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A note on D0-branes and instantons in 5d supersymmetric gauge theories

We refine a previous proposal for obtaining the multi-instanton partition function from the supersymmetric index of the 1d supersymmetric gauge theory on the worldline of D0-branes. We provide examples where the refinements are crucial for obtaining the correct result.


Introduction
The problem of counting instantons in supersymmetric gauge theories with eight supersymmetries has received some recent interest due to its appearance in the study of fivedimensional and six-dimensional superconformal field theories. In many cases these theories correspond to UV fixed points of 5d N = 1 supersymmetric gauge theories, in which the instantons, which are particles in five dimensions, provide a crucial ingredient in identifying the properties of the UV theory. To facilitate this we must be able to correctly compute the charges and degeneracies of the instanton states, namely to "count" instantons. In order to do this we must quantize the multi-instanton moduli space as provided by the ADHM construction [1]. The relevant data is contained in the multi-instanton partition function. In general this is a difficult problem since the moduli space contains singularities corresponding to instantons of vanishiing size. Another way to say this is that the corresponding supersymmetric linear sigma model [2] is not well defined in the UV, and requires a UV completion.
There have been a number of approaches for dealing with these singularities. Originally, Nekrasov used the so-called Omega-deformation to remove the singularities and compute the instanton partition function [3]. However this approach only works for U (N ). Subsequently, Nekrasov and Shadchin were able to overcome the singularities for other gauge groups by lifting to five dimensions [4,5].
An alternative approach is to find a UV completion of the ADHM sigma model, and attempt to extract from it the relevant data of the instanton moduli space. This approach is strongly motivated by string theory, in which five-dimensional supersymmetric gauge theories may be realized in terms of D4-branes in Type IIA superstring theory, and the instantons correspond to D0-branes inside the D4-branes [6]. The ADHM sigma model is naturally embedded in the 1d N = (0, 4) supersymmetric gauge theory on the D0-brane worldline. It is the low energy effective theory on the Higgs branch of the gauge theory. On the other hand the 1d N = (0, 4) gauge theory is a UV complete theory.
Recent advances in supersymmetric localization have made it possible to compute the exact partition function, or index, of such theories [7,8,9,10]. The multi-instanton partition function is then naively given by summing over instanton sectors: where I k is the index of the 1d N = (0, 4) gauge theory of k instantons, and q is the instanton fugacity. The index I k is computed using supersymmetric localization, which reduces the path integral to ordinary contour integrals over the complexified holonomies of the gauge multiplet. The general form of the multi-particle index is given by the plethystic exponent of the single-particle index f (q) (which will in general get contributions from multi-instanton bound states): By comparing with (1), one can extract information about the spectrum of one-particle states, up to a given instanton number k. Indeed one can often guess the complete f (q) by computing I k for a small number of k's. This was done for the N = 2 supersymmetric 5d U (N ) gauge theory in [7], and used to show the existence of instanton bound states at all instanton numbers, as required by the conjecture that this theory flows to the 6d A-type (2, 0) theory in the UV. However, more generally one has to overcome the following difficulty. The moduli space of the 1d gauge theory generically also has additional branches that contribute to the index. For N = 2 theories the 1d gauge theory has a Coulomb branch, and for N = 1 theories it can have both a Coulomb branch and a twisted Higgs branch parameterized by a twisted hypermultiplet. In the brane description these branches correspond to moving the D0-branes out of and away from the D4-branes. The contribution of the states on the extra branches must be removed in order to obtain the instanton partition function, which is just the contribution of the states on the Higgs branch. It seems natural to conjecture that the contribution of the extra states is accounted for by the reduced 1d gauge theory of the D0-branes in the absence of the D4-branes, namely where Z QM,red is computed for the 1d gauge theory without the D4-branes, and This was originally conjectured in [8] for the class of N = 1 theories with gauge group Sp(N ), N F ≤ 7 flavor hypermultiplets, and an antisymmetric hypermultiplet, that are realized in terms of D4-branes in the background of an orientifold 8-plane and N f D8branes [11]. It was shown in [8], by computing the 1-instanton contribution, that this correctly reveals the enhanced exceptional global symmetries for N F ≤ 5. Subsequently Z extra was computed to higher orders in these theories and an all-orders expression was conjectured in [9]. In particular this completed the work of [8] and showed the enhanced E 7 and E 8 symmetries for N F = 6 and N F = 7, respectively. The same issue was addressed in the N = 2 theories realized in terms of D4-branes in flat space or in the background of an orientifold 4-plane in [10]. For U (N ) it was shown that Z extra = 1, in line with the result of [7], which did not include this correction. However for O(N ) and Sp(N ) the contribution of the extra branches is non-trivial. Furthermore, for Sp(N ) it was found that an additional correction is required; in this case Z extra overcounts by a multiplicitive quantity, which corresponds to bound states of D0-branes that are confined to the orientifold 4-plane. This last observation can be understood from the fact that the reduced 1d gauge theory in this case still has a remnant of the original Higgs branch, whose states we do not want to remove. The additional correction adds these contributions back in.
We would like to formalize this into the following modified proposal for the contribution of the extra states: where f D0c (q) is the single-particle index of D0-branes that are confined to any 5d defect that remains after the 4-branes are removed. In addition, instead of simply removing the D4-branes by setting N = 0 in the 1d gauge theory, the reduced theory should be understood as the low-energy effective theory that results by turning on a mass for the fields charged under the 4-brane gauge symmetry and integrating them out. Geometrically this corresponds to moving the 4-branes away from the 0-branes. Depending on the theory, this can shift the values of Chern-Simons (CS) or discrete theta parameters. In all the examples studied in [7,8,9,10] the net shifts vanish, but, as we will see, in general they do not.
In this note we will present additional examples of 5d N = 1 gauge theories in support of our conjecture (5). We will consider the theories obtained by a Z 2 orbifold of the brane configuration of the N = 1 Sp(N ) + theory [12]. There is a choice in this orbifold corresponding to the action of worldsheet parity in the twisted sector. The so-called orbifold with vector structure leads to the quiver theory with Sp(N 1 )×Sp(N 2 ) and a bi-fundamental hypermultiplet, as well as flavors charged under either gauge group. The orbifold without vector structure gives an SU (N ) gauge group with two antisymmetric hypermultiplets, plus the flavors. As we will see, these two examples exhibit the need both for the second correction factor in (5), as well as for the shift in the CS or discrete theta parameters in the reduced theory.
The outline for the rest of the paper is as follows. In section 2 we will review the general structure of 1d N = (0, 4) gauge theories. In section 3 we will review the results for Z extra in the 5d N = 2 theories and N = 1 Sp(N ) + theory, and show that they are consistent with our proposal. In sections 4 and 5 we will apply our proposal to the Z 2 orbifold theories, and perform some basic checks. Section 6 contains our conclusions.
The N = (0, 4) theory for instantons in a 5d N = 1 gauge theory also has an SU (2) global symmetry, such that SU (2) × SU (2) r is the rotation symmetry of the 5d theory. Furthermore, for 5d gauge theories that are realized on D4-branes, possibly in the presence of D8 "flavor branes", the 1d (and 5d) theory has an additional SU (2) global symmetry (denoted SU (2) L in [9]). The symmetry SU (2) × SU (2) r corresponds to rotations in the directions transverse to the D4-branes and along the D8-branes. Table 1 summarizes the generic content of the theory in these cases. The subscripts R and L on the fermions denote their chirality in the lift to two dimensions. In the one-dimensional theory this corresponds to the spin. The top three rows show the 1d fields originating from the 5d gauge multiplet. The rest are associated with the 5d matter hypermultiplets. A 5d matter field in the fundamental representation gives a 1d Fermi multiplet ξ. Matter fields in higher tensor representations lead to the 1d fields shown in the bottom three rows. The four scalar fields in X correspond to the positions of the D0-branes along the D4-branes, and the four scalar fields in Y together with the vector multiplet scalar ϕ correspond to the positions of the D0-branes transverse to the D4-branes.
The schematic form of the scalar potential is given by (see for example [13,14,15]) The theory generically has three distinct branches of vacua: • A Higgs branch given by ϕ = Y = 0 and [X, X] + q 2 = 0 • A Coulomb branch given by q = 0 and [X, • A twisted Higgs branch given by ϕ = q = 0 and [X, There are also mixed branches, where some components of ϕ vanish and some components of q vanish. The reduced theory obtained by removing the fields q, ψ L and ψ R will clearly have the same Coulomb and twisted Higgs branches as the original theory. The main issue will be whether it also possesses a remnant of the original Higgs branch, namely whether the space defined by ϕ = Y = [X, X] = 0 has components not contained in the Coulomb or twisted Higgs branches. 3 Comparison with known results 3.1 The N = 2 theories The 5d N = 2 theories are completely characterized by the gauge group G D4 = U (N ), O(N ), or Sp(N ). In the N = 1 language these theories have a single matter hypermultiplet in the adjoint representation. The U (N ) theory is realized on N parallel D4-branes in Type IIA string theory. Adding an O4 − or O4 − plane gives the O(N ) theory with N even or odd, respectively. Adding an O4 + or O4 + plane gives the Sp(N ) theory with the discrete theta parameter θ = 0 or π, respectively [16]. These theories have six-dimensional UV fixed points corresponding to 6d A-type or D-type (2, 0) theories described by M5-branes in M theory.
A, Y The D0-brane gauge theory in these cases is shown in Fig. 1, with G D0 = U (k), Sp(k), and O(k) when G D4 = U (N ), O(N ), and Sp(N ), respectively. 4 The twisted hypermultiplet Y transforms in the adjoint representation of G D0 , the multiplets q and ψ R transform in the bifundamental of G D0 × G D4 , and X and χ R transform in the adjoint, antisymmetric, and symmetric representation of G D0 in the three cases, respectively. These theories really have N = (4, 4) supersymmetry: the vector multiplet and twisted hypermultiplet combine into a (4, 4) vector multiplet, and each hypermultiplet combines with a Fermi multiplet into a (4, 4) hypermultiplet. There is a Coulomb branch that combines the N = (0, 4) Coulomb and twisted Higgs branches parameterized by (ϕ, Y ), and a Higgs branch parameterized by (X, q).
The reduced 1d gauge theories are given by removing the fields {q, ψ L } and ψ R . The multi-particle indices corresponding to the reduced theories in the different cases were given in [10] as follows (the notation is reviewed in the Appendix): However it was argued in [10] that the second term in the one-particle index in the plethystic exponent for the Sp(N ) theories should be discarded to agree with the M theory viewpoint. The contribution of the extra states was therefore claimed to be given by As we will now demonstrate these identifications are consistent with our proposal in (5).
Our proposal has two new ingredients. First, one must take into account the possible shifts in the values of the gauge theory parameters as a result of integrating out rather than just removing the massive fermions ψ L and ψ R . This is relevant for the 1d U (k) and O(k) theories.
The 1d U (k) theory admits a CS term κ TrA, with κ ∈ Z, that is inhereted from the CS term of the 5d U (N ) theory (see for example [17]). This can be thought of as a background U (1) gauge charge of size κ. The effective CS parameter of the reduced theory will get a contribution from the massive fermions that are integrated out. Each fermion in the fundamental representation of U (k) contributes 1 2 sign(m), where sign of the mass is given by the product of the spin and the diagonal U (1) charge of the fermion. The net shift vanishes since the hypermultiplet fermion ψ L and the Fermi multiplet fermion ψ R have the same charge but opposite spin.
The 1d O(k) theory admits a discrete theta parameter θ taking values in {0, π}, that is inhereted from the discrete theta parameter of the 5d Sp(N ) theory [18]. This can be regarded as a background O(1) = Z 2 gauge charge. The effective theta parameter of the reduced theory will get a contribution depending on the sign of the determinant of the fermion mass matrix [22]. The mass-deformation moving the 4-branes breaks Sp(N ) to U (N ) . The fermions ψ R and ψ L each decompose into N states carrying positive charge under the diagonal U (1), and N carrying negative charge. Therefore there are 2N fermions of negative mass and 2N of positive mass, and therefore no shift in θ.
The second ingredient of our proposal has to do with the overcounting of extra states when the moduli space of the reduced theory contains a remnant of the Higgs branch of the original theory. Equivalently, the question is whether the reduced theory has a Higgs branch that is separate from its Coulomb branch. The would-be Higgs branch of the reduced theory is given by ϕ H = Y H = 0 and On the other hand on the Coulomb branch with Y C having the same form, and We see that only the orthogonal theories have a separate Higgs branch. This can be understood as the possibility for a bulk D0-brane to split into a pair of fractional D0branes confined to an O4 + -plane. The split phase, in which the two fractional D0-branes are restricted to moving on the orientifold plane, is a remnant of the original Higgs branch, in which the D0-branes were inside the D4-branes. This splitting is not possible on an O4 − -plane, which is why there is no Higgs branch in the reduced Sp(k) theory.
Our proposal is therefore consistent with the identification of Z extra for the 5d U (N ) and O(N ) theories in (8). For the Sp(N ) theory we need to divide by the correction factor corresponding to confined D0-branes, PE[f D0c (q)]. The quantity f D0c (q) is the one-particle index for D0-branes confined to the O4 + -plane. This quantity was originally studied in [19]. The O(q) term is just the index of the reduced 1d theory with k = 1, so This reproduces O(q) term in the correction factor for θ = 0 in (8). The higher order terms are more difficult to obtain. For θ = π the single D0-brane state is projected out due to the background Z 2 charge, and the lowest order contribution is at O(q 2 ), in agreement with (8). This corresponds to a state with two confined D0-branes connected by a string which cancels the background charge.

The Sp(N ) + theory
This is the theory on N D4-branes in an O8 − -plane with N f D8-branes. For N f ≤ 7 this theory has a 5d UV fixed point with an enhanced E N f +1 global symmetry [11]. For N f = 8 it corresponds to the 6d (1, 0) E-string theory. For N f = 0 there is another possibility associated to the choice of discrete theta parameter, the so-calledẼ 1 theory, in which the global U (1) symmetry is not enhanced [20,21,22]. For N f > 0 the theta parameter is not physical, since it can be removed by a transformation in the parity-reversing component The corresponding 1d N = (0, 4) gauge theory on the D0-branes is shown in Fig. 2. The hypermultiplet X and twisted hypermutiplet Y both transform in the symmetric representation of O(k). The reduced 1d gauge theory is again given by removing the fields {q, ψ L } and ψ R . The multi-particle indices corresponding to the reduced theory with 0 ≤ N f ≤ 8 were given in [9] as follows: where f denotes collectively the flavor fugacities. In all cases these were claimed to give precisely the extra factors, namely As a slight generalization we also give the result for N f = 0 and θ = π (which we checked to O(q 4 )): These results are also consistent with our proposal. First we observe that the net shift in the discrete theta parameter in the N f = 0 theory vanishes since there are an even number 2N of negative mass fermions. As to the moduli space of the reduced theory, on the Coulomb branch ϕ C has the same form as in (10) for the orthogonal theory, and X C and Y C take the same form as in (11) for the orthogonal theory. There is also a twisted Higgs branch, on which ϕ T H = 0 and The would-be Higgs branch of the reduced theory is given by ϕ H = Y H = 0 and X H = diag[x 1 , . . . , x k ], and is therefore completely contained in the twisted Higgs branch. There is no separate Higgs branch in this case, and therefore no correction factor. The reduced theory precisely accounts for the extra states.

The SU (N ) + 2 theory
This theory is obtained from the Z 2 orbifold without vector structure of the Sp(N ) + theory. The corresponding 1d N = (0, 4) gauge theory for k D0-branes in this case is shown in Fig. 3. There is an additional U (1) global symmetry in this case, since the symmetry rotating the two antisymmetrics is U (2) = SU (2) ×U (1) . The SU (2) doublets, Y , χ R , and ψ R , all carry one unit of charge under U (1) . All other mutiplets are neutral under this symmetry. The twisted hypermultiplet Y transforms in the symmetric of U (k), the Fermi multiplet χ R transforms in the antisymmetric, the hypermultiplet q transforms in (k,N), and the Fermi multiplet ψ R in (k, N). Note in particular the different representations of q and ψ R . The 1d U (k) theory admits a CS term inhereted from the 5d SU (N ) theory. As before, this corresponds to a background U (1) gauge charge κ. In addition there can be a background U (1) global charge ζ, corresponding to a CS term for a background U (1) gauge field ζ A . 5 Let us now consider the reduced theory obtained by removing q and ψ R . As in the N = 2 case in the previous section, the would-be Higgs branch of the reduced theory given by ϕ H = Y H = 0 and X H = diag[x 1 , . . . x k ] is fully contained in the Coulomb branch, and therefore does not include a remnant of the original Higgs branch. 6 Therefore the reduced theory should correctly account for the extra states without the additional correction factor.
We can see this as follows. For N = 2n, the mass deformation that moves the D4branes away from the D0-branes, and specifically away from the O8-plane, breaks the 5d symmetry SU (2n) to SU (n) × SU (n) × U (1). The fermions ψ R and ψ L decompose as (n, 1) + + (1, n) − and (n, 1) − + (1,n) + , respectively. Since the fermion mass is proportional to the U (1) charge and the spin, each one gives N positive mass fermions and N negative mass fermions. The CS parameter gets contributions from both ψ R and ψ L , and the background global charge ζ gets contributions only from ψ R . In either case the net shift vanishes since there are an equal number of negative and positive contributions. On the other hand for N = 2n + 1 the unbroken 5d symmetry is SU (n) × SU (n + 1) × U (1), and the two fermions decompose as (n, 1) + + (1, n + 1) − and (n, 1) − + (1, n + 1) + . In this case there is a mismatch of positive and negative mass fermions which shifts κ by ∆κ = ±2 × 2 × 1 2 = ±2, and shifts ζ by ∆ζ = ±2 × 1 2 = ±1. The common factor of 2 is due to the fact that ψ R and ψ L are both doublets (of SU (2) and SU (2) r , respectively). At this point it is difficult to fix the sign of the shift. However we will see below that the minus sign appears to be the correct one.
Our proposal for Z extra in this case is then given by where κ = κ + ∆κ = κ ± 2 and ζ = ζ + ∆ζ = ζ ± 1. The one-instanton contribution to Z QM,red is given by is the character of the d-dimensional representation of SU (2) , a denotes the U (1) fugacity, and we have defined This factor is associated to the translational zero modes of a D0-brane moving in C 2 × C 2 /Z 2 . The one-instanton contribution vanishes for any odd value of κ due to the nonvanishing background gauge charge. For even values of κ the background charge can be cancelled by the Y multiplet, resulting in a nonzero contribution. We claim that the full multi-particle index of the reduced theory is given by namely that there are no D0-brane bound states. As a test of this claim we have computed the two-instanton contribution for κ = 0, ±2: This agrees with the q 2 term in the expansion of (21) for κ = 0, ±2.

Adding flavors
Adding N f fundamentals (flavors) to the 5d theory adds N f Fermi multiplets ξ i R to the 1d N = (0, 4) theory, Fig. 4. These contribute an additional factor of the form where χ [m] denotes the character of the antisymmetric m-tensor representation of SU (N f ). Our proposal for Z extra is the same as in (18), namely Z extra [κ, ζ] = Z QM,red [κ , ζ ]. Here κ (and therefore also κ ) takes values in Z + N f /2 due to the parity anomaly. We expect that for a small number of flavors (21) will continue to hold, namely that Z QM,red = PE [qI 1,red ], where I 1,red is the one-instanton contribution. In the presence of N f flavors this is given by where the N f = 0 one-instanton contribution on the r.h.s. is evaluated using (19)

A consistency check
For SU (3) the antisymmetric representation is equivalent to the fundamental representation, or more precisely to the antifundamental representation, so we can check our proposal in (18) against the instanton partition function for SU (3) with two fundamentals. For simplicity we set κ = ζ = 0. Let us compare the two instanton partition functions at instanton number k = 1. Expanding to order q we get where I 1 is the k = 1 contribution to Z QM , and I 1,red is the k = 1 contribution to Z QM,red . For SU (3) + 2 we have 7 = where f 1,2 are the two flavor fugacities, and s i (i = 1, 2, 3) are the gauge fugacities constrained by i s i = 1. The counterpart in the reduced QM vanishes in this case, since there are no contributing poles, as in the N = 2 U (N ) theory [10]. For SU (3) + 2 we get and, using κ = 2ζ = ±2, Comparing with (26) we see that the two instanton partition functions agree if we identify a = 1/ √ f 1 f 2 and v = f 1 /f 2 . Note that the agreement holds in this case for both choices of sign in ∆κ and ∆ζ. However for more general values of κ and ζ the two computations agree only for the lower sign, namely κ = κ − 2 and ζ = ζ − 1.
We can further compare the two points of view with an additional flavor. Take κ = 1 2 and ζ = 0. For SU (3) + 3 we get and as before I SU (3)+3 1,red = 0. On the other hand for SU (3) + 2 + we find where we denote by f 3 the fugacity of the single flavor, and, with κ = − 3 2 and ζ = −1, The two instanton partition functions again agree once we identify a = 1/ √ f 1 f 2 and v = f 1 /f 2 .

The Sp(N 1 ) × Sp(N 2 ) + ( , ) theory
This is the theory obtained from the Z 2 orbifold with vector structure of the Sp(N ) + theory. Fig. 5 shows the structure of the corresponding 1d N = (0, 4) gauge theory for k 1 instantons of Sp(N 1 ) and k 2 instantons of Sp(N 2 ). The 1d theory admits two discrete theta parameters θ 1 , θ 2 , that are inhereted from the 5d theory. The reduced theory is obtained by integrating out the multiplets q (i) and ψ (i) R . The Coulomb branch is given by two copies of the expressions in (10) and (11) for ϕ (1,2) C and X (1,2) C in the orthogonal group cases, and Y C = 0. There is also a twisted Higgs branch on which ϕ (1,2) T H = 0 and (we assume that On the other hand the Higgs branch of the reduced theory is given by ϕ a and x (2) a are independent. This is a separate branch, and is a remnant of the Higgs branch of the original theory. It corresponds to fractional D0-branes confined to the orbifold fixed plane. We therefore need to divide by the appropriate correction factor PE[f D0c (q 1 , q 2 )]. We also need to take into account the shifted discrete theta parameters. The fermion fields ψ (36) Our proposal in this case therefore takes the form The multiparticle index of the reduced theory is given in general by Z QM,red = 1 + I red 1,0 q 1 + I red 0,1 q 2 + I red 1,1 q 1 q 2 + I red 2,0 q 2 where I red k 1 ,k 2 is the index of the O(k 1 ) × O(k 2 ) theory. First, we would like to claim that this plethystically-exponentiates to We will test this claim shortly by computing some higher order terms. Bust first notice that the first and second terms in the plethystic exponent are the contributions of the fractional D0-branes, and the third term is the contribution of the bulk D0-brane. The former are precisely cancelled by the denominator factor leaving Now let us compute. The (1, 0) contribution is given by and the (0, 1) contribution I red 0,1 is similarly given by replacing θ 1 by θ 2 . The (1, 1) contribution is given by Putting these together, our claim in (39) becomes where Z C 2 ×C 2 /Z 2 was defined in (20), and .

Another consistency check
As one more consistency check we will consider the (1, 1) instanton sector of the Sp(1) × Sp(2) theory, where we can compare our result to the result obtained by working in the SU (N ) formalism for the Sp(1) = SU (2) factor. This is the simplest instance where both aspects of our proposal are relevant. One has to account for the shift in the discrete theta parameters, as well as for the contribution of the fractional D0-branes. The resulting (1,1) instanton term is given by This agrees with the result of the SU (N ) formalism (see appendix B).

Adding flavors
Adding flavors to the 5d gauge theory corresponds to adding Fermi multiplets ξ R to the 1d gauge theory as shown in Fig. 6. These endow the states with spinor charges under the global O(2N f 1 ) and O(2N f 2 ) symmetries. There are two Weyl spinors that we denote S and S . Since under the Z 2 center of the O(k) gauge symmetry S is even and S is odd, there is an additional sign in the projection in the latter case. For example, the expression for I red 1,0 in (41) is replaced with As before, we expect that for a small number of flavors equation (39) continues to hold. In this case where for brevity we have suppressed the flavor fugacity arguments of the spinor characters.

Conclusions
In this note we have refined the procedure suggested in [8,9,10] for obtaining the 5d Nekarsov partition function using the 1d N = (0, 4) gauge theory constructed from the ADHM data. Our proposal correctly removes the contribution of the extra branches that are generically present when the 5d theory has matter in higher representations of the gauge symmetry, and are not associated to the instanton moduli space. In particular it accounts for the possible overcounting in the reduced 1d gauge theory in cases where it maintains a remnant of the original Higgs branch, and for the possible shifts in the parameters of the reduced 1d theory relative to the original 1d theory.
Our proposal is consistent with the results of [10] for the N = 2 theories and with the results of [8,9] for the N = 1 Sp(N ) + theory. We have applied it also to the theories obtained as Z 2 orbifolds of the Sp(N ) + theory: the SU (N ) + 2 theory and the Sp(N 1 ) × Sp(N 2 ) + ( , ) theory, where no previous results exist, and tested our proposal by exploiting simple isomorphisms between low-rank Lie groups.
An important application of this work is to test the dualities proposed in [23,24] that involve these theories by comparing the superconformal indices. So far this has only been done in the fractional instanton sectors, where the refinements described in this paper are absent. Our proposal makes it possible to extend the computation to include any (k 1 , k 2 ) instanton.

Appendix A The N = (0, 4) index
The (0, 2) QM index relevant for the N = (0, 4) theories that we discuss is defined as where the J's are the generators of the Cartan subgroups of the corresponding SU (2) symmetries, Π i are the generators of the Cartan subgroup of G D4 , and F a are the generators of the Cartan subgroup of G D8 . The QM Euclidean action is compactified on a circle with circumference β. The path integral then reduces in the free field limit to Gaussian integrals around zero modes of the gauge field A 0 and the vector multiplet scalar ϕ. The two are combined into a complexified holonomies φ = β(iA 0 +ϕ) taking value in the maximal torus of the complexified gauge group G D0 , The result can in general be expressed as a contour integral in φ of a product of contributions of the different N = (0, 4) multiplets: where following [9,10] we have defined the fugacities The contributions of the different (0, 4) multiplets can be determined by decomposing them into (0, 2) multiplets, whose contributions have a well known structure [9,10]. We will give the explicit expressions for the different theories below. In particular the contribution of the vector multiplet includes the Haar measure [dz I ].
Preforming the integral one must provide a prescription for the contours, namely for how to deal with the poles. This question was answered for 2d gauge theories with N = (2, 2) and N = (0, 2) supersymmetry in [25,26] using the Jeffrey-Kirwan (JK) residue rule. The quantity of interest in that case is the elliptic genus, a refinement of the partition function on the torus. Since N = (0, 4) QM may be regarded as the dimensional reduction of a 2d N = (0, 4) field theory, which is a special case of a 2d N = (0, 2) theory, the same rule can be applied to the QM index, which corresponds to a limit of the 2d elliptic genus. This is the approach taken in [9]. The authors of [9] have also noted that the JK prescription is equivalent, at least in the examples they studied, to integrating each of the complex U (1) holonomies separately on a unit circle, and applying the following residue rules: poles arising from the 5d gauge multiplet contribute when they are inside the contour, poles arising from 5d matter hypermultiplets contribute when they are outside the contour, and poles at the origin do not contribute. The final result is where W G D0 is the Weyl group of G D0 . In particular For G D0 = O one must sum over the two holonomy sectors O + and O − , with a relative weight given by the discrete theta parameter as follows [18]: In the remaining part of this appendix we will present the formulas for the contributions of the (0, 4) multiplets that arise for the 5d N = 1 theories featured in this paper. This information can also be extracted from [23], where it is organized by the 5d multiplets. For brevity we will adopt the shorthand 1 − u ±1 t = (1 − ut)(1 − u −1 t), etc.
For k = 2n we have: The content of the 1d theory in this case is shown in Fig. 4. The contributions of the different 1d (0, 4) multiplets are given by: We have included here the fugacity a of the additional U (1) symmetry.