(0,4) brane box models

Two-dimensional $\mathcal{N}=(0,4)$ supersymmetric quiver gauge theories are realized as D3-brane box configurations (two dimensional intervals) which are bounded by NS5-branes and intersect with D5-branes. The periodic brane configuration is mapped to D1-D5-D5$'$ brane system at orbifold singularity via T-duality. The matter content and interactions are encoded by the $\mathcal{N}=(0,4)$ quiver diagrams which are determined by the brane configurations. The Abelian gauge anomaly cancellation indicates the presence of Fermi multiplets at the NS-NS$'$ junction. We also discuss the brane construction of $\mathcal{N}=(0,4)$ supersymmetric boundary conditions in 3d $\mathcal{N}=4$ gauge theories involving two-dimensional boundary degrees of freedom that cancel gauge anomaly.

1 Introduction N = (0, 4) supersymmetric field theories are less understood due to the difficulty with their construction and quantization, however, they involve intriguing aspects and applications. N = (0, 4) supersymmetric sigma model has the Yukawa couplings which can obey the ADHM equations of the instanton construction under certain assumptions [1]. This indicates that there exists a certain N = (0, 4) sigma model for every instanton. Since then, there have been a lot of studies of N = (0, 4) supersymmetric field theories. The classical aspects of N = (0, 4) supermultiplets and off-shell formalism were studied in [2,3,4] and quantum properties were studied in [5]. The elliptic genera or superconformal indices of N = (0, 4) gauge theories have been computed in [6,7]. N = (0, 4) superconformal field theory also can play a significant role in string and M-theory to describe the dynamics of intersecting brane configurations and many features of the holographic dual supergravity solutions. One relevant example is the D1-D5-KK system which is a 1/8 BPS configuration in Type IIB string theory. It is dual to the triple intersection of M5-branes [8,9]. The dual N = (0, 4) SCFT is studied in [10,11]. Another relevant example is the D1-D5-D5 system whose near horizon geometry is the geometry AdS 3 × S 3 × S 3 × R [12,13,14]. The N = (0, 4) gaugel theory which lives on the common world-volume of D1-branes and intersecting D5-branes is studied in [15].
In this paper, we present a novel construction of N = (0, 4) supersymmetric quiver gauge theories from brane box configuration of D3-branes which intersect with NS5-, D5-, NS5 -and D5 -branes. N = (0, 4) supersymmetry can be preserved by half BPS boundary conditions in 3d N = 4 supersymmetric field theory [16] and these boundary conditions can be realized in brane setup by starting with Hanany-Witten setup [17] and introducing additional NS5 and D5 -branes on which half-infinite D3-branes can end. We promote this setup to define N = (0, 4) supersymmetric gauge theory by displacing D3-branes between NS5-and NS5 -branes. We call this brane configuration the D3-brane box model. The periodic D3-brane box model turns out to be T-dual to the D1-D5-D5 brane system probing orbifold singularity, which is a generalization of the D1-D5-KK system and D1-D5-D5 system.
In section 2 we start with reviewing N = (0, 4) supersymmetry in two dimensions. We make use of N = (0, 2) notation to formulate N = (0, 4) theories. In section 3 we realize N = (0, 4) supersymmetric boundary conditions for 3d N = 4 gauge theories by adding extra 5-branes in Hanany-Witten construction and promoting to brane box configuration in Type IIB string theory. In section 4 we study the D1-D5-D5 -KK-KK system which is T-dual of the brane box configuration. We determine the spectrum and interaction by using the techniques developed in [18]. We propose N = (0, 4) quiver which can be read from the brane box configuration and the D1-D5-D5 -KK-KK system. In section 5 we analyze the anomaly in brane box model. We discuss that the cancellation of the Abelian anomaly in brane box model requires the existence of tetravalent Fermi multiplet charged under the Abelian parts for quadrants of D3-branes which lives at the NS5-NS5 junction. Furthermore, we argue the brane construction of N = (0, 4) supersymmetric boundary conditions in 3d N = 4 supersymmetric gauge theory involving two-dimensional boundary degrees of freedom which cancel gauge anomaly.

They satisfy
{Q + , Q + } = Q + , Q + = 0, Q + , Q + = −2i∂ + . (2. 2) The superderivatives are Note that superfields are functions of (x ± , θ + , θ + ) with constraints in terms of D + , which annihilates combinations of y + = x + − iθ + θ + , y − = x − and θ + . N = (0, 2) gauge multiplets For simplicity, we focus on Abelian gauge theory. The N = (0, 2) gauge multiplet consists of a real adjoint valued superfield A + , whose lowest component is the rightmoving component of gauge field, and A − whose lowest component is the left-moving component of gauge field. A supergauge transformation is where Λ is a chiral superfield D + Λ = D + Λ = 0. In the Wess-Zumino gauge, the real superfields A + and A − have the component expansions Here the left-moving component A − = A 0 − A 1 of the gauge field has two real left-moving fermionic partners λ − and λ − while the right-moving one A + = A 0 + A 1 has none. D is a real auxiliary field. In gauge theories, the superspace derivatives D + and D + are extended to gauge covariant superderivatives D + = e −A+ D + e A+ and D + = e A+ D + e −A+ . In the Wess-Zumino gauge they are expressed as

N = (0, 2) chiral multiplets
The N = (0, 2) chiral superfield Φ satisfies the chirality constraint It is expanded in the (super)coordinates y + = x + − iθ + θ + , y − = x − and θ + as where φ is complex scalar and ψ + is its right-moving fermionic partner. The kinetic terms for the N = (0, 2) chiral superfield are given by (2.14) For a field Φ (Q) with U (1) charge Q, the covariant chirality constraint D + Φ (Q) = 0 can be solved by Φ = e −QA Φ (Q) . In components we have (2.15) where D α = ∂ α + iQu α . The kinetic terms for the N = (0, 2) charged chiral superfield are given by Here E determines the potential term and it can be solved by assuming that E is a holomorphic function of chiral superfields Φ [20]. The N = (0, 2) Fermi multiplet is expanded in the coordinates as where χ − is a left-moving fermion and G is a complex auxiliary field. In general E will have an expansion . (2.19) The N = (0, 2) Fermi multiplets may also transform as some representation R of the gauge group.
(2. 20) We see that the holomorphic function E (Q) (φ) appears as a potential term for chiral multiplet in (2.20), which we call an E-term potential. By definition (2.17), E-term transforms in the same way as the Fermi multiplet Γ.

N = (0, 2) superpotential
Let J a (Φ) be a superpotential which is a holomorphic function of chiral superfields Φ for a set of Fermi multiplets {Γ a }. Then a supersymmetric action can be also constructed by integrating over half of superspace as By integrating out the auxiliary field G a , one obtains a potential term ∼ |J a (φ)| 2 . We shall call this a J-term potential. It follows from gauge invariance of Γ a J a that Q Γa = −Q J a . Thus J-term transforms in the conjugate representation, namely as Γ. The bosonic potential terms specified by holomorphic functions E(φ) a and J a (φ) are associated to the Fermi multiplet Γ a . It is important to note that in N = (0, 2) theories, there is a symmetry between Fermi multiplet Γ and its conjugate Γ under an exchange of E-and J-terms.
Since N = (0, 2) Fermi multiplet Γ a is not a genuine chiral superfield obeying D + Γ = √ 2E, one needs to impose the condition E · J = a E a J a = 0 (2.22) to ensure that the J-term potential Γ a J a (Φ) is chiral, i.e. D + (Γ a J a ) = 0. It is important to note that as 3d N = 2 supersymmetric theories with a superpotential W 3d (Φ) admits N = (0, 2) supersymmetric boundary conditions with W (Φ) being constant [22], the condition (2.22) can be relaxed so that E · J = W 3d (Φ) (2.23) if N = (0, 2) theories live on a boundary of 3d N = 2 theories [23,24]. This is a 3d analogue of the Warner problem [25] so that (2.23) exhibits holomorphic factorization of 3d bulk superpotential. One simple example of the superpotential is an FI and θ term +h.c.
= Tr d 2 x −rD + θ 2π F 01 (2.24) where t = ir + θ 2π is a complex combination of a FI parameter r and θ angle. In total, there are three contributions to potential energy, i.e. D-, E-and J-terms. N = (0, 2) gauge theories are expected to flow in the IR to the non-linear sigma model whose target space is determined by the vanishing D-, E-and J-terms.
It is important to note that the condition (2.22) cannot be satisfied if a single N = (0, 4) vector multiplet is coupled to both N = (0, 4) hypermultiplet and N = (0, 4) twisted hypermultiplet since E Γ J Γ = Tr U DRL = 0. This can be solved by introducing N = (0, 4) Fermi multiplets which couple to both the N = (0, 4) hypermultiplet and the twisted hypermultiplet.
Since the N = (4, 4) vector multiplet decomposes into N = (0, 4) vector multiplet and adjoint valued N = (0, 4) twisted hypermultiplet, E Γ in (2.26) can be expressed as a holomorphic function of adjoint valued chiral multiplets U and D which constitute the N = (0, 4) twisted hypermultiplet [15]: Although they are trivial under the R-symmetry, they play a key role in defining consistent N = (0, 4) gauge theories. As discussed in 2.2.3, when both N = (0, 4) hyper and twisted hypermultiplets couple to a gauge field, it is required to introduce neutral Fermi multiplet so that the supersymmetric condition (2.22) is satisfied. The simplest situation would be the case with one hyper multiplet and one twisted hyper multiplet. In addition, as we discuss in section 2.2.5, when there are enough hyper and twisted hypermultiplets in the gauge theory, the charged Fermi multiplets are required to cancel a gauge anomaly.

Anomaly
N = (0, 4) supersymmetric gauge theory can be anomalous because left-and right-moving fermions are not necessarily paired together.
Let G be a simple compact group of which a system of right-and left-handed chiral fermions transform under a unitary representation R through coupling to a (background) gauge field whose field strength is f . We define the quadratic index C(R) of R as a sum of length-squared of weights λ C(R) = 1 rankG λ∈R λ 2 (2.32) where α 2 = 2 for long roots α. This is normalized so that C(adjoint) = 2h where h is the dual Coxeter number. The contribution to anomaly the 4-form is summarized as follows: Fermi Γ or C(R) adjoint 2h gauge Υ adjoint 2h (2.33) The left-and right-moving fermions have the opposite contributions to the anomaly whereas the fundamental and anti-fundamental representations have the same contributions. For SU (N ) we have C( ) = 1 and C(adjoint) = 2h = 2N and the anomaly contributions are summarized as (2. 34) In particular, the gauge anomaly is required to be cancelled for a consistent quantum field theory, which leads to an important constraint. Unlike the gauge anomaly, the global anomaly may remain in the theory. If the global anomaly remains in the IR, the current of the global symmetry of Lie algebra h can be holomorphic or anti-holomorphic, i.e. left-or right-moving. Then the corresponding global symmetry can be enhanced to the affine Lie algebra h of level |2A h | where A h is the anomaly coefficient. The affine Lie algebra h acts in the holomorphic or anti-holomorphic sector of the associated CFT depending on the sign of the anomaly coefficient A h . 8 3 N = (0, 4) boundary conditions 3

.1 Brane construction
We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 [17]. Let Q L and Q R be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQ L = Q L , ΓQ R = Q R where Γ = Γ 0 · · · Γ 9 .

N = (0, 4) boundary conditions
When the D3-branes are finite only in x 6 direction and are semi-infinite in the region x 2 ≥ 0, the configuration of D3-, NS5-and D5-branes leads to 3d N = 4 supersymmetric field theories and two extra 5-branes, NS5 -and D5 -branes break further half of supersymmetry to give rise to N = (0, 4) boundary conditions at x 2 = 0 in these theories [16].

3d N = 4 vector multiplet
When a D3-brane is stretched between two parallel NS5-branes along x 6 direction, the low-energy effective theory is that of 3d N = 4 Abelian vector multiplet. It contains a three-dimensional gauge fields A µ , µ = 0, 1, 2, three real scalar fields φ i , i = 3, 4, 5 which transforming as (3,1) of SU (2) C × SU (2) H , an auxiliary field D and a fermionic fields λ. The irreducible 3d N = 4 vector multiplet V decomposes into a sum of N = (0, 4) vector multiplet V and N = (0, 4) twisted hypermultiplet T : 1. NS5 and (0, 4) vector multiplet When the D3-brane further ends on an NS5 -brane, it fixes the motion of the D3-brane in (x 3 , x 4 , x 5 ) while the two-dimensional gauge field A α , α = 0, 1 is free to fluctuate and normal component A 2 of gauge field obeys the Dirichlet boundary condition. Correspondingly, the field theory admits the half BPS boundary conditions (3.8) along with the massless left-moving fermions. The boundary massless modes which are twodimensional gauge fields A α and the left-moving fermions form the irreducible N = (0, 4) vector multiplet. Therefore the NS5 -brane allows boundary conditions in which (0, 4) vector multiplet V on the boundary is free while N = (0, 4) twisted hypermultiplet T vanishes.

D5 and (0, 4) twisted hypermultiplet
Conversely, when the D3-brane terminates on a D5 -brane, the three scalar fields φ i are free to fluctuate as the D5 -brane spans in the (x 3 , x 4 , x 5 ) directions. On the other hand, the gauge field A α satisfies the Dirichlet boundary condition and the scalar field A 2 is free to move at the boundary. In field theory analysis this corresponds to the half of BPS boundary conditions in 3d N = 4 Abelian vector multiplet F αβ | ∂ = 0, D 2 φ i | ∂ = 0, (3.9) when the massless right-moving fermions survive at the boundary. The three scalar fields φ i and the scalar field A 2 can form a pair of complex scalars transforming as (2,1). They eventually combine with the right-moving fermions into N = (0, 4) twisted hypermultiplet. Hence the D5brane sets N = (0, 4) vector multiplet V to zero and leaves N = (0, 4) twisted hypermultiplet T at the boundary.
When N c D3-branes end on a D5 -brane, the half BPS boundary conditions are generalized to the Nahm pole boundary condition [16]: The pole governed by the Nahm equation would describe the D3-branes which polarize into a fuzzy funnel configuration [26,27].

3d N = 4 hypermultiplet
The dynamics of a D3-brane ending on two parallel D5-branes would be described by a theory of 3d N = 4 hypermultiplet. The bosonic massless modes in the theory are the fluctuations of the D3-brane in the (x 7 , x 8 , x 9 ) directions and the scalar field A 6 . The three real scalar fields transform as (1,3) under the SU (2) C × SU (2) H R-symmetry while the scalar field A 6 transform as (1, 1). The irreducible 3d N = 4 hypermultiplet H decomposes as the sum of N = (0, 4) hypermultiplet H and N = (0, 4) Fermi multiplet Ξ: 1. NS5 and (0, 4) hypermultiplet When the D3-brane terminating on two D5-branes further ends on an NS5 -brane, the three scalar fields describing the fluctuations in the (x 7 , x 8 , x 9 ) directions and the scalar field A 6 still remain. The corresponding half BPS boundary conditions in 3d N = 4 hypermultiplet can be found in field theory analysis when the massless right-moving fermions are left at the boundary. The pair of complex scalar fields and the right-moving fermions form N = (0, 4) hypermultiplet. Thus the NS5 -brane keeps N = (0, 4) hypermultiplet H and sets N = (0, 4) Fermi multiplet Ξ to zero.

D5 and (0, 4) Fermi multiplet
When the D3-brane between the two D5-branes further attach on an D5 -brane, all the bosonic massless modes are set to zero. These boundary conditions correspond to the half of BPS boundary conditions in 3d N = 4 hypermultiplet Altogether there are four types of boundary conditions which correspond to the four types of junctions of branes; D3-NS-NS , D3-NS-D5 , D3-D5-NS and D3-D5-D5 . These intersections respectively admit boundary local operators which are involved in N = (0, 4) vector, twisted hyper, hyper and Fermi multiplets.

Boundary anomaly and linking number
When we consider chiral supersymmetric boundary conditions in 3d supersymmetric gauge theory, there also exists an anomaly contribution from bulk fields. It is argued in the analysis [24] of (0, 2) boundary conditions for 3d N = 2 theory, that the bulk fields may have half of the contributions as those from boundary fields. For SU (N ) the anomaly contribution is given by adjoint −N (3.14) 11 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , x 6 ) directions, and es in (x 0 , x 1 , x 2 , x 6 ) directions: branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in o directions, the world-volume theories on the D3-branes therefore are macroscopically two nal.
8 D3-branes in (x , x , x , x ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional. 8 We introduce NS5-branes with world-volumes in with world-volumes in (x 0 , x 1 , x 2 , x 7 , x 8 , x 9 ) directi x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-vol D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We wil bounded by all the 5-branes in the (x 2 , x 6 ) direction these two directions, the world-volume theories on t dimensional. 8 We introduce NS5-branes with world-volumes in (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) directions, D5-branes with world-volumes in (x 0 , x 1 , x 2 , x 7 , x 8 , x 9 ) directions, NS5 ′ -branes with world-volumes in (x 0 , x 1 , x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , x 6 ) directions, and D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional. 8 x , x , x , x ) directions, D5 -branes with world-volumes in (x , x , x , x , x , x ) directions, and D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional. 8 x , x , x , x ) directions, D5 -branes with world-volumes in (x , x , x , x , x , x ) directions D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-bran bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduct these two directions, the world-volume theories on the D3-branes therefore are macroscopicall dimensional. 8 x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , x 6 ) dir D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein these two directions, the world-volume theories on the D3-branes therefore are macrosc dimensional. 8 x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza these two directions, the world-volume theories on the D3-branes therefore are m dimensional. 8 x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to th these two directions, the world-volume theories on the D3-branes theref dimensional. 8 x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider th bounded by all the 5-branes in the (x 2 , x 6 ) directions. Accordin these two directions, the world-volume theories on the D3-brane dimensional.
x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will cons bounded by all the 5-branes in the (x 2 , x 6 ) directions. Ac these two directions, the world-volume theories on the D3 dimensional. where N, D b.c. stand for the Neumann and Dirichlet boundary condition for 3d N = 2 chiral multiplet scalar fields and N , D b.c. imply the Neumann and Dirichlet boundary conditions for 3d gauge field.
As discussed in section 2.2.5, gauge anomaly needs to be cancelled. Otherwise, the Neumann boundary condition for gauge field is not consistent. For example, the boundary gauge anomaly polynomial for 3d N = 4 U (N c ) gauge theory with N f fundamental hypermultiplets obeying (N , N ) boundary condition is given by where s is the field strength of U (N ) gauge field. The boundary anomaly polynomial for 3d N = 4 n i U (N i ) linear quiver gauge theory with bi-fundamental hypermultiplets obeying (N , N ) boundary conditions is where s i is the field strength of U (N i ) gauge field. As shown in Figure 1, the 3d N = 4 U (N c ) gauge theory with N f hypers can be constructed as a world-volume theory of stack of N c D3-branes intersecting with N f D5-branes stretched between two NS5-branes. Also the 3d N = 4 n i=1 U (N i ) linear quiver gauge theory with bi-fundamental hypermultiplets can be realized as n sets of N i D3-branes suspended between (n + 1) NS5-branes. We observe that the boundary non-Abelian gauge anomaly coefficient is encoded as the difference of linking numbers of the corresponding right and left NS5-branes.
The gauge anomaly cancellation can be achieved by introducing two-dimensional boundary degrees of freedom which are charged under the gauge symmetry. It can be seen that this is naturally achieved by considering consistent linking number of boundary NS5 -branes and making use of our identification of matter multiplets in brane box configuration.

Brane box configuration
Furthermore, we introduce several NS5-and NS5 -branes which bound the D3-branes at a finite distance in both x 6 and x 2 directions. This configuration leads to two-dimensional N = (0, 4) quiver gauge theory in which each box of N D3-branes defines U (N ) gauge symmetry factor. The gauge coupling of the gauge theory is given by where g s is the string coupling. ∆x 6 and ∆x 2 are the distance of two NS5-branes along x 6 and that of two NS5 -branes along x 2 . We call this brane box model. In the following section, we study the resulting N = (0, 4) quiver gauge theory by identifying the matter content and interactions.

D1-D5-KK system
Now we would like to study N = (0, 4) quiver gauge theory which is constructed from a periodic array of D3-branes in (x 2 , x 6 ) plane. In this section we analyze the spectrum of brane tiling model by taking the T-dual configuration [28].
The T-duality along x 6 turns the k NS5-branes into k Kaluza-Klein (KK) monopoles which can be described by a multi-centered Taub-NUT metric with non-trivial geometry along the directions (x 6 , x 7 , x 8 , x 9 ) where x 6 is the T-dual coordinate of x 6 . Since the original k NS5-branes coincide in the directions (x 7 , x 8 , x 9 ), the centers of the corresponding k KK monopoles also coincide. This means that the geometry contains A k−1 singularities. Also the T-duality along x 6 translates the D3-branes, D5-branes and D5 -branes into D2-branes, D6-branes and the D4 -branes respectively.

D1-branes on
Let us firstly consider the configuration in which D1-branes probing C 2 /Z k × C 2 /Z k . This corresponds to the T-dual D3-brane box model where neither D5-nor D5 -branes exist. Generically the 13 two-dimensional low-energy effective world-volume theory would preserve N = (0, 4) supersymmetry.
To identify the gauge theory on D1-branes at singularity, we start from the N D1-branes over C 4 and use the technique developed in [29]. The world-volume theory is 2d N = (8,8) supersymmetric U (N ) gauge theory. In terms of N = (0, 2) supermultiplets, it consists of N = (0, 2) gauge multiplet Υ, four N = (0, 2) adjoint chiral multiplets R, L, U and D, and three N = (0, 2) adjoint Fermi multiplets ∆, ∇ and Λ. We introduce four complexifed coordinates In terms of N = (0, 2) supermultiplets, J-and E terms are given by Here and in the following · stands for appropriate index contraction which includes gauge and global symmetry groups.
Let us take generators of orbifold group labeled by (a, k − a, 0, 0) and (0, 0, b, k − b) where a ≤ k, b ≤ k are positive integers. They act on the complex coordinates as Since the action of orbifold on the Chan-Paton factor is R CP = N ( I R I ) where R I is onedimensional unitary representation of orbifold, the orbifold turns the gauge symmetry group into [18] where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action (4.3), they are represented as According to the form of interaction terms (4.2) and chiral fields (4.5) in the presence of orbifold, three N = (0, 2) Fermi multiplets transform under the gauge group (4.4) as Each of kk boxes involves four N = (0, 2) chiral multiplets and three N = (0, 2) Fermi multiplets so that in total there exist 4kk N = (0, 2) chiral multiplets and 3kk N = (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the N = (0, 4) quiver gauge theories in terms of N = (0, 2) quiver is depicted in Figure 2.
14 supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavo Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the ch U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi m there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the The basic building block of the (0, 4) quiver gauge theories are shown in Figu chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the hypermultiplets while the other two U i,j and D i,j corresponding to vertical arro supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the comple coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that und the orbifold action ( orb1a 2.16), they are represented as Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in tot there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theor The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine in the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into tw types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to th (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J-and E-terms 13 supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry group Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the comp coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that un the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in t there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge the The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0 chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0 hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symm Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral mu U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follo the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplet there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_ 2. T chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fund hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows wil the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be pr (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J -and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J -and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry group Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the com coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that un the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in t there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge the The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0 chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0 hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to (0, 4) Fermi multiplets. They produce J-and E-terms  The N = (0, 2) chiral multiplets always appear as a pair of arrows with opposite orientations between adjacent gauge nodes. The pair of N = (0, 2) chiral multiplets R i j and L i j corresponding to horizontal arrows will form the bi-fundamental N = (0, 4) hypermultiplets while the pair of N = (0, 2) chiral multiplets U i j and D i j corresponding to vertical arrows will combine into the bi-fundamental N = (0, 4) twisted hypermultiplets (see Figure 3).
Meanwhile, the three N = (0, 2) Fermi multiplets split into two types. Two N = (0, 2) multiplets supermultiplets, J -and E terms are given by where · stands for appropriate index contraction which include gauge and Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respect the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fe there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets i The basic building block of the (0, 4) quiver gauge theories are shown i chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form hypermultiplets while the other two U i,j and D i,j corresponding to vertic the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Ferm types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edg (0, 4) Fermi multiplets. They produce J -and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry grou Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the com coordinates as The orbifold turns the gauge symmetry group into k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that u the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge th The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two ( chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental ( hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to (0, 4) Fermi multiplets. They produce J-and E-terms ∆ ij and ∇ ij represented as diagonal red edges produce the following J-and E-terms: They give rise to the interaction terms between N = (0, 4) hypermultiplets and N = (0, 4) twisted hypermultiplets. Unlike the N = (0, 2) chiral multiplets, the N = (0, 2) Fermi multiplets do not appear as a pair of arrows but rather a pair of links forming a V-shaped configuration as shown in Figure 4. We identify them with the Fermi multiplet in N = (0, 4) quiver.
The four types of interactions (4.7) can be easily read from closed loops in quiver diagram. For each of red edges ∆ i,i+(a,−b) and ∇ i,i−(a,b) in the Fermi multiplet, one can draw two triangles sharing the corresponding edge. Given the orientation of the Fermi edge, the triangles lead to two closed loops. If the orientation is directed from i to i + (a, −b) or to i − (a, b), the two loops contribute to J-terms associated to the corresponding Fermi multiplet. If the orientation is opposite, they contribute to the E-terms. Assigning positive and negative signs for clockwise and anti-clockwise loops respectively, we obtain the E-and J-terms (4.7) for ∆ i,i+(a,−b) and ∇ i,i−(a,b) (see Figure 5).
The remaining N = (0, 2) Fermi multiplets Λ ij sketched as the circular red edges transform as adjoint representation under the corresponding gauge group. They lead to J-and E-terms The J-terms (4.8) describe the interaction between hypermultiplets while the E-terms (4.9) describe the interaction between twisted hypermultiplets. These Fermi multiplets will combine with the N = (0, 2) gauge multiplets to form the N = (0, 4) vector multiplets. We represent the N = (0, 4) vector multiplet as an orange node (see Figure 6).
Here we can summarize a simple dictionary between the N = (0, 2) quiver and N = (0, 4) quiver as follows: • A circular orange node of N = (0, 4) vector multiplet consists of N = (0, 2) gauge node and N = (0, 2) adjoint Fermi multiplet Λ. The map is shown in Figure 6. + supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J -and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the complex coordinates as The orbifold turns the gauge symmetry group into k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and in total there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge theory. The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two (0, 2) chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamental (0, 4) hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combine into the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split into two types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted to the (0, 4) Fermi multiplets. They produce J -and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include g Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0 coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge n U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( je_88 2.15) and chiral field three (0, 2) Fermi multiplets transform under the gauge group ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi m The basic building block of the (0, 4) quiver gauge theories are chiral multiplets R i,j and L i,j corresponding to horizontal arrow hypermultiplets while the other two U i,j and D i,j corresponding the bi-fundamental (0, 4) twisted hypermultiplets. The three (0 types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as dia (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor sy Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which act coordinates as The orbifold turns the gauge symmetry group into k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chira U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it f the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multi there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the qui The basic building block of the (0, 4) quiver gauge theories are shown in Figure  chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-f hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multipl types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor sy Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which ac coordinates as The orbifold turns the gauge symmetry group into k ′ label the gauge nodes. As the chira U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it f the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi mult there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the qu The basic building block of the (0, 4) quiver gauge theories are shown in Figure  chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bihypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multip types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will b (0, 4) Fermi multiplets. They produce J-and E-terms supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include Let us take orbifold group action to be (a, −a, 0, 0) and (0, coordinates as The orbifold turns the gauge symmetry group into k, i 2 = 1, · · · , k ′ label the gauge n U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( je_88 2.15) and chiral fie three (0, 2) Fermi multiplets transform under the gauge group ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and thre there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi m The basic building block of the (0, 4) quiver gauge theories ar chiral multiplets R i,j and L i,j corresponding to horizontal arrow The E-and J-terms ( je_d1d5 3.30) associated to (ξ i a , ξ a i ) and the E-and J-terms ( We give a quiver diagram for adding N f D5-and N ′ f D5 ′ -branes to D1-branes on Z k × Z k ′ Figure   fig_04quiver_1 11. Each of k ′ blue square boxes which are horizontally aligned represent the SU (N f ) flav symmetry while each of k green square boxes which are vertically aligned represent the SU .30) associated to (ξ i a , ξ a i ) and the E-and J-terms ( je_d1d5' 24 branes over C 4 and use the technique developed in [18]. The world-volume theory supersymmetric U (N ) gauge theory. In terms of N = (0, 2) supermultiplets, it cons gauge multiplet U , four N = (0, 2) adjoint chiral multiplets R, L, U and D, and adjoint Fermi multiplets ∆, ∇ and Λ. We introduce four complexifed coordinate

3.33) associated to
In terms of N = (0, 2) supermultiplets are given by where · stands for appropriate index contraction which include gauge and flavor sy Let us take generators of orbifold group labeled by (a, k − a, 0, 0) and (0, 0, b, k ′ b ≤ k ′ are positive integers. They act on the complex coordinates as Since the action of orbifold on the Chan-Paton factor is R CP = N ( I R I ) w dimensional unitary representation of orbifold, the orbifold turns the gauge sym Douglas:1996sw [19] k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chira U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it fo the orbifold action ( According to the form of interaction terms ( Let us firstly consider the configuration in which D1-branes probing C 2 /Z k × C 2 /Z k ′ . This corresponds to the T-dual D3-brane box model where neither D5-nor D5 ′ -branes exist. Generically the two-dimensional low-energy effective world-volume theory would preserve N = (0, 4) supersymmetry.
To identify the gauge theory on D1-branes at the singularity, we will start from the N D1branes over C 4 and use the technique developed in Franco:2015tna [18]. The world-volume theory is 2d N = (8,8) supersymmetric U (N ) gauge theory. In terms of N = (0, 2) supermultiplets, it consists of N = (0, 2) gauge multiplet U , four N = (0, 2) adjoint chiral multiplets R, L, U and D, and three N = (0, 2) adjoint Fermi multiplets ∆, ∇ and Λ. We introduce four complexifed coordinates z 1 = x 6 ′ + ix 7 , In terms of N = (0, 2) supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take generators of orbifold group labeled by (a, k − a, 0, 0) and (0, They act on the complex coordinates as Since the action of orbifold on the Chan-Paton factor is R CP = N ( I R I ) where R I is onedimensional unitary representation of orbifold, the orbifold turns the gauge symmetry group into Douglas:1996sw [19] where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( According to the form of interaction terms ( supersymmetric U (N ) gauge theory. In terms of N = (0, 2) supermultiplets, it consists of N = (0, 2) gauge multiplet U , four N = (0, 2) adjoint chiral multiplets R, L, U and D, and three N = (0, 2) adjoint Fermi multiplets ∆, ∇ and Λ. We introduce four complexifed coordinates z 1 = x 6 ′ + ix 7 , In terms of N = (0, 2) supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take generators of orbifold group labeled by (a, k − a, 0, 0) and (0, They act on the complex coordinates as Since the action of orbifold on the Chan-Paton factor is R CP = N ( I R I ) where R I is onedimensional unitary representation of orbifold, the orbifold turns the gauge symmetry group into Douglas:1996sw [19] where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( According to the form of interaction terms ( Let us firstly consider the configuration in which D1-branes probing C 2 /Z k × C 2 /Z k ′ . This corresponds to the T-dual D3-brane box model where neither D5-nor D5 ′ -branes exist. Generically the two-dimensional low-energy effective world-volume theory would preserve N = (0, 4) supersymmetry.
To identify the gauge theory on D1-branes at the singularity, we will start from the N D1branes over C 4 and use the technique developed in Franco:2015tna [18]. The world-volume theory is 2d N = (8,8) supersymmetric U (N ) gauge theory. In terms of N = (0, 2) supermultiplets, it consists of N = (0, 2) gauge multiplet U , four N = (0, 2) adjoint chiral multiplets R, L, U and D, and three N = (0, 2) adjoint Fermi multiplets ∆, ∇ and Λ. We introduce four complexifed coordinates z 1 = x 6 ′ + ix 7 , In terms of N = (0, 2) supermultiplets, J-and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry groups.
Let us take generators of orbifold group labeled by (a, k − a, 0, 0) and (0, They act on the complex coordinates as Since the action of orbifold on the Chan-Paton factor is R CP = N ( I R I ) where R I is onedimensional unitary representation of orbifold, the orbifold turns the gauge symmetry group into Douglas:1996sw [19] k ′ label the gauge nodes. As the chiral multiplets R, L, U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that under the orbifold action ( According to the form of interaction terms (    supermultiplets, J -and E terms are given by where · stands for appropriate index contraction which include gauge and flavor symmetry gr Let us take orbifold group action to be (a, −a, 0, 0) and (0, 0, b, −b), which acts on the co coordinates as The orbifold turns the gauge symmetry group into where i = (i 1 , i 2 ), i 1 = 1, · · · , k, i 2 = 1, · · · , k ′ label the gauge nodes. As the chiral multiplets U and D describe the motions of D1-branes in z 1 , z 2 , z 3 and z 4 respectively, it follows that the orbifold action ( orb1a 2.16), they are represented as According to the form of interaction terms ( Each of kk ′ boxes involves four (0, 2) chiral multiplets and three (0, 2) Fermi multiplets and i there exist 4kk ′ (0, 2) chiral multiplets and 3kk ′ (0, 2) Fermi multiplets in the quiver gauge t The basic building block of the (0, 4) quiver gauge theories are shown in Figure   fig_04quiver 2. The two chiral multiplets R i,j and L i,j corresponding to horizontal arrows will form the bi-fundamenta hypermultiplets while the other two U i,j and D i,j corresponding to vertical arrows will combin the bi-fundamental (0, 4) twisted hypermultiplets. The three (0, 2) Fermi multiplets split in types. The two (0, 2) multiplets ∆ ij and ∇ ij represented as diagonal edges will be promoted (0, 4) Fermi multiplets. They produce J -and E-terms decomposes as a (0, 2) gauge multiplet and a (0, 2) adjoint Fermi multiplet. T in Figure   fig_04vm_dec ??.
• Lines of (0, 4) hyper and twisted hyper multiplets in (0, 4) quiver diagram becomes a directional line in the N = 1 quiver diagram. This is shown in Figure 2.
• Lines of (0, 4) hyper and twisted hyper multiple directional line in the N = 1 quiver diagram. This In this case the the gauge group is transform as bi-fundamental repres U (N ) 2,0 and they form two bi-fundamental (0, 4) hyperm and D i 1 ,i 1 transform as adjoint representation under th adjoint (0, 4)twisted hypermultiplets: There are also six (0, 2) Fermi multiplets and ∇ i 1 ,i 1 −1 transform as bi-fundamental repr they are promoted to bi-fundamental (0, 4) Fermi mult U (N ) 2,0 and they combine (0, 2) gauge multiplet to for can be encoded by (0, 2) quiver diagram shown in Fig  each factor of gauge symmetry groups. The blue arrows (0, 4)hypermultiplets and the green loops represent U a hypermultiplets. Four red edges between two nodes descr Λ.
There are twelve pairs of vanishing J-and E-terms The T-dual configuration is 2 × 2 brane box model illustrated in Figure 9. In this case, two NS5branes and two NS5 -branes are trivially identified going around x 6 and x 2 directions respectively. 20 Figure 2: Left: A quiver diagram for D1-branes on C 2 /Z 2 × C 2 . Right: D3 brane box configuration with which is T-dual to D1-branes on C 2 /Z 2 × C 2 . fig_44quiver decomposes as a (0, 2) gauge multiplet and a (0, 2) adjoint Fermi multiplet. The map is shown in Figure   fig_04vm_dec ??.
• Lines of (0, 4) hyper and twisted hyper multiplets in (0, 4) quiver diagram becomes a bidirectional line in the N = 1 quiver diagram. This is shown in Figure 2.
In addition, (0, 4) supersymmetry is enhanced to (4, hypermultiplets (R, L) and bi-fundamental (0, 4) Fermi m hypermultiplets and adjoint (0, 4) twisted hypermultipl promoted to (4, 4) vector multiplets. In the T-dual bra This is obtained by the dimensional reduction of the 3d The matter content consists of sixteen (0, 2) chiral multiplets and twelve (0, 2) Fermi multiplets where i = (i 1 , i 2 ) are pairs of modulo 2 gauge indices with i 1 = 1, 2 and i 2 = 1, 2. The quiver is shown in Figure   fig_z2z2 6. The blue and green arrows describe bi-fundamental (0, 4) hyper and twisted hyperm respectively. The red lines correspond to bi-fundamental (0, 4) Fermi multiplets while the combine with (0, 2) gauge multiplet into (0, 4) vector multiplets.  The N = (0, 4) supersymmetric quiver gauge theory of regular N D1-branes on C 2 /Z 2 × C 2 /Z 3 has gauge group (4.16) The matter content consists of 24 chiral multiplets and 18 Fermi multiplets The vanishing E-and J terms are given by The quiver diagram is shown in Figure 10. The T-dual brane box model is 2 × 3 D3-brane box model, as depicted in Figure 10 In this case gauge group is (4.20) Figure 2: Left: A quiver diagram for D1-branes on C 2 /Z 2 × C 2 . Right: D3 brane box configur with which is T-dual to D1-branes on C 2 /Z 2 × C 2 .
In this case the the gauge group is U (N ) 1,0 × U (N ) 2,0 . There are eight (0, 2) chiral multiplets There are also six (0, 2) Fermi multiplets  Figure   fig_44quiver 2. As usual, the node correspon each factor of gauge symmetry groups. The blue arrows represent R and L forming bi-fundam (0, 4)hypermultiplets and the green loops represent U and D forming bi-fundamental (0, 4)tw hypermultiplets. Four red edges between two nodes describe ∆ and ∇ while two red loops are ad Λ.
(i) (ii) • Lines of (0, 4) hyper and twisted hyper multiplets in (0, 4) quiver diagram becomes a bidirectional line in the N = 1 quiver diagram. This is shown in Figure 2.
24 and there are 36 chiral multiplets and 27 Fermi multiplets. The quiver diagram is shown in Figure  11. Note that unlike the above examples, the Fermi multiplets cannot be drawn as a bi-fundamental single line since each of N = (0, 2) Fermi multiplets connects to distinct four gauge nodes. This always happens when both k and k are larger than two. Therefore the Fermi multiplets generically take the V-shaped or X-shaped configuration.
The T-dual configuration is 3 × 3 boxes of D3-branes shown in Figure 11.

D1-D1 strings
They transform as the fundamental representations under the U (N ) gauge group. Under the SU (N f ) global symmetry, they transform as the anti-fundamental representation. The N = (0, 4) twisted hypermultiplet is coupled to U (N ) vector multiplet through E-term They also couple to the adjoint hypermultiplets (R, L) in (4.23) through the E-and J-term potentials for the Fermi multiplets (ζ, ζ) (4.28) The presence of these interactions implies that the D1-D5 strings become massive when the D1-branes move in the directions where the D5-branes span.
To sum, the N = (0, 4) gauge theory of D1-D5-D5 brane system is encoded in the quiver diagram of Figure 12. The two square nodes represent SU (N f ) and SU (N f ) flavor symmetries.
The T-dual brane configuration is depicted in Figure 12. NS5-and D5-branes are depicted as vertical bold and dotted lines while NS5 -and D5 -branes are depicted as horizontal bold and dotted lines. A single box of D3-branes surrounded by a NS5-brane and NS5 -brane correspond to the gauge node. The D3-branes intersect with N f D5-branes and N f D5 -branes which are associated to the SU (N ) f and SU (N ) f flavor symmetry.

D1
-D5-D5 branes on C 2 /Z k ×C 2 /Z k Generic brane box configuration consisting of a grid of k NS5-branes and k NS5 -branes is T-dual of D1-D5-D5 brane system on the singularity C 2 /Z k × C 2 /Z k . For simplicity let us concentrate on the generator of orbifold group labeled by (1, 1, 0, 0) and (0, 0, 1, 1). In order for the SU (2) C × SU (2) H R-symmetry of N = (0, 4) supersymmetry to be preserved, the orbifold groups Z k and Z k should be embedded in SU (2) H and SU (2) C respectively (4.31) In addition to the restriction on fields from D1-D1 strings, which we have already seen, there is a further restriction on fields from D1-D5 strings, D1-D5 strings and D5-D5 strings. The action of Z k should be embedded in the Chan Paton factors of N f D5-branes and that of Z k should be embedded in the Chan Paton factors of N f D5 -branes so that flavor symmetry group is arise from D1-D5-KK system discussed in Behrndt:1998nt, Kutasov:1998zh The twisted hypermultiplet scalar fields (σ, σ † ) capture the motions of D1-branes along 2 ′ 345 in which D5 ′ -branes span. The The (0, 4) Fermi multiplets are neutral under the gauge group and transform as the bi-fundamental representation under the SU (N f ) × SU (N ′ f ) global symmetry.

D1-D5-D5
Given the above D1-D5-D5 ′ brane system, the presence of KK monopoles lead to orbifolding of C 2 = R 6 ′ 789 by Z k and that of C 2 = R 2 ′ 345 by Z k ′ . To preserve the SU (2) C × SU (2) H R-symmetry of (0, 4) supersymmetry, one should embed the orbifold group Z k and Z k ′ in SU (2) ′ H and SU (2) ′ It follows from ( The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 9. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 10, which is compared with
The twisted hypermultiplet scalar fields (σ, σ † ) capture the motions of D1-branes along 2 ′ 345 in which D5 ′ -branes span. The (0, 4) supermultiplets transform as the fundamental representations under the gauge group and transform as the anti-fundamental representation under the SU (N ′ f ) global symmetry.

D1-D5 strings
The massless modes coming from the open strings stretched between the D1-branes and D5-branes give the (4, 4) hypermultiplets. In our brane construction ( The hypermultiplet scalar fields φ,φ † describe the motion of D1-branes along Note that the fundamental and anti-fundamental representations have the same contributions to the 't Hooft anomalies. While gauge anomaly cancellation is required for consistent gauge theory, global symmetry may be anomalous. In the IR, the current of the global symmetry of Lie algebra f can be holomorphic or anti-holomorphic, i.e. left-or right-moving. Then the corresponding global symmetry can be enhanced to the affine Lie algebra f of level |2Af| which act in the holomorphic or anti-holomorphic sector of the associated CFT depending on the sign of the anomaly coefficient Af. **TODO: Check the Abelian anomalies. In Mohri:1997ef [11] the theories on D1-branes at singularities the non-vanishing Abelian gauge anomalies are shown to be cancelled by a generalized Green-Schwarz mechanism. In Gadde:2013lxa [12] the addition of appropriate matter cancels the non-vanishing Abelian gauge anomalies. ** When we consider (0, 2) boundary conditions in 3d N = 2 theory, the anomaly coefficient also receives contribution from bulk fields. They have half of the contributions as those from boundary fields Dimofte:2017tpi [8]:  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let QL and QR be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQL = QL, ΓQR = QR where Γ = Γ0 · · · Γ9.
We introduce NS5-branes with world-volumes in ( All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional.
The twisted hypermultiplet scalar fields (σ, σ † ) capture the motions of D1-branes along 2 ′ 345 in which D5 ′ -branes span. The The hypermultiplet scalar fields φ,φ † describe the motion of D1-branes along the 6 ′ 789 in which D5branes extend. Each of the (0, 4) supermultiplets transform as the fundamental representation under the gauge group and transform as the anti-fundamental representation under the SU (Nf ) global symmetry. Note that the fundamental and anti-fundamental representations have the same contributions to the 't Hooft anomalies. While gauge anomaly cancellation is required for consistent gauge theory, global symmetry may be anomalous. In the IR, the current of the global symmetry of Lie algebra f can be holomorphic or anti-holomorphic, i.e. left-or right-moving. Then the corresponding global symmetry can be enhanced to the affine Lie algebra f of level |2Af| which act in the holomorphic or anti-holomorphic sector of the associated CFT depending on the sign of the anomaly coefficient Af. **TODO: Check the Abelian anomalies. In Mohri:1997ef [11] the theories on D1-branes at singularities the non-vanishing Abelian gauge anomalies are shown to be cancelled by a generalized Green-Schwarz mechanism. In Gadde:2013lxa [12] the addition of appropriate matter cancels the non-vanishing Abelian gauge anomalies. ** When we consider (0, 2) boundary conditions in 3d N = 2 theory, the anomaly coefficient also receives contribution from bulk fields. They have half of the contributions as those from boundary fields Dimofte:2017tpi [8]:  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let QL and QR be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQL = QL, ΓQR = QR where Γ = Γ0 · · · Γ9.

D1-D5
We introduce NS5-branes with world-volumes in ( All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional. 8 the space-time symmetry so that they decompose into (0, 4) twisted hypermultiplets and (0, 4) Fermi multiplets The twisted hypermultiplet scalar fields (σ, σ † ) capture the motions of D1-branes along 2 ′ 345 in which D5 ′ -branes span. The The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 9. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 10, which is compared with
In addition, (0, 4) supersymmetry is enhanced to (4,4) in such a way that b (i) (ii) • Lines of (0, 4) hyper and twisted directional line in the N = 1 quiver In this case the the gauge group is U (N ) 1 The matter content consists of sixteen (0, 2) chiral multiplets and twelve (0, 2) Fermi multiplets where i = (i 1 , i 2 ) are pairs of modulo 2 gauge indices with i 1 = 1, 2 and i 2 =  ation of open strings stretched between D5-and D5 ′ -branes leads to a pair of (0, 4) Fermi multiplets 4 .
ngs The quantization of open strings stretched between D5-and D5 ′ -branes leads to mi multiplets, as a pair of (0, 4) Fermi multiplets 4 .
rmi multiplets are neutral under the gauge group and transform as the bi-fundamental n under the SU (N f ) × SU (N ′ f ) global symmetry.
he gauge group and transform as the bi-fundamental bal symmetry.
presence of KK monopoles lead to orbifolding of C 2 rise from D1-D5-KK system discussed in Behrndt:1998nt, Kutasov:1998zh From ( d1d5a_modes 3.28)-( d5d5_modes 3.34) we see that only orbifold actions on (0, 4) Fermi multiplets from D1-D5 strings and from D1-D5 ′ strings are non-trivial, that is (ξ i a , ξ a i ) transforming as 2 under the SU (2) ′ C and (ζ ĩ a , ζã i ) trasforming as 2 under the SU (2) ′ H . Letãi 1 = 1, · · · , N ′ f and ai 2 = 1, · · · , Nf stand for i1-th SU (Nf ) ′ flavor indices and i2-th SU (Nf ) flavor indices respectively. Under the gauge group ( The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 11. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 12, which is compared with Figure   fig_boxquiver 26 (or FIgure figgzkzk ??).

24
Zk′ ⊂ SU (2) C ⊂ SO(4)2′345. (3.36) emb_orb In addition to the restriction on fields from D1-D1 strings, which we have already seen, there is a further restriction on fields from D1-D5 strings, D1-D5 ′ strings and D5-D5 ′ strings. The action of Zk′ should be embedded in the Chan Paton factors of Nf D5-branes and that of Zk should be embedded in the Chan Paton factors of N ′ f D5 ′ -branes so that flavor symmetry group is The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 11. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 12, which is compared with Figure   fig_boxquiver 26 (or FIgure figgzkzk ??).

24
f k in the Chan Paton factors of N ′ f D5 ′ -branes so that flavor symmetry group is From ( d1d5a_modes 3.28)-( d5d5_modes 3.34) we see that only orbifold actions on (0, 4) Fermi multiplets from D1-D5 strings and from D1-D5 ′ strings are non-trivial, that is (ξ i a , ξ a i ) transforming as 2 under the SU (2) ′ C and (ζ ĩ a , ζã i ) trasforming as 2 under the SU (2) ′ H . Letãi 1 = 1, · · · , N ′ f and ai 2 = 1, · · · , Nf stand for i1-th SU (Nf ) ′ flavor indices and i2-th SU (Nf ) flavor indices respectively. Under the gauge group ( The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 11. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 12, which is compared with Figure   fig_boxquiver 26 (or FIgure figgzkzk ??).

24
generator of orbifold group labeled by (1, 1, 0, 0) and (0, 0, 1, 1). In order for the SU (2) C × SU (2) H R-symmetry of (0, 4) supersymmetry to be preserved, the orbifold groups Z k and Z k ′ should be embedded in SU (2) ′ H and SU (2) ′ C respectively In addition to the restriction on fields from D1-D1 strings, which we have already seen, there is a further restriction on fields from D1-D5 strings, D1-D5 ′ strings and D5-D5 ′ strings. The action of Z k ′ should be embedded in the Chan Paton factors of N f D5-branes and that of Z k should be embedded in the Chan Paton factors of N ′ f D5 ′ -branes so that flavor symmetry group is The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 11. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 12, which is compared with Figure   fig_boxquiver 26 (or FIgure figgzkzk ??).

Example subsubsec_d1d5d5zkzk_eg
We will see that the N = (0, 4) theories which arise from D1-D5-KK system discussed in Behrndt:1998nt, Kutasov:1998zh, Berenste [24,25,26,27,28] is a special case of our model. D1-D5 ′ branes on C 2 /Z k The corresponding quiver diagram is given in Figure   figd1d5zk 13 Sugawara:1999qp, Okuyama:2005gq [27,29]. We give a quiver diagram for adding N f D5-and N f D5 -branes to D1-branes on Z k × Z k in Figure 13. Each of k blue square boxes which are vertically aligned represents the SU (N f ) flavor symmetry while each of k green square boxes which are horizontally aligned represents the SU (N f ) flavor symmetry. The blue and green edges are N = (0, 4) fundamental hypermultiplets (H i a , H a i ) and twisted hypermultiplets (T ĩ a , Tã i ). The red lines connecting between gauge nodes and SU (N f ) boxes are the Fermi multiplets (ξ i a , ξ a i ) and those connecting between gauge nodes and SU (N f ) boxes are the Fermi multiplets (ζ a i , ζ ĩ a ). The red dotted line is the neutral N = (0, 4) Fermi multiplets (γ ã a , γã a ) which arise from D5-D5 strings.
The E-and J-terms (4.25) associated to (ξ i a , ξ a i ) and the E-and J-terms (4.28) associated to (ζ ĩ a , ζã i ) become J E ξ (i1,i2) These interactions can be read off from the quiver diagram as in Figure 14 and 15. Figure 14: E-and J-terms associated to (ξ, ξ). A circular node labeled by a pair of two integers (i 1 , i 2 ) corresponds to (i 1 , i 2 )-th gauge factor. A blue square box labeled by an integer a i2 corresponds to a i2 -th SU (N f ) flavor factor. Figure 15: E-and J-terms associated to (ζ, ζ). A circular node labeled by a pair of two integers (i 1 , i 2 ) corresponds to (i 1 , i 2 )-th gauge factor. A green square box labeled by an integerã i1 corresponds tõ a i1 -th SU (N f ) flavor factor. under the gauge group and transform as the anti-fundamental representation under the SU (N ′ f ) global symmetry.
D1-D5 strings The massless modes coming from the open strings stretched between the D1-branes and D5-branes give the (4, 4) hypermultiplets. In our brane construction ( The hypermultiplet scalar fields φ,φ † describe the motion of D1-branes along the 6 ′ 789 in which D5branes extend. Each of the (0, 4) supermultiplets transform as the fundamental representation under the gauge group and transform as the anti-fundamental representation under the SU (Nf ) global symmetry.
• Lines of (0, 4) hyper and twisted hyper multiplets in (0, 4) quiver diagram becomes a bidirectional line in the N = 1 quiver diagram. This is shown in Figure 2.
• Lines of (0, 4) hyper and twisted hyper multiplets in (0, 4) quiver diagram directional line in the N = 1 quiver diagram. This is shown in Figure 2.

D1-D5-KK system
When k = 1 and N f = 0, our brane setup reduces to D1-D5 brane system on C 2 /Z k 5 . The lowenergy effective theory is N = (0, 4) gauge theory which is encoded by a set of inner quiver and outer quiver diagram shown in Figure 16. The inner quiver diagram is the affine A k −1 diagram with U (N ) gauge nodes corresponding to the orbifold of N D1-branes. The outer quiver diagram is the another affine A k −1 Dynkin diagram with SU (N f ) gauge nodes corresponding to the orbifold of N f D5-branes. Each gauge node comes up with a N = (0, 4) vector multiplet, consisting of a N = (0, 2) gauge multiplet and an adjoint N = (0, 2) Fermi multiplet Λ, and an adjoint N = (0, 4) hypermultiplet (L, R). Between gauge nodes there is a bi-fundamental N = (0, 4) twisted hypermultiplet (U, D) and Fermi multiplets (∆, ∇). As discussed in section 4.1.4, these are N = (4, 4) gauge theory encoded in the inner quiver. In addition, there are flavor nodes and links between the inner and outer quiver. They represent fundamental hypermultiplets (H, H) and Fermi multiplets (ξ, ξ). It is compatible with the quiver previously studied in [11,37,7]. At low energy the D1-D5-KK system is described by N = (0, 4) SCFT of central charge [36] c = 6N c N f k . (4.35) and the microscopic states correspond to the chiral primary operators in the CFT.
The T-dual configuration is (1 × k ) D3-brane boxes including N f D5-branes illustrated in Figure  16.

Mirrors
We can also consider more general N = (0, 4) quiver gauge theories in which both D5-and D5 -branes exist at generic singularity. In this case we can study two distinct N = (0, 4) gauge theories which map to the other under the S-duality in Type IIB string theory.
For example, let us consider 2 × 2 D3-brane box model in which each of boxes intersecting with flavor N f D5-and N f D5 -branes. Making use of our dictionary, we can easily obtain the quiver diagram illustrated in Figure 17. This is the generalized model of 2 × 2 D3-brane box model in Figure  9 involving N f flavor D5 and N f D5 -branes. When N f = 1 and N f = 1, this model is interesting in that it is invariant under the S duality in type IIB string theory, which indicates that the model is self mirror. We postpone further analysis of this issue to the future.

Brane box model
Now that we have identified the matter content and interaction of N = (0, 4) gauge theory for periodic D3-brane box configuration from T-dual configuration of D1-D5-KK system, we would like to further discuss the detail of the brane box model.

Anomaly constraint
As discussed in section 2.2.5, N = (0, 4) gauge theory must be free from gauge anomaly. Consequently only appropriate choices of gauge group and matter content are admitted. To see this constraint in the brane box construction, let us consider a box of N D3-branes surrounded by eight adjacent boxes filled by D3-branes. Taking into account the N = (0, 4) boundary conditions in subsection 3.2.1, this leads to N = (0, 4) U (N ) vector multiplet. Without any matter multiplets, the N = (0, 4) U (N ) vector multiplet is anomalous.
In the absence of flavor 5-branes, an anomaly free theory is obtained by filling the adjacent D3branes in (x 2 , x 6 ) plane. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-branes displaced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bottom and bottom-right (see Figure 18). Applying the brane rules for matter multiplets as above, the horizontally aligned n L and n R D3-branes introduce n L and n R N = (0, 4) bi-fundamental hypermultiplets, the vertically aligned n T and n B D3-branes provide n T and n B N = (0, 4) bi-fundamental twisted hypermultiplets, and the diagonally aligned n T L , n T R , n b and n BR D3-branes lead to n T L , n T R , n b and n BR bi-fundamental Fermi multiplets. According to the anomaly contribution (2.34), the condition set by the f 2 su(N ) gauge anomaly is given by N = 1 2 (n L + n R + n T + n B ) − 1 4 (n T L + n T R + n BL + n BR ). The (0, 4) Fermi multiplets are neutral under the gauge group and transform as the bi-fundamental representation under the SU(Nf ) × SU(N ′ f ) global symmetry.
Given the above D1-D5-D5 ′ brane system, the presence of KK monopoles lead to orbifolding of C 2 = R 6 ′ 789 by Zk and that of C 2 = R 2 ′ 345 by Zk′. To preserve the SU(2)C × SU(2)H R-symmetry of (0, 4) supersymmetry, one should embed the orbifold group Zk and Zk′ in SU(2) ′ H and SU(2) ′ It follows from ( The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 9. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 10, which is compared with

21
Note that the fundamental and anti-fundamental representations have the same contributions to the 't Hooft anomalies. While gauge anomaly cancellation is required for consistent gauge theory, global symmetry may be anomalous. In the IR, the current of the global symmetry of Lie algebra f can be holomorphic or anti-holomorphic, i.e. left-or right-moving. Then the corresponding global symmetry can be enhanced to the affine Lie algebra f of level |2Af| which act in the holomorphic or anti-holomorphic sector of the associated CFT depending on the sign of the anomaly coefficient Af. **TODO: Check the Abelian anomalies. In Mohri:1997ef [11] the theories on D1-branes at singularities the non-vanishing Abelian gauge anomalies are shown to be cancelled by a generalized Green-Schwarz mechanism. In Gadde:2013lxa [12] the addition of appropriate matter cancels the non-vanishing Abelian gauge anomalies. ** When we consider (0, 2) boundary conditions in 3d N = 2 theory, the anomaly coefficient also receives contribution from bulk fields. They have half of the contributions as those from boundary fields Dimofte:2017tpi [8]:  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let QL and QR be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQL = QL, ΓQR = QR where Γ = Γ0 · · · Γ9.
We introduce NS5-branes with world-volumes in ( All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional.
The hypermultiplet scalar fields φ,φ † describe the motion of D1-branes along The The hypermultiplet scalar fields φ,φ † describe the motion of D1-branes along Note that the fundamental and anti-fundamental representations have the same contributions to the 't Hooft anomalies. While gauge anomaly cancellation is required for consistent gauge theory, global symmetry may be anomalous. In the IR, the current of the global symmetry of Lie algebra f can be holomorphic or anti-holomorphic, i.e. left-or right-moving. Then the corresponding global symmetry can be enhanced to the affine Lie algebra f of level |2Af| which act in the holomorphic or anti-holomorphic sector of the associated CFT depending on the sign of the anomaly coefficient Af. **TODO: Check the Abelian anomalies. In Mohri:1997ef [11] the theories on D1-branes at singularities the non-vanishing Abelian gauge anomalies are shown to be cancelled by a generalized Green-Schwarz mechanism. In Gadde:2013lxa [12] the addition of appropriate matter cancels the non-vanishing Abelian gauge anomalies. ** When we consider (0, 2) boundary conditions in 3d N = 2 theory, the anomaly coefficient also receives contribution from bulk fields. They have half of the contributions as those from boundary fields Dimofte:2017tpi [8]:  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let QL and QR be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQL = QL, ΓQR = QR where Γ = Γ0 · · · Γ9.
We introduce NS5-branes with world-volumes in ( All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional. The structure of flavor symmetries in (0, 4) quiver is illustrated in Figure   fig_04quiver_1 9. The square boxes connected by horizontal line represent the flavor D5 ′ -branes while those along vertical boxes represent the flavor D5-branes. The blue, green and red dotted edges are stretched between D1 gauge node and D5 flavor box or D1 gauge node and D5 ′ flavor box describe the (0, 4) fundamental hyper, twisted hyper and Fermi multiplets respectively. They come from the massless modes of D1-D5 string and those of D1-D5 ′ string. The pink dotted line is neutral (0, 4) Fermi multiplets which arise from D5-D5 ′ string. To illustrate such a general structure of the (0, 4) quiver, we show the effect of adding D5 and D5 ′ branes in Figure   fig_d1d5d5zkzk 10, which is compared with

21
Note that the fundamental and anti-fundamental representations have the same contributions to the 't Hooft anomalies. While gauge anomaly cancellation is required for consistent gauge theory, global symmetry may be anomalous. In the IR, the current of the global symmetry of Lie algebra f can be holomorphic or anti-holomorphic, i.e. left-or right-moving. Then the corresponding global symmetry can be enhanced to the affine Lie algebra f of level |2Af| which act in the holomorphic or anti-holomorphic sector of the associated CFT depending on the sign of the anomaly coefficient Af. **TODO: Check the Abelian anomalies. In Mohri:1997ef [11] the theories on D1-branes at singularities the non-vanishing Abelian gauge anomalies are shown to be cancelled by a generalized Green-Schwarz mechanism. In Gadde:2013lxa [12] the addition of appropriate matter cancels the non-vanishing Abelian gauge anomalies. ** When we consider (0, 2) boundary conditions in 3d N = 2 theory, the anomaly coefficient also receives contribution from bulk fields. They have half of the contributions as those from boundary fields Dimofte:2017tpi [8]:  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let QL and QR be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQL = QL, ΓQR = QR where Γ = Γ0 · · · Γ9.

8
Note that the fundamental and anti-fundamental representations have the same contributions to the 't Hooft anomalies. While gauge anomaly cancellation is required for consistent gauge theory, global symmetry may be anomalous. In the IR, the current of the global symmetry of Lie algebra f can be holomorphic or anti-holomorphic, i.e. left-or right-moving. Then the corresponding global symmetry can be enhanced to the affine Lie algebra f of level |2Af| which act in the holomorphic or anti-holomorphic sector of the associated CFT depending on the sign of the anomaly coefficient Af. **TODO: Check the Abelian anomalies. In Mohri:1997ef [11] the theories on D1-branes at singularities the non-vanishing Abelian gauge anomalies are shown to be cancelled by a generalized Green-Schwarz mechanism. In Gadde:2013lxa [12] the addition of appropriate matter cancels the non-vanishing Abelian gauge anomalies. ** When we consider (0, 2) boundary conditions in 3d N = 2 theory, the anomaly coefficient also receives contribution from bulk fields. They have half of the contributions as those from boundary fields Dimofte:2017tpi [8]:  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let QL and QR be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQL = QL, ΓQR = QR where Γ = Γ0 · · · Γ9.
We introduce NS5-branes with world-volumes in ( All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional. 8 Figure 2: Left: A quiver diagram for D1-branes on C 2 /Z 2 × C 2 . Right: D3 brane box confi with which is T-dual to D1-branes on C 2 /Z 2 × C 2 . fig_44quiver decomposes as a (0, 2) gauge multiplet and a (0, 2) adjoint Fermi multiplet. The map in Figure   fig_04vm_dec ??.

Examples sec_tiling_eg1
As a simple example let us consider two sets of horizontal tilings of a single D3-branes in which e tiling intersects with a single D5-brane and a single D5 ′ -brane ( Figure   fig_tiling1 13). Two tilings of a sin D3-brane lead to G = U (1) 1 × U (1) 2 gauge symmetry and intersecting D5-brane and D5 introd U (1) F and U (1) F ′ global symmetry for each node. There is a single bi-fundamental hypermultip H between two U (1) gauge nodes. For each U (1) gauge node, there is a set of a hypermultiplet twisted hypermultiplet t and Fermi multiplet γ which transform as the fundamental representat under the U (1) gauge group. Besides, there are two neutral Fermi multiplets ξ's under the ga group. There is a complex mass parameter m charged under the flavor U (1) F ′ 's. The matter fie 28 fig_tiling1 Without any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge anom free theory from non-periodic configuration, one can take further D3-branes with infinite exten either x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-br displaced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bot and bottom-right. The anomaly coefficient is related to the linking number of NS5-branes.

Examples ssec_tiling_eg1
As a simple example let us consider two sets of horizontal tilings of a single D3-branes in which tiling intersects with a single D5-brane and a single D5 ′ -brane ( Figure   fig_tiling1 13). Two tilings of a s D3-brane lead to G = U (1) 1 × U (1) 2 gauge symmetry and intersecting D5-brane and D5 intro U (1) F and U (1) F ′ global symmetry for each node. There is a single bi-fundamental hypermult H between two U (1) gauge nodes. For each U (1) gauge node, there is a set of a hypermultiple twisted hypermultiplet t and Fermi multiplet γ which transform as the fundamental representa under the U (1) gauge group. Besides, there are two neutral Fermi multiplets ξ's under the g group. There is a complex mass parameter m charged under the flavor U (1) F ′ 's. The matter fi 28 fig_tiling1 Without any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge ano free theory from non-periodic configuration, one can take further D3-branes with infinite exte either x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-b displaced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bo and bottom-right. The anomaly coefficient is related to the linking number of NS5-branes.

Examples sssec_tiling_eg1
As a simple example let us consider two sets of horizontal tilings of a single D3-branes in which tiling intersects with a single D5-brane and a single D5 ′ -brane ( Figure   fig_tiling1 13). Two tilings of a s D3-brane lead to G = U (1) 1 × U (1) 2 gauge symmetry and intersecting D5-brane and D5 intro U (1) F and U (1) F ′ global symmetry for each node. There is a single bi-fundamental hypermul H between two U (1) gauge nodes. For each U (1) gauge node, there is a set of a hypermultip twisted hypermultiplet t and Fermi multiplet γ which transform as the fundamental represent under the U (1) gauge group. Besides, there are two neutral Fermi multiplets ξ's under the g group. There is a complex mass parameter m charged under the flavor U (1) F ′ 's. The matter 28 Figure 13: U (1) × U (1) gauge theory with U (1) 4 flavor symmetry.
Without any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge anomaly free theory from non-periodic configuration, one can take further D3-branes with infinite extent in either x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-branes displaced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bottom and bottom-right. The anomaly coefficient is related to the linking number of NS5-branes.

Examples
As a simple example let us consider two sets of horizontal tilings of a single D3-branes in which each tiling intersects with a single D5-brane and a single D5 ′ -brane ( Figure   fig_tiling1 13). Two tilings of a single D3-brane lead to G = U (1) 1 × U (1) 2 gauge symmetry and intersecting D5-brane and D5 introduce U (1) F and U (1) F ′ global symmetry for each node. There is a single bi-fundamental hypermultiplet H between two U (1) gauge nodes. For each U (1) gauge node, there is a set of a hypermultiplet h,  Without any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge anoma free theory from non-periodic configuration, one can take further D3-branes with infinite extent either x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-bran displaced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, botto and bottom-right. The anomaly coefficient is related to the linking number of NS5-branes.  Without any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge anomaly free theory from non-periodic configuration, one can take further D3-branes with infinite extent in either x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-branes displaced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bottom and bottom-right. The anomaly coefficient is related to the linking number of NS5-branes. ut any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge anomaly eory from non-periodic configuration, one can take further D3-branes with infinite extent in x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-branes ced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bottom ottom-right. The anomaly coefficient is related to the linking number of NS5-branes. hout any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge anomaly theory from non-periodic configuration, one can take further D3-branes with infinite extent in er x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-branes laced in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bottom bottom-right. The anomaly coefficient is related to the linking number of NS5-branes.

Examples
Examples simple example let us consider two sets of horizontal tilings of a single D3-branes in which each g intersects with a single D5-brane and a single D5 ′ -brane ( Figure   fig_tiling1 13). Two tilings of a single brane lead to G = U (1) 1 × U (1) 2 gauge symmetry and intersecting D5-brane and D5 introduce ) F and U (1) F ′ global symmetry for each node. There is a single bi-fundamental hypermultiplet etween two U (1) gauge nodes. For each U (1) gauge node, there is a set of a hypermultiplet h, ted hypermultiplet t and Fermi multiplet γ which transform as the fundamental representation er the U (1) gauge group. Besides, there are two neutral Fermi multiplets ξ's under the gauge p. There is a complex mass parameter m charged under the flavor U (1) F ′ 's. The matter fields 28 Figure 13: U (1) × U (1) gauge theory with U (1) 4 flavor symmetry. thout any matter content, (0, 4) U (N ) vector multiplet is anomalous. To obtain gauge anomaly e theory from non-periodic configuration, one can take further D3-branes with infinite extent in her x 2 or x 6 direction. Let n T L , n T , n T R , n L , n R , n BL , n B , n BR be the numbers of D3-branes placed in the neighboring infinite regions in top-left, top, top-right, left, right, bottom-left, bottom d bottom-right. The anomaly coefficient is related to the linking number of NS5-branes.

Examples
a simple example let us consider two sets of horizontal tilings of a single D3-branes in which each ng intersects with a single D5-brane and a single D5 ′ -brane ( Figure   fig_tiling1 13). Two tilings of a single -brane lead to G = U (1) 1 × U (1) 2 gauge symmetry and intersecting D5-brane and D5 introduce 1) F and U (1) F ′ global symmetry for each node. There is a single bi-fundamental hypermultiplet between two U (1) gauge nodes. For each U (1) gauge node, there is a set of a hypermultiplet h, isted hypermultiplet t and Fermi multiplet γ which transform as the fundamental representation der the U (1) gauge group. Besides, there are two neutral Fermi multiplets ξ's under the gauge up. There is a complex mass parameter m charged under the flavor U (1) F ′ 's. The matter fields 28 Figure 18: A box of N D3-branes surrounded by eight adjacent boxes filled by D3-branes. Blue, green and red solid edges correspond to hyper, twisted hyper and Fermi multiplets respectively. Blue and green dotted edges represent 3d hyper and twisted hypermultiplets respectively. Red dotted edges are neutral Fermi multiplets.
In the absence of D5-or D5 -branes, this constraint on the occupation numbers of D3-branes in the adjacent boxes is required for gauge anomaly cancellation.

Junction tetravalent Fermi multiplet
Although the condition (5.1) fixes the f 2 su(N ) gauge anomaly, the Abelian part is still anomalous because the Abelian gaugino has no contribution to the gauge anomaly. To see this precisely, let us compute the Abelian gauge anomaly polynomial by setting all the numbers of D3-branes to be one. We denote the field strength for the gauge and flavor symmetries which are associated to the multiplicities of D3-branes in Figure 18 as The two-dimensional fields which are illustrated as solid lines in Figure 18 have the following contributions to the Abelian gauge anomaly polynomial: The non-vanishing terms which include the field strength s of the Abelian gauge field, provide the anomalous contributions. In addition, the three-dimensional bulk matter fields which are illustrated as dotted blue and green lines obeying the boundary conditions lead to the following contributions to the Abelian global anomaly: D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3 bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein these two directions, the world-volume theories on the D3-branes therefore are macrosc dimensional. 8 with world-volumes in (x 0 , x 1 , x 2 , x 7 , x 8 , x 9 ) directions, NS5 ′ -branes with wo x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in w bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Ka these two directions, the world-volume theories on the D3-branes therefore a dimensional. Figure 19: The tetravalent Fermi multiplet living at NS5-NS5 junction. This is required to cancel the Abelian gauge anomaly in such a way that it couples to the Abelian parts of quadrant of the gauge or/and global symmetries. It has the same charges along diagonal boxes.
Furthermore, there are four neutral Fermi multiplets illustrated as red dotted lines in Figure 18 which contribute to the Abelian global anomaly Collecting the contributions (5.3)-(5.5), we have The first and second lines are the Abelian global anomaly while the last line is the (mixed) Abelian gauge anomaly. It is quite remarkable that the remaining Abelian gauge anomaly and Abelian global anomaly can be completely canceled by taking into account the additional tetravalent Fermi multiplets which are charged under the quadrant of D3-branes. They can give rise to anomaly contributions This beautifully cancels the anomaly polynomial (5.6) ! This anomaly computation strongly indicates that there are additional Fermi multiplets living at the NS5-NS5 intersection in the brane box configuration 6 . As in Figure 19, the charges are equal on diagonal boxes and opposite on boxes which share a line. They will only couple to the Abelian parts of gauge and/or global symmetries for quadrant of D3-branes separated by NS5-and NS5 -branes. For non-Abelian gauge and/or global symmetries, we can easily check that the Abelian part of the anomaly is cancelled under the balancing condition of equation (

Brane box with flavor 5-branes
Let us consider a non-periodic array of D3-brane boxes. For non-periodic brane box configuration a pair of linking numbers can be introduced. It is defined as the number of 5-branes of the opposite kind that are to the left (bottom) of the given 5-branes plus net number of D3-branes ending on the 5-brane from the right (top) minus the number ending from the left (bottom). It is important to note that the linking numbers can be read off for each segment of 5-branes.
To consider the effect of adding D5-and D5 -branes in the box configuration, we take a box of N c D3-branes surrounded by two NS5-branes and two NS5 -branes which intersect with N f D5-branes and N f D5 -branes (see Figure 20). By keeping the linking numbers of 5-branes, we rearrange 5branes in such a way that all the D5-branes are located on the right and the D5 -branes are located on the top. The resulting configuration is illustrated in Figure 20.
Moving Similarly, moving N f horizontal D5 -branes to the top leads to N f D3-branes which are created in the adjacent boxes at the top-left, top, and top-right. Correspondingly, three types of edges give rise to N f fundamental Fermi multiplets ζ, N f fundametnal N = (0, 4) twisted hypermultiplets (T, T ), and N f fundamental Fermi multiplets ζ, respectively.
When N f D5-branes and N f D5 -branes intersect in the central box of N c D3 branes, neutral N f N f Fermi multiplets can be read off from dotted diagonal edge between the top box and the right box. They are coupled to N f hypers and N f twisted hypermultiplets.
The above brane rearrangement supports the analysis of the T-dual brane configuration in subsec-tion 4.2. In fact, the corresponding quiver diagram is again given in Figure 13. Therefore the quiver diagram of Figure 13 obtained from the computation from D1-D5-D5 brane system on Z k × Z k can be also derived by the Hanany-Witten move in the non-periodic box model. We present a dictionary between the quiver and the brane box model as follows:

2d-3d coupled system
Let us go back to the issue of N = (0, 4) boundary condition in section 3.3 and consider the 2d-3d coupled system realized in the brane box configuration. As we have discussed, the boundary gauge anomaly must be cancelled and it is encoded by the linking numbers. Remembering that Neumann boundary condition for a gauge field is realized by NS5 -brane, we would like to require that linking numbers of NS5 -branes defining the boundary should share the same linking numbers.
In addition to the above requirement, we further assume that linking numbers of NS5 -branes defining the boundary is not larger than the numbers of D3-branes terminating on the boundary NS5 -brane.
with world-volumes in (x , x , x , x , x , x ) directions, NS x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes i D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consi bounded by all the 5-branes in the (x 2 , x 6 ) directions. Acc these two directions, the world-volume theories on the D3dimensional. ll the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are ounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in hese two directions, the world-volume theories on the D3-branes therefore are macroscopically two imensional. All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction these two directions, the world-volume theories on the D3-branes therefore are macroscopically dimensional. All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein re these two directions, the world-volume theories on the D3-branes therefore are macroscop dimensional. All the branes share the (x 0 , x 1 ) directions. We will consider the case in which bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza these two directions, the world-volume theories on the D3-branes therefore are m dimensional. All the branes share the (x 0 , x 1 ) directions. We will consider the case in bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the K these two directions, the world-volume theories on the D3-branes therefore dimensional. All the branes share the (x 0 , x 1 ) directions. We will consider the bounded by all the 5-branes in the (x 2 , x 6 ) directions. According these two directions, the world-volume theories on the D3-branes t dimensional. This suggests that the N = (0, 4) boundary conditions in 3d N = 4 gauge theory can be labeled by the linking numbers of the boundary 5-branes. Now that we have a recipe to read off the N = (0, 4) matter content from the brane box configuration, we can determine what types of boundary degrees of freedom may appear in the boundary conditions for given linking numbers. It can be seen that the above simple requirement and our identification of matter multiplets consistently provides the appropriate boundary degrees of freedom which cancel the gauge anomaly.
Let us start with the brane construction of 3d N = 4 n i=1 U (N i ) linear quiver gauge theory with bi-fundamental hypermultiplets as in Figure 1. Using the above rules, we can choose a non-positive linking number −L ≤ 0 for the NS5 -brane in each segment. These linking numbers can be realized when a certain set of D3-branes exist across the boundary NS5 -brane as shown in Figure 21.
Using the dictionary (5.8), we can read off the matter fields which couple to the 3d bulk gauge fields. For gauge nodes U (N i ) with i = 2, · · · , n − 1, there are (N i + L) N = (0, 4) fundamental twisted hypermultiplets and (N i−1 + L) + (N i+1 + L) N = (0, 2) fundamental Fermi multiplets. Gauge nodes U (N 1 ) and U (N n ) are the ends of the quiver and the numbers of Fermi multiplets are smaller. For U (N 1 ) the number of Fermi multiplets is L + (N 2 + L), while for U (N n ) the number of Fermi multiplets is (N n−1 + L) + L. Recalling (2.34), we find f 2 su(Ni) anomaly contributions from the 2d boundary fields nes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are y all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in irections, the world-volume theories on the D3-branes therefore are macroscopically two l. ll the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are unded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in ese two directions, the world-volume theories on the D3-branes therefore are macroscopically two mensional. 8 We introduce NS5-branes with world-volumes in (x , x , x , x , x , x ) directions, D5-with world-volumes in (x 0 , x 1 , x 2 , x 7 , x 8 , x 9 ) directions, NS5 ′ -branes with world-volumes in ( x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , x 6 ) direction D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-bran bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduc these two directions, the world-volume theories on the D3-branes therefore are macroscopical dimensional. All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes ar bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction i these two directions, the world-volume theories on the D3-branes therefore are macroscopically tw dimensional. This shows that the set of boundary fields determined by the linking number of the boundary 5-brane and the dictionary (5.8) consistently produces a gauge anomaly free boundary condition. For 3d N = 4 U (N c ) gauge theory with N f hypermultiplets, we have f 2 su(N ) gauge anomaly (3.15). Making use of the rules above and rearranging the 5-branes, the boundary NS5 -brane with linking number −L introduces (N c + L) twisted hypermultiplets and L + (N f + L) Fermi multiplets (see Figure 22). The f 2 su(N ) anomaly contributions from these 2d boundary fields are given by Again this demonstrates that the rules above and the dictionary (5.8) consistently leads to gauge anomaly free boundary conditions with appropriate boundary degrees of freedom. We should note that the Abelian gauge anomaly again can be resolved by tetravalent Fermi multiplet living at the NS5-NS5 junction. Let us consider the simplest example of SQED with one hypermultiplet. The bulk hypermultiplet obeying the Neumann boundary condition has boundary U (1) gauge anomaly. Taking L = 0 for the linking number of NS5 -brane and rearranging 5-branes, one can read off one charged twisted hypermultiplet as vertical edge, one charged Fermi multiplet as diagonal edge and tetravalent Fermi multiplet living at the NS5-NS5 junction from Figure 23.
We denote the field strengths for 3d gauge symmetry and 3d global symmetry by f and a. Also we let b be the field strength for additional global symmetry under which the charged twisted hyper is charged and c be the field strength for the other under which the charged Fermi is charged. The 39 and n R D3-branes introduce n L and n R (0,4) fundamental hypermultiplets, vertically aligned n T and n B D3-branes provide n T and n B (0, 4) fundamental twisted hypermultiplets, and diagonally aligned n T L , n T R , n b and n BR D3-branes lead to n T L , n T R , n b and n BR (0, 2) fundamental Fermi multiplets. According to the anomaly contribution ( t_Anom2a 1.40), we find f 2 su(N ) gauge anomaly free condition N = 1 2 (n L + n R + n T + n B ) − 1 4 (n T L + n T R + n BL + n BR ). (4.1)

Brane ordering q
We will define a pair of net numbers of D3-branes ending on a 5-brane. One is the number of D3branes ending on the 5-brane from the right minus the number ending from the left while the other is the that of D3-branes ending on the 5-brane from the top minus the number ending from the bottom.
where N H is the number of (0, 4) fundamental hypermultiplet and N T is that of (0, 4) twisted hypermultiplets. When these conditions are obeyed at each node, the (0, 4) quiver gauge theories will be good or balanced.  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let Q L and Q R be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQ L = Q L , ΓQ R = Q R where Γ = Γ 0 · · · Γ 9 .
We introduce NS5-branes with world-volumes in (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) directions, D5-branes with world-volumes in (x 0 , x 1 , x 2 , x 7 , x 8 , x 9 ) directions, NS5 ′ -branes with world-volumes in (x 0 , x 1 , x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , x 6 ) directions, and D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional. 8 gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let Q L and Q R be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQ L = Q L , ΓQ R = Q R where Γ = Γ 0 · · · Γ 9 .
We introduce NS5-branes with world-volumes in (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) directions, D5-branes with world-volumes in (x 0 , x 1 , x 2 , x 7 , x 8 , x 9 ) directions, NS5 ′ -branes with world-volumes in (x 0 , x 1 , x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , x 6 ) directions, and D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional.  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let Q L and Q R be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQ L = Q L , ΓQ R = Q R where Γ = Γ 0 · · · Γ 9 .
We introduce NS5-branes with world-volumes in (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) directions, D5-branes with world-volumes in (x 0 , x 1 , x 2 , x 7 , x 8 , x 9 ) directions, NS5 ′ -branes with world-volumes in (x 0 , x 1 , x 6 , x 7 , x 8 , x 9 ) directions, D5 ′ -branes with world-volumes in (x 0 , x 1 , x 3 , x 4 , x 5 , x 6 ) directions, and D3-branes in (x 0 , x 1 , x 2 , x 6 ) directions: All the branes share the (x 0 , x 1 ) directions. We will consider the case in which the D3-branes are bounded by all the 5-branes in the (x 2 , x 6 ) directions. According to the Kaluza-Klein reduction in these two directions, the world-volume theories on the D3-branes therefore are macroscopically two dimensional.  [13,14] configurations of D3-branes and 5-branes were used to construct 3d N = 4 supersymmetric gauge theories. In this section we will generalize these brane configurations to construct 2d N = (0, 4) supersymmetric gauge theories. We consider Type IIB superstring theory in Minkowski spacetime with time coordinate x 0 and space coordinates x 1 , · · · , x 9 . Let Q L and Q R be the supercharges generated by left-and right-moving world-sheet degrees of freedom. They satisfy the chirality conditions of Type IIB superstring theory: ΓQ L = Q L , ΓQ R = Q R where Γ = Γ 0 · · · Γ 9 .

Brane ordering subsec_04ineq
We will define a pair of net numbers of D3-branes ending on a 5-b branes ending on the 5-brane from the right minus the number endi the that of D3-branes ending on the 5-brane from the top minus the where N H is the number of (0, 4) fundamental hyperm hypermultiplets. When these conditions are obeyed at e will be good or balanced.

28
To obtain gauge anomaly free theory from non-periodic c branes in (x 2 , x 6 ) plane. Let n T L , n T , n T R , n L , n R , n BL , displaced in the neighboring infinite regions in top-left, top, t and bottom-right (see Figure   fig_singlebox   13). From the dictionary ( dic_ 3.37 and n R D3-branes introduce n L and n R (0,4) fundamental hy n B D3-branes provide n T and n B (0, 4) fundamental twisted n T L , n T R , n b and n BR D3-branes lead to n T L , n T R , n b and n According to the anomaly contribution ( We will define a pair of net numbers of D3-branes ending o branes ending on the 5-brane from the right minus the numb the that of D3-branes ending on the 5-brane from the top min 1. Any D5-branes with non-zero net numbers of D3-brane NS5-branes and any D5 ′ -branes with non-zero net num the NS5 ′ -branes. This constraint requires that we shou described by 2d gauge theory, i.e. Nahm pole ρ : su(2) constraint is imposed in Gaiotto:2008ak [13] for NS55-and D5-branes. and D5 ′ -branes.
2. The linking numbers are nondecreasing from left to rig constraint is satisfied for D5-brane, the moduli space involve extra decoupled 3d N = 4 hypermultiplets. A imposed from D5 ′ -brane Chung:2016pgt [14], we also impose a simila additional 5-branes. Following the same line in Gaiotto: [13], thi leads to conditions in (0, 4) U (N ) gauge theory: where N H is the number of (0, 4) fundamental hyperm hypermultiplets. When these conditions are obeyed at e will be good or balanced.

28
To obtain gauge anomaly free theory from non-periodic configu branes in (x 2 , x 6 ) plane. Let n T L , n T , n T R , n L , n R , n BL , n B , n displaced in the neighboring infinite regions in top-left, top, top-righ and bottom-right (see Figure   fig_singlebox   13). From the dictionary ( dic_box 3.37) of b and n R D3-branes introduce n L and n R (0,4) fundamental hypermu n B D3-branes provide n T and n B (0, 4) fundamental twisted hyper n T L , n T R , n b and n BR D3-branes lead to n T L , n T R , n b and n BR (0 According to the anomaly contribution ( We will define a pair of net numbers of D3-branes ending on a 5branes ending on the 5-brane from the right minus the number end the that of D3-branes ending on the 5-brane from the top minus the 1. Any D5-branes with non-zero net numbers of D3-branes are lo NS5-branes and any D5 ′ -branes with non-zero net numbers o the NS5 ′ -branes. This constraint requires that we should first described by 2d gauge theory, i.e. Nahm pole ρ : su(2) → g a constraint is imposed in Gaiotto:2008ak [13] for NS55-and D5-branes. We im and D5 ′ -branes.
2. The linking numbers are nondecreasing from left to right and constraint is satisfied for D5-brane, the moduli space of so involve extra decoupled 3d N = 4 hypermultiplets. As Nah imposed from D5 ′ -brane Chung:2016pgt [14], we also impose a similar cons additional 5-branes. Following the same line in Gaiotto:2008ak [13], this cons leads to conditions in (0, 4) U (N ) gauge theory: where N H is the number of (0, 4) fundamental hypermultiple hypermultiplets. When these conditions are obeyed at each no will be good or balanced.
This shows that the boundary condition (N , N ) with linking number L = 0 for SQED with one hypermultiplet is free from gauge anomaly. The remaining global anomaly is beautifully resolved by further taking into account the anomaly contributions from 3d hyper and twisted hypermultiplets which live across the boundary with the Neumann boundary condition and the uncharged boundary Fermi multiplet which can be read off from vertical, horizontal and diagonal edges respectively: