The moduli space of vacua of N=2 class S theories

We develop a systematic method to describe the moduli space of vacua of four dimensional $\mathcal{N}=2$ class ${\cal S}$ theories including Coulomb branch, Higgs branch and mixed branches. In particular, we determine the Higgs and mixed branch roots, and the dimensions of the Coulomb and Higgs components of mixed branches. They are derived by using generalized Hitchin's equations obtained from twisted compactification of 5d maximal Super-Yang-Mills, with local degrees of freedom at punctures given by (nilpotent) orbits. The crucial thing is the holomorphic factorization of the Seiberg-Witten curve and reduction of singularity at punctures. We illustrate our method by many examples including ${\mathcal N}=2$ SQCD, $T_N$ theory and Argyres-Douglas theories.

1 Introduction The structure of moduli space of vacua plays a crucial role in studying the dynamics of supersymmetric field theory. Sometimes one can find exact solutions by analyzing various properties of the moduli space, such as the singularity structure, asymptotical behavior, etc; and the typical example is the Seiberg-Witten solutions of N = 2 theory [1,2]. The full moduli space of four dimensional N = 2 theory has the general structure where α labels the components of branches, and C α are parametrized by Coulomb moduli fields and H α are parametrized by Higgs moduli fields. The C α are special Kahler manifolds and the H α are hyperkahler manifolds, and their metrics do not mix due to N = 2 supersymmetry [3,4]. There is usually a pure Coulomb branch, which we simply denote as C; sometimes, there is also a pure Higgs branch H which touches with the Coulomb branch at a single point (such as SCFT points). Mixed branches are emanating from special loci of the Coulomb branch as shown in figure 1, and the corresponding submanifolds on the Coulomb branch are called the Higgs or mixed branch roots [4]. The Coulomb branch of N = 2 theory has been studied extensively, and the Seiberg-Witten solution can often be found easily by using the brane construction [5][6][7], or using the relation to integrable systems [8,9]. The pure Higgs branches are much less studied, see e.g., [4,[10][11][12][13][14][15] for some examples. The full structure of moduli space is known for very few examples, i.e. the full vacua structure of N = 2 SQCD has been determined in [4,10] using the non-renormalization theorem and exact solution on the Coulomb branch. See also the discussion using type IIA brane construction [5,[16][17][18].
We are going to study the full moduli space of N = 2 theories obtained by twisted compactification of 6d N = (2, 0) A N −1 theory on a Riemann surface, called class S theories [19][20][21][22][23] 1 . Instead of using field theory method, we are going to use the geometric method developed in [26][27][28] (see also [29]) to determine various branches.
Let's first recall some of the basic features of M5 brane construction which leads to this class of theories. There are five scalars ϕ I (I = 1, 2, 3,4,5) in the adjoint representation of A N −1 = SU (N ) which describe the transverse fluctuation of M5 branes. Four dimensional theory is derived by twisted compactification of the 6d theory on a Riemann surface C, and one get a Higgs field Φ in the canonical (or cotangent) bundle of C which describes the Coulomb branch deformations. There are other real scalars ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) which are in the trivial bundle of C and describe the Higgs branch deformations. Generically, the Coulomb branch and Higgs branch are determined as follows: • The Coulomb branch is described by turning on only the deformations in Φ, and the Seiberg-Witten curve is then given by the spectral curve of Φ: here φ i are degree i holomorphic differential on Riemann surface.
• The pure Higgs branch is described by turning on only ϕ deformations (we can get another scalar from B µν field on M5 brane [16].).
• Mixed branches are described by turning on both ϕ and Φ.
Let's discuss the mixed branch in more detail. The deformation in ϕ is simple: they can be diagonalized with constant eigenvalues : ϕ = diag( a (1) I n 1 , · · · , a (k) I n k ), (1.3) where n s are integers such that k s=1 n s = N , I ns are n s ×n s the unit matrices, and a (s) are constants. Now the crucial thing is that brane dynamics requires the commuting condition [Φ, ϕ] = 0, which impies that the Seiberg-Witten curve has to be factorized as: here φ s,i (z) are various holomorphic differentials, and n s are the same as above.
There are more variations due to punctures on the Riemann surface C. Regular punctures are classified by Young diagrams with total number of boxes N , and one can define an ordering between two Young diagrams. In the mixed branch, a Young diagram Y D of SU (N ) theory is split into a collection of Young diagrams Y D s of SU (n s ) theory, see figure 2. (The superscript D will be explained later in this paper.) One can form a new Young diagram Y D with N boxes by assembling Y D s together, and the constraint on Y D is that Y D ≤ Y D , (1.5) where we used the ordering between two Young diagrams. There is a mixed branch corresponding to each collection Y D s . Unlike regular singularity, the constraint from irregular singularities is more rigid, as the boundary condition of Φ is fixed at the puncture and sometimes the curve can not be factorized in an arbitrary way. The factorization pattern of Seiberg-Witten curve of equation (1.4) has to be consistent with the boundary condition on the irregular puncture.
This paper is organized as follows. In section 2, we give a general algorithm of how to find mixed branch roots, and how to count the dimensions of the Coulomb and Higgs factors. In section 3 we derive those rules following the strategy outlined above. In section 4 and appendix A, we reproduce the results of N = 2 SQCD by our method. In section 5, theories defined using regular singularities are discussed. In section 6, we study Argyres-Douglas theories. Finally in section 7, we give a short discussion. Figure 2. The Higgs and mixed branch roots can be described as: the SU (N ) Hitchin system is decomposed into SU (n 1 ) × SU (n 2 ) × . . . × SU (n k ) × U (1) k−1 Hitchin systems, and the regular punctures of SU (N ) are decomposed into sums of regular punctures of lower rank Hitchin systems.

General rules for finding mixed branches
In general, the moduli space of N = 2 supersymmetric field theory has the general form where α runs over all the components of the moduli space, and C α and H α are Coulomb factor and Higgs factor, respectively. In other words, C α is parametrized by vevs of vector multiplets, and H α is parametrized by vevs of hypermultiplets. For Lagrangian theory, the Higgs branch can be found using classical equations of motion due to non-renormalization theorem [4]. Coulomb branch is much more difficult to determine because one needs to take into account various quantum corrections, and the result can be elegantly summarized using Seiberg-Witten curve.
In this section we would like to state the rules of how to find the roots of these mixed branches C α × H α and give the formulas for the dimensions of the Coulomb factor C α and the Higgs factor H α . Derivations of the rules are given in the next section 3. One of the branches is the pure Coulomb branch C × {0}, which we denote as just C. Then, the root of the branch C α × H α is the intersection of it with C.

Pure Coulomb branch of class S theories
The Seiberg-Witten curve describing the Coulomb branch C is given by the general form [19,20,23], where z is the coordinate of the base Riemann surface C, and x is the coordinate of the fiber of the canonical (or cotangent) bundle K = T * C. The φ i (z) is a section of the line bundle K ⊗i , i.e., it is an i-th differential on C. Neglecting punctures, its contribution to the dimension of Coulomb moduli can be found using the Riemann-Roch theorem, where we used the vanishing theorem H 0 (C, K 1−i ) = 0 for i > 1.
On the Riemann surface C, we can also add various regular and irregular punctures. Let us recall some properties of them. We consider local properties near one puncture, and we take the coordinate so that the puncture is at z = 0 for regular singularity.
Regular singularities. Regular (or tame) singularities [19] are classified by a partition of N , [m 1 , · · · , m ], which satisfies N = m 1 + · · · + m . Without loss of generality we take m 1 ≥ m 2 · · · ≥ m . This partition can be represented by a Young diagram Y such that the a-th column has m a boxes (i.e., its hight is m a ). In our convention, a full puncture is represented by [1, 1, . . . , 1], and a simple puncture is represented by The order of pole p i at the regular singularity of the differential φ i (z) can be easily found from the Young diagram 2 The local contribution of a regular puncture to the Coulomb branch dimension is where Y D is the Young diagram obtained by transposing Y , i.e., the rows of Y D are columns of Y . The reason for using Y D in (2.5) will become clearer in the next section. For later use, let us define partial ordering of two Young diagrams. We define The second equivalence can be checked by using (2.4). Notice that this is only a partial ordering: it is possible that neither Irregular singularities. We can also have irregular (or wild) singularities, which are needed for asymptotically free theories and Argyres-Douglas theories. The irregular singularities which are relevant for SQCD will be discussed in detail in appendix A. The irregular singularities which are relevant for Argyres-Douglas theories can also be summarized by a Newton polygon, see figure 3, and the local contribution to the Coulomb moduli can also be found from Newton polygon, for details, see [23]. The leading order eigenvalues of x can be read from the slop of the Newtwon polygon: here k is the height of the marked point in the Newton polygon of figure 3, ω = exp(2πi/ ), and the eigenvalue degeneracy of A i is determined by the Young diagrams Y i . The major feature of irregular singularity is that the leading order behavior is fixed as the UV data, and we can not change it by tuning the Coulomb branch moduli. For regular singularities, one can actually tune the Coulomb branch moduli to change the singularity.

Mixed branches
Now we can state our general rules for finding the roots of mixed branches C α × H α .
Step 1. The first step is to choose a partition of N : X = [n 1 , n 2 , · · · , n k ]. Then we take the curve (2.2) to be of factorized form where φ s,i are holomorphic differentials on the Riemann surface, and we also have s φ s,1 = 0 due to the traceless condition. The existence of type I and type II irregular singularities would constrain the possible factorization X: • Type I irregular singularity: define d as the maximal common divisor of (k, N ). Then the maximal partition X is and other possible factorization is derived by combining the columns of X.
• Type II irregular singularity: define d as the maximal common divisor of (k, N − 1), then the maximal partition X is and other partitions are found by combining columns expect the last column with height one.
When there are more than one irregular singularity, X is constrained to be compatible with all irregular singularities.
Step 2. The second step is to take into account local punctures. After the above global factorization, the SU (N ) theory is split into a sum of SU (n s ) (and U (1)) theories. For each regular puncture, a dual Young diagram Y D of the SU (N ) theory is split into a sum of diagrams Y D s for SU (n s ) theories. The important point is that these Y D s are not uniquely determined from Y D . A choice of the collection Y D s corresponds to the choice of a mixed branch we consider.
One can form a Young diagram of N boxes Y D by combining Y D s . The columns of Y D are taken to be the columns of Y D s . Then the only constraint on Y D s is Step 3. We consider every possible set of X and Y D s , and find the curve consistent with the above conditions. If (2.8) cannot be further factorized in a generic curve, this is the root of a branch C α × H α , where the label α is specified as α = {X, Y D s }. If (2.8) is factorized in a generic curve, this is just a boundary of some other mixed branch and we discard this case. Now let us give the formulas for the dimensions of the branches.
Coulomb factor. Once the factorization and local data are given, the dimension of the Coulomb factor C α can be easily found from the factorized Seiberg-Witten curve: each factor is the spectral curve of a lower rank SU (n s ) or U (1) Hitchin system. Each U (1) Hitchin system contributes dimension g to the Coulomb branch dimension.
Here we give an explicit formula for the Coulomb branch factor if there are only regular punctures. First, let us define local contribution of a puncture p as where d(Y D s ) is defined as in (2.5). Then, the total dimension is given by where k is the number of elements in the partition X = [n 1 , · · · , n k ]. The local contribution of the irregular singularity can be found using the method discussed in [23].
Higgs factor. The dimension of the Higgs factor H α has several contributions. There is global contribution due to the factorization of the Seiberg-Witten curve. The quaternionic dimension of this part is simply k − 1.
There is also contribution from local punctures, and an explicit form of the regular puncture contribution can be easily written down. Defining Y as the dual Young diagram of Y D , the local Higgs branch dimension is given by (2.14) This local contribution is always non-negative due to the constraint on Y given by (2.11), which is actually equivalent to Y ≥ Y . There is no local contribution to Higgs branch from type I and type II irregular singularity. The total dimension of the Higgs branch is given by (assuming there is only regular puncture): 3 Vacuum structure of class S theories In this section, we derive the rules for finding mixed branches stated in the previous section. The reader who is only interested in applications of the rules can skip this section. The essential ingredients are generalized Hitchin's equations and 5d maximal Super-Yang-Mills (SYM) twisted on the Riemann surface C [26][27][28], T ρ [SU (N )] theories of Gaiotto and Witten [30] placed at punctures [22], and irregular singularity classified in [23].

5d maximal SYM and generalized Hitchin's equations
The class S theories are defined as the low energy limit of 6d N = (2, 0) theory compactified on a Riemann surface C with punctures. If we further compactify it on a circle S 1 so that the 6d theory is placed on R 3 × S 1 × C, we get an S 1 compactification of four dimensional theory (in the small area limit of C). The same configuration can be looked at in a different way. We first compactify the theory on S 1 and get 5d maximal SYM, then we study the 5d theory on R 3 × C. Actually, the 5d maximal SYM might be enough to compute (BPS or protected) quantities of the 6d N = (2, 0) theory on S 1 (see e.g., [31,32]). Then, denoting the are of the Riemann surface as A, we claim (see [28] for more discussions and the scope of validity of the claim), 3 Vacuum moduli space of 4d class S theory on Furthermore, in this relation, the Coulomb branch and Higgs branch are exchanged, Coulomb (Higgs) branch of 4d class S theory on R 3 × S 1 = Higgs (Coulomb) branch of 5d maximal SYM on R 3 × C.
The Coulomb branch of 4d theory on S 1 is a Hyperkahler manifold. The Coulomb branch has a distinguished complex structure which we denote as I. This complex structure does not depend on the radius of S 1 [33], so we can learn the moduli space of 4d theory using complex structure I of Coulomb branch of 4d theory on S 1 . 4 Moreover, in complex structure I there is a fibration structure which can be identified with the Seiberg-Witten fibration over the Coulomb moduli space. Now we are going to describe the structure of the moduli space of vacua using the 5d theory. The 5d maximal SYM contains gauge fields A M (M = 0, 1, 2, 3, 4) and five real adjoint scalars ϕ I (I = 1, 2, 3, 4, 5). It must also be twisted on the Riemann surface so that half of the supersymmetry is preserved by the compactification. Then, taking the coordinate of R 3 × C as (x µ , z) where µ = 0, 1, 2 and z is a complex coordinate on C, the fields of the 5d SYM are given by Here, Φ(= ϕ 4 +iϕ 5 ) is a complex adjoint field, which takes values in K ⊗ad(E), where K = T * C is the canonical bundle and ad(E) is the vector bundle in the adjoint representation of the gauge group. The fact that Φ is a section of the canonical bundle K comes from the twisting of SO(2) R ⊂ SO(5) R R-symmetry to preserve eight superchages. The ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) are real adjoint fields in ad(E), A µ is the gauge field on R 3 and Az is the gauge field on C. The ϕ is a triplet of the SU The combinations ( ϕ, A µ ) and (Φ, Az) are taken so that each of them form a single multiplet of the supersymmetry. An important point is that (Φ, Az) corresponds to the Coulomb moduli fields of the 4d class S theory, and ( ϕ, A µ ) corresponds to the Higgs moduli fields of the 4d theory. The classical equations for supersymmetric vacua of the 5d SYM are given by generalized Hitchin's equations [26] (see also [29]), 5 which in general can describe theories with four superchages. In the case of eight supercharges (i.e., 4d N = 2), they are simplified as [28] The Coulomb branch of the 4d theory (or equivalently the Higgs branch of 5d theory) is described by the first two equations which are precisely the Hitchin's equations [34,35]. This branch can be determined using the classical Hitchin's equations due to nonrenormalization theorem.
The classical configuration of the Higgs branch of the 4d theory (or Coulomb branch of 5d theory) can be described by the expectation values of the scalar fields ϕ i , whose solutions are just constant matrices, and there are various quantum corrections to the metric of the moduli space which are difficult to calculate. However, for our purpose of studying the mixed branch structure and counting dimensions of moduli space, we can still use the classical equations due to the nonrenormalization theorem discussed in [28]. Due to the commuting condition between Φ and ϕ i , we get the breaking pattern on the Hitchin's equations, and therefore determine the mixed branch roots.

Factorization of the Seiberg-Witten curve
Let us solve these equations in detail. From (3.8), the ϕ are simultaneously diagonalized by SU (N ) gauge transformations. Let us assume that this is done. Then we get Here N = n 1 + n 2 + · · · + n k defines a partition of N , X = [n s ] = [n 1 , · · · , n k ]. The I ns are n s × n s unit matrices, and a (s) = (a 3 ) are vectors with a (s) = a (t) for s = t and k s=1 n s a (s) = 0. These a (s) are constants because of (3.6). The SU (N ) gauge symmetry is broken to a subgroup H = SU (n 1 ) × · · · × SU (n k ) × U (1) k−1 . Furthermore, (3.6) and where Φ (s) and A for each s = 1, · · · , k. Therefore, each pair (A (s) z , Φ (s) ) satisfies Hitchin's equations. If the vevs of (A (s) z , Φ (s) ) completely break the SU (n s ) gauge symmetries, the unbroken gauge group is U (1) k−1 . Then, taking the zero mode of A µ (µ = 0, 1, 2) on C and dualizing these gauge fields to scalar fields on R 3 , we get dual scalars b (s) (s = 1, · · · , k) with s n s b (s) = 0.
For the moment, we neglect punctures. Then, the general structure is as follows. Mixed branches are labelled by the partition of N , X = [n s ]. The Higgs factor H bulk α consists of the moduli fields ( a (s) , b (s) ) which contribute to the quaternionic dimension of H α as k − 1 (or real dimension 4(k − 1)). Here we added the word "bulk" to H bulk α because we are neglecting punctures. The Coulomb factor C α consists of the solutions of Hitchin's equations with the gauge group If generic solutions of Hitchin's equations for (A (s) z , Φ (s) ) do not break SU (n s ) completely, the above branch is just a boundary of a more larger branch. For example, consider a simple case where the only solution to the SU (n s ) Hitchin's equations is trivial, (A (s) z , Φ (s) ) = 0. This can happen e.g., when the genus is g = 0 and there are no punctures. Then, we can smoothly go to another branch as a (s) I ns → diag( a (s,1) , · · · , a (s,ns) ) (3.12) while satisfying (3.4)- (3.8). This means that we can go to the branch in which n s → 1 + · · ·+1. More generally, if (A (s) z , Φ (s) ) breaks SU (n s ) to a nontrivial subgroup H s ⊂ SU (n s ), we can smoothly turn on a vev of ϕ in the Cartan subalgebra of the unbroken group H s . The integer n s is further partitioned to n s = n s,1 + n s,2 + · · · . So we do not consider the original partition [n s ] as a separate mixed branch. The spectral curve of the original Hitchin system is given as Under the decomposition (3.10), the curve is factorized as: Constraint on factorization from Irregular singularity. If there are irregular singularities, the factorization pattern of the SW curve is constrained because the irregular singularity can not factorize in an arbitrary way. The reason is that the leading order term is fixed and define the UV theory, so we can not tune the leading order behavior of the SW curve. Near the irregular singularity, the curve has the following form (we put the singularity at z = ∞): where in the Type I (or type II) case, d is the maximal common divisor of k and N (or N − 1), and we have neglected all the coefficients. From these, one can easily see the rules stated in section 2.

Local moduli from regular singularities
Regular punctures are realized [22,28] by coupling the 3d N = 4 T ρ [SU (N )] theories [30] to the 5d SYM. These theories are extended in the R 3 direction, and they are localized at punctures on the Riemann surface C.

T ρ [SU (N )] theories
We first review some properties of T ρ [SU (N )] necessary for our purposes; see [22,30] for more details.
where m a is the m a dimensional (spin (m a − 1)/2) representation of SU (2). Similarly, we can consider the dual Young diagram of Y which we call Y D = [m a ], and we can define another embedding ρ D : SU (2) → SU (N ). Define e = ρ (σ + ) and e D = ρ D (σ + ), i.e., the image of the raising operator σ + of SU (2). Then, they are nilpotent matrices (because acting raising operators too many times gives zero). Mixed branches are classified by Y , and the branch labeled by Y is given as where, for a given matrix A, O A is the orbit of A by the action of the complexified group where c in the last equation runs over irreducible representations which appear in the decomposition. Corresponding to this decomposition, the generators of SU (N ) are given as The subtraction of dim C O e in dim H H α comes from the fact that we have eliminated the Goldstone multiplets in (3.21).
For example, the pure Coulomb branch C ×{0} is given by the orbit O e D corresponding to the partition Y D = [m a ] dual to Y = [m a ]. This will reproduce the usual rule of regular singularities given in (2.4).
Notice that the constraint Y ≥ Y is equivalent to Y D ≤ Y D . 6 So in all the branches, we must have the constraint (3.26)

T ρ [SU (N )] coupled to 5d SYM
Now let us couple the T ρ [SU (N )] theory to the 5d SYM at a point p ∈ C which will be a puncture. We gauge the Coulomb branch SU (N ) C symmetry of T ρ [SU (N )] by the gauge group of the 5d SYM. In particular, this coupling includes a superpotential coupling between the holomorphic moment map ν h and the complex adjoint field ϕ h = ϕ 1 + iϕ 2 , where ϕ h (p) is the value of ϕ h evaluated at the point p.
There are two important effects of this coupling. First, by giving the vev (3.9), this coupling gives a mass term of ν h . Then the Higgs branch moment map µ h is no longer a nilpotent matrix, but it is such that its characteristic polynomial is given by [28] det(xI N − µ h ) = det(xI N − ϕ h ), (3.28) where x is an arbitrary variable. (When ϕ h = 0, this equation means that all the eigenvalues of µ h are zero, and hence µ h is nilpotent.) The other effect of (3.27) is that this term gives a source of Φ. In the bulk superpotential of the 5d SYM, there is also a coupling of the form W ⊃ d 2 z tr(ϕ h DzΦ) and the equation of motion of ϕ h gives DzΦ ∼ δ 2 (z)ν h , where we have assumed that the point p is located at z = 0. Then we get a pole of Φ as 7 Suppose that the ν h is in the orbit ν ∈ O e D corresponding to the partition [m a ]. Then, combining the pole term ν h /z and also the order one term O(1) in the above equation, one can check that the singularity of the spectral curve is precisely given by (2.4). The constraint (3.26) ensures that the singularities cannot be stronger than (2.4). Now we can study the local contribution C α (p) × H α (p) from the puncture p to the mixed branch C α × H α of the 4d field theory. 6

If two partitions [ma] and [m a ] satisfy [m a ] ≤ [ma], then [m a ] ≥ [ma]. This claim is proved as follows.
Examining the Young diagram, one can see that a b=1 m b = N − b max{m b −a, 0}. Then by taking c such thatm c = a andm c+1 < a, we get  (3.4) and dividing by the gauge group is equivalent to dividing by complexified gauge group. Then, using complexified gauge transformation to set Az = 0, we get (3.29).
Coulomb factor. The commutation relation [Φ, ϕ] = 0 requires that the residue ν h of the pole of Φ at the puncture as given in (3.29) must be of the form Furthermore, each of the block ν  h . However, the following point must be taken into account. As explained in section 3.1, we compactified the 4d theory on S 1 . Then, in the low energy 3d theory, a U (1) massless gauge multiplet is dualized to two real scalars (or equivalently one complex scalar). Therefore, the dimension of Coulomb moduli space C (3d) α is doubled from the 4d Coulomb moduli space C (4d) α . Having this distinction between the 3d and 4d theory in mind, we conclude that This is what we have written in (2.12).
Higgs factor. The µ h must satisfy (3.28). Furthemore, the ν h satisfies (3.30) and (3.31). From these facts, it is natural to expect the following (see section 2.3 and 2.4 of [22]). For each s, let Y s = [m s,a ] be the partition of n s dual to Y D s , ρ (s) : SU (2) → SU (n s ) be the corresponding embedding, and e(s) = ρ (s) (σ + ). Let e = s e(s) be the block diagonal N × N matrix. Then we define the matrix  h for s = t. In this case, we propose that µ h is in the orbit of ϕ h + e , The complex dimension of the orbit O ϕ h +e is given by We can perform a consistency check by taking the limit ϕ h → 0. In this case, the T ρ [SU (N )] theory at the puncture has no (holomorphic) mass term, and hence we have where e = ρ (σ + ) is a nilpotent element corresponding to the partition Y = [m a ]. Therefore, we should get Actually, the dimension of O ϕ h +e computed above is the same as the dimension of O e . This is required because, even if ϕ h = ϕ 1 + iϕ 2 → 0, this limit should be smooth as long as ϕ 3 is generic. 8 We conclude that the quaterionic dimension of H α (p) is given as This is the formula written in (2.14).

N = 2 SUSY QCD
In this section, we reproduce the results of the moduli space of vacua in SQCD obtained by Seiberg and Witten for SU (2) [2] and by Argyres, Plesser and Seiberg for SU (N ) [4].

SU (2)
We study the SU (2) SQCD with N f flavors. Although these models are simple, they illustrate our method without technical complication.
We consider massless cases. In addition to the regular singularities discussed in the previous sections, we also need some irregular singularities. The curve is x 2 + φ 2 (z) = 0, and the singularities we use are given as where Λ is a parameter which roughly corresponds to the dynamical scales of field theory. The first singularity A is the regular singularity corresponding to the partition Y = [m a ] = [1,1] or equivalently its dual partition is Y D = [m a ] = [2]. The second and third singularities B and C are irregular singularities which are obtained [20] from the M-theory uplift of type IIA brane configurations [5]. More details on those types of irregular singularities are discussed in appendix A, but here we simply claim that (4.2) and (4.3) have no local contribution to the Higgs branch. Let us summarize the important points.
• If the curve is factorized as , there is a contribution to the quaternionic dimension of the Higgs branch as +1 coming from the term k − 1 in (2.15), i.e., the curve is factorized according to the partition X = [n s ] = [1,1].
• If there is a singularity of type B, then it is impossible to factorize the curve as x 2 + φ 2 = (x + f (z))(x − f (z)) since we cannot have Λ 2 /z 3 + · · · = (f (z)) 2 for some single valued function f (z). On the other hand, in the case of the singularity of type C, there is no such local obstacle.
• For a singularity of type A, we can have local contribution to the Higgs branch if and only if the O(z −1 ) term vanishes. In this case we have Y D = [m a ] = [1,1], and the local contribution to the quaternionic dimension, calculated by the formula (2.14), is Zero flavor N f = 0. The theory with no flavor is realized by putting two irregular singularities of type B, (4.2), on a Riemann sphere parametrized by z ∈ C ∪ {∞}. Putting the singularities at z = 0 and z = ∞, the curve is where u is the Coulomb modulus. (Note that near z → ∞, we must use the coordinates z = 1/z and x = −z 2 x.) We cannot factorize the curve, and there are no local contributions to the Higgs branch from the punctures. Thus there is no Higgs branch, reproducing the result of field theory.
One flavor N f = 1. The theory with one massless flavor is realized by putting one irregular singularity of type B, (4.2), and one irregular singularity of type C, (4.3), on a Riemann sphere. Putting singularities at z = 0 and z = ∞, the curve is In this case, there is no Higgs branch as in the case of N f = 0.
Two flavors N f = 2. The theory with two massless flavors can be realized in two different ways. The first way is to put two type C singularities. The second way is to put one type B irregular singularity and two type A singularities, (4.1). First, let us study the first realization. We use two type C singularities. Putting singularities at z = 0 and z = ∞, the curve is The curve can be factorized if and only if u = ±2Λ 2 . In this case, we have Therefore, the Higgs branch is emanating from the Coulomb branch points u = ±2Λ 2 and its quaternionic dimension is 1. This is exactly as was found by Seiberg and Witten [2]. Next, let us study the second realization. We take regular punctures of type A at z = 0, 1 and a puncture of type B at z = ∞. The curve is .
We cannot factorize the curve in this case because of the existence of the singularity of type B. However, there are contributions to the Higgs branch dimension from regular punctures. Notice that Then, we can have local contribution to the Higgs branch from the puncture at z = 0 when u = Λ 2 , and from the puncture at z = 1 when u = 0. At each of these points, we have the Higgs branch with the quaternionic dimension 1. Therefore, we find the same structure as the case of the above first realization if we identify the moduli parameters as u = 2u − Λ 2 . It is interesting that the way the Higgs branch appears is quite different in the two realizations. In the first realization, the Higgs branch comes from the "bulk" contribution, while in the second realization the Higgs branch comes from local contributions from punctures. However, the final results are the same in both cases.
Three flavors N f = 3. The theory with three massless flavors is realized by putting two regular singularities of type A at z = 0, 1 and one irregular singularity of type C at z = ∞. The curve is . (4.10) Note that the local contributions to the Higgs branch from the punctures at z = 0, 1 are both turned on when u = 0. Furthermore, only in this case the curve is factorized, Therefore, the Higgs branch is emanating from the single point u = 0 and the quaternionic dimension is 3. This is exactly as in [2].
Four flavors N f = 4. The theory with four massless flavors is realized by putting four regular singularities of type A at z = 0, q, 1, ∞. The curve is . (4.12) All the local contributions to the Higgs branch and the factorization of the curve is possible if and only if u = 0. Therefore, the Higgs branch is emanating from u = 0 and has the quaternionic dimension 5. This point is the superconformal point of the theory.
Mass deformation and AD point. Let us briefly describe mass deformation. Generic deformation would typically eliminate the Higgs branch. Consider the first realization of SU (2) theory with two flavors. The curve turning on generic mass deformation is (4.13) The above curve can be factorized if and only if (4.14) So for non-zero m = m 1 = ±m 2 , there is only one Higgs branch of dimension 1, which is consistent with the fact that the flavor symmetry is broken to U (2) by the mass term. The above curve can be further factorized for the following special value of m: There should be new massless particles here, but our method tells us that the Higgs branch dimension does not change at this point. This suggests that the new massless particles might be nonlocal relative to the massless particles which are already present in (4.14). This point is actually the Argyres-Douglas point as discussed in [10].

SU (N )
In this section we study the SU (N ) SQCD. First, let us summarize the result of Argyres, Plesser and Seiberg [4] obtained by field theory methods.
• There is a baryonic Higgs branch for N f ≥ N , and the quaternionic dimension is N (N f − N ) + 1. There is no Coulomb direction in this branch.
• There are non-baryonic mixed branches with a label 0 ≤ r ≤ min([ . The complex dimension of the Coulomb factor C r is N − r − 1, and the quaternionic dimension of the Higgs factor H r is r (N f − r). The pure Coulomb branch is the case r = 0.
Here we only treat the conformal theory N f = 2N which only requires regular singularities. Other cases are discussed in appendix A.
Non-baryonic branch. Suppose that some of the u i is nonzero. Then we can define r (0 ≤ r ≤ N − 2) such that u N −r = 0 and u N −r+1 = · · · = u N = 0. Then the curve is factorized as In the second step, we should choose the local puncture type on the SU (N − r) factor. However, it is easy to see that to have nonzero u N −r , the punctures have to be chosen as in the figure 4: the full (simple) puncture of SU (N ) becomes the full (simple) puncture of SU (N − r). We have the following data on the split of punctures: Full puncture : Using the formula (2.14), local contribution to the Higgs component from a full or simple puncture is given by Then, the formula (2.15) gives the total Higgs branch dimension as where we have used k = r + 1. This is exactly what was found in [4] with N f = 2N .
Baryonic branch. In the above we have assumed u N −r = 0 for some r. Local contribution from a full or simple puncture is given by This is exactly what was found in [4] with N f = 2N .

Theories defined using regular singularities
Our method can be also applied to theories whose Lagrangian description is not yet know. One of such theories is the so called T N theory. Pure Higgs branch H (i.e., branch where there are no Coulomb moduli fields) are studied in [11,14]. But it is not so easy to study the mixed branch structures directly by field theory. Here we use our method to study these structures. The case of the partition X = [n s ] = [1 N ] was studied in [28].

T N theory
The T N theory is realized by using a Riemann sphere with three full punctures. Each full puncture has SU (N ) H Higgs branch symmetry, so the T N theory has SU (N ) A ×SU (N ) B × SU (N ) C flavor symmetry.
In this case, what are discussed in sections 2 and 3 suggest the following interpretations. First, the factorization of the curve (2.8) is realized by giving vevs to Higgs branch operators of the T N theory so that the low energy theory is given by smaller T N theories [14] T N → T n 1 + T n 2 + · · · + T n k + (free fields). (5.1) Here the free fields come from some k − 1 moduli fields responsible for the above breaking which roughly represent the relative "positions" of T ns on moduli space, 9 and also the Goldstone multiplets associated to the spontaneous breaking [SU (N )] 3 → [S(U (n 1 ) × · · · × U (n k ))] 3 . There are 3× 1 2 (N 2 − k s=1 n 2 s ) Goldstone multiplets (in quaternionic dimension). Next, we consider the partition n s → Y D s = [m s,a ] at three punctures, which we call A, B and C respectively. Thus, for each s, we have Y D s,A , Y D s,B and Y D s,C , and their dual partitions Y s,A , Y s,B and Y s,C . Then, we can Higgs the punctures of the T ns theory by giving nilpotent vevs to the moment maps to get more general three-punctured theories specified by these partitions [22,36,37] T ns → T ns (Y s,A , Y s,B , Y s,C ) + (Goldstone multiplets), (5.2) where the Goldstone multiplets are the ones associated to the spontaneous breaking of [SU (n s )] 3 by nilpotent vevs. As discussed in section 3, each puncture contributes d(Y s ) = This process gives the mixed branches of the T N theory. For this process to define a mixed branch of the T N theory, T ns (Y s,A , Y s,B , Y s,C ) must be such that its generic curve cannot be factorized. If some of n s is 1, we interpret the T n=1 theory to be empty.
The above picture of the vacua structure suggests that we can find all the lower three punctured sphere theory on the Coulomb branch moduli space of the T N theory. [25,38] are defined by a genus g Riemann surface with no punctures at all. Therefore, we only need to consider a partition X = [n s ] of the curve

Maldacena-Nunez theory
If g = 0, the only possible curve is just 0 = x N , since there are no nontrivial holomorphic sections of φ i . Hence the only branch allowed by our rule is the case X = [1 N ]. This is the Higgs branch with quaternionic dimension N − 1. There is no Coulomb branch.
If g = 1, then the most general curve is given by 0 Therefore, also in this case, the only possible partition allowed is X = [1 N ]. The quaterionic dimension of the Higgs branch is N − 1. In fact, the Coulomb branch and Higgs branch are combined into a manifold with complex dimension 3(N − 1), which agrees with the field theory result as the theory is just N = 4 SYM. If g > 1, we can have the following factorization: This means all the partitions [n a ] are possible, and one can easily find the dimensions of the Higgs and Coulomb factor using our general formula. Let's remark an interesting point for the branch labeled by the partition X = [1 N ]. In this branch, we have the maximal Higgs branch deformations. Unlike the sphere case, we still have Coulomb branch deformations which are described by the curve (5.5) and h i are sections of the canonical bundle. This means that there is no pure Higgs branch for this class of theories. See [28] for more detailed discussions on this case.

Argyres-Douglas theory
Here we use our method to determine the Higgs branch of Argyres-Douglas theory [23,24,39]. We will recover and extend the results obtained in [13]. The theories we consider are discussed in [23,40,41].

(A 1 , A N ) theory
This theory is described by a sphere with one irregular singularity which we put at z = ∞.
N=2n-1. The Seiberg-Witten curve is x 2 = z 2n + c 2 z 2n−2 + · · · + c n z n + c n+1 z n−1 + u n+2 z n−2 + . . . + u 2n (6.1) where c i (i = 2, · · · , n) are coupling parameters of the theory, c n+1 is a mass parameter, and u i (i = n + 2, · · · , 2n) are Coulomb branch moduli. To find the Higgs branch, we need to consider the factorization of the Seiberg-Witten curve. In fact, there is a factorization happening at a single point on the moduli space: There are 2n − 1 equations for 2n − 1 parameters c n+1 , u i (i = n + 2, · · · , 2n) and a i (i = 2, · · · , n) when c i (i = 2, · · · , n) are fixed. By tuning c n+1 appropriately, factorization occurs for certain values of u i determined by c i uniquely. For example, for the SCFT case (c i = 0), we find a one dimensional Higgs branch coming out from u i = 0.
N=2n. The curve for this theory is It is impossible to factorize the curve because of the existence of the leading term z 2n+1 . Therefore there is no Higgs branch.
This theory is described by one irregular singularity at z = ∞ and one regular singularity at z = 0.
We only consider the case in which m = 0 and also c n+1 is tuned appropriately so that the curve can be factorized. Then, if u 2n+1 = 0, we get a local contribution to the Higgs factor H α from the puncture at z = 0. Therefore, we have a mixed branch at u 2n+1 = 0. The Coulomb factor C α is the same as the (A 1 , A 2n−1 ) theory discussed above, and the Higgs factor H α has quaternionic dimension 1.
At a single point on the Coulomb moduli space, we can factorized the curve as At this point, we must also set u 2n+1 = 0. There is a Higgs branch coming out of this point. The quaternionic dimension is 2 by counting the contributions from the bulk and the puncture. In particular, the Higgs branch at the SCFT point (a i = 0) has dimension 2, which agrees with the result in [13].
N=2n+3. The curve is Again we consider the case m = 0. Then, there is a mixed branch defined by u 2n+2 = 0. The contribution to the Higgs factor H α comes from the puncture at z = 0, and the Coulomb factor C α is the same as the (A 1 , A 2n ) theory.

(A 1 , E N ) theory
The Seiberg-Witten curve at the E N type SCFT point is It is easy to see that only the curve for E 7 theory is factorizable, and there is only one type of factorization with X = [2,1], so the dimension of Higgs branch is one.

Conclusion
We have developed a general method to find the vacua structure of class S theory using generalized Hitchin equation. We derived our results using the geometric method from six dimensional (2, 0) theory. For those 4d theories with lagrangian description, it is interesting to check our result using field theory calculations. The holomorphic factorization of Seiberg-Witten curve and reduction of singularity at punctures play a crucial role in our method, and a field theory interpretation of this fact would be welcome. Let's summarize the possible moduli space structure of N = 2 theories: • Typically, there is a Coulomb branch, a Higgs branch and mixed branches. Such examples include the theory defined by a sphere with regular punctures.
• Sometimes there is only a Coulomb branch. The typical example is the pure SU (2) SYM and (A 1 , A N −1 ) AD theory with even N .
• There is only a Higgs branch for the free field theory such as bifundamentals.
• There is no pure Higgs branch, namely only a Coulomb and mixed branches exist. This situation happens e.g. for the theory defined on a higher genus Riemann surface.
We focused on the theories obtained from A N −1 type N = (2, 0) theory in this paper, but our method is also applied to other theories using D [42,43] and E [44] type (2, 0) theory. It would be interesting to extend our study to those other types of theories.
The singularity structure on the Coulomb branch of N = 2 theory is crucial for the discovery of Seiberg-Witten solution [1,2]. New massless particles appear at those singularities. The loci where Higgs or mixed branches appear must have massless particles, but the appearance of new massless particles does not necessary mean that there is a mixed branch. The method described in this paper detects mixed branches rather than massless particles. It would be very interesting to develop a systematic way to find massless particles and their types (e.g., mutually local or nonlocal, abelian or nonabelian, etc.). The study of effective field theory on the Higgs branch roots is particular interesting, as those singularities might survive under N = 1 deformation [4]. We leave the study of those effective theories and N = 1 deformation to the future.
where Y is the partition of K dual to Y D , and d(Y ) is defined as We stress that the relevant partitions are partitions of K instead of N in this irregular case.
Derivation. The above rules are derived as follows. If we use complex gauge transformations Φ → gΦg −1 , g ∈ SU (N ) C , to make the Higgs field of the Hitchin system Φ block diagonal, the irregular singularity is given as (assuming that the singularity is at z = 0), where ν h is a K × K nilpotent matrix in the orbit ν h ∈Ō Y D , and W (z) is an (N − K) × (N − K) matrix whose eigenvalues are given as where ω N −K = exp(2πi/(N −K)). The first K×K block behaves just as regular singularity. The commutation relation [Φ, ϕ] = 0 requires that the block which commutes with W (z) must be proportional to an identity matrix. Along the lines of the discussion around (A.4), this block of ϕ is an n r × n r unit matrix with n r ≥ N − K.
In [28], it was proposed that the holomorphic moment map µ h of the U (K) flavor symmetry is related to ϕ h as Also from this, we can see that φ h must be proportional to the unit matrix in the block which contains W (z). Now the matrices ν h and µ h are just as in the regular case. The rules stated above can be found as in the regular singularity.

A.2 Mixed branches of SQCD
Now we study mixed branches of SQCD using the irregular singularities discussed above.
Before going to the general cases, let us consider the case N f < N in a simple realization as a warm-up. We proceed as if N f < N − 1, but the case N f = N − 1 can be obtained just by shifting x to cancel the x N −1 term in the curve.
On a Riemann sphere, we put the singularity (A.1) with K = N f at z = 0, and with K = 0 at z = ∞. The curve is Then the curve is given as The dimension of the Coulomb branch C r is N − r − 1 spanned by u 2 , · · · , u N −r . The quaternionic dimension of the Higgs component H r is computed by using (A.5) as This is exactly as was found by Argyres, Plesser and Seiberg [4].
Now let us consider more general cases. We take N f = N 1 +N 2 , and assume that N 1,2 < N . Using the singularity (A.7) for K = N 1 , N 2 at z = 0, ∞ respectively, the curve is (A.13) When factorizing the curve as in (2.8), there are two possibilities.
(A. 15) This is possible only if N f = N 1 + N 2 ≥ N .
We call the first branch as mesonic branch and the second branch as baryonic branch.
Mesonic branch. Assume that the curve is factorized as in (A.14). Then the factors which do not contain the higher pole terms have only regular singularities at z = 0 and ∞. Then, one can see that the only possibility is the factorization of the form 16) Then the partition specifying the factorization is given by X = [n s ] = [n 1 , 1 N −n 1 ].
First, let us consider the case n 1 > max{N − N 1 , N − N 2 }. In this case, we can assume u n 1 = 0, since otherwise the curve would be further factorized according to the partition [n s ] = [n 1 − 1, 1 N −n 1 +1 ] if u n 1 = 0. Then, the condition u n 1 = 0 requires that the partitions Y D = [m a ] of N 1 and N 2 at z = 0 and z = ∞ must be [N 1 − N + n 1 , 1 N −n 1 ] and [N 2 − N + n 1 , 1 N −n 1 ], respectively. Thus we get where we have defined r = N − n 1 and N f = N 1 + N 2 . From the constraint max{N − N 1 , N − N 2 } < n 1 ≤ N , we get 0 ≤ r < min{N 1 , N 2 } Next, let us consider the case n 1 = max{N − N 1 , N − N 2 }. Without loss of generality, we assume N 1 ≥ N 2 and n 1 = N − N 2 . The curve is (A.18) Here we do not assume that u N −N 2 is nonzero. Now, the situation is somewhat similar to section A.2.1. The existence of the last term Λ N −N 2 /z −1+N −N 2 makes it impossible to factorize the curve further. Furthermore, because of this last term, the partition Y D = [m a ] of N 1 at z = 0 must be of the form [N 1 − N 2 − r , r , 1 N 2 ], where 0 ≤ r ≤ [(N 1 − N 2 )/2]. The partition Y D = [m a ] of N 2 at z = ∞ is given by [1 N 2 ]. The nonzero Coulomb moduli are given by u 2 , · · · , u N −N 2 −r , so the dimension of the Coulomb factor C α is N − N 2 − r − 1. The dimension of the Higgs component H α is where we have defined r = r + N 2 . It satisfies min{N 1 , Summarizing what we have found above, mesonic mixed Higgs-Coulomb branches are labelled by r, 0 ≤ r ≤ [N f /2]. The Higgs branch dimension is r(N f −r), while the Coulomb branch dimension is N − r − 1. This is exactly as was found in [4].
Baryonic branch. Next let us consider the case (A.15). In this case, one can check that the only possible curve allowed by the singularities is given by This reproduce the result of [4]. It is remarkable that the final results do not depend on N 1 and N 2 separately, but depend only on the combination N f = N 1 + N 2 , although each bulk and local contributions to dim H H α are different for different pairs (N 1 , N 2 ). This means that different brane constructions lead to the same low energy 4d theory as expected.
There are also constructions of SQCD with N f ≥ N by using two regular punctures (one of them is full and the other is simple), and one irregular puncture. We leave that case for the reader.