6d strings and exceptional instantons

We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5d and 6d supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study $G_2$ theories with $n_{\bf 7}\leq 3$ matters in ${\bf 7}$, and $SO(7)$ theories with $n_{\bf 8}\leq 4$ matters in the spinor representation ${\bf 8}$. We also study the elliptic genera of self-dual instanton strings of 6d SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank $2$ or $3$ tensor branches. Some of them are tested with topological vertex calculus. We also explore a D-brane-based method to study instanton particles of 5d $SO(7)$ and $SO(8)$ gauge theories with matters in spinor representations, which further tests our ADHM-like proposals.


Introduction
Instantons are semi-classical representations of nonperturbative quantum phenomena. In Yang-Mills gauge theories, instantons given by self-dual gauge fields on R 4 play important roles in various contexts. For gauge theories in higher dimensions, d > 4, they can be solitonic objects rather than vacuum tunneling, being particles in 5d and strings in 6d.
Self-dual Yang-Mills instanton solutions have continuous parameters, which form a moduli space. Understanding the moduli space dynamics is often an important step towards better understanding the physics of instantons. For gauge theories with classical gauge groups, the self-dual solutions and the moduli space dynamics can be described by the ADHM formalism [1]. The ADHM formalism provides gauge theories on the worldvolume of instantons (1d for particles, 2d for strings), which at low energy reduce to the nonlinear sigma models on the instanton moduli space. In string theory, such gauge theories come from open fundamental strings between Dp-D(p+4) branes. This is why only classical gauge groups admit such constructions. Including matters to gauge theories also affects the moduli space dynamics of instantons. Open strings can engineer matters in the fundamental or rank 2 product representations. This gives the notion of 'classical matters,' whose inclusion to the instanton moduli space dynamics still admits ADHM description. But matters in other representations, like higher rank product representations or spinor representations of SO(N), cannot be engineered using open strings. We shall call them 'exceptional matters.' Yang-Mills instantons also play crucial roles in supersymmetric gauge theories. Among others, in 4d N = 2 theories, the Seiberg-Witten solution [2] in the Coulomb branch acquires all order multi-instanton contributions. It can be microscopically derived by computing the instanton partition function [3] on R 4 with the so-called Omega deformation. The uplifts of this partition function to 5d N = 1 gauge theories on R 4 × S 1 , and to 6d N = (1, 0) gauge theories on R 4 × T 2 , are also important observables of 5d/6d SCFTs. The computation of this partition function in [3] relies on the ADHM method, applicable only to classical gauge theories. However, exceptional gauge theories are also important in various situations. For instance, one often uses various (p, q) 7-branes wrapped on 2-cycles to engineer 6d N = (1, 0) SCFTs, which admit exceptional gauge groups and matters rather generically.
In this paper, we develop ADHM-like formalisms of instantons for a small class of exceptional gauge groups or matters, which can be used to study the instanton partition functions in the Coulomb phase. Namely, we provide 1d/2d gauge theories which we suggest to describe certain aspects of exceptional instanton particles and strings. We managed to find such formalisms for G 2 theories with matter hypermultiplets in 7, and for SO(7) theories with exceptional matters in the spinor representation 8. We also expect their 0d reductions to describe exceptional instantons of 4d gauge theories, although we do not study them here. In 5d, we can describe G 2 theories with n 7 ≤ 3 matters in 7, and SO(7) instantons with n 8 ≤ 4 matters in 8. In 6d, gauge anomaly cancelations restrict our set-up to n 7 = 1 and n 8 = 2.
Our constructions have the following features. The correct instanton moduli space has G r isometry for gauge group G r of rank r. However, our gauge theories realize the symmetry only in a subgroup H r with same rank. We always take H r to be a classical group, say H r = SU(r + 1). The moduli space of G r instantons contains that of H r ⊂ G r instantons as a subspace. Our 1d/2d gauge theories only realize the latter correctly. Away from this subspace, only the dimensions of moduli space agree. So we do not expect that our gauge theories capture the full moduli space dynamics of the exceptional G r instantons. In the Coulomb branch, the instanton size and the G r gauge orbit parts of the moduli space are lifted to a set of isolated points (which are non-degenerate after Omega deformation). We propose that our gauge theories correctly compute the Coulomb branch observables of G r instantons. In fact the SUSY partition functions of our models in the Coulomb branch exhibit full G r symmetry. More precisely, since we turn on r Coulomb VEVs, we find that the Weyl symmetry of H r enhances to that of G r . Since the UV gauge theory flows to the non-linear sigma model on the moduli space, this is a kind of IR symmetry enhancement.
Our ADHM-like descriptions are found based on trials-and-errors. We start from H r classical ADHM construction, and add more worldvolume matters and interactions to suit the physics. In particular, they are motivated by [4], where 2d gauge theories were found for the instanton strings of non-Higgsable 6d SU(3) gauge theories. Although SU(3) is a classical group, 6d non-Higgsable SU(3) gauge theory cannot be engineered by using just D-branes. Rather, it is engineered by using mutually non-local 7-branes wrapping a 2-cycle, in precisely the same way as engineering exceptional gauge theories. Incidently, the naive ADHM description for SU (3) instantons is sick, by having 2d gauge anomalies. The correct gauge theories for the SU(3) instanton strings were found in [4]. Its reduction to 1d provides a novel alternative ADHMlike description for SU (3) instantons. The ADHM-like descriptions of this paper extend these results, related by Higgsings SO(7) In 6d, G 2 and SO(7) theories with matters are somewhat important, as they appear in the 'atomic' constituents of 6d SCFTs [5,6]. 1 Roughly speaking, in the list of atomic 6d SCFTs, there are 9 of them with rank 1 tensor branches. They are constructed by putting F-theory on an elliptic Calabi-Yau 3-fold, whose base is given by the O(−n) → P 1 bundle with n = 1, 2, · · · , 8, 12 [7,8,9]. Also, there are 3 more atomic SCFTs with higher rank tensor branches called '32,' '322' and '232' [10]. Constructing more complicated 6d SCFTs is basically forming 'quivers' of these atoms [10,6,5]. The last 3 atomic SCFTs with higher rank tensor branches contain G 2 or SO(7) gauge theories with half-hypermultiplets in 7, 8, respectively. See our section 4 for a summary. Our descriptions of G 2 and SO(7) instantons allow us to study the instanton strings of such SCFTs. Collecting recent studies and new findings of this paper, one now has 2d gauge theory methods to study the strings of the following 6d atomic SCFTs: O(−1) theory [11,12], O(−2) theory [13], O(−3) theory [4], O(−4) theory [14], and all three higher rank theories (this paper). Quivers of these 2d theories are also explored in [13,15,12,4]. Among others, these gauge theories can be used to study the elliptic genera of the strings. These elliptic genera were also studied using other approaches, including the topological vertex methods [16] and the modular bootstrap-like approach [17,18,19].
We test the BPS spectra of our ADHM-like models using various alternative approaches. Among others, in section 3, we develop a D-brane-based approach to study exceptional instanton particles in 5d gauge theories. This approach is applicable to G 2 instantons, and SO (7) or SO (8) instantons with matters in spinor representations. For G 2 and SO(7) cases, the Witten indices computed from this approach test our ADHM-like proposals. Also, a 5d description for the circle compactified '232' SCFT [10] has been found in [16], using 5-brane webs. This allows us to compute the BPS spectrum of its strings using topological vertices. We do this calculus in section 4 and find agreement with our new gauge theories.
We leave a small technical remark on SO (7) instantons. The canonical ADHM description for k SO (7) instantons (without matters in 8) is given by an Sp(k) gauge theory. Its partition function is given by a contour integral [20,21], yielding a complicated residue sum. Unlike SU(N) instanton partition functions, in which case closed form expressions for the residue sums are known [3], such expressions have been unknown for SO(N). In this paper, using our alternative ADHM-like description, we find a closed form residue sum expression for SO (7).
The rest of this paper is organized as follows. In section 2, we sketch the basic ideas. We then present the ADHM-like descriptions of instantons for SO (7) theories with matters in 8, and G 2 theories with matters in 7. In section 3, we explore alternative D-brane descriptions to study certain exceptional instanton particles, and use them to test our proposals. In section 4, we construct the 2d gauge theories for the strings of 6d atomic SCFTs with rank 2 or 3 tensor branches, with G 2 × SU(2), G 2 × SU(2) × {}, SU(2) × SO(7) × SU(2) gauge groups. We test the last one with topological vertices. Section 5 concludes with various remarks.
We note that a paper [22] closely related to some part of our work will appear.

Exceptional instanton partition functions
Our proposal is based on the following ideas: (1) We are interested in the Coulomb phase partition functions of exceptional instantons, not in the symmetric phase. (2) In the Coulomb phase, the instanton moduli space is lifted by massive parameters, to saddle points lying within the moduli space of instantons with classical subgroups. (3) Thus we only seek for a formalism to study the massive fluctuations around the last saddle points, accomplished by extending ADHM formalisms for classical instantons. We elaborate on these ideas in some detail.
Coulomb phase: We are interested in the gauge theory in the Coulomb branch. Suppose that the gauge group G r has rank r. We turn on nonzero VEV v of the scalar in the vector multiplet, which breaks G r to U(1) r . In 6d, vector multiplet does not contain scalars. In this case, we consider the theory compactified on circle, with nonzero holonomy playing the role of Coulomb VEV. In the symmetric phase, instantons develop a moduli space, part of which being gauge orientations and instanton sizes. In the Coulomb phase, there appears nonzero potential on the instanton moduli space, proportional to v 2 . This potential lifts the size and orientation 0-modes. There are extra 4k position moduli of k instantons on R 4 , which will also be lifted in the Omega background. The moduli space is then completely lifted to points. So we expect that it suffices to understand the quantum dynamics of instantons near these points.
ADHM on a subspace: The second idea is that one can use the ADHM formalism of instantons when G r is a classical group. In d dimensional gauge theory, the ADHM formalism can be understood as a d − 4 dimensional gauge theory living on the instanton solitons. For classical G r , the low energy moduli space of d − 4 dimensional gauge theory is the instanton moduli space, so one expects in IR to get non-linear sigma models on the instanton moduli space. When G r is exceptional, no such formalisms are known. However, it is often possible to find a classical subgroup H r ⊂ G r of the given exceptional group G r with same rank. Then, we try to describe the (massive) quantum fluctuations around the saddle points by expanding the H r ADHM formalism, adding more d − 4 dimensional fields. This is where we need educated guesses, in the spirit of model buildings. We want a subgroup H r with same rank as G r , partly because we wish our formalism to see all U(1) r in the Coulomb phase. Possible G r and H r are given in Table 1, when H r is a simple group. To study 'exceptional matters' of SO (7), we shall also consider H = SU(4) for G = SO (7).
For example, consider the case with H r=N −1 = SU(N). The SU(N) ADHM description of k instantons has U(k) gauge symmetry, and the following fields, chiral : (q, ψ) ∈ (k, N) , (q,ψ) ∈ (k, N) , (a, Ψ), (ã,Ψ) ∈ (adj, 1) vector ∼ Fermi : (A µ , λ 0 ) ∈ (adj, 1) , (λ) ∈ (adj, 1) . (2.1) The fields are organized into 2d N = (0, 2) supermultiplets, and we have shown the representations in U(k) × SU(N). Fields in a parenthesis denote bosonic/fermionic ones in a multiplet, while (λ) denotes a Fermi multiplet. These fields combine to N = (0, 4) vector multiplet and hypermultiplets. The instanton moduli space is obtained from the scalar fields, subject to the complex ADHM constraint and the D-term constraint (real ADHM constraint) where R i (H) are suitable representations of H r in Table 1. Vector multiplet in adj(H) induces the standard instanton moduli, described in UV by the above ADHM description. Vector multiplets in R i introduce further moduli, whose real dimension is 4kT (R i ). T (R) is the Dynkin index of R. When R i is a fundamental representation or rank 2 product representations, we managed to find the extra fields. We are technically motivated by the mathematical constructions of [23], but will simply present them as our 'ansatz' for the UV uplift of these zero modes.
From Table 1, one finds that R i 's are product representations with ranks less than or equal to 2 only for G 2 ⊃ SU(3) and SO(7) ⊃ SU (4). For these, the adjoint representations of G r decompose as We shall present extra chiral and Fermi multiplets with suitable interactions in the next subsections, which extends the moduli space in i 4kT (R i ) new directions.
When it fails: We made similar trials with other exceptional gauge groups and matters, which failed. It may be worthwhile to briefly report the reasons of failure. A typical reason is that the UV theory has extra branch of moduli space that does not belong to our instanton moduli space at low energy. Namely, apart from the exceptional instanton's moduli space, one sometimes has extra branch which cannot be lifted by supersymmetric potentials.
For instance, we tried to extend the ADHM construction of H = SU(8), SU(9) to get those for G = E 7 , E 8 . In these cases, R i 's are product representations of rank 4 and 3, respectively. We made several trials to realize the extra moduli with right dimensions, especially without unwanted extra branches of moduli space which may spoil the instanton calculus. We however failed to get the precise descriptions, despite finding models which partly exhibit the right physics of instantons and instanton strings. See section 5 for more discussions.
We also suspect that some choice of H r ⊂ G r may miss certain small instanton saddle points. To clearly understand this issue, we should find more examples than we have now.
There are simpler examples in which our new formalism fails. For instance, we consider our alternative SO(7) ADHM (section 2.1) and try to add zero modes from matters in 7 (vector representation). Zero modes of 7 are well known in the standard SO(N) ADHM, which form an Sp(k) fundamental Fermi multiplet. In our SU(4) × U(k) formalism, 7 is regarded as 7 → 6 + 1. We can ignore the singlet if the gauge orientation of instantons is along SU(4). 6 is the rank 2 anti-symmetric representation. According to [23], and in D-brane engineerings, the ADHM fields induced by matters in bulk anti-symmetric representation include scalars in rank 2 symmetric representation of U(k). This creates an extra branch of moduli space which is unphysical in the instanton calculus, but is present only in the UV uplifts. Even in ADHM models engineered by string theory, there are often such extra branches. In [21], the contributions from these branches are factored out, mainly guided by string theory. However, including this extra branch in our SO(7) ADHM-like model, we find it difficult to properly identify and separate the extra contributions. Similarly, we cannot do an ADHM-like calculus for the 5d G 2 N = 1 * theory. Now we explain examples that turn out to work.

SO(7) instantons and matters in 8
The adjoint representation of SO(7) decomposes in SU(4) as 21 → 15 + 6. We first seek for an alternative ADHM-like formalism of pure SO (7) instantons, extending SU(4) ADHM. We explain it as the quantum mechanics of instanton particles in 5d N = 1 Yang-Mills theory.
With extra vector multiplet fields in 6, there are extra bosonic zero modes. A vector multiplet in rank 2 antisymmetric representation of SU(N) induces 2kT (anti 2 ) = k(N − 2) complex bosonic zero modes in k instanton background. So we should add extra fields in UV and modify interactions, to get extra 2k complex bosonic modes at N = 4. We find that the following extra fields, taking the forms of N = (0, 2) chiral or Fermi multiplets, yield the right physics 2 (only bosonic fields shown for the chiral multiplets): Fermiλ : (sym 2 , 1) J=−1 .
(2.6) 2 Technically, we took the equivariant index of the so-called universal bundle [23], and made an antisymmetrized product of two of them (for 6): this is the character for the fields we list in (2.6). Since we lack physical explanations of this procedure, we just present them as our ansatz for the UV theory. Table 2: Charges/representations of fields in our SO(7) ADHM-like model sym 2 , anti 2 denote rank 2 (anti-)symmetric representations of k, and the charge J in the subscript will be explained shortly. We introduced extra 4k + 2 · k 2 −k 2 complex bosonic fields. Using the extra Fermi fieldsλ,λ, we introduce the following interactions. As noted in [4], the desired interactions should be non-holomorphic in the chiral multiplet fields, which is possible only with N = (0, 1) SUSY. Therefore, we regard all these fields as (0, 1) superfields, as explained in [4], and turn on the following N = (0, 1) superpotential, The subscripts S denote symmetrization of the k × k matrices. We want (2.7) to be the only source of breaking (0, 2) SUSY to (0, 1) in the classical action. The D-term is given by Then, since |J| 2 appear in the bosonic potential for each J, one imposes at low energy extra k 2 + k complex constraints from the new superpotentials. Collecting all, one finds complex bosonic zero modes. This agrees with the dimension 2kc 2 of SO(7) instanton moduli space, where c 2 (SO(2N + 1)) = 2N − 1. The SU(4) ADHM quantum mechanics with extra charged fields given by (2.6) is our proposed ADHM-like formalism for SO (7) instantons.
We explain the symmetries of this model. All symmetries we explain below are compatible with the superpotentials. It first has SU(4) symmetry. There is also a U(1) J symmetry, whose charges J we already listed above when we introduced fields. There is also SU(2) l , which rotates a,ã and also b,b as doublets. The charges and representations are summarized in Table  2. This system has only N = (0, 1) supersymmetry and SU(4) global symmetry in UV. We assert that they enhance to (0, 4) SUSY and SO (7) in IR, when we compute the Coulomb branch partition functions. SO(7) enhancement will be visible as SO(7) character expansions of the partition functions. In the context of SUSY enhancement, we claim that U(1) J enhances to SU(2) r × SU(2) R , where SO(4) = SU(2) r × SU(2) l rotates the spatial R 4 on which particles can move, and SU(2) R is the 5d R-symmetry. J is identified as J = Jr+J R 2 , where J r , J R are the Cartans of SU(2) r , SU(2) R . One Cartan is not visible in UV. The index of our models will agree with different computations whose settings manifestly preserve (0, 4) SUSY.
We study the moduli space, with and without 5d Coulomb VEV. For technical reasons, let us just consider the case with k = 1. At k = 1, the fields b,b are absent, and a,ã are free fields for the center-of-mass motion. First consider the symmetric phase at v = 0. One should solve the following equations: At φ i = 0, this is the equation for the SU(4) instantons. This subspace is the cone over SU(4)/U (2). Away from φ i = 0, although the dimension of the moduli space is same as the relative moduli space of an SO(7) instanton, the two moduli spaces are different. The proper SO(7) instanton moduli space is the cone over the SO(7)/(SU(2) × SO (3)) coset, whose metric is given by the homogeneous metric. However, we find no SO(7) isometry on our moduli space.
(2.11) This is because ϕ, v are scalars in the 1d vector multiplet of U(k) × SU(4), where v is a background field, and fields couple to them according to their representations in U(k)×SU (4). Note that the relative − signs for q,q appear because they are in the bifundamental representations (k,4) or its conjugate, while relative + sign for φ is because it is in (k,4). We set all complete square terms to zero at energies lower than 1d gauge coupling. One should also minimize the following D-term potential at k = 1, Here, since we have a U(k) gauge theory, we have turned on a Fayet-Iliopoulos (FI) parameter, ξ, which we take to be positive ξ > 0 for convenience. ξ can also be taken to be negative, without changing the Coulomb phase partition function, as we shall see below. However, physics is easier to interpret with ξ > 0. So we set (at k = 1) where i = 1, 2, 3, 4 indices are not summed over, and 14) The equations (2.13) are eigenvector equations for the matrix v, whose eigenvalues are ϕ for q,q to be nonzero, and −ϕ for φ to be nonzero. From (2.14), one should have q i = 0, which means that ϕ is set equal to one of the v i 's. Then one can have nonzero q i at the saddle point, whose value is tuned to meet (2.14). At generic values of v i 's, one should set φ i = 0, meaning that we are forced to stay in the SU(4) instanton moduli space. 3 So in the Coulomb branch calculus, φ provides massive degrees of freedom living on the SU(4) instanton moduli space.
The Witten index of the quantum mechanics preserving (0, 4) SUSY is defined by where trace is over states in the k instanton sector. J 1 , J 2 are the two Cartans of SO(4) which rotate the spatial R 4 , where they rotate mutually orthogonal R 2 factors. They are related to J l,r by J r = J 1 +J 2 2 , J l = J 1 −J 2 2 . J R is the Cartan of SU(2) R coming from the 5d R-symmetry. Note that only the combination J r + J R = 2J appears, so our UV model can fully detect them. q i are the r electric charges in U(1) r ⊂ G r , which is SO (7) here. F a denote other flavor symmetries, which is absent now but introduced for later purpose. The measures are chosen to commute with two Hermitian supercharges Q Aα = Q +− , Q −+ . See, e.g. [21] for the notations. These two supercharges are mutually Hermitian conjugate, which we write as Q, Q † . They form a pair of fermionic oscillators, pairing a set of bosonic and fermionic states. Such a pair of states is not counted in the index, as their contributions cancel due to the factor (−1) F . Such a Hilbert space interpretation will hold with as little as (0, 2) SUSY. In our UV (0, 1) system, we abstractly interpret the partition function as a SUSY path integral of the Euclidean QFT on T 2 . 1 Hermitian SUSY in UV is enough to derive the formula for Z k available in the literatures. With IR SUSY enhancement, Z k acquires the interpretation of an index.
3 At this stage,q i can also be nonzero by solving the same eigenvector equation as q i . However, as shown in the appendix of [24], the eigenvector equations for q i andq i become different with nonzero Omega background parameter. Therefore, in the fully Omega-deformed background, only q i is nonzero. 4 The overall signs of Z k are fixed by requiring agreement with the index for the Sp(k) ADHM theory [20].
The integrand on the first line comes from the SU(4) ADHM fields q,q, a,ã and U(k) vector multiplet fermions. The second line comes from the extra fields.
The integral can be performed as follows. The nonzero residue contributing to Z k is called the JK residue. To define this, one first picks up an auxiliary vector η in the k dimensional charge space ('conjugate' to the integral variables φ I ). Possible poles in the integrand are given by hyperplanes of the form ρ α ·φ+· · · = 0, where the expression on the left hand side comes from the argument of the sinh factors 2 sinh ρα·φ+··· 2 in the denominator of (2.16). One can in general pick d(≥ k) charge vectors ρ α , α = 1, · · · , d and hyperplanes to specify a pole. In our systems, all relevant poles satisfy d = k. With chosen η, JK-Res may be nonzero only if η is spanned by the k charge vectors ρ 1 , ρ 2 , · · · , ρ k with positive coefficients. Here, the choice η = (1, · · · , 1) simplifies the evaluation [21]. Since the charges appearing in the denominator of the second line are all negative in (2.16), one can show (combined with the fact that charges on the first line take the form of e I or e I − e J ) that JK-Res should always be zero by definition if one of the charges from the second line are chosen in ρ α . This implies that the poles with nonzero residues are always chosen from the first line only, which are already classified in [3,29,30,21]. The pole locations for φ I are classified by the colored Young diagrams with k boxes, meaning a collection of 4 Young diagrams Y = (Y 1 , · · · , Y 4 ) whose box numbers sum to k. Let us denote by s = (m, n) the box of a Young diagram Y i , which is the box on the m'th row and n'th column of Y i . s running over possible k boxes replaces I = 1, · · · , k index of φ I . We specify the pole location associated with Y as φ(s). The result is [3,29,30,21] φ(s) = v i − ǫ + − (n − 1)ǫ 1 − (m − 1)ǫ 2 , s = (m, n) ∈ Y i (i = 1, · · · , 4) .
(2.17) (This corrects a typo in [21], exchanging m ↔ n.) Had there been only the first line in (2.16), the residues were computed in [29,30,21]. Plugging in φ(s) into the second line of (2.16), one obtains an extra factor for each residue. The residue sum is given by Here and below, s i < s j means (i < j) or (i = j and m i < m j ) or (i = j and m i = m j and n i < n j ). h i (s) denotes the distance from s to the right end of the diagram Y i by moving right. v j (s) denotes the distance from s to the bottom of the diagram Y j by moving down. See, e.g. [24].
(2.18) is our proposal for the partition function of k SO (7) instantons. This is quite novel for the following reason. SO(7) instantons have standard ADHM formulation, using Sp(k) gauge theories for k instantons. The pole classification is unknown for the Sp(k) index. On the other hand, (2.18) is an explicit formula.
Before adding matters in 8, we first check that (2.18) is indeed the correct SO(7) instanton partition function. We checked the equivalence of (2.16), or (2.18), and the index of Sp(k) ADHM gauge theory [20,21], up to k ≤ 3 (turning off all chemical potentials except ǫ + at k = 3). Here we explain the case with k = 1 in detail, which is already nontrivial. For the purpose of illustration, we directly start from the contour integral. At k = 1, one finds from our model. Taking the residues at φ = v i − ǫ, for η > 0, one finds This is a special case of (2.18). To check this result is correct, we study the SO(7) single instanton partition function obtained from the standard Sp(1) ADHM formalism [20,21] 2 sinh Residues are taken at φ = ±u a − ǫ + and −ǫ + for η > 0, but the last residue is 0. u a and v i are related by The residue sum is given by (2.24) Despite very different looks, one can show (say, by using computer) that after the identification (2.23). This identity and similar ones at higher k's imply that Z k exhibits SU(4) → SO(7) enhancement, since Z standard k has manifest SO(7) Weyl symmetry.
Now we discuss the inclusion of ADHM fields coming from the hypermultiplet matters in 8. We continue to study the instanton particles of 5d SYM. 8 decomposes in SU(4) as In the original ADHM formalism of SO(7) instantons, it is unclear how to UV-uplift the fermion zero modes caused by these hypermultiplets in the instanton background. One may even feel it impossible, since the standard SO(7) ADHM cannot see 4π rotations in Spin (7). However, viewing it as SU(4) instantons with certain extensions, each hypermultiplet in 4 (or4) induces a Fermi multiplet which is fundamental (or anti-fundamental) in U(k). So in our new description, we naturally guess that the effect of n 8 hypermultiplets is adding n 8 pairs of Fermi multiplets of the following form: Ψ a ,Ψ a : (k, 1) + (k, 1) (a = 1, · · · , n 8 ) . (2.27) It has been known [31] that the 5d SO(7) SYM has a UV completion to a 5d SCFT for n 8 ≤ 4. Recently, it was discussed that 5d SCFTs can exist till n 8 ≤ 6 [32]. See also [33]. Our construction provides good descriptions of instantons for n 8 ≤ 4. It will be easiest to explain this point after we discuss the index below. The flavor symmetry for Ψ a ,Ψ a may naively appear to be U(2n 8 ). This is because we do not have any superpotential for these Fermi fields.
They interact with other fields through gauge coupling only, so that one can rotate Ψ a ,Ψ † a with U(2n 8 ). However, these fermions can couple to 5d background bulk fields, including the hypermultiplet fields in 8. (Even in ADHM models based on D-brane engineerings, it sometimes happens that the soliton quantum mechanics is ignorant on the bulk symmetry, in a similar manner.) These couplings will only preserve U(n 8 ) ⊂ Sp(n 8 ). See the beginning of the next subsection for this coupling to the bulk fields.
Adding these fermions, our ADHM-like description can be easily generalized. Namely, the extra Fermi fields are given standard kinetic term, whose derivatives are covariantized with 1d U(k) vector multiplet fields. Its Witten index Z n 8 k (ǫ 1,2 , v i , m a ) with a = 1, · · · , n 8 is defined with extra factors e −maFa inserted in its definition, where F a are the Cartans of Sp(n 8 ). The contour integral expression for the Witten index takes the form of (2.16), with the following extra integrand multiplied for the new Fermi fields: (2.28) The extra factor (2.28) does not create new poles at finite φ, but may create new poles at infinity φ I → ±∞. We first discuss the last possibility.
Here, first note that φ I originate from the eigenvalues of the 1d U(k) vector multiplet fields, φ ≡ ϕ + iA τ , where A τ is the vector potential on the Euclidean time. The contour integrand Z 1-loop comes from 1-loop path integral of 1d fields in the background of constant φ I . So V (φ) ∼ − log Z 1-loop is the 1-loop potential energy for φ I . Before multiplying (2.28), the integrand of (2.16) converges to zero at |φ I | → ∞ for any I, since there are more bosonic fields than fermionic fields. More concretely, consider the case with k = 1. One obtains Z n 8 =0 1-loop ∼ e −4|φ| , implying that the linear potential V (φ) = 4|φ| confines the eigenvalues to φ = 0. In other words, although φ classically develops a continuum to φ → ±∞, 1-loop effect lifts this continuum by an attractive force. In ADHM models with brane engineering, this can be visualized as the instantons being attracted to the locations of 5d SCFTs [21]. The U(k) vector multiplet fields are clearly extra degrees of freedom that enter while making a UV completion of the nonlinear sigma model. If there is a continuum created by φ, this represents states that do not belong to 5d QFT. The confinement from V (φ) = 4|φ| signals that such obvious extra states may not be present in the quantum system. Now we extend these studies to n 8 > 0. At k = 1, one obtains V (φ) = (4 − n 8 )|φ|. So at n 8 ≤ 3, the quantum potential still confines the instanton. At n 8 = 4, φ generates a flat direction. This branch has extra states which is an artifact of UV completion, not belonging to the 5d QFT Hilbert space. So strictly speaking, n 8 ≤ 3 is the bound in which our ADHM-like model is reliable. Fortunately, there are well developed empirical ways of factoring out such extra states' contribution to the index. So we believe that our approach will be useful till n 8 = 4. At n 8 ≥ 5, the quantum potential is repulsive, and it is not clear whether one can use this theory to study 5d QFT at all. (However, see [34] for some progress.) In the contour integral like (2.16) or its extension with (2.28), the absence of continuum means the absence of poles at infinity. This implies that the choice of η in the JK-residue evaluation does not change the final result [21,25]. This is the case for n 8 ≤ 3.
For n 8 ≤ 3, the pole classification that we explained earlier for pure SO(7) instantons still holds, labeled by SU(4) colored young diagrams. We only need to multiply the value of (2.28) at the pole to the residue. The resulting index is given by The partition functions (2.29) will be tested in sections 3 and 4 at n 8 = 1, 2 using alternative descriptions, which include no guess works but are more elaborate in calculations. For instance, the indices at k = 1 divided by the center-of-mass factorẐ 1 ≡ 2 sinh Here means that all 4 factors with different signs are multiplied. The convention on representations (e.g. primes) all follows [35]. The numerators are invariant under SO(7) × Sp(n 8 ) Weyl symmetry, being character sums. Since the denominators are products with all possible ± signs, they are also invariant under SO (7) Weyl group which flips We expect our quantum mechanics to work also at n 8 = 4. Here, the 1d Coulomb branch with nonzero φ I has a continuum. There may appear extra contribution from this continuum to the index [21], apart from (2.29). (For conceptual simplicity, we consider the problem at zero FI parameter ξ = 0.) The extra contribution from the 1d Coulomb continuum is neutral in SO(7) × Sp (4). This is because the extra states in the 1d Coulomb branch come from the region with large φ I , where all U(k) charged fields acquire large masses. The charged fields are those which see SO(7) × Sp (4). So the extra continuum does not see these charges. Here we shall only test the SO(7) × Sp(4) symmetry enhancements at n 8 = 4. So we simply ignore the extra contribution, and show that (2.29) exhibits SO(7) × Sp(4) Weyl symmetry. The result at k = 1, showing SO(7) × Sp(4) Weyl symmetry, is given bŷ with f 0,1,2,3 (v) given by (2.33), and (7) 378 +2χ SO (7) 616 +3χ SO (7) 693 +2χ SO (7) 819 −χ SO (7) 1617 −χ SO (7) 1911 − 1 . (2.35) Now we consider the instanton strings of 6d super-Yang-Mills theories with SO(7) gauge group and matters in 8. The number n 8 of hypermultiplets cannot be arbitrary, due to gauge anomalies [36,6]. Without matters in other representations, one should have n 8 = 2 [6]. Incidently, the 6d consistency requirement n 8 = 2 is also reflected in our ADHM-like construction, uplifted to 2d for instanton strings. This comes from 2d U(k) gauge anomaly cancelation. First consider the SU(k) anomaly, proportional to D R = ±2T (R) for right/left moving fermions.
As a basic test of our 2d gauge theories, we study the 't Hooft anomalies of global symmetries. The full 2d symmetry is expected to be SO (7) There is an alternative way of computing the anomalies on the strings, using anomaly inflow [4,37]. By comparing two calculations, we shall provide a test of our gauge theories.
Using the inflow method, the 2d anomaly can be computed as follows. We first compute the anomaly polynomial 8-form of the 6d SCFT with a tensor multiplet, SO(7) vector multiplet, and half-hypermultiplets in 1 2 (8, 4) of SO (7) ×Sp (2). The anomaly polynomial in the tensor branch consists of 1-loop contribution I 1-loop , coming from massless tensor/vector/hyper-multiplets, and the classical Green-Schwarz contribution I GS [38,39]. The two contributions should partly cancel for the terms containing SO(7) gauge fields [40,41]. I 1-loop is given by where · · · denote terms independent of the SO(7) field strength F SO (7) . Following [41], we use the notation Tr ≡ 1 h ∨ tr adj , and tr adj ( for SO (7). To cancel the 1-loop SO (7) anomaly, one should have the following Green-Schwarz 8-form [41]: (2) ) .

(2.39)
This takes the form of I GS = 1 2 Ω ij I i I j with i, j running over just 1, so that I 1 = I and Ω 11 = 3. Ω 11 may be fixed from the fact that it comes from O(−3) → P 1 geometry in F-theory, with self intersection number of P 1 being 3. Knowing I i appearing in I GS = 1 2 Ω ij I i I j , one can determine the 2d anomaly 4-form on the strings, from inflow. The formula is [4,37] where k i is the string number in the i'th gauge group (or i'th tensor multiplet). We decomposed the 6d tangent bundle T to T 2 × T 4 , along/normal to the strings. From this formula, one finds for the SO (7) instanton strings at n 8 = 2, with topological number k.
Now we compute I 4 from our gauge theory. A chiral fermion's anomaly 4-form is given by where ± signs are for left/right-moving fermions, respectively, in our convention. F collectively denotes all background gauge fields for the global symmetries acting on the fermion. Here it is for R is the U(1) Cartan of SU(2) R . Here and later, we shall often use expressions like c 2 (r), c 2 (R) = F 2 R 4 assuming symmetry enhancement, but only the U(1) J part is to be kept in UV. Namely, one first keeps the Cartan parts of the field strengths, for J r , J R . Then they are all replaced by J and its field strength F J . We present the results using c 2 (R) and c 2 (r) since this may suggest possible patterns of IR symmetry enhancement. (See also [4].) The fields charged under SU(4) × U(2) contribute to I 4 as follows: Adding all, and using p 1 (T ) = p 1 (T 2 ) − 2c 2 (l) − 2c 2 (r), one obtains Here and below, we shall frequently use the fact that Tr(F 2 ) remains the same after restricting F to a subalgebra if a long root of the original algebra is kept, so that unit instanton charge 1 4 Tr(F 2 ) remains the same [41]. Here it applies to SU(4) ⊂ SO (7). As for U(2) ⊂ Sp (2), or more generally U(n) ⊂ Sp(n), the embedding is such that tr 2n → 2tr n . Taking these into account, (2.45) agrees with (2.41) upon restricting (2.41) to SU(4), U(2), SU(2) r ×SU(2) R → J, and using χ(T 4 ) = c 2 (l) − c 2 (r). Their mixed anomalies with U(k) also vanish.
One can study the elliptic genera Z k of k instanton strings, whose spatial direction wraps S 1 . The definition is almost identical to (2.15), except that there is another factor e 2πiτ P inside the trace, where P is the left-moving momentum on S 1 . The basic formula is given in [27,28]. The result is obtained by simply replacing all 2 sinh functions in (2.16), (2.28), (2.29) Some tests of these formulae will be given in section 4.2.

G 2 instantons and matters in 7
With a hypermultiplet in 8, one can Higgs SO(7) to G 2 . Decomposing the scalar to 8 → 7 ⊕ 1 in G 2 , 1 is given VEV and decouples in IR. 7 is eaten up by the broken part of the SO(7) gauge fields, since 21 → 14 ⊕ 7. The matter consists of two half hypermultiplets, forming a doublet of flavor symmetry Sp(1) F . The scalar can be written as [Φ Aa ] (s 1 ,s 2 ,s 3 ) , where A = 1, 2 is the doublet index of SU(2) R R-symmetry, a = 1, 2 is that of Sp(1) F , and s 1,2,3 = ± 1 2 label the components of 8. It satisfies the reality condition with a pure imaginary VEV Φ. This preserves a diagonal subgroup of SU(2) R × Sp(1) F , which is the SU(2) R symmetry after Higgsing. At general n 8 , we give VEV to the last hypermultiplet scalar, a = n 8 . One should lock the chemical potentials as with both signs, not to rotate the scalar VEV. So we should take m last − ǫ + = 0, v 4 = 0. The former condition turns off the Sp(1) ⊂ Sp(n 8 ) chemical potential m last , and the latter reduces the rank of gauge group by 1. As the index is invariant under the RG flow triggered by the scalar VEV, one can get the IR G 2 index by constraining the SO(7) index.
The index Z k for k G 2 instantons can either be obtained from the Witten index of the above gauge theory, or by taking the Higgsing condition of the SO(7) index, m n 8 = ǫ + , v 4 = 0. It may be more illustrative to write both the contour integral expression and the residue sum. The contour integral expression for the index is given by The residue sum, labeled by SU(3) colored Young diagrams, is given by 6 One may more generally take J Ψ ∼ αQ iq i + βQ i φ i , compatible with U (k) × SU (4). However, with SU (4) broken to SU (3),q 4 and φ 4 have same charges in unbroken symmetries, and α, β does not affect the IR physics. 6 The factors on the first line of (2.53) containing E ij are the residues for pure SU (3) theory. In this type of expression, one finds an overall (−1) N k factor for pure SU (N ) instantons. This is why we have (−1) k in (2.53).
We first study the case with n 7 = 0. We can test the results against known G 2 instanton partition functions of [42]. We tested (2.53) till k ≤ 3. Firstly, at k = 1, it will be illustrative to make a basic presentation, directly from the contour integral. (2.52) at k = 1 is given by However, the sum of three residues exhibits enhanced Weyl symmetry of G 2 , the dihedral group D 6 of order 12. The extra transformation generating full is the character of the irrep (0, n) of G 2 , which is the n'th symmetric product of the adjoint representation 14. (2.56) is known as the correct G 2 instanton partition function at k = 1 [42].
, and g n (v i , u)'s are given by As the numerator is manifestly arranged into G 2 characters, it shows enhanced G 2 Weyl symmetry. The denominator is also invariant under the extra generator v i → −v i of D 6 , being invariant under G 2 Weyl symmetry. One can also check the agreement with the known G 2 partition function at k = 2. For the simplicity of comparison, let us turn off all v i = 0 and ǫ − = 0. Then, (2.57) becomes where the omitted terms · · · can be restored by the t → t −1 Weyl symmetry of SU(2) (i.e. the coefficients of t p and t 22−p are same on the numerator). The overall t 7 factor is like a zero point energy factor, and is needed to have this Weyl symmetry. Apart from this factor, (2.59) agrees with eqn.(9.5) of [43] after correcting a typo there, as noted in [42].
Now as for the indices at n 7 ≥ 1, these observables have not been computed or studied in the literature, to the best of our knowledge. Here we simply note that, making expansions of the indices in t = e −ǫ + , one observes that the coefficients are characters of G 2 × Sp(n 7 ). At least at k = 1, this does not need independent calculations, since we already illustrated the symmetry enhancement of SO(7) × Sp(n 8 ) in the previous subsection. Also, whenever we provide concrete tests of some SO(7) results in section 3 and section 4, this implies similar tests of the G 2 results at n 7 = n 8 − 1 by Higgsing. At n 7 = 1, 6d SCFT exists with G 2 gauge group. This can be obtained from 6d SO(7) theory at n 8 = 2 by Higgsing. Our 2d gauge theories on G 2 instantons can also be uplifted to 2d gauge theories. As in the previous subsection, this gauge theory is free of U(k) gauge anomaly. The 2d anomaly of G 2 × Sp(n 7 ) × SU(2) R × SU(2) r × SU(2) l global symmetries, computed from anomaly inflow, is also compatible with the SU(3) × U(n 7 ) × U(1) J × SU(2) l anomalies of our 2d gauge theories. To see this, one first restricts F SO (7) . Since we lock Sp(1) ′ and SU(2) R during Higgsing, one identifies F Sp(1) ′ = F R . Then, both anomaly 4-forms (2.41), (2.45) reduce to with restrictions to UV symmetry understood for gauge theory anomalies. So the inflow anomaly and 2d gauge theory anomaly continue to agree with each other.
The elliptic genera for the strings can be computed similarly. One takes the formulae (2.52) or (2.53), and replace 2 sinh ≡ θ(z) for all 2 sinh functions. The G 2 symmetry of this elliptic genus at k = 1 is systematically discussed in [19]. At n 7 = 1, one has a pair Ψ,Ψ of Fermi multiplets. One can again investigate the effect of bulk Higgsing G 2 → SU(3). In the bulk, one decomposes 7 → 3 +3 + 1, where scalar in 1 assumes VEV and breaks G 2 into SU(3). The other hypermultiplet fields are eaten up by vector multiplets for the broken symmetry. The constant VEV of the bulk scalar ≡ Q in 1 will behave as a background field in 1d/2d ADHM-like models. With foresight on the SU(3) instantons studied in [4], we propose that the coupling of the background bulk field Q to the G 2 ADHM-like gauge theory is given by where φ ≡ φ 4 . The N = (0, 1) superpotential JΨ is compatible with symmetries, but at this stage it may not be obvious why we should turn it on in this way. Ψ and the chiral multiplet φ become massive due to J Ψ , and decouple at low energy. However,Ψ does not decouple at low energy, since it does not acquire mass. In fact, the remaining system (includingΨ, which was called ζ in [4]) with the above cubic superpotential was studied in [4], which showed various nontrivial physics of the SU(3) instanton strings. In 1d, this provides a novel alternative ADHM-like description for SU(3) instanton particles. In 2d, this is (by now) the uniquely known SU(3) ADHM construction of instanton strings without matters. All models presented so far in this paper, for SO (7) and G 2 instantons, were initially constructed by guessing the un-Higgsing procedures from SU (3). See [4] for further discussions on the last SU(3) model. Figure 1: Brane realizations of (a) SO(2N) and (b) SO(2N + 1) gauge theories

Brane setup and quantum mechanics
In this section, we test some indices of the previous section, using 5-brane webs for the 5d N = 1 gauge theories with SO(N) gauge groups and matters in spinor representations [44]. A type IIB 5-brane web on x 5 , x 6 plane consists of (p, q) 5-branes stretched along lines with slope q/p: e.g. D5-branes (1, 0) along x 5 and NS5-branes (0, 1) along x 6 directions. They occupy x 0 , · · · , x 4 directions for the 5d QFT. SO(N) gauge theories are realized by 5-brane webs with orientifold 5-planes. An NS5-brane crossing the O5-plane bends to a suitable (p, 1)-brane, and changes the types of O5 across NS5. An SO(2N) theory is engineered by suspending N D5-branes between two NS5-branes, also with an O5 − , as shown in Fig. 1(a). SO(2N + 1) theory is realized by N D5-branes and an O5 − plane, which is an O5 − with a half D5. See Fig. 1(b). Dasheddotted line is a monodromy cut, to have (p, q) 5-branes at right angles with properly quantized charges [44]. In these constructions, instanton particles are D1-branes stretched between two NS5-branes, as shown in Fig. 2. In this setting, a 5d hypermultiplet in the spinor representation is introduced as follows [44]. One introduces another NS5-brane as shown in Fig. 3. D1 ′ -branes suspended between NS5 1 and NS5 2 are the particles obtained by quantizing the hypermultiplet in the SO(N) spinor representation. (See [44] for the chirality of the SO(2N) spinor.) The mass of this field is proportional to the distance between NS5 1 and NS5 2 . To introduce two hypermultiplets in the spinor representation, one puts another NS5-brane on the right side, as shown in Fig. 4. Note that for SO(N) gauge theory with N ≤ 6, NS5 1 and NS5 2 -branes do not intersect. For N = 7, 8, NS5 1 and NS5 2 are parallel to each other. In the last cases, there Figure 3: Hypermultiplet in the spinor representation of (a) SO(2N) and (b) SO (2N + 1). are extra continua of D1 ′ branes, orthogonally suspended between these parallel NS5-branes, which can escape to infinity and do not belong to 5d QFT. In section 3.2 we discuss this extra sector in more detail. When N ≥ 9, NS5 1 and NS5 2 meet at a certain point. In this case, we do not know how to use this setting to study the 5d QFT. So in the rest of this paper, we focus on SO(N) QFTs with N ≤ 8.
We discuss the quantum mechanical gauge theory, with given numbers of SO(N) instantons k and hypermultiplet particles n a . Their Witten indices will be used to test some results of section 2. In section 2, we did not fix the numbers of hypermultiplet particles, but instead had chemical potentials m a for Sp(n 8 ). Expanding the indices of section 2 in e −ma , the coefficients will be the indices with fixed k, n a , studied in this section.
We start from the case with 1 hypermultiplet, and consider the quantum mechanics of the D1 and D1 ′ branes. We first explain the symmetries. There is SO(4) ∼ SU(2) l × SU(2) r rotating x 1 , · · · , x 4 , and SO(3) ∼ SU(2) R rotating x 7 , x 8 , x 9 . The quantum mechanics preserves 4 real SUSYQα A , whereα and A are doublet indices of SU(2) r and SU(2) R . It can be regarded as the 1d reduction of 2d N = (0, 4) SUSY. There are symmetries associated with D-branes and orientifolds. For r D1's and N D5's on various O5-planes, the symmetries are given as follows: Here r is a half-integer r = n/2 for O5 + , O5 + . So D1 and D1 ′ in Fig. 3 have Sp(k) × O(n) gauge symmetry, while D5's induce SO(2N) or SO(2N + 1) global symmetry.  Fig. 3. The Lagrangian of this system preserving N = (0, 4) supersymmetry can be written down in a canonical manner. We focus on the bosonic part here. Along the strategy of [45], we first construct the Lagrangian in N = (0, 2) formalism, specifying the two possible types of superpotentials E and J for each Fermi multiplet [46]. Our (0, 4) multiplets decompose to (0, 2) multiplets as follows: Figure 6: (a) 1d quiver and (b) matters for SO(2N + 1).
The scalars in rank 2 symmetric or antisymmetric representations are real. It decomposes to two (0, 2) chiral multiplets whose scalars are complexified as a cβ = a 1β + ia 2β and likewiseâ cβ .
In (0, 2) theories, one can turn on two types of holomorphic 'superpotentials' for each Fermi multiplet Ψ, J Ψ and E Ψ . (0, 2) supersymmetry demands the superpotentials to satisfy We first consider the SO(2N) theory, in which case E and J for Fermi multiplets are given by 3) The first two lines, for (0, 4) gauginos of Sp(k) × O(n), are required by demanding (0, 4) SUSY enhancement [45]. Namely, gaugino fields' J and E acquire contributions only from hypermultiplets and twisted hypermultiplets, respectively. But with the first two lines only, (3.2) is not met. The next three lines are fixed (up to sign choices) by demanding (3.2) to hold, as illustrated in [45] in different models. D-terms are given by With these superpotentials and D-terms, the bosonic potential energy is given by [45,46], One can show that (3.5) exhibits enhanced SO(4) = SU(2) r × SU(2) R R-symmetry, Since SO(4) is the N = (0, 4) R-symmetry, this is a strong indication that the classical action indeed has (0, 4) SUSY. We content ourselves with this observation, rather than checking (0, 4) SUSY of the full action. The fields in the last expression satisfy the pseudo-reality condition of One can repeat the analysis for the SO(2N + 1) quiver. One point to note here is that there is no superpotential for the Fermi multiplet Λ. So despite the presence of 2N + 2 O(n) fundamental Fermi multiplets Λ l , Λ, their flavor symmetry is SO(2N + 1), as we expect from 5d bulk.
When there are two 5d hypermultiplets in the spinor representation of SO(N), we can consider a sector with n 1 and n 2 particles and k instantons. The 1d quivers and fields are shown in Figs. 7, 8. The Lagrangians can be constructed by following the completely same procedures, which we do not present here.

The instanton partition functions
We shall compute the Witten indices of the quantum mechanics presented in the previous subsection. They count BPS states preserving Q = −Q 12 and Q † =Q 21 , and is defined by (3.7) J l , J r and J R are Cartans of SO(4) = SU(2) l × SU(2) r and SU(2) R respectively, while q i are the SO(N) electric charges. F , z denote other charges and their chemical potentials. Figure 7: The 1d quiver (a) and matters (b) for 5d SO(2N) theory with two hypermultiplets.
We compute (3.7) using the contour integral formula of [21,11,25]. The zero modes in the path integral appear as the contour integral variables. They are the eigenvalues of the scalar ϕ and A τ in the vector multiplet. For O(n), the flat connections on S 1 have two disconnected sectors O(n) ± . U = e Rφ ≡ e R(ϕ+iAτ ) , where R is the radius of the temporal circle, is given by  contribute as respectively. ρ runs over the weights of Sp(k), O(n) in the representation R Φ , R Ψ , and J is defined by J = J r + J R . F collectively denotes the remaining charges. A (0, 2) vector multiplet V contributes similarly as Z V = α∈root 2 sinh α(φ) 2 , where we used the formula for a Fermi multiplet at J = 0, F = 0. Collecting all, the Witten index is given by The O(n) holonomy has two discrete sectors. The Witten index is given by [47] Z = Z + + Z − 2 . (3.11) The Weyl factors |W | of O(2n) ± , O(2n + 1) ± , Sp(k) are given by [47] |W O(2n) (3.12) For SO(N) with odd N, one can show that Z 1-loop = 0 in O(2n) − and O(2n + 1) + sectors, since the fermionic zero modes from Λ (in Table 6(b)) provide factors of 0's.
Let us call Z k,n the index of the Sp(k)×O(n) quiver. Being a multi-particle index, it acquires contribution from n hypermultiplet particles either bound or unbound to k instantons. Also, as we shall explain in more detail below, Z k,n for n ≥ 2 also contains a spurious contribution from particles not belonging to the 5d QFT. To explain these structures clearly, we first discuss the indices Z 0,n before considering the instanton partition functions at k = 0. At n = 1, k = 0, the O(1) indices do not contain integrals. The results are given by (3.13) The overall factors (2 sinh ǫ 1 2 · 2 sinh ǫ 2 2 ) −1 in (3.13) come from the center-of-mass motion on , and that of SO(2N +1) spinor N l=1 2 cosh v l 2 , respectively. They are the perturbative partition functions of matters in SO(N) in spinor representations. Next, Z 0,2 is given by . (3.14) For SO(2N + 1) and the first term of SO(2N) index, one should evaluate JK-Res. With the choice η > 0, one keeps the residues at χ = − ǫ 1,2 2 and χ = − 15) while for N = 7, 8 one obtains (3.16) 7 For O(n) and Sp(k) gauge theories, the choice of η does not affect the results due to Weyl symmetry [21]. NS5 2 NS5 1 Figure 9: D1 ′ -brane escaping the 5d QFT (3.15) and the first term of (3.16) are the indices of two non-interacting identical particles, whose single particle index is given by Z SO(N ) 0,1 . There are no bound states formed by these perturbative hypermultiplet particles, as expected. The second term of (3.16) requires more explanations, which we now turn to.
The second term of (3.16) comes from extra states in the brane system that do not belong to the 5d QFT. In particular, the fractional coefficient in the fugacity expansion implies that it comes from a sector which has a continuum unlifted by our massive deformations. In fact, following the arguments presented between (2.28) and (2.29), one finds that the linear 1-loop potential from (3.14) vanishes for N = 7, 8, implying continua. Physically, this comes from a D1 ′ -brane moving away from 5d QFT, suspended between two parallel 5-branes as in Fig. 9. Although we are not aware of fully logical arguments, it has been empirically observed that the last term − 1 2 2 cosh ǫ + 2 sinh ǫ 1, 2 2 is the contribution from the escaping particle for strings suspended between parallel 5-branes: e.g. see eqn.(3.62) of [21]. See also [48,49,50] for related results. The suspended string of Fig. 9 carries the same spacetime and R-symmetry quantum numbers as a 5d vector multiplet particles, since the configuration of Fig. 9 is locally dual to a fundamental string suspended between two D5-branes (a 5d vector W-boson). Indeed, the chemical potential dependence ∼ 2 cosh ǫ + 2 sinh ǫ 1, 2 2 is precisely that of a 5d W-boson and its superpartners. Such extra states start to appear at n ≥ 2, since at n = 1, one only has fractional D1 ′ stuck to O5.
Collecting all, we expect that the partition function at k = 0 is given by for N ≤ 6, while for N = 7, 8 we expect that it is given by Here, Z 0,0 ≡ 1 by definition, and P E [f (x, y, · · · )] ≡ exp ∞ n=1 1 n f (nx, ny, · · · ) is the multiparticle index for the single particle index f . Expanding Z inst (q) = ∞ k=0 Z k q k where Z 0 ≡ 1, (3.19) implies at given q k order that

The full partition function would factorize as
When there are two 5d hypermultiplets as in Fig. 4, the full partition function is q k e −n 1 m 1 −n 2 m 2 Z k 1 ,n 1 ,n 2 (3.21) where Z k,n 1 ,n 2 is the index for k D1, n 1 D1 ′ and n 2 D1 ′′ . m 1,2 are the Sp(2) flavor chemical potentials. The contributions from perturbative and extra degrees of freedom in this case are where Z pert and Z extra take the same forms as in (3.18).
Although our methods apply well to both SO(8) and SO (7), we only study the cases with SO(7) in this paper. We start from the case with one hypermultiplet field. From the field contents of Fig. 6, Z 1-loop for k instantons and 2n(=even) hypermultiplet particles is given by 2 )) n · n I=1 2 sinh( while Z 1-loop for k instantons and 2n + 1 hypermultiplet particles is given by . The partition function at k = 1, n = 0 is given by .
Collecting the residues, one obtains where t = e −ǫ + and u = e −ǫ − . Here χ R is the character of SO (7) representation R. This is simply the well-known 1-instanton partition function of SO(7) gauge theory. E.g. see [51] for the above character expansion form.
Next, consider the sector at k = 1, n = 1. Z SO (7) 1,1 is given by . (3.28) Hat denotes the instanton partition function at level (k, n), while Z k,n is simply the Witten index of our Sp(k) × O(n) quantum mechanics. From this formula, one obtainŝ where ∞ p=0 χ (0,p,1) t 2p+5 = χ 8 (v l ) + χ 112 (v l ) t 2 + χ 720 (v l ) t 4 + · · · . Then consider the sector at k = 1, n = 2. Z SO(7) 1,2 is given by the Sp(1) × O(2) contour integral . [+πi] means that there are two cases with and without +πi addition. Subtracting the contribution from Z pert Z extra in (3.20), the instanton partition functionẐ SO (7) 1,2 at this order is given byẐ 1,2 = Z 1,2 −Ẑ 1,1 Z 0,1 − Z 1,0 Z 0,2 . One finds after computations that (3.31) For n ≥ 3, we findẐ 1,n = 0. We checked this exactly for n = 3. For n = 4, to save time, we plugged in random numbers in the chemical potentials and checked thatẐ SO (7) 1,4 is very small. (Below, we present an argument for this phenomenon.) Collecting all the computations at n = 0, 1, 2, one obtains Here Here we discuss more about the maximal value of n withẐ SO(7) k,n = 0, at given k. Note that (7) k,n (ǫ 1,2 , v) , (3.33) refining the previous definition by the zero point energy-like factor. Note that m is the flavor chemical potential for the 5d hypermultiplet. Since a hypermultiplet only adds fermion zero modes on the instanton moduli space, the rotation parameter m acts only on these fermions. So unlike the chemical potentials v i , ǫ 1,2 which act on noncompact zero modes, the coefficient Z k of Z inst at given q k order should not have any poles in m. Since Z k admits fugacity expansions, this implies that Z k is a finite polynomial in e m and e −m . So the sum over n should truncate to 0 ≤ n ≤ n max for some finite n max , also with a suitable m dependent ǫ 0 to ensure the Weyl symmetry of Sp(1). One can also naturally infer the value of n max . To see this, note that a 5d hypermultiplet in the spinor representation induces kD(8) = 2kT (8) = 2k complex fermion zero modes on the moduli space, where we used 2T = 2 N −2 for SO(2N + 1) spinor representation. Quantizing them into 2k pairs of fermionic harmonic oscillators, each oscillator raises/lowers the particle number n by 1. This means that the charge difference between the lowest and highest states is 2k, implying n max = 2k. Then Sp(1) Weyl symmetry implies n → −n symmetry, demanding ǫ 0 = −km andẐ k,2k−n =Ẑ k,n . These completely agree with our empirical findings around (3.31). Below, we shall proceed with these properties assumed.
As explained in section 2, one can Higgs the SO (7) gauge theory with a matter hypermultiplet in 8, to pure G 2 Yang-Mills theory by giving VEV to the hypermultiplet. In the index, this amounts to setting m n 8 = ǫ + , v 4 = 0. See section 2.2. Since we have provided concrete tests of SO (7) instanton partition functions of section 2 using our D-brane-based methods, Higgsing both sides do not yield any further significant information or tests. Namely, calculations in this section at n 8 = 1, 2 already tested our G 2 instanton calculus of section 2 at n 7 = 0, 1. Therefore, we shall not repeat the analysis of Higgsings to G 2 in our D-brane-based formalism.

Strings of non-Higgsable 6d SCFTs
In this section, we study the strings of non-Higgsable 6d SCFTs containing G 2 theories or SO(7) theories with matters in 8. In particular, we shall construct the 2d gauge theories for the strings of 6d atomic SCFTs with 2 and 3 dimensional tensor branches [10].
We first briefly review the 'atomic classification' [5,6,10] of 6d N = (1, 0) SCFTs. This is based on F-theory engineering of 6d SCFTs, on elliptic Calabi-Yau 3-fold (CY 3 ). Elliptic CY 3 admits a T 2 fibration over a 4d base B, which is non-compact and singular. The singular point on B hosts 6d degrees of freedom which decouple from 10d bulk at low energy. In 6d QFT, resolving this singularity corresponds to going to the tensor branch. Namely, there is a 6d supermultiplet called tensor multiplet, consisting of a self-dual 2-form potential B µν (whose field strength H = dB + · · · satisfies H = ⋆ 6 H), a real scalar Φ, and fermions. Giving VEV to Φ, one goes into the tensor branch. Geometrically, the singularity of B is resolved into a collection of intersecting 2-cycles P 1 . Associated with the i'th P 1 , there is a tensor multiplet B i , Φ i , and sometimes a non-Abelian vector multiplet A i with simple gauge group G i . The VEV of Φ i is proportional to the volume of the i'th P 1 . Depending on how the 2-cycles intersect, the vector multiplets form a sort of 'quiver' possibly with charged hypermultiplet matters.    [5,6,10] proceeds by first identifying possible non-Higgsable theories, and then considering possible 'un-Higgsings.' Non-Higgsable theories are constructed by first taking a finite set of 'quiver nodes' and connecting them with certain rules. Technically, the nodes are connected by suitably gauging the E-string theory and identifying them with the gauge groups of the quiver nodes. See [5] for the detailed rules. Roughly speaking, the possible 'quiver nodes' are given in Tables 3 and  4. More precisely, the SCFTs at n = 1 and n = 2 play different roles: see [5,6] for the precise ways of using the SCFTs in Table 3, 4. The SCFTs in Table 3 are called 'minimal SCFTs' in [14]. Here, the numbers on the first rows denote the self-intersection numbers of P 1 . Thus in Table 4, there are two or three 2-cycles (tensor multiplets). We are interested in the self-dual strings, which are charged under B i µν with equal electric and magnetic charges. If a node has gauge symmetry, the string is identified as an instanton string soliton. See, e.g. [4] and references therein for a review. In this section, we are interested in the strings of the SCFTs given in Table 4. Since they involve G 2 gauge group with matters in 7 or SO(7) gauge group with matters in 8, the gauge theories on these strings will be constructed using our gauge theories of section 2 as ingredients.

2, 3, 2: SU (2) × SO(7) × SU (2) gauge group
Since this QFT has three factors of simple gauge groups, one can assign three topological numbers k 1 , k 2 , k 3 for the instanton strings in SU(2) 1 , SO (7), SU(2) 2 . To construct the 2d quiver for these strings, we proceed in steps. We first consider the case in which two of the three gauge symmetries are ungauged in 6d, when only one of k 1 , k 2 , k 3 is nonzero. They are instanton strings of either SU (2) or SO(7) gauge theory with certain matters. After identifying three ADHM(-like) gauge theories, we then consider the case with all k 1 , k 2 , k 3 nonzero, and form a quiver of the three ADHM(-like) theories.
We first consider the case with k 1 = k 3 = 0, when SU(2) 1 × SU(2) 2 is ungauged. Then SU(2) 2 ∼ Sp(1) 2 becomes a flavor symmetry rotating the hypermultiplets, which in the strict ungauging limit enlarges to Sp (2). This is because the matters in 1 2 (2, 8, 1) + 1 2 (1, 8, 2) will arrange into 1 2 (8, 4) of SO(7) × Sp(2) in the ungauging limit. This theory was discussed in section 2.1, the 6d SO(7) theory at n 8 = 2. So as the ADHM-like description, we take this theory with U(k 2 ) gauge symmetry and reduced SU(4) × U(2) ⊂ SO(7) × Sp(2) global symmetry. Note that in section 2, our 2d gauge theory can have U(4) global symmetry rotating 4 Fermi multiplets, but it reduced to U(2) after coupling to the 5d/6d background fields, especially the hypermultiplet scalar VEV. So the relevant global symmetry of this model (as describing higher dimensional QFT's soliton) depends on the bulk information. Here, since we shall use this model for the strings of the non-Higgsable 2, 3, 2 SCFT, with SU(2) 2 gauged, one cannot turn on such a background hypermultiplet field. Instead, SU(2) 2 ⊂ U(4) global symmetry will remain in 2d after 6d gauging. 4 Fermi fields are divided into 2 pairs, and we can rotate them only within a pair.
Now when all k 1 , k 2 , k 3 are nonzero, one can form a quiver of the above three ADHM(like) theories. We shall add more 2d matters to account for the zero modes coming from 6d hypermultiplets, and introduce extra potentials. Between adjacent SU(2) 1 × SO(7) or SO(7) × SU(2) 2 pair of nodes, one has bi-fundamental hypermultiplet in 1 2 (8, 2). Since we seek for a 2d UV description seeing SU(2) × SU(4) subgroup only, this hypermultiplet is in (2,4) bi-fundamental representation of the latter. Usually in D-brane models with bi-fundamental matters, the induced (0, 4) matters on the U(k 1 ) × U(k 2 ) ADHM construction of instantons are Ψ α = (Ψ 1 , Ψ 2 ) : two Fermi multiplet fields in (k 1 , k 2 ) and Ψ a : Fermi multiplets in (k 1 , 4) of U(k 1 ) × SU(4) (a = 1, · · · , 4) See, e.g. [12,15] for the details. Although our construction is not guided by D-brane models, we advocate the same field contents as our natural ansatz. The fields Ψ a with a = 1, · · · , 4 are not new, but come from the last line of (4.1). This is natural because the 6d SU(4) ⊂ SO(7) gauge symmetry is obtained by gauging the global symmetry in the setting of (4.1). Also, Ψ i with i = 1, 2 can also be found in the ADHM-like quiver in section 2. Namely, in section 2, we had four Fermi multiplet fields in k 2 representation of U(k 2 ), at n 8 = 2. Ψ i of (4.3) is obtained by taking two of these four. (The other two will be associated with the SO(7)×SU(2) 2 pair.) The bi-fundamental fields in (4.2) are new, and link the two ADHM(-like) gauge nodes. Similarly, between the second and third nodes, bi-fundamental fields of the form of (4.2), replacing k 1 → k 3 , are added. The remaining Fermi fields in the second and third nodes take the form of (4.3), with k 1 → k 3 . The flavor symmetry of these Fermi multiplets in an ADHM node is locked with the 6d gauge symmetry of the adjacent ADHM node. The resulting quiver is shown schematically in Fig. 10.
In the previous paragraph, and in Fig. 10, we locked some 6d flavor symmetries of an ADHM theory with 6d gauge symmetries of adjacent ADHM theories. This has to be justified by writing down the interactions which lock the symmetries as claimed. Now we explain such superpotentials. In the (0, 2) off-shell description [45] of (0, 4) theories, one can introduce interactions by two kinds of superpotentials J Ψ , E Ψ given for each Fermi multiplet. There are some constraints on J Ψ 's and E Ψ 's to be met, either for (0, 2) SUSY or for (0, 4) enhancement of the classical action. These conditions are all mentioned in section 3, when we discussed models with manifest (0, 4) SUSY. In our current ADHM-like models, some part of the matters and interactions inevitably break manifest (0, 4) SUSY. However, most of the fields still take the form of (0, 4) multiplets, so that we find it convenient to turn on classical interactions in two steps. We first turn on manifestly (0, 4) supersymmetric classical interactions for the fields shown in Fig. 10 with black lines/nodes. Then we rephrase these interactions in N = (0, 1) language, after which we turn on further (0, 1) interactions for the fields shown as red lines in Fig. 10. We find that securing the partial (0, 4) SUSY structure plays important roles for the correct physics, e.g. yielding the right multi-particle structures of the elliptic genus, etc.
Here, notations like 2 1 , 2 3 mean representations of SU(2) on the first (leftmost) and the third (rightmost) nodes, respectively. Then, using the results of [53], eqns. (3.3) and (3.4), we find the following superpotentials after mapping our fields with those in Table 4 of [53]: (We correct overall normalization of [53] by √ 2 factors.) These are part of the interactions, and we shall add more interactions later preserving less SUSY. Only with the interactions shown above, one can check the (0, 4) SUSY of the classical action, for instance in the bosonic potential [45,53]. The rearrangement of the potential energy with SU(2) r × SU(2) R symmetry can be made similar to eqn.(3.6) of [53]. In particular, the flavor symmetries which rotate Fermi multiplets are locked by these interactions as shown in Fig. 10.
With these potentials, one can show that the moduli space is that of each ADHM-like quiver, at Φ i−1,i = 0,Φ i−1,i = 0. In particular, no extra branch is formed by Φ i−1,i ,Φ i−1,i .
One can compute the 2d anomalies from our gauge theory, and compare with the result known from anomaly inflow. The 6d 1-loop anomaly 8-form in the tensor branch is given by We only showed the terms containing SU(2) 1 × SO (7) × SU(2) 2 gauge fields. This can be written as Using (2.40), one finds the following anomaly 4-form I 4 on the instanton strings with string numbers k i = (k 1 , k 2 , k 3 ).
We now compute the anomaly from our gauge theory. We first compute the anomalies of three ADHM quivers I Tr(F 2 SO (7) ) + 5c 2 (R) + where we replaced tr 4 (F 2 . As in section 2.1, F SO(7) is restricted to SU(4) in our UV gauge theory, and fields in c 2 (R), c 2 (r) are also restricted to F J . I can be computed from the known anomaly polynomial for the instanton strings of 6d SU(2) theory at n 2 = 4. The result is eqn. (5.19) of [4] at N = 2, with k replaced by k 1 or k 3 : Tr(F 2 SO (7) )−2k 1 c 2 (R) , I Here we replaced F = SU(4) of [4] by SO (7), assuming symmetry enhancement. Finally, I bif 4 is also computed in [4], eqn. (3.58), which for our model is One finds that I 4 = 3 i=1 I (i) 4 + I bif 4 agrees with (4.10), providing a check of our gauge theory.
It is important to note that Φ i−1,i ,Φ i−1,i do not provide extra JK-Res, for the following reason. For instance, suppose that we take the 'pole' from θ(−ǫ + + u I 1 − u I 2 ) −1 on the fourth line, at −ǫ + + u I 1 − u I 2 = 0 to determine u I 1 , with u I 2 determined from (4.15). Suppose that u I 2 is determined by ǫ + +u I 2 −v i = 0. Then on the first line of (4.14), a Fermi multiplet contribution θ(v i −u I 1 ) vanishes at the pole, because u I 1 −v i = (−ǫ + + u I 1 −u I 2 ) + (ǫ + + u I 2 −v i ) = 0. On the other hand, suppose that u I 2 is determined by one of ǫ 1,2 + u I 2 − u J 2 = 0, with u J 2 determined by other equations. Then, from Ψ 12 ,Ψ 12 's contributions θ(ǫ − ± (u I 1 − u J 2 )) on the fourth line, one again finds that one of the two θ factors vanishes at the pole location. Therefore, one finds that the residue vanishes due to the vanishing determinant from certain Fermi multiplet. This idea turns out to hold most generally, so that one can show that the fourth line of (4.14) never provides a pole with nonzero JK residue. Based on these observations, one can make a recursive proof of this statement, similar to that made for the 5d N = 1 * instanton partition function in [21]. Note that the symmetry locking provided by the (0, 4) potentials (4.6) played crucial roles for the vanishing of these residues.

Tests from 5d descriptions
In this subsection, we test the elliptic genera of section 4.1, using a recently proposed 5d description for the 6d 2, 3, 2 SCFT compactified on S 1 [16]. The description is available when the elliptic CY 3 in F-theory admits an orbifold description, of the form [B×T 2 ]/Γ with a discrete group Γ. One can dualize F-theory to M-theory on same CY 3 . The small S 1 limit (together with suitably scaling other massive parameters) on the F-theory side corresponds to the large T 2 limit on the M-theory dual. There may be fixed points of Γ on T 2 , as it decompactifies into R 2 . Near each fixed point, there exists an interacting 5d SCFTs. So in this 5d limit, one obtains factors of decoupled 5d SCFTs. The 6d KK momentum degrees of freedom can be restored by locking certain global symmetries of these 5d SCFTs and gauging it, so that the instanton quantum number of this 5d gauge theory provides the 6d KK momentum. See [54,55,56,16] for the details.
If a 6d SCFT admits a 6d gauge theory description, an obvious 5d limit is given by the 5d gauge theory with same gauge group. This is obtained by a scaling limit with 6d tensor multiplet scalar VEV, v = Φ → ∞. Namely, v = g −2 6d and Rv = g −2 5d are the 6d and 5d inverse gauge couplings, respectively, where R is the circle radius. If one takes R → 0, v → ∞ with g −2 5d kept fixed, one often gets a 5d SCFT with a relevant deformation made by g −2 5d = 0 [56]. The 5d factorization limit described in the previous paragraph takes different scaling limit of massive parameters when taking R → 0. The latter 5d limit scales other massive parameters like the holonomies of gauge fields on S 1 . From the viewpoint of former 5d limit, the latter 5d limit keeps a different slice of 5d states, which contains states with nonzero KK momenta from the former viewpoint. For our 2, 3, 2 SCFT, the new 5d limit consists of three 5d SCFTs. The three 5d SCFTs admit IIB 5-brane web engineerings, given by Fig. 11 [16]. Each factor in Fig. 11(a) is a non-Lagrangian theory, in that it does not admit a relevant deformations to 5d Yang-Mills theory. (Fig. 11(a) is related to that in [16] by a flop transition.) To have states with general KK momenta, one locks the three SU(2) g flavor symmetries associated with gauge symmetries on the blue-colored parallel 5-branes of Fig. 11, and gauge it. The relations between 6d parameters and the Kahler parameters of 5-brane web are shown below Fig. 11, which will be (empirically) justified. [16] also discusses the gauging of SU(2) g in the brane web context as trivalent gluing, with some prescriptions for computations. But here we shall only discuss computations in the factorization limit.
We want to test our elliptic genera (4.14), (4.16) using this 5d description. The test will be made in the 5d factorization limit in which SU(2) g is ungauged, as in Fig. 11 with semi-infinite blue lines. In some sectors with special values of k 1 , k 2 , k 3 , the BPS spectrum of the brane configuration is well known, so our elliptic genera in these sectors will be tested against known results. More generically, we shall do topological vertex calculus. Technically, identifying the parameters of 6d gauge theory (and our elliptic genus) and those in the 5-brane web is t 4 is roughly the Kahler parameter for the blue line in Fig. 11, which is sent to zero in the factorization limit, with t 1,2,3 fixed. This limit requires q ∼ y 2 3 → 0 with fixed y 1 , y 2 . To fully specify this 5d limit, we still have to specify the scaling of other parameters in q → 0. The remaining parameters are: two SU(2) inverse gauge couplings (or tensor VEV's) which we call e −v , e −ṽ in this subsection, two SU(2) fugacities w,w (related to ν,ν of section 4.1 by w = e −ν ,w = e −ν ), SO(7) inverse gauge coupling e −u . All the scaling rules except that of e −u will be determined below by considering an SU(2) subsector. The scaling of e −u will then be determined next by considering the SO(7) subsector, at which stage we shall already make some tests of our elliptic genera. Then we consider more general sectors for further tests. SU(2) subsector: We first study the limit in which SO(7) is ungauged, or equivalently, when k 2 = 0. The limit u → ∞ should yield two 6d SU(2) theories at n 2 = 4, decoupled to each other. So in this limit, the brane web of Fig. 11 (with SU(2) g gauged) should factorize into two. The natural identification of u → ∞ in the web is to take the distance between the parallel blue lines to infinity. (Assuming the identification of Kahler parameters in Fig. 11, the distance between two blue lines is proportional to (11) = (2)(3) 2 (4) = (7)(8) 2 (9) ∝ e −2u .) The string suspended between the two parallel blue lines is infinitely heavy in this limit. So the 5d description suggests that the 6d SU(2) theory at n 2 = 4 is given by U(1) g (⊂ SU(2) g ) gauging of three factors, where two of them take the form of Fig. 13(a), and one takes the form of Fig. 13(b). Upon a suitable SL(2, Z) transformation, Fig. 13(a) is the standard 5-brane web for the 5d N = 1 pure SU(2) theory. Similarly, Fig. 13(b) describes the 5d 'SU(1) theory.' The SU(1) theory simply refers to the brane configuration of Fig. 13(b), not containing an interacting 5d SCFT. This sector will be void. So we shall take a suitable 5d scaling limit of the elliptic genera of 6d SU(2) theory at n 2 = 4, and find the parameter map which exhibits two copies of 5d pure SU(2) theories.
The 5d SU(2) theory's BPS spectrum can be computed from its instanton partition function. It contains two fugacities, Q for the instanton number, and W for the SU(2) electric charge in the Coulomb branch. It also contains Omega deformation parameters ǫ 1,2 . Here, we only  Table 5: BPS spectrum of 5d N = 1 pure SU(2) theory consider the unrefined single particle spectrum, defined as follows. The partition function Z SU (2) (Q, W, ǫ 1,2 ) is written as Z SU (2) = exp ∞ n=1 1 n f (Q n , W n , nǫ 1,2 ) , where f is the single particle index. Then one considers the limit and we are left with the remaining finite numbers in the tables. One can then show that the remaining numbers in the tables are two copies of Table 5. Namely, one finds where Q ≡ q w 2 y 3 , W ≡ wy 3 . The first two lines yield a 5d pure SU(2) index, with the identification of Kahler parameters (4.26) The last two lines yield another copy of 5d SU(2) index, with parameters Note that the identifications of Q 1 , Q 2 are consistent with our previous findings based on affine SO(7) structure. This identifies the parameters (2), (7) of Fig. 11, and similarly (4), (9). SO(7) subsector: We now consider another subsector with k 1 = k 3 = 0, k 2 = 0. We start from the elliptic genus of the SO(7) instanton strings at n 8 = 2, studied in section 2. In the 5d scaling limit, e.g. at k 2 = 1, we found the following exact factorization, where U ≡ e −u− v+ṽ 2 q − 1 2 y 1 y 2 y 2 3 . So f rel decomposes into two factors. To have such a factorization, one should scale e −u → ∞ so that e −u q −1/2 y 2 3 ∼ e −u q 1/2 is finite, which guarantees that U is finite. Here, one can show that each factor takes the form of the instanton partition function for the 5dẼ 1 SCFT, upon identifying U Q as the instanton number fugacity and Q i as the electric charge fugacity (Coulomb VEV), for i = 1, 2 respectively. To understand this from the brane web description, we take the 5d factorization limit q ∼ y 2 3 → 0, and also consider the limit v,ṽ → ∞ to realize the sector with k 1 = k 3 = 0. One finds that Fig. 11 Table 9: N 4,p,m in the limit into two SU(1) theories in this limit since (11) → 0, thus void. Each factor of Fig. 11(a) becomes the left side of Fig. 14, since (2), (4), (7), (9) → 0. After an SL(2, Z) transformation, it becomes the right side of Fig. 14. This is the standard brane configuration for the 5dẼ 1 theory [57]. It is the 5d U(2) theory at Chern-Simons level 1. From these studies, one can identify the Kahler parameters (3), (8) of Fig. 11. Note that in (4.28), the leading term at U 1 order is U W iWi (with i = 1, 2) for the two 5dẼ 1 factors. This is the Kahler parameter for the bottom horizontal line on the right side of Fig. 14 ww √ q , which were already shown in Fig. 11. Once we know (3) and (8), one can determine (11) from the gluing condition (11) = (2)(3) 2 (4) = (7)(8) 2 (9), again already shown in Fig. 11. Thus we fixed all Kahler parameters of Fig. 11 in terms of our 6d fugacities.
We have in fact made a nontrivial test of our elliptic genera of section 2, for the SO(7) in-SL(2; Z) Figure 14: Brane web for the 5dẼ 1 SCFT stanton strings at n 8 = 2, using the 5-brane web description, from (4.28). Although apparently we tested the elliptic genera in a 5d factorizing limit, this is different from the tests made in section 3. This is because the '5d limit' here scales other massive parameters and keeps a different slice of BPS states in its zero momentum sector. Indeed, using the original 6d variables, (4.28) is a nontrivial series in Q 1 = q y 1 y 2 y 2 3 ∼ q, acquiring contributions from the 6d KK tower. So this provides an independent nontrivial test of our results in section 2.
The topological vertex [58] computes all genus topological string amplitudes, which is equivalent to the logarithm of the 5d Nekrasov partition function on Omega-deformed R 4 × S 1 [59].
3, 2, 2 SCFT strings: The strategy is similar to that of section 4.1. We first consider the limits in which all except one gauge symmetry are ungauged in 6d, and take three factors of ADHM(like) quivers. We then combine these quivers by locking certain symmetries, and introducing  Figure 15: Brane configurations for: (a) 6d (2, 0) SCFT of type A 1 ; (b) 6d 3, 2, 2 SCFT in the limit with ungauged G 2 bi-fundamental matters of the form of (4.2). To be more precise, we have no 6d gauge group associated with the '2' node on the right. Although the notion of ungauging is absent for this node, we can still take the tensor VEV associated with this node to infinity. Whenever a node has a 6d gauge group, its inverse coupling is proportional to the tensor VEV Φ , so taking Φ → ∞ ungauges the symmetry.
If one takes all tensor VEVs to infinity except the '3' node, one obtains the 6d G 2 theory at n 7 = 1. This is because the 6d matter in 1 2 (7, 2) behaves like one full hypermultiplet in 7, while 1 2 (1, 2) is neutral in G 2 and invisible in the gauge dynamics. So with a G 2 theory at n 7 = 1, its k 1 G 2 instanton strings are described by the 2d U(k 1 ) gauge theory explained in section 2.2, with fields given by (2.49), (2.50), (2.51) at n 7 = 1. The ungauged SU(2) ∼ Sp(1) acts as the flavor symmetry of the 6d hypermultiplet. In the ADHM-like quiver at general n 7 , one may have as big as U(2n 7 ) flavor symmetry which rotates Fermi multiplets. But the coupling to bulk fields only allowed U(n 7 ) part, which we further expected to enhance to Sp(n 7 ). This is similar to the flavor symmetries of SO(7) ADHM-like theory at n 8 = 0. In the current context, again like the 2, 3, 2 quiver, we should couple the system to different bulk fields. At n 7 = 1, one can classically have as big as U(2n 7 ) → U(2) flavor symmetry. We restrict it to SU(2) which rotates Ψ,Ψ † of (2.51) as a doublet. Also, as explained in section 2.2, only SU(3) ⊂ G 2 is visible in this quiver. More formally, it will be convenient to regard the fields q i ,q i , φ i , φ 4 as transforming in SU(3) × SU(1) ⊂ SU(4).
When G 2 is ungauged and the tensor VEV for the right '2' node is sent to infinity, we have 6d SU(2) theory at n 2 = 4. Its ADHM quiver is explained around (4.1). In this limit, G 2 is enhanced to SO(7) flavor symmetry rotating the four hypermultiplets in 8 of SO (7), but only SU(4) ⊂ SO (7) is visible in the UV ADHM, as explained in section 4.1. SO(7) will later be broken to G 2 by gauging. In our ADHM-like quiver, which only sees SU(3) ⊂ G 2 , SU(4) will be broken to SU(3) × SU(1), locked with the G 2 ADHM of the previous paragraph.
We finally ungauge G 2 × SU(2), leaving one tensor VEV for the right '2' node finite. One then obtains the 6d N = (2, 0) SCFT of A 1 type, geometrically engineered on the O(−2) → P 1 base with no associated gauge group. Although the strings of this SCFT in the tensor branch lacks the instanton string interpretation, one still knows the UV 2d gauge theory description where a = 1, 2. We showed the representation and charge of the classical symmetry U(k)×U (1), where one should further restrict U(1) → SU(1) due to mixed anomaly. This formally takes the form of the ADHM instanton strings of '6d SU(1) theory' with two charged quarks. The SU(2) F flavor symmetry which rotates Ψ a is identified with the enlarged R-symmetry group of the 6d (2, 0) theory. Namely, we expect that SU(2) R of 6d (1, 0) SCFT enhances to SO(5) R . In the tensor branch, this is broken to SO(4) ∼ SU(2) R × SU(2) L , where the latter SU(2) L is realized as SU(2) F in the 2d quiver. The 6d A 1 (2, 0) theory and the above 2d gauge theory admit D-brane engineerings. Using D2-D6-NS5, one can use either of Fig. 15(a), in IIA or massive IIA string theory [60,61].
Before fully combining the three ADHM(-like) quivers, we note that the combination of two '2' nodes (with G 2 ungauged) is dictated by a D-brane setting. This is given by the brane configuration of Fig. 15(b) in the massive IIA theory. The 2d quiver is given by Fig. 16 at k 1 = 0. The quiver and the brane system only has manifest SU(3) × SU(2) × U(1) symmetry, where the last U(1) is a combination of three overall U(1)'s in U(3) ×U(2) ×U(1) which survive the mixed anomaly cancelation with U(k 2 ) × U(k 3 ). More precisely, taking the overall U(1) generators Q i for SU(i), i = 1, 2, 3, only Q 1 + Q 2 + Q 3 is free of the mixed anomaly. (This U(1) is not shown in Fig. 16, as it will be irrelevant generally at k 1 , k 2 , k 3 = 0.) One can see that the 2d quiver exhibits SU(3) × U(1) → SO(7) symmetry enhancement, say by studying the elliptic genera. This should be the case since one has 6d SU(2) theory at n 2 = 4. Just to be sure, we tested the SO(7) enhancement of the elliptic genus at k 2 = k 3 = 1. Now we keep k 1 = 0, with G 2 gauged. In our UV GLSM, we can only see SU(3) ⊂ G 2 , which we lock with the SU(3) symmetry of the quiver in the previous paragraph. The resulting unaffected by dynamical χ and its superpartner H χ . Now consider the equation of motion for n 7 ≤ 3 matters in 7. At n 8 = 2 for SO(7) and at n 7 = 1 for G 2 , where anomaly-free 6d gauge theories exist, our gauge theories uplift to 2d for instanton strings. These ADHM strings can be used to construct the 2d quivers for the 'atomic' non-Higgsable 6d SCFTs of Table 4. These gauge theories do not describe the symmetric phase physics of instantons, but we propose them to compute the Coulomb phase partition functions correctly. Although the worldvolumes of instantons host N = (0, 4) SUSY (or its 1d reduction), our gauge theories are made of N = (0, 2) supermultiplets, and some of their interactions only exhibit N = (0, 1) SUSY. We expect various symmetry enhancements in 1d/2d. We tested our 1d/2d gauge theories by computing their Witten indices or elliptic genera using various other methods. Firstly, for 5d G 2 theory without matters, n 7 = 0, we used the results of [42], which uses 3d Coulomb branch techniques. We tested our results for instanton numbers k ≤ 3, but the comparisons can in principle be made for arbitrary high k's. In section 3, we developed another D-brane-based method to study the instantons of 5d SO(7) theories at n 8 = 1, 2, and related G 2 theories at n 7 = 0, 1. This method provides a much more elaborate computational procedure, which however does not require guesswork. We used this method to successfully test our results for n 8 = 2 at k = 1, and for n 8 = 1 at k = 1, 2. Finally, we used the 5-brane web description of [16] to test the SO(7) instanton strings at n 8 = 2, and also the strings of the 6d 2, 3, 2 SCFT of Table 4. All the methods that we used to test our results exhibit manifest (0, 4) SUSY. So the agreements of our indices with these alternative calculus are indirect signals that our systems exhibit (0, 4) SUSY enhancement.
The coefficient of t 2n for the correct resultẐ E 7 k=1 is the n'th symmetric product of the E 7 adjoint representation 133. The indexẐ trial k=1 is close toẐ E 7 k=1 at low orders in t (e.g. exact at t 2 ), but slightly deviates and converges to ≈ −1.928% error asymptotically at large orders. We think this failure provides helpful lessons on our ADHM-like trials, and possible subtle points (some stated at the beginning of section 2).
First of all, the gauge theory we constructed which yieldsẐ trial k=1 has two branches of moduli spaces. The first branch has the SU(8) instanton moduli space as a subspace, and has the right complex dimension 2kc 2 (E 7 ) = 36k for E 7 instantons. The second branch meets the first one at a point, the small instanton singularity, and arises from the extra matters we added to the SU(8) ADHM. We find that the second branch cannot be eliminated (with given extra matters) by turning on N = (0, 1) potentials. We suspect that there may be a contribution tô Z trial k=1 from the second branch, which spoils the results. Since we are doing a UV computation in which two branches are not separated, we do not know how/whether one can separate the contributions from two branches. Certainly this may be one reason for the deviation. In case this is the dominant reason for the deviation, there should be large enough contributions from massless fermions on the second branch, asẐ trial k=1 is always no greater thanẐ E 7 k=1 .
Secondly, there is another reason why we suspect the deviation happens. To explain this, note that 133 at t 2 order is correctly reproduced in our trial gauge theory. From the branching rule 133 → 63 ⊕ 70, two different contributions make this to happen. 63 is nothing but the t 2 order contribution from the SU(8) ADHM fields. 70 comes from gauge invariant operators of the SU(8) ADHM fields and extra fields we added. The next order t 4 contains the irrep. 7371 of E 7 , which is rank 2 symmetric product of 133. We are missing some states inẐ trial k=1 , which is 70 ⊕ 1, the rank 4 antisymmetric representation and a singlet of SU(8). If we blindly take the operators in our trial gauge theory which successfully reproduced 133, and take rank 2 symmetric product of them, we find that the representation 70 ⊕ 1 is missing due to the compositeness of 70 that appeared at t 2 order. This is like the quarks of QCD accounting for the plethora of gauge-invariant mesons at low energy, with less microscopic degrees of freedom, while quarks manifest themselves at high energy. So it appears that we should add more extra fields to make up for these missing states. We have found several possible combinations of extra supermultiplets which one may add to the SU(8) ADHM, satisfying very strong constraints of the correct 2d anomaly. But we have not managed yet to construct the model which exhibits the right Hilbert series of (5.1).
Finally, while restricting to SU(8) ⊂ E 7 subset of moduli space, we deleted one simple root so that we lost extra possibility of embedding SU(2) single instanton. Note that the deleted root has same length as the roots kept in SU (8), so that we might have lost extra small instanton saddle points residing in the deleted SU(2). This may be related to the observations in the previous paragraph. In any case, if the basic idea of this paper is applicable to other exceptional instantons, we have many strong constraints which may eventually guide us to the correct ADHM-like models. We hope to come back to this problem in the future.
One important spirit of our construction is to take advantage of symmetry enhancements of gauge theories after RG flow. One is the enhancement of global symmetries from a classical group to an exceptional one. Another is the SUSY enhancement to N = (0, 4). Allowing less number of SUSY in UV provides more room to engineer the desired system. Being in 1d or 2d, we can have as little as 1 Hermitian SUSY in UV, the minimal number which admits any computation relying on SUSY. Still the requirement of using gauge theories put some constraints. However, we are not fully aware of whether we have overlooked the possibilities of subtler supermultiplets or more general interactions, even within gauge theory.
It may also be interesting to further study the physics of exceptional instanton strings from recent 4d N = 1 gauge theory descriptions for N = 2 SCFTs, by suitably reducing them to 2d [62,63]. Many aspects of these constructions are different from ours. For instance, different symmetries are manifest in UV, so it may be helpful to compare the two approaches.
Some of our 1d/2d gauge theories are not tested with their Witten indices and elliptic genera, simply because we have not thoroughly thought about alternative approaches. The recent engineering of 5d SCFTs (e.g. see [33,56]) will allow more geometric/brane realizations. We may be able to test our models relying on these developments, perhaps using topological vertices. Also, for studying 6d strings, one can use the topological string approach (e.g. see [14] and references therein) or the modular bootstrap like approach [17,18,19].
The strategy of this paper was just to write down UV models and test them empirically when data from alternative descriptions are available. It will be nice to have a more conceptual understandings of the ADHM-like models, either from string theory or by other means. Here we feel that, compared to the simple Young diagram sums for the indices and elliptic genera, the microscopic explanations in terms of N = (0, 1) UV gauge theories look less elegant, although practically useful and flexible. In particular, at least at the moment, it is hard for us to imagine a viable string theory engineering of our gauge theories. It is not clear to us whether we are making UV uplifts intrinsically beyond the territory of string theory, or whether there are nicer reformulations which may allow string theory embeddings.
Finally, it will be interesting to see if our ADHM-like gauge theories can be used to study other observables in the Coulomb branch. For instance, study of the Wilson loops or other defect operators will be interesting. See, e.g. [64,65,66,67] and references therein. As many of these constructions rely on D-brane settings, one should see if employing similar prescriptions without D-brane engineering will work (as we did for the partition function in this paper).