Instanton Operators and the Higgs Branch at Infinite Coupling

The richness of 5d $\mathcal{N}=1$ theories with a UV fixed point at infinite coupling is due to the existence of local disorder operators known as instanton operators. By considering the Higgs branch of $SU(2)$ gauge theories with $N_f \leq 7$ flavours at finite and infinite coupling, we write down the explicit chiral ring relations between instanton operators, the glueball superfield and mesons. Exciting phenomena appear at infinite coupling: the glueball superfield is no longer nilpotent and the classical chiral ring relations are quantum corrected by instanton operators bilinears. We also find expressions for the dressing of instanton operators of arbitrary charge. The same analysis is performed for $USp(2k)$ with an antisymmetric hypermultiplet and pure $SU(N)$ gauge theories.


Introduction
The dynamics of five-dimensional supersymmetric gauge theories has many interesting features. From the Lagrangian perspective these field theories are not renormalisable. However, by using string theoretic methods along with field theory analysis, it was demonstrated that a number of such field theories can be considered as flowing from certain non-trivial superconformal field theories in the ultraviolet (UV) [1][2][3][4]. Such UV fixed points at infinite gauge coupling may exhibit an enhancement of the global symmetry. In particular, in the seminal work [1], it was pointed out that the UV fixed point of 5d N = 1 SU (2) gauge theory with N f ≤ 7 flavours exhibits E N f +1 flavour symmetry, which enhances from the global symmetry SO(2N f )×U (1) apparent in the Lagrangian at finite coupling. Since then a large class of five dimensional supersymmetric field theories have been constructed using webs of five-branes [5][6][7] and the enhancement of the global symmetry of these theories has been studied using various approaches, including superconformal indices [8][9][10][11][12][13][14][15][16][17][18][19], Nekrasov partition functions and (refined) topological string partition functions [20][21][22][23][24][25][26][27][28].
In five dimensions, instantons are particles charged under the the U (1) global symmetry associated with the topological conserved current J = 1 8π 2 Tr * (F ∧ F ); this global symmetry is denoted by U (1) I in the rest of the paper. In the UV superconformal field theory, the instanton particles are created by local operators known as instanton operators, that insert a topological defect at a spacetime point and impose certain singular boundary conditions on the fields [29][30][31]. These operators play an important role in enhancing the global symmetry of the theory. For 5d N = 1 field theories at infinite coupling, it was argued that instanton operators with charge I = ±1, form a multiplet under the supersymmetry and flavour symmetry [31]. In 5d N = 2 Yang-Mills theory with simply laced gauge group, it is believed that the instanton operators constitute the Kaluza-Klein tower that enhances the Poincaré symmetry and provides the UV completion by uplifting this five dimensional theory to the 6d N = (2, 0) CFT [29,32,33].
Standard lore says that the Higgs branch of theories with 8 supercharges in dimensions 3 to 6 are classically exact, and do not receive quantum corrections. In 5 dimensions, this statement turns out to be imprecise, and should be corrected. In fact, one of the main points of the paper, is that there are three different regimes, given by 0, finite, and infinite gauge coupling. The hypermultiplet moduli space, which we always refer to as the Higgs branch, turns out to be different in each of these regimes, and hence our analysis corrects and sharpens the standard lore. The main goal of the paper is to understand how, at infinite coupling, instanton operators correct the chiral ring relations satisfied by the classical fields at finite coupling.
In order to perform such an analysis we start from the known Higgs branch at infinite coupling and write the Hilbert series of such a moduli space for various 5d N = 1 theories. We mostly focus on the SU (2) gauge theories with N f flavours, for which string theory arguments show that the Higgs branch at infinite coupling is the reduced moduli space of one E N f +1 instanton on C 2 [1]. The Hilbert series counts the holomorphic functions that parametrise the Higgs branch, graded with respect to the Cartan subalgebra of the (enhanced) flavour symmetry and the highest weight of the SU (2) R-symmetry of the theory: (1. 1) where H is the Hilbert space of chiral operators of the SCFT, R the SU (2) R isospin and H A the Cartan generators of the enhanced global symmetry. Such a Hilbert series can then be expressed in terms of the global symmetry of the theory at finite coupling -the latter is a subgroup of the enhanced symmetry at infinite coupling: H(t, y(x, q)) = Tr H t 2R q I x Ha a , (1.2) where I is the topological charge and H a the Cartan generators of the SO(2N f ) flavour symmetry. This decomposition allows us to extract the contributions of the classical fields and the instanton operators to the Higgs branch chiral ring and explicitly write down the relations they satisfy.
The paper is organised as follows. In section 2 we study the Higgs branch of SU (2) gauge theories with N f ≤ 7 flavours, spell out the relations in the chiral ring in terms of mesons, glueball and instanton operators, and discuss the dressing of instanton operators. We generalise the analysis to pure U Sp(2k) Yang-Mills theories with an antisymmetric hypermultiplet in sections 3 and 4, and to pure SU (N ) Yang-Mills in section 5. We close the paper with a discussion of our results and an outlook in section 6. Several technical results are relegated to appendices.
The analysis presented in this paper focuses on how the Higgs branch of these 5d theories changes along the RG flow. In particular we take care in distinguishing three different regimes for these theories, the operators that contribute to the chiral ring on the Higgs branch 1 and the defining equations that these operators satisfy: • In the classical regime, where fermions are neglected, these 5d theories have the usual Higgs branch which is just given by M 1,SO(2N f ) , the centred (or reduced ) moduli space of one SO(2N f ) instanton. The gauge invariant operators that generate this space are mesons M ab , constructed out of chiral matter superfields in the bifundamental of the SU (2) gauge group and SO(2N f ) flavour group. The relations that these generators satisfy on the moduli space can be extrapolated from its description as the minimal nilpotent orbit of SO(2N f ) [34]. They are the usual Joseph relations [35] and their transformation properties can be read off from the decomposition of the second symmetric product of the adjoint, the representation in which the generator transforms. Let V (θ) denote the adjoint representation. The decomposition prescribes that the relations transform in the representation I 2 .
For SO(2N f ) We can construct these representations from the adjoint mesons M ab as follows. Take M to be an antisymmetric ., 2N f . Then the two terms of (2.2) correspond respectively to: We call the last equation the rank 1 condition, since for an antisymmetric matrix it is equivalent to the vanishing of all degree 2 minors.
• When the coupling is finite, one needs to take into account the contribution from the gaugino sector. In particular, the glueball superfield S, which is a chiral superfield bilinear in the gaugino superfield W, is now no longer suppressed and will de jure appear in the chiral ring. This operator satisfies a classical relation in the chiral ring as in four dimensions [36], namely hence S is the only extra operator that one needs to consider at finite coupling.
Geometrically we interpret this operator as generating a 2-point space, which by a slight abuse of notation we denote by Z 2 . Algebraically the Hilbert series for this space is simply written as where 1 signifies the identity operator and the t 2 term is associated to the quadratic operator S. The fugacity t grades operators by their SU (2) R representation and the normalisation is chosen so that the power is twice the isospin. The meson M ab and the glueball superfield S obey the chiral ring relation [36,37] SM ab = 0 .
This signifies that the spaces M 1,SO(2N f ) and Z 2 intersect only at the origin.
From an algebraic perspective, when two moduli spaces X and Y intersect, the Hilbert series of their union is given by the surgery formula where the subtraction is done to avoid double counting [38]. Thus, when Z 2 is glued to M 1,SO(2N f ) , the net effect on the Hilbert series is simply that of adding a t 2 to the Hilbert series of M 1,SO(2N f ) .
The plethystic logarithm 2 of this newly obtained expression is interesting: it shows that at order t 4 there are two extra relations compared to the classical regime, one transforming in the singlet and one transforming in the adjoint of SO(2N f ). The singlet relation is (2.5). For the adjoint relation the only possible extra operator that one can construct in such a representation is SM ab . The adjoint relation is then precisely (2.7).
• At infinite coupling, the moduli space is a different space altogether. Instanton operators, carrying charge under U (1) I , contribute to the chiral ring and are responsible for prompting symmetry enhancement: the Higgs branch in this regime becomes isomorphic to the reduced moduli space . In order for this to happen a crucial event on the chiral ring takes place: instanton and anti-instanton operators I and I of U (1) I charge ±1 correct the relation (2.5). 3 This is the most dramatic dynamical mechanism happening at infinite coupling: the operator S is no longer a nilpotent bilinear in the vector multiplet and it becomes, for all intents and purposes, a chiral bosonic operator on the Higgs branch. The contribution of S to the chiral ring will no longer amount to (2.6), but instead an infinite tower of operators will appear generating a factor (1 − t 2 ) −1 in the Hilbert series.
The purpose of this paper is to explore these statements quantitatively for known cases of UV-IR pairs of theories. We do this as follows. Starting with the Hilbert series of the reduced one E N f +1 instanton moduli space written in terms of representations of E N f +1 [39], we decompose the representations under which the holomorphic functions on this space transform into those of SO(2N f ) × U (1) I . For all theories of our interest, the Hilbert series after this decomposition admits a very simple expression in terms of the highest weight generating function [40]. This allows us to analyse the generators of the moduli space in terms of instanton operators and clas- 2 The plethystic logarithm of a multivariate function f (x 1 , ..., x n ) such that f (0, ..., 0) = 1 is where µ(k) is the Moebius function. The plethystic logarithm of the Hilbert series encodes generators and relations of the chiral ring. 3 We call the instanton operator I of topological charge −1 "anti-instanton operator", even though it is mutually BPS with the positively charged instanton operator I. sical fields, and in many cases the relations between such generators are sufficiently simple to be written down explicitly.

E 0
The E 0 theory is the trivial case. There is no hypermultiplet moduli space. Consequently the Hilbert series for this theory is just given by 1, corresponding to the identity operator. The theory has no RG flow. Its interest lies in it being the limiting case of all the theories we consider in this section since none of the operators (M ,S,I, I) makes an appearance.
2.2 N f = 0 A pure SU (2) SYM theory with N = 1 supersymmetry in 5d can be obtained by flowing from two UV fixed points which have different global symmetry. The existence of these two theories is dictated by a discrete θ parameter taking value in π 4 (Sp(1)) = Z 2 [2]. For the non-trivial element the global symmetry at infinite coupling is E 1 = U (1) whilst for the identity element the global symmetry is E 1 = SU (2).

The E 1 theory
For the theory with θ = π no enhancement of the global symmetry occurs: the global symmetry at finite and infinite coupling is the instanton charge symmetry U (1) I . Here instanton operators are absent and the generator of the moduli space is just S obeying S 2 = 0, both at infinite and finite coupling. The moduli space generated by this operator is simply Z 2 . Classically the moduli space is trivial.

The E 1 theory
For the theory associated to the trivial element of the Z 2 valued θ parameter the U (1) I topological symmetry is enhanced to SU (2) by instanton operators at infinite coupling. In this regime the Higgs branch of the theory is isomorphic to the reduced moduli space of one-SU (2) instanton M 1,SU (2) , which is the orbifold C 2 /Z 2 . This theory is the prototypical example of the class we study. Since there is no flavour symmetry, we can understand the three regimes by means of simple physical arguments.
As we flow away from the UV fixed point, the Higgs branch is lifted and its only remnant is a discrete Z 2 space generated by S. Classically, even this contribution can be neglected and the Higgs branch is completely absent. This is a remarkable effect whereby from no Higgs branch in the classical regime a full Higgs branch opens up at infinite coupling.
Algebraically we start from the Hilbert series for C 2 /Z 2 and decompose it in representations of U (1) I so that we can identify the contribution from instanton operators, as well as the finite coupling chiral operators, and their relations.
The Hilbert series for C 2 /Z 2 can be written as , (2.10) where t is the fugacity for the SU (2) R symmetry, x is the fugacity for the SU (2) global symmetry acting on C 2 /Z 2 , and [2n] x stands for the character, as a function of x, of the representation of SU (2) with such a Dynkin label. Identifying the Cartan subalgebra of the SU (2) symmetry with U (1) I , we obtain (2.11)

The generators and their relations
Eq. (2.11) has a natural interpretation in terms of operators at infinite coupling: • Each term in the sum t 2|j| q j corresponds to an instanton operator I +|j| for j > 0 and an anti-instanton operator I −|j| for j < 0 that is the highest weight state of the SU (2) R representation with highest weight 2|j|. 4 q is the fugacity for the instanton number U (1) I . The plethystic logarithm of the Hilbert series shows that the instanton operator I +|j| is generated by the charge 1 operator I +1 ≡ I through the relation I +|j| = (I) j . Similarly I −|j| = ( I) j where I ≡ I −1 .
• The tower of operators generated by S can be identified with the factor (1 − t 2 ) −1 . This enhancement in the number of operators constructed from powers of S is crucial: at infinite coupling S is a full-on operator on the Higgs branch and, together with the instanton and anti-instanton operators I, I, forms a triplet of the SU (2) that generates C 2 /Z 2 .
From this form of the Hilbert series we can also give another interpretation to the Higgs branch at infinite coupling. Instanton operators on the Higgs branch in 5d N = 1 theories play a similar role to monopole operators in 3d N = 4 [41] and N = 2 theories [41,42]: in this sense (2.11) can be interpreted as the space of dressed instanton operators, where the factor 1 1−t 2 is the dressing from the operator S and it is freely generated.
The numerator in the rational function of (2.11) signifies a relation quadratic in the operators which can only be given by the defining equation for C 2 /Z 2 . At finite coupling, where I, I = 0, we recover the known chiral ring relation (2.5), i.e. the nilpotency of the operator S. As we have explained, the only remnant of C 2 /Z 2 is a residual Z 2 generated precisely by S.
Classically, we can set S = 0 and lift the Higgs branch entirely.
The Hilbert series of M 1,E 2 can be written using (2.8) as: where H[Z 2 ] = 1 + t 2 is generated by an operator P such that P 2 = 0. The additional t 2 term in (2.13) introduces an extra generator P compared to This addition is necessary to saturate the four generators of U (2). The generator of the C 2 /Z 2 factor is Φ ij , i = 1, 2, with Φ ij = Φ ji . Moreover the two generators satisfy where ij is defined by its antisymmetry property and 12 = 1. The first relation is the usual Joseph relation for the SU (2) minimal nilpotent orbit C 2 /Z 2 whilst the last equation encodes the fact that the two spaces, C 2 /Z 2 and Z 2 , only intersect at one point, the origin of the moduli space.

The generators and their relations
The theory at finite coupling has a Higgs branch which consists of the union of two Z 2 spaces that meet at the origin, i.e a space with three points. This becomes clear by taking the infinite coupling relations on the Higgs branch and setting the instanton operators to zero. This can be seen as follows. Decompose the generators Φ ij and P of M 1,E 2 by letting where S is the lowest operator in the glueball superfield, M is an SO(2) mesonic operator and I, I are the instanton and anti-instanton operators respectively. The relations in (2.14) can then be rewritten as: The finite coupling limit is obtained by setting I, I = 0 whence we recover the finite coupling relations S 2 = SM = M 2 = 0. The Higgs branch at finite coupling is then precisely the space generated by S and M subject to these relations, i.e Z 2 ∪ Z 2 .
In the classical regime, where we neglect the contribution from S, we recover the space Z 2 , the reduced moduli space of 1 SO(2) instanton generated by M , such that M 2 = 0. 5

Expansion in the instanton fugacity
It is instructive to rewrite (2.13) as an expansion in q, the U (1) I fugacity. Replacing x, the fugacity for SU (2), by q 1/2 we have that: (2.24) Hence a bare instanton operator with U (1) I charge n is the highest weight state of the spin |n| representation of the SU (2) R symmetry. For n = 0, the tower of states originating from the glueball (1 − t 2 ) −1 , i.e the space C, acts as a dressing for the instanton operators. For n = 0, the dressing is a different space, due to the presence of an extra piece of the moduli space unaffected by instantons. It is in fact the space generated by S and M , subject to the relations SM = 0 and M 2 = 0, i.e C ∪ Z 2 .

N f = 2
The reduced moduli space of one E 3 = SU (3) × SU (2) A instanton 6 is isomorphic to the union of two hyperKähler cones, the reduced moduli space of one SU (3) instanton, M 1,SU (3) , and the reduced moduli space of one SU (2) A instanton M 1,SU (2) A , meeting at a point. As an algebraic variety it is generated by operators transforming in the reducible adjoint representation subject to the Joseph relations, which can be extracted from (2.1). The Hilbert series can again be written using the surgery formula (2.8) as where x = (x 1 , x 2 ) are the fugacities for SU (3) and y is the fugacity for SU (2) A . The SU (3) factor of the enhanced global symmetry E 3 is broken to SU (2) B × U (1) I when one flows away from the fixed point. The U (1) factor is identified with the topological symmetry U (1) I , up to a normalisation of charges that is explained below. The SU (2) B factor instead combines with the SU (2) A factor in E 3 , which acts as a spectator for the breaking, and together they form a global symmetry SO(4). Hence, we decompose the representations of SU (3) in (2.25), whilst keeping the representations of SU (2) A , i.e we break: be the fugacities of SU (3); z and w be those of SU (2) B and U (1) respectively (the fugacity w for the U (1) factor will be related to the fugacity q for U (1) I shortly). Under the action of this matrix, the weights of the fundamental representation of SU (3) are mapped as follows: In other words, we have The character of the fundamental representation of SU (3) is mapped to that of while the adjoint representation decomposes as The U (1) charge is a multiple of 3 for states in the root lattice. To obtain integer instanton numbers I ∈ Z, we set w 3 = q, where q is the fugacity for U (1) I .

Under this map, the Hilbert series of the reduced moduli space of one SU (3) instanton becomes
where z is the SU (2) B fugacity and q is the U (1) I fugacity.
The highest weight generating function 7 [40] associated to this Hilbert series is where µ is the fugacity for the highest weight of SU (2) B . Thus, the highest weight generating function for (2.25) becomes where µ and ν are the fugacities corresponding to the highest weights of SO (4) The highest weight generating function (2.35) provides five dominant representations that generate the highest weight lattice in a simple way. The information can be read as follows. Inside the first PE we can identify the SU (2) R spin 2 generators: the singlet S, the instanton operator µq which we denote by I ≡ I 1 , the anti-instanton operator µq −1 which we denote by I ≡ I −1 , and the meson transforming in the adjoint of SU (2) B , µ 2 , which we denote by T αβ and is subject to the traceless condition T αβ αβ = 0. We also identify a relation quadratic in the generators and transforming in the adjoint representation of SU (2) B , the term −µ 2 t 4 . The second PE is the contribution from the spectator SU (2) A , with the only representation ν 2 , the inert meson that we denote by Tαβ.
Eq. (2.35) is an expression that carries information about the representation theory more concisely than the Hilbert series and furthermore the lattice it encodes is a complete intersection. However in order to write the relations between the operators on the chiral ring explicitly, we consider what the Joseph relations for M 1,E 3 imply.

The generators and their relations
For the M 1,E 3 case, the generators are Φ i j , with i = 1, 2, 3 and Φ i i = 0, transforming in the [1, 1; 0] of SU (3) × SU (2) A , and Tαβ with Tαβ αβ = 0, transforming in the 7 The highest weight generating function for group of rank r is defined as follows:  Hence the generator Φ i j obeys a quadratic relation transforming in the reducible representation [1, 1; 0] + [0, 0; 0] whilst Tαβ obeys a singlet relation. This is to be expected, since the minimal nilpotent orbit of traceless 2 × 2 matrix is the subset of matrices with zero determinant. There is also a quadratic relation mixing Φ i j and Tαβ transforming in the [1, 1; 2]. We can write these relations as follows: 8 The glueball operator, the instanton and anti-instanton operators and the meson are embedded into the generator Φ i j since this is the one transforming nontrivially under the SU (3) factor that breaks into SU (2) B × U (1). We choose the following embedding: The aim is to decompose the relations in the first and third equations of (2.37). (2.39) Thus the relations in the first equation of (2.37) decompose into the five relations (2.40) 8 For T a symmetric 2 × 2 matrix, T 2 = 0, det T = 0 and Tr T 2 = 0 are equivalent statements.
The relations in the second line of (2.39) can be explicitly written as: Recall also from (2.37) that In total there are thus 10 equations, namely (2.40), (2.41) and (2.42). 9 The finite coupling result that S be nilpotent is obtained by virtue of the last equation of (2.40) when we set I, I = 0. Consequently we also restore the condition Tr(T 2 ) = 0, which, for a traceless 2 × 2 matrix, is equivalent to T 2 = 0, the classical relation. Moreover (2.7) is also recovered.
Another approach to see these 10 relations between the operators at infinite coupling is to rewrite (2.25) in terms of characters of representations of SO(4)×U (1) and compute its plethystic logarithm. For reference, we present such a Hilbert series up to order t 4 as follows:

The plethystic logarithm of this Hilbert series is
Indeed, the 10 relations listed in (2.40), (2.41) and (2.42) are in correspondence with the terms at order t 4 in (2.44). We emphasise here that the computation of the plethystic logarithm provides an efficient way to write down the relations that are crucial to describe the moduli space. This method is applied for the cases of higher N f in subsequent sections. 9 Notice that the meson Tαβ, the generator for the spectator SU (2) A , is made up of the same fundamental fields (quarks) as the meson T αβ . Before considering gauge invariant combinations, the quarks Q αα a , with α,α = 1, 2 and a = 1, 2, transform in the vector representation of the global symmetry SO(4) ∼ = SU (2) A × SU (2) B and in the fundamental representation of the gauge group SU (2). Out of these quarks the following gauge invariant mesons can be constructed: The difference between these two mesons is in the relations they satisfy at infinite coupling, one being quantum corrected whilst the other being unaffected: Tr( T 2 ) = 0 vs 2 Tr(T 2 ) = S 2 = I · I.
We can rewrite these relations in terms of a 4 × 4 adjoint matrix M ab , with a, b, c, d = 1, . . . , 4 vector indices of SO(4), such that as follows: The gamma matrices γ a for SO(4) take the following index form: (γ a ) αα (2.53) and the product of two gamma matrices is defined as: where the spinor indices are raised and lowered with the epsilon tensor.

Expansion in the instanton fugacity
It is useful to rewrite (2.35) in terms of an expansion in q: From here, we can extract the transformation properties of instanton operators of charge n under the U (1) I . They transform as spin |n| highest weight states for SU (2) R and as spin |n|/2 representations of SU (2) B . The classical dressing for each q n instanton operator, the factor outside the sum, is, for n = 0, the moduli space of the one SU (2) B instanton with the (1−t 2 ) −1 term providing the glueball contribution. For n = 0 the contribution from SU (2) A , the second term in (2.55), enhances the classical dressing to the glueball corrected moduli space of the centred one SO(4) instanton.

N f = 3
The moduli space of the reduced one E 4 = SU (5) instanton, M 1,E 4 =SU (5) , is the nilpotent orbit generated by the adjoint representation of SU (5). Its associated Hilbert series can thus be written as . In order to proceed with a decomposition from weights of SU (5) representations to those of SO(6) × U (1), we choose the projection matrix which gives the fugacity map States in the root lattice carry a charge multiple of 5 for the U (1) associated to the fugacity w, hence we set w 5 = q in the following, where q is the fugacity for the integer quantized instanton number U (1) I . Then (2.56) can be written in terms of the character expansion of SO(6) × U (1) ⊃ SU (5) as where [p 1 , p 2 , p 3 ] y is the character of a representation of SO(6) as a function of fugacities y = (y 1 , y 2 , y 3 ). The information contained in this equation can be carried compactly by means of the associated highest weight generating function where at t 2 we can again recognise the contribution of S, a singlet of SO (6), the instanton and the anti-instanton operators in the spinor [0, 1, 0] and cospinor [0, 0, 1] representations, and the meson in the adjoint representation [0, 1, 1], while at order t 4 is the basic relation between the operators. Notice that (2.60) is a generating function for a lattice with conifold structure.

The generators and their relations
The generators and the relations can be extracted from the plethystic logarithm of the Hilbert series. The Hilbert series of the reduced moduli space of 1 E 4 instanton can be written in terms of characters of SO (6)    Below we write down the generators corresponding to the terms at t 2 and the explicit relations corresponding to the terms at order t 4 of (2.62).
For SO (6), we use a, b, c, d = 1, . . . , 6 to denote vector indices and use α, β, ρ, σ = 1, . . . , 4 to denote spinor indices. Note that the spinor representation of SO(6) is complex. The delta symbol carries has one upper and one lower index: The gamma matrices γ a can take the following forms: where the α, β indices are antisymmetric. The product of two gamma matrices has one lower spinor index and one upper spinor index: As can be seen, the classical relations are corrected by instanton bilinears and this is a recurrent feature for all number of flavours. These relations can also be rewritten in terms of an SU (4) matrix M α β using the following relation (2.72)

Expansion in the instanton fugacity
We rewrite (2.60) as an expansion in q as follows: (2.73) Two very interesting features emerge from the q expansion. Firstly, an instanton operator of charge n has SU (2) R spin |n| and it transforms as an |n|-spinora representation with |n| on a spinor Dynkin label -of the global flavour group SO (6). Whilst in [31] it was found that this result holds for n = 1, here we find a prediction for all n.
Secondly the instanton operators are dressed by a factor, the one in front of the sum, which is generated by S and M ab , subject to the following relations: 2.6 N f = 4 The Higgs branch at infinite coupling for an SU (2) theory with N f = 4 flavours is isomorphic to the reduced moduli space of one E 5 = SO(10) instanton M 1,E 5 =SO(10) , which is given by the minimal nilpotent orbit of SO(10). Its Hilbert series is [0, n 1 , 0, n 2 + n 3 ] y q n 2 −n 3 t 2n 1 +2n 2 +2n 3 , (2.77) where we decompose representations of SO(8) × U (1) ⊂ SO(10) using a projection matrix that maps the weights of SO(10) representations to those of SO(8) × U (1) as follows (2.78) Under the action of this matrix, the fugacities x of SO(10) are mapped to the fugacities y of SO(8) and w of U (1) as follows: In (2.77) we set w 2 = q to have integer instanton numbers, rather than even. The corresponding highest weight generating function is where we recognise the usual SU (2) R spin-2 generators: the glueball superfield S, a singlet of SO(8), the instanton operators I α and I α associated to µ 4 q and µ 4 q −1 , both transforming in the same spinor representation of SO(8) with opposite U (1) charge, as well as the meson M ab , associated to µ 2 . The highest weight lattice is freely, generated as we see from the lack of relations at order t 4 .

The generators and their relations
The expansion of (2.77) up to order t 4 is given by   The gamma matrices γ a can take the following forms: The product of two gamma matrices has the following forms: and similarly for both upper indices; the indices α, β andα,β are antisymmetric. The product of four gamma matrices has the following forms: and similarly for both upper indices; the indices α, β andα,β are symmetric. The generators of the moduli space are M ab , which is a 8 × 8 antisymmetric matrix; the instanton operators I α and I α ; and the glueball superfield S.
The relations corresponding to terms at order t 4 of (2.82) can be written as

Expansion in the instanton fugacity
In terms of an expansion in q, (2.80) can be written as Here again we find that instanton operators of charge n are spin |n| of SU (2) R and transform in |n|-spinor representations of SO (8).
However the interpretation of the classical dressing is more subtle than in previous cases. The prefactor in the q expansion signifies a space which is algebraically determined by some of the conditions that define the moduli space of one SO (8) instanton; in particular it is a space generated by two operators, M ab , in the adjoint representation [0, 1, 0, 0] of SO (8)  (2.98) A projection matrix that maps the weights of E 6 to those of D 5 × U (1) is given by Under the action of this matrix, the fugacities of x of E 6 are mapped to the fugacities y of SO(10) and w of U (1) as follows: [0, n 1 , 0, n 2 , n 3 ] y q n 2 −n 3 t 2n 1 +2n 2 +2n 3 , (2.101) The corresponding highest weight generating function is

The generators and their relations
The expansion of (2.101) up to order t 4 is given by   For SO(10), we use a, b, c, d = 1, . . . , 10 to denote vector indices and α, β, ρ, σ = 1, . . . , 16 to denote spinor indices. Note that the spinor representation of SO(10) is complex. The delta symbol has the following form: The gamma matrices γ a can take the following forms: (γ a ) αβ and (γ a ) αβ , (2.106) where the α, β indices are symmetric. The product of two gamma matrices has the following form: The product of four gamma matrices has the following form: The generators of the moduli space are M ab , which is a 10 × 10 antisymmetric matrix; the instanton operators I α and I α ; and the gaugino superfield S.
The relations appearing in the plethystic logarithm (2.104) are as follows:

Expansion in the instanton fugacity
The highest weight generating function (2.102) can be expanded in the instanton number fugacity q as

N f = 6
The Hilbert series of M 1,E 7 can be written as (2.118) The E 7 representations can be decomposed into those of SO(12) × U (1) using the projection matrix: Under the action of this matrix, the fugacities x of E 7 are mapped to the fugacities y of SO (12) and the fugacity q of U (1) as (2.120) We then have the following highest weight generating function: where at order t 2 we recognise the contributions of: S, which is a singlet of SO (12); the instanton and the anti-instanton operators with U (1) I charge ±1 in the spinor representation [0, 0, 0, 0, 1, 0]; the instanton and the anti-instanton operators with U (1) I charge ±2 which are singlets of SO (12); the meson in the adjoint representation [0, 1, 0, 0, 0, 0]. In addition there is a fourth-rank antisymmetric tensor of SO(12) at order t 4 .

The generators and their relations
The expansion up to order t 4 of (2.121) is given by  The product of two gamma matrices has the following forms: where the spinor indices are symmetric. The product of four gamma matrices has the following forms: where the spinor indices are antisymmetric. The generators of the moduli space are M ab , which is a 12 × 12 antisymmetric matrix, the instanton operators I α 1+ , I α 1− and I 2+ , I 2− , and the glueball superfield S. From (2.123), we have the following sets of relations: To aid computations it is useful to rewrite (2.122) and (2.123) in terms of characters of SO(12) × SU (2). The reader can find the relevant formulae in Appendix B.

Expansion in the instanton fugacity
The highest weight generating function (2.121) can be expanded in powers of the instanton number fugacity q as (2.135) The first equality is a q expansion in terms of a double sum. This separates the classical dressing from the one and two instanton contributions. It is precisely the presence of both types of instantons as quadratic generators that, for N f > 5, complicates the features of the q expansion in terms of a one sum only. We still write such an expansion in the second equality, splitting it into odd and even terms.
2.9 N f = 7 The Hilbert series of M 1,E 8 can be written as (2.136) The E 8 representations can be decomposed into those of SO(14) × U (1) using the projection matrix We then have the following highest weight generating function:  The gamma matrices γ a can take the following forms: where the α, β indices are antisymmetric. The product of two gamma matrices is The product of three gamma matrices has the forms symmetric in the spinor indices. The product of four gamma matrices is The generators of the moduli space are M ab , which is a 14×14 antisymmetric matrix; the instanton operators I α and I α ; and the gaugino superfield S.
The relations corresponding to order t 4 of (2.141) are as follows:

Expansion in the instanton fugacity
The highest weight generating function (2.139) can be rewritten in terms of an implicit expansion in q involving 5 sums: (2.168)

U Sp(4) with one antisymmetric hypermultiplet
In this theory, we pick the trivial value of the discrete theta angle for the U Sp(4) gauge group. The Higgs branch at infinite coupling of this theory is identified with the reduced moduli space of 2 SU (2) instantons on C 2 [4], whose global symmetry is SU (2) × SU (2). The Hilbert series is given by (3.14) of [43]. For reference, we provide here the explicit expression of the Hilbert series up to order t 6 :   The corresponding highest weight generating function is (see (4.25) of [40]) where µ 1 and µ 2 are respectively the fugacities for the highest weights of the SU (2) acting on the centre of instantons and the SU (2) associated with the internal degrees of freedom. Let us use the indices a, b, c, d = 1, 2 for the first SU (2) and i, j, k, l = 1, 2 for the second SU (2). The generators of the moduli space are as follows.
• Order t 2 : The rank two symmetric tensors P ab and M ij in the representation [2; 0] and [0; 2] of SU (2) × SU (2): The relations at order t 5 are The relations at order t 6 are Let us now rewrite the above statements in SU (2)×U (1) language. Up to charge normalisation, we identify the Cartan subalgebra of the latter SU (2) associated with µ 2 with the U (1) I symmetry. More precisely, if w is the fugacity associated to the Cartan generator of the latter SU (2), then q = w 2 is the fugacity for the topological symmetry. The highest weight generating function can then be written as (3.11) This can be written as a power series in q as (3.12) The Hilbert series up to order t 6 can be written explicitly as follows: The plethystic logarithm of this Hilbert series is given by 14) The generators. At order t 2 , the generators are [2] : P ab with P ab = P ba , (3.15) q, q −1 , 1 : The generators P ab are identified as a product of two antisymmetric tensors: where the generators T a are identified as a product of two gauginos and one antisymmetric tensor The relations. The relation at order t 4 can be written as [0]t 4 : Tr(P 2 ) + S 2 = I I . (3.20) The relations at order t 5 can be written as

U Sp(2k) with one antisymmetric hypermultiplet
As in the previous sections, we pick the trivial value of the discrete theta angle for U Sp(2k) gauge group. The Higgs branch of the conformal field theory at infinite coupling is identified with the moduli space of k SU (2) instantons on C 2 [4]. Below we consider the moduli space of the theory at finite coupling. For k = 1, the Higgs branch at finite coupling is where C 2 is the classical moduli space of a U Sp(2) gauge theory with 1 antisymmetric hypermultiplet and Z 2 is the moduli space generated by the glueball superfield S such that S 2 = 0. The Hilbert series is then given by where the fugacity w corresponds to the number of gaugino superfields. For higher k, the theory in question can be realised as the worldvolume theory of k coincident D4-branes on an O8 − plane. Hence, the moduli space is expected to be the k-th symmetric power of C 2 × Z 2 , whose Hilbert series is given by where H[Sym n C 2 ](t, x) is the Hilbert series for the n-th symmetric power of C 2 : . We tested the result for k = 2 directly from the field theory side using Macaulay2; the details are presented in Appendix A.
Note that this result also holds for U Sp(2k) gauge theory with 1 antisymmetric hypermultiplet and 1 fundamental hypermultiplet. This is because the classical moduli space of this theory is the moduli space of k SO(2) instantons on C 2 -this space is in fact the k-symmetric power of the moduli space of 1 SO(2) instanton on C 2 , which is identical to C 2 .
Since the symmetric product Sym k (C 2 × Z 2 ) has a C 2 component that can be factored out, it is natural to define the Hilbert series H k (t; x, w) of the reduced moduli space as follows: Examples. For k = 2, we have these follow from the generators of the moduli space of two instantons, given by section 8.5 of [44]. Explicitly, these generators are Tr(X a 1 X a 2 ), Tr(X a 1 X a 2 X a 3 ), . . . , Tr(X a 1 X a 2 · · · X a k ), (4.10) Tr(WW), Tr(X a 1 WW), Tr(X (a 1 X a 2 ) WW) , . . . , Tr(X (a 1 · · · X a k−1 ) WW) where a 1 , a 2 , . . . , a k = 1, 2. The set of relations with the lowest dimension transform in the representation [k − 2] at order t k+2 .
In the limit k → ∞, the moduli space is thus freely generated by (4.10). 11 A similar situation was considered in [45], where it was pointed out that the generating function of multi-trace operators for one brane is equal to that of single trace operators for infinitely many branes. 5 Pure SU (N ): C 2 /Z N In the 5-brane web for 5d N = 1 SU (N ) pure Yang-Mills theory, one can use an SL(2, Z) transformation to make the charges of the external 5-brane legs to be (p 1 , q 1 ) = (N, −1) and (p 2 , q 2 ) = (0, 1). In this basis, the web can be depicted as follows (this example is for N = 3): At infinite coupling, the two 5-branes intersect and move apart, giving a one quaternionic dimensional Higgs branch. The intersection number is given by The Higgs branch at infinite coupling is therefore C 2 /Z N . The generators of the Higgs branch at infinite coupling are I, S, I, singlets under SU (N ), and with U (1) I charge +1, 0 and − 1 respectively. For N > 2, the isometry group of C 2 /Z N is U (1), under which the operators have charge +N, 0 and −N . For N = 2, the isometry is SU (2) and the operators form a triplet (I, S, I) with weights given by +2, 0 and − 2 respectively. These generators satisfy the relation S N = I I .

Discussion
A coherent picture of the Higgs branch of 5d N = 1 theories for all values of the gauge coupling emerges from this paper. In particular, we have presented explicit relations that define the chiral ring at infinite coupling and are consistent with those at finite coupling. A crucial result of this paper is the correction to the glueball superfield, S, which at finite coupling is a nilpotent bilinear in the gaugino superfield and at infinite coupling becomes an ordinary chiral operator on the Higgs branch. For pure SU (2) theories with N f ≤ 7 flavours a nice pattern was established. The finite coupling relations involving mesons and the glueball operator are corrected at infinite coupling by bilinears in the instanton operators, in the obvious way dictated by representation theory. New relations also arise which exist uniquely at infinite coupling.
By expanding the highest weight generating function of the Hilbert series at infinite coupling in powers of q, we have analysed the dressing of instanton operators by mesons and gauginos. For N f ≤ 5 the defining equations for the space associated to the dressing can be obtained by keeping the relations at infinite coupling which are not corrected by the instanton operators. For N f = 6, 7, the presence of charge ±2 instanton operators as generators independent from the charge ±1 ones complicates the picture and leaves the interpretation of the classical dressing in a preliminary and unsatisfactory stage.
The techniques developed in this paper could also be applied to other 5d N = 1 theories with known Higgs branch at infinite coupling. We leave this to future work. The long term goal is to better understand supersymmetric instanton operators and their dressing from first principles and use such knowledge to derive a general formula for the Hilbert series associated to the Higgs branch at infinite coupling. We hope that the results of this paper can shine some light in this direction.

A Hilbert series of chiral rings with gaugino superfields
In this appendix we present a method to compute the Hilbert series of the Higgs branch at finite coupling. In this computation we include the classical chiral operators as well as the gaugino superfield W.
In five dimensions, the gaugino λ A I carries the U Sp(4) spin index A = 1, . . . , 4 and the SU (2) R index I = 1, 2. Since we focus on holomorphic functions, which are highest weights of SU (2) R representations, we restrict ourselves to I = 1. In 4d N = 1 language, which we adopt throughout the paper, the fundamental representation of U Sp(4) decomposes to [1; 0] + [0; 1] of SU (2) × SU (2). These are usually denoted by undotted and dotted indices, respectively. Since the latter correspond to non-chiral operators in the 4d N = 1 holomorphic approach, we adhere to the undotted SU (2) spinor index. The gaugino superfield is henceforth denoted as W α .
We will see that the 4d N = 1 formalism adopted in this appendix yields results for the Hilbert series that are consistent with the chiral ring obtained by setting instanton and anti-instanton operators to zero in the five dimensional UV fixed point, which is discussed in the main body of the paper. The analysis is similar to the previous subsection. Let us denote the antisymmetric fields by X ij a , where a = 1, 2 and i, j = 1, . . . , 2k are the U Sp(2k) gauge indices. The F -terms associated to the classical Higgs branch is where J ij is the symplectic matrix associated with U Sp(2k).
For the gaugino superfield W ij α (with α = 1, 2), we impose the conditions Here we write Q i a as Qai and (W α ) ab as wabα. The ring R is multi-graded with respect to the following charges (in order): 1. the R-charge associated with the fugacity t, 2. the number of gaugino superfields associated with the fugacity w, 3. the weights of the SU (2) gauge group, and 4. the weights of the SU (2) symmetry associated with the index α.
After integrating over the U Sp(2k) gauge group and restricting to the scalar sector under the Lorentz group, we obtain the Hilbert series of the space Sym k C 2 × Z 2 , (A. 15) In particular, for k = 2, we recover the Hilbert series (4.6). The representation [0, 0, 0, 0, 0, 0; 2] corresponds to I 2+ , I 2− and S, [0, 0, 0, 0, 1, 0; 1] to I 1± and [0, 1, 0, 0, 0, 0; 0] to M . In the Hilbert series (B.1) there is only one independent singlet at order t 4 : this means that the singlets coming from these three sets of operators must be proportional to each other. These indeed correspond to the trace part of (2.129) and the relation (2.131).