eigenvalues. As a result, it is sufficient to specify the instanton configuration just by the framing (N 1 , . . . , N n ). In this case, the monodromy breaks P U (N ) ∼ = U (N )/U (1) into the residual symmetry (U (N 1 ) × U (N 2 ) × · · · × U (N n ))/U (1).

The Coulomb branch of the Kronheimer-Nakajima quiver
So far we have discussed about the Higgs branch of the KN quiver. Let us now study the Coulomb branch of such a quiver. A simple way to do so is to apply three dimensional mirror symmetry [45] to quiver (2.1) and study the Higgs branch of the resulting quiver. The former amounts to apply an S-duality on the configuration (2.3), under which the NS5-branes and the D5-branes are exchanged [44]. The result is a circular quiver with N U (k) gauge groups with one flavour of fundamental hypermultiplet under each of the N 1 -th, N 2 -th, . . ., N n -th gauge groups. By mirror symmetry, the Coulomb branch of the KN quiver (2.1) is therefore the Higgs branch of this resulting quiver and it describes the moduli space of k P U (n) instantons on C 2 /Z N with framing (0 N 1 −1 , 1, 0 N 2 −1 , 1, . . . , 0 Nn−1 , 1). (2.7) Comparing (2.1) and (2.7), one observes that the roles of the gauge group and the orbifold type get exchanged [7,8,11,36] under mirror symmetry. The framing in (2.7) determines the residual symmetry of P U (n). If N i > 0 for all i = 1, 2, . . . , n, then P U (n) is broken to its maximal abelian subgroup U (1) n−1 .
There are some interesting special cases to consider: • If one of the N i 's is equal to one and the other N i 's are zero, the Coulomb branch of can be identified with the moduli space of k SU (n) instantons on C 2 ; in agreement with [27].
• If one of the N i 's is equal to N and the other N i 's are zero, the symmetry of the Coulomb branch is U (1) × SU (n) for N ≥ 3 and SU (2) × SU (n) for N = 1, 2, where the U (1) and SU (2) are associated with the isometry of C 2 /Z n .
If in addition we set k = 1, the Coulomb branch of (2.1) is isomorphic to 8) where N SU (n) is the reduced moduli space of one SU (n) instanton on C 2 , which is is the minimal nilpotent orbit of SU (n) [46][47][48][49]. On the other hand, the Higgs branch of (2.1) in this case is The feature (2.9) of the moduli space was in fact pointed out in (2.69) of [43].
2.2 A simple group G and Γ = Z n or G = P U (N ) and Γ = A n , D n , E 6,7,8 This result can in fact be generalised to the moduli space of a simple gauge group G instantons on C 2 /Z n and that of P U (n) instantons on C 2 /Γ, with Γ = D n , E 6 , E 7 , E 8 .
In each case, we specify the monodromy of the gauge field at infinity by stating the residual symmetry of the instanton gauge group. As a natural generalisation of the KN quiver, we focus on the following quiver diagram: The affine Dynkin diagram of the group G with the gauge groups U (ka ∨ i ), where a ∨ i (i = 0, 1, . . . , rank G) is the dual Coxeter label of the i-th node, and with n i flavours of fundamental hypermultiplets attached to the i-th node such that where a i are the Coxeter label of the i-th node. 3 For G a simply laced group of types A, D or E, this is a conventional quiver diagram possessing a known Lagrangian description, whose moduli space was studied in various papers, e.g. [7,8,23,25]. On the other hand, if G is a non-simply laced group of type B, C, F or G, the Lagrangian description of this quiver is not currently known, due to the presence of multiple-laces in the Dynkin diagram, and we refer to such a diagram as a generalised quiver diagram. As was discussed in [24,26,27], the Coulomb branch of such quiver diagrams has been proven to be useful in the study of the moduli space of instantons on flat space. In the following, we generalise the previous results to the moduli space of instantons on a singular orbifold.
Given quiver diagram (2.10), we state the following results regarding its Coulomb and Higgs branch: 1. The Coulomb branch of the above quiver describes the moduli space of k G instantons on C 2 /Z n with monodromy at infinity such that the symmetry G is broken to a subgroup H of G, where the non-abelian factors of H are constructed by removing the simple roots α i associated with the nodes where n i = 0 and the abelian factors of H are present such that the total rank of H is equal to the rank of G.
For n = 1, 2, the isometry of the Coulomb branch is SU (2) × H, where the SU (2) factor corresponds to the isometry of C 2 or C 2 /Z 2 . For n ≥ 3 the isometry of the Coulomb branch is U (1)×H, where the U (1) factor corresponds to the isometry of C 2 /Z n with n ≥ 3.
2. For G = A, D, E, the Higgs branch of the above quiver describes the moduli space of P U (n) instantons on C 2 / G, where G is the discrete group that is in McKay correspondence to G, with monodromy at infinity such that the group P U (n) is broken to its subgroup determined by the flavour nodes in the quiver diagram.
Observe that the instanton gauge group and the orbifold type get exchanged as one goes from the Higgs branch to the Coulomb branch and vice versa [7,8,11,36]. Many of these statements can be conveniently checked using Hilbert series. For a simply laced group G, the Hilbert series of the Coulomb branch mentioned in item 1 can be computed using the method described in [24]. For a non-simply laced group G, the method of [27] can be applied. We review such methods in Appendix A and provide several explicit examples in section 2.3.
The quaternionic dimension of the Coulomb branch of quiver (2.10) is where h ∨ G is the dual coxeter number of G. This is to be expected as the dimension of the moduli space of k G instantons on C 2 /Z n . On the other hand, for G = A, D, E, the quaternionic dimension of the Higgs branch of quiver (2.10) is which is to be expected as the dimension of the moduli space of k P U (n) instantons on C 2 / G. In certain special cases, the moduli space of quiver (2.10) possesses the following important features: • If n 0 = 1 and n i = 0 for i = 0, the Coulomb branch of can be identified with the moduli space of k G instantons on C 2 . This is discussed in detail in [27].
If in addition we take k = 1, the Coulomb branch is isomorphic to C 2 × N G , where N G is the minimal nilpotent orbit of G [46][47][48][49], and the Higgs branch • If n 0 = n and n i = 0 for i = 0, the symmetry of the Coulomb branch is U (1)×G for n ≥ 3 and SU (2) × G for n = 1, 2.
If in addition we set k = 1, the Coulomb branch of (2.10) is isomorphic to and, for G = A, D, E, the Higgs branch of (2.10) is isomorphic to (2.14) In the following, we present some examples and compute the Hilbert series to demonstrate the above general rule.

Examples
Below we provide three examples involving D 4 , B 3 and G 2 affine Dynkin diagrams. In the first example, the moduli space of SO(8) instantons on C 2 /Z n and that of P U (N ) instantons on C 2 / D 4 are discussed. In the second and the third examples, we demonstrate the use of generalised quiver diagrams, which involving non-simple laces, in the study of moduli spaces of SO(7) and G 2 instantons on C 2 /Z n .

Flavoured D 4 affine Dynkin diagrams
Below we present the quivers diagrams whose Coulomb branches correspond to the moduli space correspond to k SO(8) instantons on C 2 /Z 2 with various monodromies at infinity. The symmetry of the Coulomb branch is indicated below each quiver diagram. The SO(8) symmetry is broken to SO(8), SO(2) × SO(6) and SO(4) × SO(4), respectively from left to right. The factor of SU (2) that appears under each diagram corresponds to the isometry of C 2 /Z 2 . These symmetries can also be read off from the Coulomb branch Hilbert series 4 , whose computations are presented below. They are in accordance with the general rule stated earlier.
The rightmost diagram of (2.15) can be redrawn as with the overall U (1) decoupled (for example, from the middle U (2k) gauge node). It should be noted that this theory is actually a 3d mirror theory [52] of the 3d Sicilian theory (the class S theory compactified on S 1 ) of type A 2k−1 with punctures: The Coulomb branch Hilbert series for k = 1 Let us compute the Coulomb branch Hilbert series of (2.15) in the case of k = 1. For the sake of brevity in writing the formulae below, let us define where ∆ m denotes the contribution from the fundamental matter attached to the D 4 affine quiver. It may depend on u's and n's. The Coulomb branch Hilbert series of the left quiver of (2.15) is where This is the Hilbert series of (C 2 /Z 2 ) × N SO (8) , where N SO (8) is the minimal nilpotent orbit of SO (8).
The Coulomb branch Hilbert series of the middle quiver of (2.15) is This is in agreement with (4.53) of [43]. The Coulomb branch Hilbert series of the right quiver of (2.15) is If we set z 0 = . . . = z 4 = 1, we obtain the unrefined Hilbert series In addition to the quivers depicted in (2.15), let us consider the following diagram The Coulomb branch of this quiver describes the moduli space of k SO(8) instantons on C 2 /Z 3 with monodromy at infinity such that SO(8) is broken to U (1) × SU (2) × SO (4). According to the general rule stated above, the SU (2) × SO(4) factor follows from the removal of the simple roots associated with the top left and the middle nodes and the U (1) factor is precisely the abelian factor whose existence makes the rank add up to 4. The symmetry of the Coulomb branch is U (1)×(U (1)×SU (2)×SO (4)), where the first U (1) is associated with the isometry of C 2 /Z 3 . This symmetry can be confirmed using the Coulomb branch Hilbert series. For example, for k = 1, this is given by For brevity, we present the Hilbert series when z i are set to 1 for all i = 0, . . . , 4: Note that the coefficient 11 of t 2 is indeed the sum of the dimensions of the adjoint representations of each factor in U (1) × (U (1) × SU (2) × SO (4)), i.e. 1 + 1 + 3 + 6.
Mirror theories of (2.15) and (2.26) The above instanton moduli spaces of instantons can be realised from the Higgs branch of the following quiver diagram (see section 4.2 of [9] and section 4.1 of [43]): where p = 0, 2, 4. For p = 2, 4, this quiver is indeed a three-dimensional mirror theory for the quivers depicted in (2.15), respectively from left to right. Note that the symmetry of the Higgs branch of this quiver is indeed SU (2)×SO(p)×SO(8−p). We check that the Coulomb branch Hilbert series computed from (2.15) are in agreement with those computed from the Higgs branch of (2.29). The case of p = 0 of the deserves a special discussion 5 . In this case, the left U Sp(2k) gauge node has 2k flavours of hypermultiplets transforming under its fundamental representation and so the Higgs branch of (2.29) is expected to be the union of two cones [53]. For simplicity, let us discuss explicitly both cones in the case of k = 1. (The discussion can be easily generalised to the higher values of k.) One of the cone can be identified with the product of C 2 /Z 2 and (the closure of) the minimal nilpotent orbit of SO (8); this is identified with the Coulomb branch of the leftmost quiver of (2.15). The other cone turns out to be the next-to-minimal nilpotent orbit of SO (8); this is identified with the Coulomb branch of the quiver depicted in [54, Figure 9 where the first two terms are the contributions of the two cones and −1 takes into account of the intersection. To obtain (2.29) with p = 0 and k = 1, we gauge the U Sp(2) subgroup of the flavour symmetry of the said U Sp(2) gauge theory and couple it to 4 flavours of hypermultiplets in the fundamental representation. The Hilbert series of the first cone of (2.29) with p = 0 and k = 1 is where this is indeed the Hilbert series of the product of C 2 /Z 2 and the minimal nilpotent orbit of SO (8). The Hilbert series of the other cone of (2.29) with p = 0 and k = 1 is As a result of the integration, we obtain the Hilbert series of the next-to-minimal nilpotent orbit of SO (8), whose highest weight generating function and the unrefined Hilbert series (i.e. all components of f are set to 1) are given by [59, Row 4, Table  15], namely , It can also indeed be checked that this Hilbert series is also equal to the Coulomb branch Hilbert series of (2.30). Note that for the middle quiver of (2.15) has two mirror theories (see sections 4.1 and 4.2 of [9], section 4.2 of [43] and Fig. 22 on Page 47 of [51]): where A 1 and A 2 denote antisymmetric hypermultiplet under gauge group U (2). The Higgs branch of either theory describes k SO(8) instantons on C 2 /Z 2 with SO(8) broken to SO(2) × SO (6), whose algebra is isomorphic to U (4). A mirror theory of (2.26) is depicted below.
where A denotes the rank two antisymmetric hypermultiplet under the U (2k) gauge group. Indeed, the symmetry of the Higgs branch is U (1) × U (2) × SO(4), where U (1) is associated with the antisymmetric hypermultiplet. This symmetry agrees with the Coulomb branch symmetry of quiver (2.26). 7 The Higgs branch of (2.15) The Higgs branch of the quivers in (2.15) corresponds to the moduli space of k P U (2) instantons on C 2 / D 4 with monodromy at infinity such that P U (2) ∼ = U (2)/U (1) symmetry is broken to P U (2), (U (1) × U (1))/U (1) and empty, respectively from left to right. 8 These symmetries are manifest from the quiver diagrams 9 . In the following, we examine the Higgs branches of the leftmost and the rightmost quivers in detail.
The leftmost quiver of (2.15). For k = 1, the Higgs branch Hilbert series can be 7 Indeed one can also check that the Higgs branch Hilbert series of (2.37) agrees with (2.28). 8 These moduli spaces can also be realised from the Coulomb branch of (2.29) with p = 0, 2, 4 respectively. 9 For the rightmost quiver in (2.15), one can see this from quiver (2.16).

computed as follows:
H Higgs (2.15),left (t, y) where (z, q) are U (2) ∼ = SU (2) × U (1) gauge fugacities associated with the middle node; w is the U (1) gauge fugacity associated with top left node; and y is the fugacity of the SU (2) flavour symmetry. The first factor in the numerator denotes three copies of the Higgs branch of U (1) gauge theory with 2 flavours (i.e. three copies of C 2 /Z 2 ). The second and the third factors in the numerator denote the contribution from the remaining bi-fundamental hypermultiplets. The factors in the denominators denote the contributions of the F -terms associated with the middle U (2) gauge node and the top left U (1) gauge node. Evaluating the integrals, we find the following refined and unrefined Hilbert series We can therefore conclude that The Higgs branch of leftmost quiver in (2.15) for k = 1 where C 2 / D 4 is the orbifold singularity in question and C 2 /Z 2 is the reduced moduli space of 1 SU (2) instanton on C 2 , which is the minimal nilpotent orbit N SU (2) of SU (2). It is interesting to compare this with the Coulomb branch of the same quiver, which is isomorphic to (C 2 /Z 2 ) × N SO(8) (see (2.19)). Notice that the group associated with the orbifold and that associated with the nilpotent orbit get exchanged.
The rightmost quiver of (2.15). The Higgs branch of the rightmost diagram of (2.15) has an interesting feature that worth discussing in detail. Some properties of this space for k = 1 were actually studied in section 4 of [60]. In the following, we compute the Higgs branch Hilbert series of such a diagram for k = 1. 10 The fact that the coefficient of t 2 vanishes implies that the flavour symmetry is empty. As a matter of fact, the Higgs branch of this quiver is isomorphic to the orbifold C 4 /Γ 32 , where Γ 32 refers to extraspecial group of order 32 and type − (also known as Γ 5 a 2 ) in https://groupprops.subwiki.org/wiki/Central_product_of_D8_and_ Q8. It is isomorphic to the central product of the dihedral group D 8 of order 8 and the quaternion group Q 8 . 11 . In order to check this, we compute the Hilbert series of C 4 /Γ 32 using the discrete Molien formula: 12 where the elements of Γ 32 are the Kronecker products D i ⊗ Q j (with i, j = 1, . . . , 8 labelling the elements of D and Q defined below) such that the repeated elements (2.42) 11 The group Γ 5 a 2 has a close cousin, namely the extraspecial group of order 32 and type + (also known as Γ 5 a 1 ). The latter is isomorphic to the central product of two dihedral groups of order 8, D 8 * D 8 , and to the central product of two quaternion groups, Q 8 * Q 8 ; see https: //groupprops.subwiki.org/wiki/Groups_of_order_32. The Hilbert series for C 4 /Γ 5 a 1 can be computed using the discrete Molien formula similarly to (2.45). The result is The plethystic logarithm of (2.45) is indicating that there are 5 generators at order t 4 and 4 generators at order t 6 subject to 4 relations at order t 10 and 10 relations at order t 12 .

Flavoured B 3 affine Dynkin diagrams
Below we present the generalised quivers diagrams whose Coulomb branches correspond to the moduli space of k SO(7) instantons on C 2 /Z 2 with various monodromies at infinity specified by the residual symmetry of SO (7). The symmetry of the Coulomb branch of each quiver can be read off from the Hilbert series computed using the prescription given in [27]. For each quiver, SO(7) symmetry is broken to SO (7)  Let us now compute the Coulomb branch Hilbert series of the above quivers. For the sake of brevity in writing the formulae below, let us define where ∆ m denotes the contribution from the fundamental matter attached to the B 3 affine quiver. It may depend on u's and n's. The Coulomb branch Hilbert series of the top left quiver in (2.51) is χ SO (7) [0,p 2 ,0,0] (y 1 , . . . , y 4 )t 2p 2 , This is the Hilbert series of (C 2 /Z 2 ) × N SO (7) , where N SO (7) is the minimal nilpotent orbit of SO (7). where  where The Coulomb branch Hilbert series of the bottom right quiver in (2.51) is [0,1,1] (y) + 2)t 4 + . . . , (2.59) where The above moduli spaces of instantons can be realised from the Higgs branch of the following quiver diagram (see section 4.1 of [43]): where p = 0, 2, 4, 6. The symmetry of the Higgs branch of this theory is SU (2) × (SO(p) × SO(7 − p)), where the SU (2) factor is associated with the bifundamental hypermultiplet under U Sp(2k) × U Sp(2k); this is in agreement with those written in (2.51). The Higgs branch Hilbert series of this quiver can be computed as in [43]. This yields the same results as the Coulomb branch Hilbert series of (2.51), and hence provides a non-trivial check of the proposal in [27].

Flavoured G 2 affine Dynkin diagrams
Below we present the generalised quivers diagrams whose Coulomb branches correspond to the moduli space correspond to k G 2 instantons on C 2 /Z 2 with various monodromies at infinity. For each quiver, the monodromy at infinity breaks G 2 symmetry to G 2 and to SU (2) × SU (2), respectively from left to right. These can be seen from the Coulomb branch Hilbert series computed below. For the sake of brevity in writing the formulae below, let us define where ∆ m denotes the contribution from the fundamental matter attached to the G 2 affine quiver. It may depend on u's and n's. The Coulomb branch Hilbert series of the left quiver in (2.62) can be computed using the prescription given in [27] as follows: where we donate by [0, 1] the adjoint representation of G 2 , and Indeed, this is the Hilbert series of (y 2 ) , (2.67) and

Instantons on a smooth ALE space
Hitherto we have discussed the moduli spaces of instantons on orbifold singularities and their 3d field theory realisations. In this section, we consider the situation in which such singularities are resolved and the study moduli space of instantons on the smooth ALE space. The goal is to describe such a space using the moduli spaces of 3d N = 4 gauge theories. For instantons in a unitary gauge group on the ALE space of type A n−1 , the blow-up parameters of C 2 /Z n are given by the Fayet-Iliopoulos (FI) terms [6] of the KN quiver (2.1). Under mirror symmetry, these FI parameters are in correspondence with the mass parameters [23] in the mirror gauge theory described around (2.7).
On the other hand, for instantons in an orthogonal gauge group, one may not be able to turn on such an FI term, for example, in quiver (2.29). Thus, the blow-up parameter may not be apparent from the weakly coupled Lagrangian description. In particular, for SO(8) instantons on the smooth ALE space C 2 /Z 2n with monodromy at infinity such that SO(8) is broken to SO(4) × SO(4), it was pointed out in [37] that the blow-up parameter of C 2 /Z 2n can be made apparent in the field theoretic description by exploiting theories of class S and their dualities.
The purpose of this section is to reformulate such a description using 3d gauge theories and present a quiver whose Coulomb branch describes the moduli space of SO(2N ) instantons on the smooth ALE space C 2 /Z 2n . For simplicity, we restrict our presentation to the case in which the monodromy at infinity breaks SO(2N ) to SO(2N − 4) × SO (4). For N = 4, we compare our results with those in [37], where a different approach was adopted. In fact, it was proven in [5] that the Higgs branch of such quiver describes the moduli space of P U (2n) instantons on C 2 / D N with monodromy at infinity such that P U (2n) is unbroken. Note that this 3d N = 4 quiver is a bad theory in the sense of [30] for all k ≥ 1, because the U Sp(2k + 2) gauge group has 2k + 2 flavours of fundamental hypermultiplets charged under it 13 . Motivated by (2.15) and by the brane configuration in [11], we propose that a mirror theory of quiver (3.1) is Thus, the Coulomb branch of (3.2) describes the moduli space of SO(8) instantons on C 2 /Z 2 with monodromy at infinity such that SO(8) is broken to SO(4) × SO(4).
The quaternionic dimension of the Coulomb branch of (3.2) is and the quaternionic dimension of the Higgs branch of (3.2) is The former is indeed the dimension of the moduli space of k SO(8) instantons on C 2 /Z 2 [9,37], and the latter is equal to with the total rank of the gauge groups in quiver (3.1). In quiver (3.2), there are two flavours of fundamental hypermultiplets charged under gauge group U (k + 1). Hence the theory admits one mass parameter that cannot be eliminated by shifting any of the dynamical fields (this is actually the difference between the masses of the two flavours). This mass parameter is indeed identified with the blow-up parameter of C 2 /Z 2 . Note that, under mirror symmetry, this mass parameter corresponds to a "Fayet-Iliopoulos parameter" that is not visible in the Lagrangian of (3.1); this is an example of the "hidden" FI parameter discussed in [61].
It is also instructive to compare (2.16) with (3.2). Recall that the Coulomb branch of former describes the moduli space of k SO(8) instantons on C 2 /Z 2 . Observe that theory (2.16) does not admit a (non-trivial) mass parameter, whereas theory (3.2) admits a mass parameter that corresponds to the blow-up parameter of C 2 /Z 2 .

SO(2N ) instantons on C 2 /Z 2n
It is straightforward to generalise the previous result to a general SO(2N ) gauge group and to a general smooth ALE space of type A 2n−1 .
We propose a quiver theory whose Coulomb branch describes the moduli space of k SO(2N ) instantons on C 2 /Z 2n with monodromy at infinity such that SO(2N ) is broken to SO(2N − 4) × SO(4) to be as follows: We claim that this theory is a 3d mirror of the following quiver Note that, for N = 4, quiver (3.9) was discussed in Figure 9 of [37]. Notice that the quaternionic dimension of the Coulomb branch of (3.8) is dim H Coulomb of (3.8) = (2N − 2)k + n = kh ∨ SO(2N ) + n , (3.10) as expected. The quaternionic dimension of the Higgs branch of (3.8) is dim H Higgs of (3.8) = 2nk + n 2 = kh ∨ SU (2n) + n 2 , (3.11) in agreement with the dimension of the Coulomb branch of (3.9). Note that the latter is independent of N . Theory (3.8) admits 2n − 1 non-trivial mass parameters for the 2n flavours of the fundamental hypermultiplets under gauge group U (k + n). These mass parameters are identified with the blow-up parameters of C 2 /Z 2n .
In order to check this proposal, let us compute the Coulomb branch Hilbert series of (3.8) for some values of k, N and n using the method described in [24] and Appendix A. The unrefined Hilbert series for k = 1, N = 5 and n = 1 is given by 3.3 P U (2n) instantons on C 2 / D N It was proven in [5] that the Higgs branch of (3.8) describes the moduli space of k P U (2n) instantons on C 2 / D N with monodromy at infinity such that P U (2n) remains unbroken. As a result of mirror symmetry, the Coulomb branch of (3.9) also describes the same moduli space. Indeed, the N non-trivial FI parameters of the former and the N non-trivial mass parameters in the latter can be identified with the blow-up parameters of C 2 / D N . Let us examine the Higgs branch Hilbert series of (3.8). For k = 1, n = 1 and N = 4, namely one P U (2) instanton on C 2 / D 4 , the unrefined Hilbert series is H Higgs (3.8) (t) = 1 + 2t 4 + 4t 6 + 4t 8 + 4t 10 + 4t 12 + 2t 14 + t 18 (1 − t 2 ) 6 (1 + t 2 + t 4 ) 3 = 1 + 3t 2 + 8t 4 + 23t 6 + 52t 8 + 105t 10 + 204t 12 + 363t 14 + . . . (3.14) The coefficient 3 of t 2 is indeed the dimension of P U (2) ∼ = U (2)/U (1). It is worth contrasting this result with the the moduli space of one P U (2) instanton on singular orbifold C 2 / D 4 with monodromy at infinity such that P U (2) is unbroken. The latter is the product of two orbifolds (C 2 / D 4 ) × (C 2 /Z 2 ) according to (2.41), whereas the space corresponding to (3.14) is not.

Conclusions and open problems
In this paper, we study 3d N = 4 field theories whose Higgs branch and/or Coulomb branch describe the moduli space of instantons on a singular orbifold or on a smooth ALE space.
In the first part of the paper, we show that for instantons in a simple gauge group G on C 2 /Z n , the Hilbert series of such an instanton moduli space can be computed from the Coulomb branch of the quiver given by the affine Dynkin diagram of G with flavour nodes of unitary groups attached to various nodes of the Dynkin diagram. The techniques presented in [24,27] can thus be applied to compute the Coulomb branch Hilbert series of such quivers. For G a simply laced group of type A, D or E, the Higgs branch of such a quiver describes the moduli space of P U (n) instantons on C 2 / G, where G is the discrete group that is in McKay correspondence to G.
In the second part of the paper, the moduli space of instantons on a smooth ALE space is discussed. We present a quiver whose Coulomb branch describes the moduli space of SO(2N ) instantons on a smooth ALE space C 2 /Z 2n with monodromy at infinity such that SO(2N ) is broken to SO(2N − 4) × SO(4). Various special cases of our results are checked against those presented in [37] and yield an agreement. We leave the study of other monodromies at infinity for future work.
Our work leaves a number of open questions. In the case of singular orbifolds, one may naturally ask for a field theory that describes the moduli space of instantons in a non-unitary gauge group on orbifold C 2 /Γ, where Γ = D n or E 6,7,8 . In fact, a quiver description for the moduli space of SO(2N ) instantons on C 2 / D n was proposed in Figure 17 in section 4.4 of [11]. However, upon considering the special case in which SO(2N ) remains unbroken by the monodromy at infinity (i.e. when there is exactly one flavour node attached to the affine node of the Dynkin diagram), the quiver contains a U Sp(2N c ) gauge group with 2N c flavours. This renders the quiver a bad theory in the sense of [30] and all known methods for computing the Coulomb branch Hilbert series yield infinity. Hence we cannot use the Coulomb branch Hilbert series to check that the Coulomb branch of the quiver in this case has a desired property, namely being either C 2 / D n × N SO(2N ) or C 2 / D N × N SO(2n) ; cf. (2.8), (2.9), (2.13) and (2.14). On the other hand, by computing the Higgs branch Hilbert series of such a quiver, we find that the Higgs branch does not have such a property either. Given such a situation, we do not go into detail of this quiver and leave the study of such a moduli space of instantons for future work.
In the case of smooth ALE spaces, our knowledge is far from complete. So far in this paper we have only provided a few examples that can be checked explicitly against the known results. It would be nice to study such instanton moduli spaces in a systematic fashion.

A The Coulomb branch Hilbert series
In this appendix, we review the computation of the Hilbert series for the Coulomb branch of 3d N = 4 quiver gauge theories whose gauge group is a product of unitary groups. As is discussed in the main text (and in [27]), for suitable generalised quivers, possibly with non-simple laces, this method computes the Hilbert series of the moduli space of instantons.
A weak coupling description of a 3d N = 4 theory is specified by vector multiplets in the adjoint representation and matter fields (hypermultiplets or half-hypermultiplets) in some representation of the gauge group. At a generic point on the Coulomb branch the scalars in the vector multiplet acquire non-zero vacuum expectation values, breaking the gauge group G of rank r to U (1) r . As a result, matter fields and W-bosons gain masses and are integrated out, while the r massless gauge fields, the photons, can be dualised to scalars. Hence the low energy effective theory of the Coulomb branch consists of r abelian vector multiplets which, by virtue of the gauge field dualisation to a scalar, can be themselves dualised to twisted hypermultiplets.
The previous description breaks down at the origin of the Coulomb branch, which corresponds to a strongly coupled superconformal fixed point in the infrared (IR). Moreover, when the residual gauge group is non-abelian, the dualisation of a nonabelian vector multiplet is not understood. A suitable description of the Coulomb branch at the fixed point involves disorder operators that cannot be described in terms of a polynomial in the microscopic degrees of freedom [62]. These operators are known as monopole operators and are defined by specifying a Dirac monopole singularity at an insertion point in the Euclidean path integral [63]. Monopole operators are classified by embedding U (1) into the gauge group G, and are labeled by magnetic charges which, by a generalised Dirac quantization [64], take value in the weight lattice Γ G ∨ of the GNO or Langlands dual group G ∨ [65,66]. The monopole flux breaks the gauge group G to a residual gauge group H m by the adjoint Higgs mechanism. Restricting to gauge invariant monopole operators is achieved by modding out by the Weyl group W G of G, thus restricting m ∈ Γ G ∨ /W G .
In a 3d N = 2 theory, half-BPS monopole operators are contained in the chiral multiplets. There is a unique BPS monopole operator V m for each choice of magnetic charge m [67]. If the theory has N = 4 supersymmetry, the N = 4 vector multiplet decomposes into an N = 2 vector multiplet V and a chiral multiplet Φ in the adjoint representation. To describe the Coulomb branch, V is replaced by monopole operators V m , which now can be dressed by the classical complex scalar φ inside Φ. This dressing preserves the same supersymmetry of a chiral multiplet [68] if and only if φ is restricted to a constant element φ m of the Lie algebra of the residual gauge group H m [24]. The monopole operators that parametrise the Coulomb branch of an N = 4 field theory are therefore polynomials of V m and φ m , which are made gauge invariant by averaging over the action of the Weyl group [24].
Let us focus on a gauge group G that is a product of U (N i ) unitary groups, which are self-dual. For U (N ) monopole operators V m , with magnetic charge m = diag(m 1 , ..., m N ), the weight lattice of the dual group is given by Γ U (N ) = Z N = {m i ∈ Z, i = 1, .., N }. Modding out by the Weyl group S N restricts the lattice to the Weyl chamber Γ U (N ) /S N = m ∈ Z N |m 1 ≥ m 2 ≥ ... ≥ m N . For U (N ) gauge groups, which are not simply connected, its centre U (1) corresponds to a topological U (1) J symmetry group. Classically, monopole operators are only charged under this symmetry. To each such U (N i ) gauge group, we associate a fugacity z i for the topological U (1) J i symmetry with conserved current * Tr F i , where F i is the field strength of the i-th gauge group.
Other charges are acquired quantum-mechanically. In particular, monopole operators become charged under the Cartan U (1) C of the SU (2) C R-symmetry acting on the Coulomb branch. For a Lagrangian N = 4 gauge theory with gauge group G, where the first contribution, arising from vector multiplets, is a sum over the positive roots of G, while the second contribution is a sum over the weights of representations of the gauge group G of the hypermultiplets. The fugacity for this R-symmetry is called t 2 in the following. The dimension formula (A.1) was conjectured in [30] based on a weak coupling computation in [67], and later proven in [69,70]. For the theories that we study in this paper, which are good or ugly in the sense of [30], (A.1) is believed to equal the scaling dimension in the IR CFT.
The main focus of this paper is on field theories described by Dynkin diagrams, possibly non-simply laced and possibly with flavours. We propose the following prescription for computing the R-charge of a monopole operator, generalising the Lagrangian formula (A.1). Each diagram is constructed from two basic building blocks: a node and a line. They contribute to (A.1) as follows: • A U (N ) node, with magnetic charge m, contributes to the Coulomb branch Hilbert series as follows: • A line connecting the nodes U (N 1 ) and U (N 2 ) can be either a single line (−), a double line (⇒) or a triple line ( ), which we take to be oriented from node 1 to node 2. Let us assign magnetic charges m (1) and m (2) to U (N 1 ) and U (N 2 ) respectively. We propose that the contribution from a line is: ∆ line (m (1) , m (2) ) = 1 2 If one of the nodes, say U (N 1 ), is a flavour node (i.e. donated by a square), then the corresponding m (1) should be treated a background magnetic flux (see e.g. [71]). In this paper, we turn off such background fluxes and hence the contribution to the dimension formula of a single line connecting a square node and a circular node is given by Let us enumerate gauge invariant chiral operators on the Coulomb branch of nonsimply laced quivers according to their quantum number J i and ∆. The generating function, also known as the Coulomb branch Hilbert series, of such operators is given by [24] H(t, z) = . Each component of the formula can be explained as follows: • The sum is over GNO magnetic sectors [65], restricted to a Weyl chamber to impose invariance under the gauge group G. There is precisely one bare monopole operator per magnetic charge sector [67].
• The factors z J(m) t 2∆(m) account for the topological charges and conformal dimension of bare monopole operators of magnetic charge m.
• The factor P G (t; m) reflects the dressing of a bare monopole operator V m by polynomials of the classical adjoint scalar φ m which are gauge invariant under the residual gauge group H m left unbroken by the monopole flux. The contribution of this dressing factor to the Hilbert series is given by the generating function of independent Casimir invariants under the residual gauge group H m : where d i (m) are the degrees of the Casimir invariants of H m . We refer the readers to Appendix A of [24] for more details on these classical dressing factors.
We demonstrate the use of formula (A.5) to compute the Coulomb branch Hilbert series of various quiver theories in Section 2.3.