Argyres-Douglas Theories and S-Duality

We generalize S-duality to N=2 superconformal field theories (SCFTs) with Coulomb branch operators of non-integer scaling dimension. As simple examples, we find minimal generalizations of the S-dualities discovered in SU(2) gauge theory with four fundamental flavors by Seiberg and Witten and in SU(3) gauge theory with six fundamental flavors by Argyres and Seiberg. Our constructions start by weakly gauging diagonal SU(2) and SU(3) flavor symmetry subgroups of two copies of a particular rank-one Argyres-Douglas theory (along with sufficient numbers of hypermultiplets to guarantee conformality of the gauging). As we explore the resulting conformal manifold of the SU(2) SCFT, we find an action of S-duality on the parameters of the theory that is reminiscent of Spin(8) triality. On the other hand, as we explore the conformal manifold of the SU(3) theory, we find that an exotic rank-two SCFT emerges in a dual SU(2) description.


Introduction and Summary
N = 2 superconformal field theories (SCFTs) often have exactly marginal deformations that preserve N = 2 supersymmetry (SUSY). Such deformations are descendants of dimension two operators that we can add to the prepotential where the integration is taken over the N = 2 chiral Grassmann parameters. The λ i parameterize spaces commonly referred to as "conformal manifolds. The simplest isolated SCFTs we can consider gauging are just collections of free hypermultiplets. For example, taking a collection of eight hypermultiplets and gauging an SU(2) ⊂ Sp (8) flavor subgroup, we construct the SU (2) theory with N f = 4 and SO (8) flavor symmetry. As we vary the resulting exactly marginal coupling, τ = θ π + 8πi g 2 , the theory becomes strongly coupled. However, if we tune the coupling appropriately, a new weakly coupled S-dual description emerges at another cusp [2] which looks like the original theory up to an S 3 triality outer automorphism of the flavor Spin (8). The duality group in this case is SL(2, Z), and this construction extends the notion of N = 4 duality [3] to an N = 2 theory.
More generally, if one starts from a Lagrangian theory and tunes the gauge coupling to another cusp, one often finds that a new isolated interacting SCFT emerges. For example, in [4], Argyres and Seiberg found that, by starting with the weakly coupled SU(3) gauge theory with six fundamental flavors and varying the gauge coupling, a new cusp emerges with an S-dual description in which the Minahan-Nemeschansky (MN) theory with E 6 global symmetry [5] is weakly coupled to a doublet of SU(2) via an SU(2) ⊂ E 6 gauging.
This type of duality has been generalized by Gaiotto [6] and many other authors (see, 1 On general grounds, such conformal manifolds are Kähler [1]. 2 They can also couple sectors with co-dimension one or higher conformal manifolds. However, we can often continue this process iteratively until we have a collection of isolated theories. e.g., [7] and [8]).
All other examples of S-duality discussed in the literature essentially share the general characteristics of the above two cases, but with varying numbers of cusps and isolated sectors of varying ranks (i.e., varying dimensions of their Coulomb branches). In particular, all the instances of S-duality that we are aware of involve N = 2 scalar chiral primaries (we mean operators annihilated by all the anti-chiral Poincaré supercharges; these operators are often called "Coulomb branch" operators) of integer dimension.
In this paper, we will generalize S-duality to theories with non-integer scaling dimension Coulomb branch primaries. Since Lagrangian theories have only integer dimension N = 2 chiral operators, our theories of interest are never completely weakly coupled. Instead, we will find various cusps where weakly coupled gauge fields emerge and couple various isolated strongly coupled sectors that are related to each other in interesting ways. 3 The original examples of theories with non-integer dimension chiral operators were discovered as special points in the Coulomb branch of SU(3) super Yang-Mills by Argyres and Douglas [9] and in SU (2) SQCD with N f = 1, 2, 3 flavors in [10] (the N f = 1 SCFT is the same as the one in [9]). Following the notation of [11], we will refer to these theories as the I 2,3 , I 2,4 , and I 3,3 SCFTs respectively. 4 These theories are believed to be the only rank-one SCFTs with non-integer dimension N = 2 chiral operators. 5 Of course, there are also many higher-rank Argyres-Douglas (AD) theories (e.g., see the review in [14]).
Although the above AD theories are isolated, they typically inherit some flavor symmetry from the UV gauge theories in which they are embedded. For example, the I 2,4 and I 3,3 theories have SU(2) and SU(3) flavor symmetry respectively (the I 2,3 theory has no flavor symmetry). Therefore, we can try gauging the flavor symmetries of the I 2,4 or I 3,3 theories in an exactly marginal fashion (adding additional sectors charged under a diagonal combination of flavor symmetries as necessary), studying the resulting conformal manifold, 3 Note that there can be conformal manifolds with only integer dimension Coulomb branch operators that do not have a Lagrangian limit because they have some exceptional flavor symmetry (for example, one can gauge an SU (3) subgroup of the flavor symmetry of the E 8 SCFT as in [7]). 4 These SCFTs also go under many different names. For example, they are sometimes referred to as the (A 1 , A 2 ), (A 1 , A 3 ), and (A 1 , D 4 ) theories [12] (this notation arises from the fact that the BPS quivers of these theories are the products of the corresponding ADE Dynkin diagrams). 5 More precisely, Kodaira's classification of elliptic fibrations over the complex plane [13] implies that the only consistent non-integer scaling dimensions of N = 2 Coulomb branch generators describing a rank one theory are 6/5, 4/3, or 3/2. These scaling dimensions are realized, respectively, by the I 2,3 , I 2,4 , and I 3,3 theories [10]. While it is not inconceivable that other inequivalent theories have the same spectrum, no such theories have been found to date. and finding the various S-dual frames.
To that end, in the first part of this paper, we will study a particular rank three theory, which we denote as T 2, 3 2 , 3 2 . This theory consists of SU(2) gauge fields coupled to two I 3,3 theories and a doublet of hypermultiplets. As a result, T 2, 3 2 , 3 2 has one marginal coupling.
We will see that this marginal coupling parameterizes a conformal manifold with three S-dual cusps, and that, at each of these cusps, SU(2) gauge fields emerge and couple two I 3,3 theories and a doublet of hypermultiplets (with the parameters of the theory mixed in interesting ways). After appropriately taking into account the mixing of the different parameters, we will find an analog of the triality discussed in [2]. Furthermore, subject to some assumptions, we will prove that the T 2, 3 2 , 3 2 theory is the minimal (i.e., lowest-rank) theory with non-integer dimensional Coulomb branch operators that has a marginal gauge coupling and exhibits S-duality. As such, our discussion of the T 2, 3 2 , 3 2 SCFT represents the minimal generalization of Seiberg and Witten's analysis of SU (2) with N f = 4 [2].
In the second part of the paper, we look for the lowest-rank generalization of Argyres-Seiberg duality. We will argue that the simplest generalization is given by a rank four theory we call T 3,2, has an SU(2) gauge theory realization in which the gauge group is coupled to a single I 3,3 theory and a more exotic theory of rank two with spectrum 3, 3 2 that we will call T 3, 3 2 (this theory plays the role of the E 6 SCFT in our duality). The latter has a G T 3, 3 2 ⊃ SU(3) × SU(2) flavor symmetry, of which we gauge the SU(2) factor. T 3, 3 2 has not been explicitly discussed in the literature (although it appears implicitly in the classification of [11,15]), and our analysis will elucidate some of its interesting properties. For example, our results imply that the flavor symmetry does not suffer from Witten's anomaly [16]. Moreover, it follows from our analysis that the SU(2) and SU(3) flavor central charges are The result for k is somewhat unconventional, since it does not follow from the usual rule of thumb for relating flavor central charges to (in our normalization) twice the scaling dimension of some Coulomb branch operator in the theory; indeed, the T 3, 3 2 theory has no dimension- 5 2 operator. Also, using the results of [17], we can immediately conclude that since the I 3,3 theory does not have exotic N = 2 chiral primaries, neither does the T 3, 3 2 theory.
The rest of this paper is organized as follows: In Section 2 we describe the tools that let us identify the AD building blocks in the various S-dual frames. In Section 3 we give the details of the rank three example generalizing the S-duality of SU (2) with N f = 4, while in Section 4 we discuss the rank four generalization of Argyres-Seiberg duality. We briefly conclude in Section 5. In Appendix A, we sketch out the Hitchin system derivation of the various Seiberg-Witten curves we use in the main part of the paper. Appendix B exhibits the equivalence of the (III , F ) theory to I 3,3 plus a triplet of hypermultiplets.

The Strategy
The idea of using isolated sectors to construct conformal manifolds of N = 2 SCFTs by weakly gauging flavor symmetry subgroups is rather general. In order to make sense of the vast set of possible building blocks and the S-dual cusps that can emerge, we should find some simple, universal, and invariant characterizations of the physics on an N = 2 conformal manifold, M. For example, we can study: (i) The a and c conformal anomalies.
(ii) The set of flavor symmetries (in our conventions, these are symmetries commuting with the N = 2 superconformal algebra and not related by supersymmetry to higherspin symmetries), G = i G i , and the corresponding flavor central charges, k i .
(iii) The spectrum, S, of Coulomb branch operators.
These quantities do not change as we travel along M. 6 As we go between different cusps of the conformal manifold, the various quantities in (i), (ii), and (iii) are "partitioned" among the different emergent sectors. One interesting aspect of the Argyres-Seiberg-like dualities is that, unlike in the case of SU(2) gauge theory 6 The a and c central charges are invariant under exactly marginal deformations by the usual anomaly matching arguments (conformal symmetry is unbroken as we move along M). The flavor symmetries are also invariant (at the cusps, where weakly coupled gauge fields emerge, we also have emergent flavor symmetries; however, these symmetries are arbitrarily weakly gauged) since the exactly marginal primaries, O i , are uncharged under the flavor symmetries (this follows, e.g., from the analysis of the O i O † j OPE in [18]). As a result, by anomaly matching arguments, the k i are constant on M. The invariance of the N = 2 chiral spectrum follows from [19], which shows that the number of such operators cannot change as we traverse the conformal manifold, and from [20], which shows that the dimensions of these operators do not change either. Note that this reasoning applies also to the "exotic" higher-spin N = 2 chiral primaries considered in [17].
with N f = 4, these quantities are generally distributed differently at the different cusps.
For example, in the case of [4], at the SU(3) cusp we have (2.1) The first contributions in a and c come from the SU(3) gauge sector, while the remaining contributions come from the six flavors (this partition reflects the fact that there are seven corresponding N = 2 stress tensor multiplets and hence seven different N = 2 sectors). 7 Finally, the flavor symmetry comes from the hypermultiplets, and the gauge sector gives all the contributions to S (the elements of S are the scaling dimensions of the generators of the N = 2 chiral ring-in this case the Casimirs of SU (3)). On the other hand, at the SU(2) cusp, we find three distinct N = 2 sectors (with three independent N = 2 stress tensor multiplets) (2. 2) The first contributions in the above partitions are from the gauge sector, the second contributions come from the MN theory, which has rank one (its Coulomb branch chiral ring has a single generator of dimension three), and the third contributions come from the doublet of hypermultiplets. Now let us turn to theories with fractional-dimensional operators. In the case of the T 2, 3 2 , 3 2 theory mentioned in the introduction, we have In our conventions, a = 3 32 3TrR 3 − TrR and c = 3 32 3TrR 3 − 5 3 TrR , whereR = 1 charge, and I 3 is the Cartan of SU (2) R (a free N = 2 U (1) vector multiplet scalar primary has I 3 = 0 and where the first contributions are from the SU(2) gauge sector, the second and third contributions are from the two I 3,3 SCFTs, and the final contributions are from the hypermultiplets. The global symmetry group is U(1) 3 since we gauge a diagonal SU(2) ⊂ (2), where the SU(3) factors come from the I 3,3 sectors and the Sp(2) factor comes from the two hypermultiplets. This gauging is marginal since k SU (2) = 2k I 3,3 SU (2) + k 2⊕2 = 2 · 3 + 2 = 8. 8 On the other hand, in the case of the T 3,2, 3 conformal manifolds is simple. We first take the data in (2.3) and (2.4) and match it to data for the corresponding theories in the infinite class of AD SCFTs described in [11,15]. In particular, we will argue that T 2, where the theories listed on the RHS of (2.5) are defined in [11,15]. 9 Using our methods, it is clear that one can explore infinitely many generalizations of the conformal manifolds we will discuss in this text.
In ( , since the embedding index of SU (2) ⊂ SU (3) is unity. 9 Evidence for the first equality in (2.5) was presented at the level of the BPS spectra in [22] (note that the methods in [23] are also useful for finding the BPS spectrum in this case). We will describe how S-duality works in this theory. Note also that, as we explain in more detail below, the superscript "3 × [2, 2, 1, 1]" in the second equality refers to certain Young tableaux that define the III 3×[2,2,1,1] 6,6 SCFT.
ifications of the A k (2, 0) theory and are therefore referred to as being of class S). 10 These theories can be succinctly described in terms of Hitchin systems, 11 and the corresponding Seiberg-Witten (SW) curves come from the spectral covers of these Hitchin systems. Using the resulting curves, we can then explore the various cusps of the conformal manifolds and find new S-dual frames. As an alternate derivation, we will also show how to obtain the SW curves directly from certain UV-complete linear quiver theories.
Crucially, the Hitchin systems also give us direct access to the quantities (i)-(iii) without the need to fully analyze the SW curves. 12 As a result, we can immediately generate conjectures about different S-dual frames and perform some checks on our guesses before verifying them by analyzing the SW curve. Indeed, in the examples below, we will essentially be able to conjecture the S-dualities from studying the different ways in which the quantities in (i)-(iii) can be partitioned. To confirm these guesses, we then study various limits of the SW curve.
The reasons we can proceed in this way are as follows: • The Casimirs of the adjoint Higgs field in the Hitchin system description allow us to find the Coulomb branch spectrum, S = {∆ 1 , · · · , ∆ N }. By the results of [25], this data also fixes 13 • Using the recipes in [11,15,26] (see also the discussion in [27] and [28]), we can give a Lagrangian description of the three-dimensional mirror of the S 1 compactification of our theory, T 3dm . Although this description is not always "good" (in the sense that the IR superconformal R-symmetry can mix with accidental symmetries), we can unambiguously compute the dimension of the corresponding Coulomb branch, dimM 3dm C , and hence a − c via the relation (2.7) 10 In fact, there is some redundancy in this description, and, as we will see, both the T 2, theories can also be realized as the IR description of M 5 branes wrapping a sphere with one irregular and one regular puncture. 11 See [24] for a beautiful account of the relationship between theories of class S and Hitchin systems. 12 One apparent exception to this statement is the set of flavor anomalies. 13 A condition for using the results in [25] is that our theory has a freely generated Coulomb branch. All the theories we study in this paper satisfy this condition.
We expect (2.7) to hold in all theories that have a genuine Higgs branch (all the superconformal theories of class S discussed in [11,15] with non-integer dimension Coulomb branch operators come from genus zero compactifications of the (2, 0) theory and therefore have Higgs branches). 14 • The three-dimensional mirror often allows us to fix the precise flavor symmetry of the theory via the monopole analysis of [29] or, sometimes, from applying mirror symmetry again and reading off the flavor symmetry directly. Note that we can essentially always find the number of mass parameters of the theory in this way (we can also do this by studying Casimirs of the adjoint Higgs field), and we can read off the full flavor symmetry as long as the IR behavior is under sufficient control. 15 We should note that from the perspective of the compactification of the A k (2, 0) theory, it may be somewhat surprising that we have an exactly marginal parameter at all. Indeed, in the case of Gaiotto's theories [6], marginal parameters in the four-dimensional field theory are identified with complex structure deformations of the Riemann surfaces on which the parent six-dimensional theory is compactified. Clearly, the punctured spheres we consider do not have any complex structure deformations. Instead, it turns out that the exactly marginal deformations in our theories arise from certain dimensionless parameters of the co-dimension two defects used in defining the six-dimensional parent theory. 16 Finally, before we proceed, we should also note that in studying the behavior of our theories at different cusps in the marginal coupling space, we will often find it necessary to renormalize some of our parameters by multiplying them by functions that either vanish or diverge at a given cusp. The reason we do this is simple. We must demand that our parameterization of the Coulomb branch is non-singular so that the BPS masses are finite and non-trivial functions of the Coulomb branch coordinates. Presumably this criterion can be also understood as the necessity of renormalizing the operators whose vevs parameterize the Coulomb branch as we traverse the conformal manifold. In [20], this renormalization was interpreted as the statement that operators can pick up non-trivial phases or mix in interesting ways as we travel along closed loops in the marginal coupling space (i.e., 14 The first equality in (2.7) is a natural generalization, to strongly coupled theories with a Higgs branch, of the weakly coupled result that a − c = − 1 24 (n H − n V ), where n H is the number of hypermultiplets and n V is the number of vector multiplets. The second equality in (2.7) follows from mirror symmetry (in particular, the exchange of Higgs and Coulomb branches under this duality) and the fact that the Higgs branch does not receive quantum corrections as we go to long distances compared to the S 1 radius. 15 As we will discuss, in the case of the T 3, 3 2 theory, this analysis is much more subtle. 16 We thank G. Moore for a discussion of this point. operators transform as sections of certain bundles over the conformal manifold). We will find some evidence for this picture, since our normalizations introduce monodromies in the marginal parameter space.

A minimal generalization of Seiberg and Witten's S-duality
In this section, we will study the T 2, 3 2 , 3 2 theory introduced above. In the first subsection, we find the invariant quantities (i)-(iii) of the I 4,4 theory [11] and show that they match . 17 We also argue that, subject to some assumptions, the only potential cusps of the T 2, 3 2 , 3 2 theory involve an SU(2) gauge sector coupled to two I 3,3 sectors and a doublet of hypermultiplets (in other words, we argue that there is no emergent rank-two sector with Coulomb branch spectrum 3 2 , 3 2 ). We then find further evidence for this picture by analyzing the SW curve of the I 4,4 theory. Moreover, we find an S-duality action on the parameters of the theory that is reminiscent of the Spin(8) triality of the SU(2) gauge theory with N f = 4. As a result, this discussion represents a simple generalization of Seiberg and Witten's analysis [2]. In the final subsection, we show how T 2, 3 2 , 3 2 can be derived from a UV-complete linear quiver.
Before proceeding to the calculations, let us show that our theory is the simplest (i.e., lowest-rank) example of an S-duality with non-integer dimension Coulomb branch operators under certain reasonable assumptions: (a) the only rank-zero theories are collections of free hypermultiplets and (b) the only rank-one theories with non-integer scaling dimension primaries are the I 2,3 , I 2,4 , and I 3,3 theories. 18 Under these assumptions, it follows from the fact that I 2,3 has no flavor symmetry and the fact that k that the lowest rank theory we can imagine constructing-let us call it T rk2 -involves an SU(2) gauge theory coupled to one copy of the I 3,3 theory (via a gauging of the SU(2) ⊂ SU(3) flavor symmetry) and five hypermultiplets (via a gauging of SU(2) ⊂ Sp (5)) so that k SU (2) = k I 3,3 SU (2) + 5k 2 = 8. However, T rk2 is inconsistent, because the gauged SU(2) suffers from Witten's SU(2) anomaly [16].
To understand this last statement, note that the I 3,3 theory cannot have such an anomaly. Indeed, as we described above, the I 3,3 theory can be obtained as the IR endpoint of an RG flow from the asymptotically free limit of SU(2) SQCD with N f = 3 [10] (the 17 Note that we can also realize our theory in terms of the (I 3,3 , S) Hitchin system. This system has lower rank than the I 4,4 Hitchin system, but it also has an additional regular singularity. 18 It might be possible to prove assumption (a) by generalizing [25] and using the N = 2 version of the arguments presented in [30].
short-distance limit clearly has vanishing Witten anomaly since we can give SU(2) ⊂ SO(6)preserving masses to the squarks). This flow preserves an SU(3) ⊂ SO(6) flavor symmetry of the gauge theory, and, moreover, this symmetry is identified with the flavor symmetry of the I 3,3 theory in the deep IR. Since the RG flow does not leave any additional massless matter besides the I 3,3 theory at long distances, it must be the case that the Witten anomaly for the I 3,3 theory matches the (vanishing) Witten anomaly for the UV theory.
Therefore, T rk2 has the same Witten anomaly as the five half-hypermultiplet doublets. This anomaly is clearly non-vanishing, and so the T rk2 theory is inconsistent. On the other hand, theory has an even number of hypermultiplet doublets, it is a consistent theory. theory match the corresponding quantities for the I 4,4 theory (evidence for the equivalence of the BPS spectra of these theories was given in [22]). To that end, we first note that, as desired, the I 4,4 SCFT has the following N = 2 chiral spectrum [11,15]

Evidence that T 2,
As a result, using (2.6), we find [15] 2a Next, we can write down a good UV description of the three dimensional mirror theory. 19 According to [15], this theory is described by a quiver involving four U(1) nodes with a bifundamental between each node and the overall U(1) decoupled. 20 Deleting a redundant 19 By this we mean a theory in which the IR superconformal R symmetry is visible in the UV. More precisely, we have in mind a theory in which the IR superconformal R-symmetry (or R-symmetries if there are multiple sectors) descends from a symmetry (or symmetries if there are multiple sectors) of the RG flow. 20 In the prescription of [11,15], this statement follows from the fact that the irregular singularity of the corresponding Hitchin system has boundary conditions specified by three 4×4 matrices whose eigenvalues are generically different (and whose degeneracies are therefore in one-to-one correspondence with three Young tableaux of the form [1, 1, 1, 1]). Note also that we have written the remaining U (1) factors in certain linear combinations that are convenient for applying the mirror symmetry algorithm in [31].
U(1) factor, we find the theory As a result, we conclude that dimM 3dm C = 3 and therefore [15] Finally, we can check that the flavor symmetries match. One way to do this is to take the mirror transform of the above theory (using the algorithm in [31]) We see that this theory has a U(1) 3 flavor symmetry, and so Alternatively, we can find the same result directly in the mirror theory by noting that there are three U(1) Coulomb branch symmetries that shift the three independent dual photons by constants. Any additional symmetries would correspond to currents that sit in monopole multiplets of dimension one [29]. However, the monopole multiplets have dimension where − → a = (a 1 , a 2 , a 3 ) ∈ Z 3 is a magnetic U(1) 3 charge vector. Note also that (3.6) is consistent with the claim that we have a good description of the IR theory since there are no free (or unitarity bound violating) monopole operators in our microscopic description.
These results strongly indicate that T 2, 3 Let us now ask about possible S-dual descriptions. One possibility is that we have various dual descriptions involving an SU(2) gauge group coupled to two I 3,3 sectors and a doublet of hypermultiplets. A more exotic possibility would involve a dual description with an SU(2) gauge group coupled to a rank-two theory with Coulomb branch spectrum 3 2 , 3 2 . While we cannot prove that this second possibility does not occur without the SW analysis of the next section, we can already see it is unlikely. Indeed, it is reasonable to assume that any of the sectors that emerge at the cusps of the conformal manifold are also of class S and can be realized as compactifications of the (2, 0) A k theory (since the parent I 4,4 theory is in this class). However, there are no rank-two theories with spectrum 3 2 , 3 2 that can be built from the recipes in [11,15] (besides two decoupled copies of the I 3,3 theory). In the next subsection, we will demonstrate that the first option described in this paragraph is indeed realized.

Analysis of the SW curve
We begin by writing down the Seiberg-Witten curve for the I 4,4 theory 0 = x 4 + qx 2 z 2 + z 4 + c 30 x 3 + c 03 z 3 + c 20 x 2 + c 11 xz + c 02 z 2 + c 10 x + c 01 z + c 00 . (3.7) The Seiberg-Witten 1-form is given by λ = xdz. Since the mass of a BPS state is given by λ, the 1-form λ has scaling dimension one. This observation fixes the scaling dimensions of x, z, c ij and q as In order to make contact with the T 2, 3 2 , 3 2 theory discussed above, we should first show that an SU(2) gauge symmetry emerges. To that end, let us turn off all the c ij except for c 00 . The SW curve is given by In terms of y = −i(c 00 ) is expressed as with the 1-form now λ = u dx/y. The equation (3.10) is precisely the curve for SU (2) with [2,32], where u is the Coulomb branch parameter of dimension 2. The parameter f is related to the exactly marginal gauge coupling τ = θ π + 8πi g 2 . 21 The equivalence of (3.9) and (3.10) suggests that the I 4,4 curve contains a sector described by a conformal SU (2) vector multiplet.
The above SU(2) gauge theory has cusps at q = ∞ and q = ±2 where the curve (3.10) degenerates and different S-dual descriptions of the theory become weakly coupled. We can go between the cusps via the transformations T : τ → τ + 1 and S : τ → −1/τ [2]. In terms of q, these are expressed as T : q → 12−2q 2+q , S : q → −q. It turns out that S and T can be extended to the full I 4,4 curve (3.7). To that end, first consider The equation (3.7) is invariant under this transformation after we perform a one-form- x. Next, consider the T transformation. We first shift x → x + c 30 /(2q − 4) and z → z + c 03 /(2q − 4) so that the curve (3.7) is 30 x +c 03 z) +c 20 x 2 +c 11 xz +c 02 z 2 +c 10 x +c 01 z +c 00 .

(3.12)
This shift keeps λ invariant up to an exact term. While the relation between c kℓ andc kℓ is generically complicated, it reduces toc kℓ = c kℓ when q → ∞. Now consider the following transformation:T where g is a linear map defined by g(  21 Without loss of generality, we can take keeps the 1-form invariant up to an exact term. Hence, the I 4,4 curve is invariant under the transformations generated byS andT . As we will show in the remaining parts of this subsection, the cusps of the conformal SU(2) gauge theory persist in the presence of the fractional dimensional operators, and, at each of the cusps q = ∞, ±2, a weakly coupled SU(2) gauge group couples two I 3,3 theories and a doublet of hypermultiplets. We go between the cusps via theS andT transformations (and we use this freedom to study the cusp at q = ∞ and then study the q = ±2 cusps via these symmetries).
Moreover, we see in (3.11) and (3.13) that these transformations act non-trivially on the various parameters and vevs. Note that theS andT transformations take a particularly simple form when acting on the independent physical mass parameters (i.e., the independent residues of the one-form), m i (i = 1, 2, 3), of the theorỹ S : where the m i are the independent eigenvalues of the simple poles in the Hitchin field at z = ∞. 22 As a result, we see that the duality group acts on the residues via S 3 .
This situation is somewhat reminiscent of the action of the SL(2, Z) duality group of the SU(2) N f = 4 gauge theory on the mass parameters via triality [2] (although here we only have a U(1) 3 flavor symmetry instead of SO(8), and we have a non-trivial action of the duality group on the various non-integer dimension parameters of the theory). Indeed, it would be interesting to make this analogy more precise. 23

Cusp at q = ∞
Consider the I 4,4 curve (3.7) near q = ∞. Since one of the coefficients is divergent in this limit, it is not clear whether our parameterization of the curve describes the Coulomb branch in a non-singular fashion. As discussed in the introduction, we should normalize the c ij so that the masses of BPS states are non-trivial functions of these quantities.
Let us first consider the Coulomb branch parameter c 10 of dimension 3 2 . When all the other deformations of the conformal point are turned off, the curve is given by 0 = x 4 + qx 2 z 2 + z 4 + c 10 x . To evaluate the periods of this curve, let us change variables as (x, z) → (x, w) with w = z/x. Neglecting a trivial branch (x = 0), we find (3.14) The 1-form is λ = 1 2 x 2 dw up to exact terms. The curve (3.14) is a triple covering of the w-plane with branch points at the roots of 1 + qw 2 + w 4 and at w = ∞. Let us define the roots w ± = ± 1 2 (−q + q 2 − 4). In the limit q → ∞, the 1-cycle with the largest absolute value of the period of the one-form is the one around w = ∞ and w = w + (or w − ). Its period behaves in the limit as with a q-independent constant κ. Since so that the period with the largest absolute value remains finite and non-vanishing in the limit q → ∞. We renormalize all the c ij except for c 00 in the same way (i.e., we demand that the largest period created by each c ij = 0 remains finite and non-vanishing in the limit q → ∞).
The only deformation we need to study more carefully is c 00 . When only c 00 is turned on, the curve is the genus one curve (3.9). With an appropriate choice of two independent 1-cycles A and B, their periods behave in the limit q → ∞ as Since the ratio of the two periods is divergent, the curve is pinched in the limit q → ∞.
This is the signature of a light W-boson and an infinitely massive monopole. A natural normalization in this case is c 00 → qc 00 so that 1 2πi A λ ∼ √ c 00 and 1 2πi B λ ∼ 1 πi √ c 00 log q. 24 24 We could also normalize c 00 as c 00 → q(log q) 2 so that 1 2πi A λ ∼ √ c 00 / log q and 1 2πi B λ ∼ 1 πi √ c 00 . Here we use the traditional normalization in which the period of the pinched cycle is finite and non-vanishing.
Note that for c ij = c 00 there is a unique renormalization up to q-independent rescaling. The reason for this is that no 1-cycle created by c ij = c 00 is pinched in the limit q → ∞.
As a result, the curve near q ∼ ∞ is written as 0 = x 4 + qx 2 z 2 + z 4 + q Let us now study the behavior of this curve in the limit q → ∞. It turns out that the curve splits into three sectors.
• In the region |z/x| ∼ 1/ √ q, the curve is well-described by the new set of variables In the limit q → ∞, the curve reduces to By shiftingz →z − c 11 /(2x), the curve is written as . • In the region |z/x| ∼ √ q, the curve is well-described by the new variablesz = q − 1 4 z andx = q 1 4 x. The 1-form is now λ =xdz. After shiftingx →x − c 11 /(2z), the curve in the limit q → ∞ is written as . • In the region |z/x| ∼ 1, the curve in the limit q → ∞ is given by 0 = x 2 z 2 + c 11 xz + c 00 . This curve describes the SU(2) superconformal QCD in the weak coupling limit with c 11 a mass parameter for a fundamental hypermultiplet. We can eliminate this term by shifting x → x−c 11 /(2z). The curve after the shift is 0 = x 2 z 2 +(c 00 −c 2 11 /4), which describes the pinched W-boson cycle of the weak-coupling SU(2) curve. The mass of the W-boson is proportional to c 00 − c 2 11 /4. 26 The monopole cycle is overlapping between |z/x| ∼ √ q ∼ ∞ and |z/x| ∼ 1/ √ q ∼ 0; its period is divergent.
To recapitulate: the first two sectors describe two I 3,3 theories while the third sector describes an SU(2) vector multiplet coupled to a fundamental hypermultiplet. The W-boson mass implies that the SU(2) sector is gauging the SU(2) flavor subgroups of the I 3,3 sectors.
Hence, the I 4,4 curve (3.7) near q ∼ ∞ describes the Coulomb branch of the weak coupling limit of the T 2, 3 2 , 3 2 theory defined in the introduction.

Cusps at q = ±2
Let us briefly discuss the other cusps at q = ±2. Since they are mapped to q = ∞ by the where ǫ = q ∓ 2. It is straightforward to show that, in the limit q → ±2, the curve splits into two I 3,3 curves connected by an SU(2) curve. A difference from the previous cusp is that the parameters c ij are now mixed among the three sectors. In terms of the linear map g defined below (3.13), one of the I 3,3 curves is characterized by g(c 30 ), g(c 20 ), g(c 10 ) and g(c 00 ) − g(c 11 ) 2 /4 while the other is governed by g(c 03 ), g(c 02 ), g(c 01 ) and g(c 00 ) − g(c 11 ) 2 /4.
The SU(2) vector multiplet and a fundamental hypermultiplet are characterized by g(c 00 ) and g(c 11 ).

The linear quiver
In this section, we would like to demonstrate how the T 2, 3 2 , 3 2 theory can be engineered from a UV-complete linear quiver. To that end, consider the theory in Figure 1. Following [33], 26 The shift of the W-boson mass squared by a hypermultiplet mass squared is a common phenomenon.
See for example [2]. we can write the corresponding SW curve as follows: (3.24) The SW differential has the form λ = v t dt. In the above formula u i , u ′ 2 andũ 2 are the Coulomb branch coordinates of the theory, m 1 and m 3 encode the mass parameters for the two SU(2) doublets, m 2 is related to the mass of the fundamental hypermultiplet of SU(3), µ 1 and µ 2 are associated with the mass parameters of the bifundamental hypermultiplets. 27 q 1 , q 2 and Λ are, respectively, the marginal couplings of the SU(2) gauge groups and the dynamical scale of the SU(3) group. If we send one of the q i couplings to zero, the curve reduces to that of the linear quiver with the SU(2) group replaced by two hypers in the fundamental of SU(3), which is indeed the expected degeneration in the "ungauging" limit.
Setting q 1 = q 2 = 0 the curve reduces to that of SU(3) SQCD with N f = 5. If we send to zero Λ, thus ungauging SU(3), the quiver breaks into two pieces, each describing a scale invariant SU(2) theory. Depending on how we write the curve, in the degeneration limit we are left with the curve for one of these two sectors. For example, in the above formula, only the terms proportional to a positive power of t remain. We can change description and keep the other sector simply with the redefinition t → t/Λ. With a constant shift of v, which does not affect the form of the SW differential, and a suitable redefinition of the parameters, we can bring the curve to the following form, which is more convenient for our later discussion: (3.25) 27 Notice that the above curve is schematic: the parameters m i are not the physical masses (i.e., the residues of the SW differential) but are instead combinations of the mass parameters and the dynamical scale of the theory.
We are interested in the origin of the moduli space of this theory (i.e., the point in the moduli space we get by setting all the parameters in (3.25) to zero except q i ) where the curve reduces to The resulting curve is singular and, as usually happens in N = 2 theories, the degeneration of the curve signals the presence of a superconformal fixed point, whose SW curve can be extracted starting from (3.25) by taking a suitable scaling limit. First of all we define new variables (3.27) In terms of these variables, (3.25) becomes (3.28) The SW differential is λ = (v/z)dz. Then, sending Λ to infinity, we get the curve To obtain this formula we divided the whole curve by a constant and rescaled z to set to one the coefficient of z 2 v and to −1 − g and g the coefficients of the terms v 2 and v 3 /z 2 respectively. This manipulation is also accompanied by the proper redefinition of the parameters. Notice that this transformation does not affect the SW differential.
Since we are discussing a superconformal theory, all the parameters appearing in (3.29) should have a definite scaling dimension. This can be read from (3.27) using the UV dimension of the parameters appearing in (3.25). Notice that the above curve is homogeneous, in the sense that assigning dimension one to v (which is consistent with the constraint on the SW differential) and 1/2 to z we find that all the terms in (3.29) have dimension two. This is precisely the property we expect for the curve describing an SCFT.

A minimal generalization of Argyres and Seiberg's S-duality
In this section, we turn our attention to the rank four T 3,2, gauge fields coupled to an I 3,3 theory and an exotic rank two theory we call T 3, 3 2 (in the language of [15], this theory can be written as III can be embedded in a UV-complete linear quiver.
Before proceeding to the calculations, let us show that-under the same assumptions we used at the beginning of Section 3 to demonstrate the minimality of our first examplethere are no rank three theories that exhibit Argyres-Seiberg-like duality.
We can prove this statement as follows. Let us consider the possible rank three theories.
They break up into two cases: (a) a rank one gauge theory coupled to either a rank two sector or to two rank one sectors, and (b) a rank two gauge theory coupled to a rank one sector. Let us consider (a) first. In this case, the gauge theory must be SU(2). Let us suppose that it is coupled to two rank one sectors. Vanishing of the one-loop beta function implies that the only possibility is that SU(2) is coupled to two copies of the I 3,3 theory with an additional doublet. This is the T 2, 3 2 , 3 2 = I 4,4 theory we studied in Section 3 and showed did not exhibit Argyres-Seiberg-like behavior. Next let us suppose that the SU (2) gauge theory is coupled to a rank two sector. In order to have an Argyres-Seiberg-like duality, such a theory must be dual to a rank two gauge theory coupled to a rank one sector with a non-integer dimension Coulomb branch operator as in (b). The possible rank two gauge groups are: SU(2) × SU(2), SU(3), Sp (2), and G 2 . We can rule out Sp(2) and theory we are about to study). We will now compute the quantities (i)-(iii) described in Section 2 for the III 3×[2,2,1,1] 6,6 theory and show that they match the quantities given in (2.4) for the T 3,2, 3 2 , 3 2 SCFT. We will then motivate the existence of an SU(2) gauge theory cusp and demonstrate how these quantities are partitioned at such a point on the conformal manifold.

Preliminary evidence that T 3,2,
We first note that the Hitchin system description of III 3×[2,2,1,1] 6,6 is specified by three 6 × 6 matrices whose eigenvalue degeneracies are encoded in three copies of the Young tableaux [2, 2, 1, 1] (i.e., each matrix has two sets of two-fold degenerate eigenvalues, see Appendix A.2). 28 Next, from the three Young tableaux [2, 2, 1, 1], we use the rules described in [11] to write down the three-dimensional mirror where the subscripts in the U (2) To read off the symmetries of the theory, we can apply mirror symmetry again and find This theory has a U(3) flavor symmetry, and so we conclude that Alternatively, we can work directly in the mirror theory. Clearly, there is a Coulomb branch symmetry that shifts the three dual photons by independent constants. To see the symmetry enhancement to U(3) in the IR, we should study the monopole operators with dimension one [29]. The general formula for the dimensions of the monopole operators is where − → a = (a 1,1 , a A,1 , a A,2 , a B,1 , a B theory. From the discussion in [4], we then expect that there should be a degeneration limit where an SU (2) gauge group emerges. As we will see, the presence of fractional dimensional operators does not spoil this picture, although the emergent sectors that appear are quite different than in [4].
What can this cusp look like? We again expect a decomposition into sectors of class S (of type A k ). One possibility is an SU(2) gauge group coupled to a rank one theory with a dimension three Coulomb branch operator, a rank two theory with spectrum 3 2 , 3 2 , and some number of fundamentals. However, as we argued in the previous section, such a rank two sector is unlikely to exist in the A k theories of class S, and, since our parent theory is of this type, such an option should not be realized. Another possibility is an SU(2) gauge group coupled to a rank three theory with spectrum 3, 3 2 , 3 2 . However, just as in the case of the rank two theory with spectrum 3 2 , 3 2 , such a theory cannot be constructed from the recipes in [11,15]. As a result, the last possibility is an SU(2) gauge group coupled to a rank two theory with Coulomb branch spectrum, 3, 3 2 , and a copy of the I 3,3 theory. We denote this rank two theory, T 3,  for the total theory is supplied by the I 3,3 sector. 29 We will now argue that the .
In other words, we claim that the irregular singularity of the Hitchin system describing this theory has three 6 × 6 matrices with the first two (i.e., those multiplying the third and second order poles at z = ∞ in the Higgs field) having three doubly degenerate eigenvalues theory is quite subtle. From the Hitchin system, we can deduce the following UV description of the three dimensional mirror where the subscripts in the U(2) B,C representations denote charges under the corresponding U(1) subgroups. 29 One nice check of our discussion is the following. If our conjecture is correct, then the fundamental hypermultiplets at the SU (3) cusp are monopoles in the SU (2) gauge theory description (our argument is similar in spirit to the argument in [4]). With this understanding, let us consider k U(1) . On the SU (2) gauge theory side of the duality, it is natural to take k U(1) = k Note that all of the nodes in this description are "good" in the sense of [29]. In particular, the U(1) 1 node has N f − 2N c = 0 and so too do the U(2) B,C nodes. The U(2) A node is also good since it has N f − 2N c = 1. Therefore, is is natural to guess that this theory should have no monopole operators of dimension ∆ ≤ 1 2 and that the flavor symmetry should be SU(3) × SU(2). 30 We also find dimM 3dm C = 6 and therefore a − c = − 1 4 . 31 This result is certainly compatible with what we expect from (4.8) and what we obtain in the next section. However, there is a wrinkle (note that we do not expect the discussion that follows to affect dimM 3dm C or therefore a − c). Indeed, we can compute the dimensions of the monopole operators [29] where − → a = (a 1,1 , a A,1 , a B,1 , a B,2 , a C,1 , a C,2 ) ∈ Z 6 is a U(2) 3 magnetic charge vector (we have used the freedom of shifting the flux by a charge corresponding to the overall U(1) in order to set the magnetic flux from the U(1) 1 node to zero). It is easy to check that, up to Z 3 2 permutations, the dimension one monopole operators are M ± 1 = (0, 0, ±1, 0, 0, 0), M ± 2 = (0, 0, 0, 0, ±1, 0), M ± 3 = ±(0, 0, 1, 0, 1, 0), M ± 4 = ±(1, 1, 1, 1, 1, 1), M ± 5 = ±(2, 0, 2, 0, 2, 0), M ± 6 = ±(1, −1, 1, −1, 1, −1). However, there is also a dimension half monopole operator, M ± = ±(1, 0, 1, 0, 1, 0). The heuristic reason for this result is that the extra U(2) A we have added to connect the two linear quivers that produce the SU(3) and SU(2) symmetries gives large quantum corrections to the theory. Therefore, even though the quiver is "good" by the usual tests, it actually has an apparent dimension half free monopole operator! It is not immediately clear to us how to proceed (in particular, we are not sure how to deduce the flavor symmetry group from the three dimensional perspective). 32 30 If we regard the theory as an N = 1 theory, then the flavor symmetry would, intriguingly, be SU (3) × SU (2) × U (1). 31 Note that this value of a − c rules out another potential candidate for describing T 3, theory. 32 One naive way of analyzing the theory is to turn on a vev Φ A = Φ B = Φ C = diag(v 1 , 0) and move out onto the Coulomb branch. The remaining massless theory splits into two sectors: one is identical to the three-dimensional reduction of I 3,3 (with the (Q AB ) 1 1 , (Q BC ) 1 1 , and (Q CA ) 1 1 hypermultiplets as matter

Analysis of the SW curve
The SW curve for the III 3×[2,2,1,1] 6,6 theory is given by The theory has an exactly marginal coupling q and three mass deformation parameters m i .
The b i are relevant couplings associated with two Coulomb branch operators of dimension 3 2 , whose vevs are identified with c i . There are also Coulomb branch operators u, v of integer dimensions.
In order to make contact with the T 3,2, In terms ofũ = u/[2(q + 1 q )],ṽ = v/[2 √ 2(q + 1 q )], f = 4/(q + 1 q ) 2 ,x = xz/ √ 2 and y = x 3 + √ 2z 2x2 /(q + 1 q ) +ũx +ṽ, the curve is expressed as (4.14) content), and the other looks like the I 3,3 quiver but with one node having an extra flavor attached to it (with the (Q AB ) 2 2 , (Q BC ) 2 2 , (Q CA ) 2 2 , and (Q A1 ) 2 hypermultiplets as matter content). This second quiver again leads to an apparently free monopole operator which we can again attempt to decouple by moving out onto the Coulomb branch of this reduced theory. Proceeding in this way, we ultimately find two decoupled U (1)'s and two copies of the S 1 reduction of the I 3,3 theory (the (Q A1 ) 2 hypermultiplet becomes massive and is integrated out). While we find the correct a − c in this case, we also find an Sp(2) × SU (3) 2 flavor symmetry. This result is clearly incompatible with the four-dimensional discussion below, and so we predict that the three-dimensional behavior of the T 3, 3 2 theory is more complicated. We might try to rescue this interpretation by gauging an SO(3) ⊂ Sp(2) × SU (3) × SU (3) diagonal subgroup, where the first SU (3) is from the T 3, 3 2 theory and the second is from the I 3,3 sector. This would leave an SO(2) × SU (3) flavor symmetry (the commutant of SO(3) in SU (3) has dimension zero). However, as we will see below, we have a mass parameter in the I 3,3 sector and so this interpretation is not correct.

The 1-form is written as
up to exact terms, where P =x 3 +ũx +ṽ.
These are the one-form and curve for the SU(3) gauge theory with N f = 6 [32]. The parameter f is identified with a modular function of the exactly marginal gauge coupling τ = θ π + 8πi g 2 . 33 The emergence of (4.14) suggests that the III 3×[2,2,1,1] 6,6 curve contains a sector described by a conformal SU(3) vector multiplet.
The curve (4.14) is known to be invariant under Γ(2) ⊂ SL(2, Z), which is generated by T 2 : τ → τ + 2 and S : τ → −1/τ [32]. In terms of q, these correspond to as long as we also send x → ix, z → −iz (which keeps the 1-form invariant). The two transformations q → 1/q and q → −q will be important later in this subsection.
The SU(3) superconformal QCD described by (4.14) has a weak-coupling cusp at τ = i∞ and a strong coupling cusp τ = 1. In terms of q, these correspond to q = 0, ∞ and q = ±1, respectively. Below we study the behavior of the full III We first renormalize all the deformations of the curve so that the largest period created by each deformation is finite and non-vanishing in the limit q → 0. The renormalized curve is written as which turns out to split into three sectors as follows. 33 Once again, we take • In the region |z/x| ∼ q, we definez = q − 1 2 z andx = q 1 2 x so that |z/x| ∼ 1. In terms ofx andz, the curve in the limit q → 0 is written as 17) and the 1-form is given by λ =xdz up to exact terms. Let us shiftz →z − 1 3 (x + b 1 + m 3 /x). This curve can be identified with that of the (III 27 . This means that the sector near |z/x| ∼ q describes the Coulomb branch of the (III • In the region |z/x| ∼ 1/q, we definez = q 1 2 z andx = q − 1 2 x. The curve in the limit q → 0 is now written as • In the region |z/x| ∼ 1, the curve is On the other hand, the above discussion shows that the (III

Cusps at q = ±1
We now turn to the points q = ±1 in the marginal coupling space. Since these two points are related by the symmetry transformation q → −q, we will, without loss of generality, focus on q = 1. Note that there is no symmetry transformation which maps q = ±1 to q = 0, ∞. Therefore, we expect to have a different weak coupling description in this case.
To understand the above statement, we first renormalize the deformations so that the largest period created by each deformation is finite and non-vanishing in the limit q → 1.
The correct renormalization turns out to be where ǫ = 1 − q. Therefore the renormalized curve is written as x + v , (4.21) x−z . In the limit q → 1, or equivalently ǫ → 0, the curve splits into three sectors, depending on |ζ|.
• In the region |ζ| ∼ 1, the curve in the limit q → 1 is given by theory.
• Finally, let us look at the region |ζ| ∼ ǫ 1 2 , which is between the above two regions.
. It follows that finitex andz correspond to |ζ| ∼ ǫ 1 2 in the limit q → 1. The curve in terms of these variables reduces to 0 =x 2 (x 2z2 +û) , (4.24) in the limit q → 1. Apart from the trivial branchx 2 = 0, this is the weak coupling limit of the SU(2) curve. The period of the pinched cycle is proportional to √û , which is identified with the central charge of the SU(2) W-boson.
Hence, in the limit q → 1, the III theory and an I 3,3 theory; see Figure 3.

SCFT.
We see that these results immediately imply that k

The linear quiver
In this subsection, we will show that the T 3,2, 3 2 , 3 2 theory can be embedded in a UV-complete linear quiver theory. To understand this claim, let us consider the theory in Figure 4. The SW curve can be written as follows [33]: The notation is identical to that of Section 3.3. 34 The SW differential is again λ = (v/t)dt.
After the shift v → v − m 3 and a suitable redefinition of the parameters we find the curve By setting all the parameters to zero (apart from q 1 and q 2 ) in (4.26) the curve becomes singular. Our next task is to extract the SW curve describing the effective low-energy theory at this singular point. As in the previous example, we extract the curve starting from (4.26) and taking a scaling limit. We change variables as follows:  Rewriting (4.26) in terms of the new variables and taking the limit Λ → ∞ we find the curve (4.28) In the above formula we have divided everything by a constant and rescaled z to set to one the coefficient of the terms z 2 v 2 and v 4 /z 2 . This transformation does not change the SW differential λ = (v/z)dz. We are then left with a single marginal parameter that we call q.
We claim that the above curve describes the theory III 3×[2,2,1,1] 6,6 . Indeed, setting x = v/z we bring the SW differential to the canonical form λ = xdz. The resulting curve is precisely (4.11) with the identification u 2 = u and u 3 = v. The only difference is a factor of two in the definition of m 1 and m 2 .
34 As in Section 3.3, the above curve is schematic, and the parameters m i , µ i do not correspond to the physical mass parameters of the theory.

Conclusions
In this paper, we found minimal generalizations of Seiberg and Witten's S-duality in SU (2) gauge theory with four fundamental flavors and Argyres and Seiberg's S-duality in SU (3) gauge theory with six fundamental flavors to theories with non-integer dimensional Coulomb branch operators. Along the way, we found an S-duality action on the parameters of the

Appendix A. Hitchin system perspective
In this appendix we briefly review how one obtains the SW curves of various Argyres-Douglas type theories from the corresponding Hitchin system [11,24]. A class of Argyres-Douglas theories are obtained by compactifying the 6d (2,0) theory on a punctured sphere.
The Coulomb branch of such a 4d theory (or more precisely its reduction to 3d) is described by At the punctures on P 1 , we impose BPS boundary conditions. Since we can trivialize the gauge bundle around the puncture, the boundary condition is given by specifying the singular behavior of Φ near the puncture. For a trivialized gauge field,∂ A Φ = 0 implies Φ is meromorphic. The singularity at a puncture is called "regular" or "irregular" if Φ has a simple or higher-order pole there, respectively. It was shown in [11] that the resulting 4d theory is an Argyres-Douglas type theory only if there is a single irregular singularity on P 1 with at most one additional regular singularity. Below, we review the SW curves of several Argyres-Douglas theories of this type.
A.1. I n,n theory The I n,n theory is obtained from the A n−1 Hitchin system on P 1 with an irregular singularity. 35 Suppose that the singularity is at z = ∞. The boundary condition of the Higgs 35 Here we use the notation of [15]. The same theory is called (A n−1 , A n−1 ) in the language of [12].
field Φ(z) is given by where M i are traceless n-by-n matrices. By using gauge transformations, M i can be simultaneously diagonalized. For the I n,n theory, the matrices M i can be any diagonal traceless matrices. The lower-order terms of O(z −2 ) are not fixed by the boundary condition at z = ∞ but are subject to the constraint that Φ(z) is not singular at z = ∞. The SW curve of the I n,n theory is then given by the spectral curve det(xdz − Φ(z)) = 0. The second non-trivial example is the I 4,4 theory. The boundary condition (A.3) is now given by 4 × 4 matrices M i . Up to coordinate changes, the spectral curve is written as 0 = x 4 + qx 2 z 2 + z 4 + c 30 x 3 + c 03 z 3 + c 20 x 2 + c 11 xz + c 02 z 2 + c 10 x + c 01 z + c 00 . (A.5) Here the dimensions of the parameters are given in (4.12). In particular, this theory has a single exactly marginal coupling, q. theory, which is obtained from the A 5 Hitchin system on P 1 with an irregular singularity. Suppose that the singularity is at z = ∞. The boundary condition for the Higgs field is characterized by (A.3) with three six-by-six matrices M i .
In the case of a type III theory, we specify the number of coincident eigenvalues of M i by Young tableaux [11]. Since our Young tableaux are now [2, 2, 1, 1], we demand that M i are of the form M 1 = diag(ã 1 ,ã 1 ,ã 2 ,ã 2 ,ã 3 ,ã 4 ) , up to gauge equivalence. We implicitly assume the tracelessness of these matrices. The resulting spectral curve is written as 0 = x 2 z 2 (z + x) 2 + x 2 z 2 (z + x)b + xz m 1 z(x + z) + m 2 x(z + x) + b 2 4 xz + xz (c + bm 1 2 )z + (c +  In the above table, the subscripts in the representations signify the charges of the fields under the corresponding U(1) subgroups.
Note that the existence of the M ± 3 dimension half monopole operator follows immediately from the fact that the U(2) C node is "ugly" in the classification of [29] (it has N f − 2N c = −1).
To find the remainder of the theory in the IR, we can follow [29] and move along the Coulomb branch of the U(2) C node by taking Φ C = diag(v 1 , 0) (where the Φ C is the adjoint chiral multiplet of U(2) C ) and examine the remaining massless theory. 36 Turning on this vev in the N = 4 superpotential leaves (besides a decoupled U(1) parameterizing the moduli space of the free M ± 3 theory [29]) a massless theory with U(2) C → U(1) C and the following matter multiplets: (Q AB ) a , (Q BC ) 2 , (Q CA ) a 2 , (Q A1 ) a (along with the corresponding hypermultiplet partners; here a is an SU(2) A index and 2 is a U(2) C index).