Singular BPS boundary conditions in $\mathcal{N} = (2,2)$ supersymmetric gauge theories

We derive general BPS boundary conditions in two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories. We analyze the solutions of these boundary conditions, and in particular those that allow the bulk fields to have poles at the boundary. We also present the brane configurations for the half- and quarter-BPS boundary conditions of the $\mathcal{N}=(2,2)$ supersymmetric gauge theories in terms of branes in Type IIA string theory. We find that both A-type and B-type brane configurations are lifted to M-theory as a system of M2-branes ending on an M5-brane wrapped on a product of a holomorphic curve in $\mathbb{C}^2$ with a special Lagrangian 3-cycle in $\mathbb{C}^3$.

In particular supersymmetric boundary conditions in 2d N = (2, 2) supersymmetric field theories which describe D-branes [27,28,29] have attracted much attention in both physics and mathematics. For example, they can provide a physical setup to address the homological mirror symmetry conjecture [30] and gauge theoretic definition of knot homology [31,32]. The 2d N = (2, 2) supersymmetric field theory admits two types of half-BPS boundary conditions, that is the A-type and B-type boundary conditions [33] which preserve 1d N = 2 A and N = 2 B supersymmetries respectively. The D-brane of type B in a Calabi-Yau manifold X is argued to be equivalent to the bounded derived category of coherent sheaves on X [34,35,36,37,38,39] while that of type A is closely related to the Fukaya category [40,41].
The detailed analysis of B-type boundary conditions for Abelian gauge theories was presented in [42] and the basic B-type boundary conditions associated to the Neumann boundary conditions on the gauge field for non-Abelian gauge theories were studied in [43,44]. On the other hand, to our knowledge, A-type boundary conditions for gauge theories have been much less studied in the literature.
In this paper we examine more general A-type and B-type boundary conditions as well as quarter-BPS boundary conditions in N = (2, 2) supersymmetric non-Abelian gauge theories for which the gauge field may not satisfy the Neumann boundary condition. Although the problem of describing all half-BPS boundary conditions is enormously involved and our understanding of the boundary conditions for the non-Abelian gauge theories is far from complete, we find new types of BPS boundary conditions which admit singular solutions. Singular boundary conditions were found in higher dimensional BPS boundary conditions; e.g. the half-BPS boundary conditions [45,46] in 5d SYM theory, the BPS boundary conditions [1] and the quarter-BPS boundary conditions [3,4] in 4d N = 4 SYM theory and the N = (0, 4) boundary conditions [15] in 3d N = 4 gauge theories. Such higher dimensional and highly supersymmetric cases are described by Nahm's equation [47] as the Nahm pole boundary condition. We argue that the B-type boundary conditions in N = (2, 2) gauge theories, which are distinguished from the Nahm pole boundary condition, allow the bulk fields to have singularities even for the Abelian gauge theory when the gauge fields are not subject to the ordinary Neumann boundary condition. In particular, when the gauge field is subject to the Dirichlet boundary condition and the chiral multiplet scalar field satisfies the Neumann-type boundary condition, such singular solutions naturally arise without any boundary degrees of freedom. We further discuss the quarter BPS boundary conditions which have many more solutions, but much richer than the half-BPS case. They also admit singular solutions and mixed Dirichlet-Neumann boundary conditions.
We also construct the BPS boundary conditions in N = (2, 2) supersymmetric gauge theories by using brane configurations in Type IIA string theory by introducing additional branes to the Hanany-Hori brane setup [48]. For each of A-type and B-type boundary conditions, one can additionally introduce two kinds of NS5-branes and two kinds of D4-branes. The Neumann and Dirichlet boundary conditions for the vector multiplet are realized by the NS5-branes and D4-branes respectively and the two possible choices for each brane corresponds to those of boundary conditions for the chiral multiplet. We find singular solutions to some of the boundary conditions, as could be anticipated from our brane configurations which include systems of D2 and D4 branes orineted such that they are T-dual to the D1-D3 system realizng the Nahm equation [49,50]. Furthermore, the presence of both kinds of branes can preserve 1d N = 1 supersymmetry, which realizes the quarter-BPS boundary conditions. We find that the boundary conditions on fermions crucially determine the boundary branes as well as the boundary conditions on the bosonic fields. Both A-type and B-type boundary conditions can be lifted to M-theory as a system of a single M5-brane wrapped on a product of a holomorphic curve in C 2 with a special Lagrangian 3-cycle in C 3 as well as M2-branes.
The organization of the paper is straightforward. In section 2 we compute the half-BPS boundary conditions for 2d N = (2, 2) supersymmetric gauge theories and argue that they admit singularities. In section 3 we analyze the quarter-BPS boundary conditions. In section 4 we construct brane configurations realizing these BPS boundary conditions in Type IIA string theory which generalizes the Hanany-Hori brane setup [48]. We also describe the M-theory lift of these Type IIA brane configurations, showing that there is a rich set of duality relations, unifying all the configurations we consider. In appendix A we give our conventions and notations of superspace and supermultiplets.

Half-BPS boundary conditions
We consider 2d N = (2, 2) supersymmetric gauge theories on a half-space R × R + of the Minkowski space with coordinated (x 0 , x 1 ) whose boundary is at x 1 = 0. We analyze the supersymmetric boundary conditions imposed on the bulk fields without any boundary degrees of freedom. It would also be interesting to explore boundary conditions in the presence of coupling to boundary fields but we leave that for future work. Similarly it would be interesting to derive our results within the framework of supersymmetry without boundary conditions by including boundary interactions, developed by Belyaev and van Niewenhuizen [23].
The 2d N = (2, 2) supersymmetric gauge theories can be constructed in terms of three supermultiplets, the chiral, twisted chiral and vector multiplets. The twisted chiral multiplets arise in 2d, but the other multiplets come from direct dimensional reduction of hypermultiplet and vector multiplets in 4d N = 1 theories. These 2d theories have R-symmetry group U (1) A × U (1) V which arises from the 4d U (1) R-symmetry and the dimensional reduction to 2d. This R-symmetry group may be broken depending on the field content and superpotential.
The chiral multiplet consists of a complex scalar field φ and Dirac fermions ψ ± , ψ ± . The scalar φ has no U (1) A charge while the fermions ψ − , ψ + carry charge +1 and ψ − , ψ + have charge −1. The U (1) V charges of the fields can be shifted by a constant.
The vector multiplet has a two-dimensional gauge field A µ , a complex scalar field σ and gauginos λ ± , λ ± as Dirac fermions. They transform as the adjoint representation under the gauge group G. The gauge field is neutral under the R-symmetry group. The scalar field σ has no charge under the U (1) V but carries the U (1) A R-charge +2. The gauginos λ ± and λ ± carry the U (1) V R-charge −1 and +1. The gauginos λ + and λ − have the U (1) A R-charge +1 while the other gauginos have the U (1) A R-charge −1.
We use notation for 2d spinors ψ α where α can take values α = 1 = − or α = 2 = +, and ψ α denotes the complex conjugate of ψ α , or Hermitian conjugate for vector or matrix valued spinors. Our conventions for spinors are summarized in appendix A. 1.
In order to have normalizable supersymmetric ground states, the gauge theory must have large enough numbers of matter multiplets. For G = U (N c ) gauge theory with N f chiral multiplets transforming in the fundamental representation there are N f Nc supersymmetric ground states and supersymmetry is broken for N c > N f . 3 For G = SU (N c ) gauge theory with N f fundamental chiral multiplets, there is no supersymmetric ground state for 1 ≤ N f ≤ N c [51]. 4 The Lagrangian densities of 2d N = (2, 2) gauge theory are given by where L gauge and L chiral are the kinetic terms of the vector multiplet and those of the chiral multiplet which arise from the dimensional reduction of the 4d N = 1 Lagrangians (in the WZ gauge), and L FI are the 2d FI and theta angle terms (with θ here not to be confused with the Grassmann coordinates). When the gauge group G has an Abelian factor, the FI parameter ζ appears and we can include the FI term in 2d. The final (total derivative) terms in the gauge and chiral Lagrangians are required for the Lagrangians to be real in the presence of a boundary. More generally we can include N f fundamental chiral superfields Φ i (and also consider other representations) and include a superpotential W(Φ) and twisted superpotential W( Φ) which are holomorphic functions of chiral and twisted chiral superfields. These contribute to the action as For the supercurrent we find the following result for the gauge multiplet so explicitly 14) while for the fundamental chiral multiplet we have We can easily include twisted mass m for the chiral multiplet by simply shifting σ → σ − m and σ † → σ † −¯ m in the contributions from the chiral multiplet.
Finally, for the fundamental twisted chiral multiplet one finds We now have the condition to preserve supersymmetry at a boundary x 1 = 0. In the absence of boundary the conservation law ∂ 0 Q = 0 can be obtained by integrating the continuity equation ∂ µ J µ = 0 over the large volume. However, in the presence of boundary it may be violated due to the net supercurrent through the surface. Thus we demand that the component of the supercurrent orthogonal to the boundary vanishes at the boundary where ± and ± are the supersymmetry parameters.
We can now build general N = (2, 2) theories by combining these multiplets, choosing the gauge groups and matter field representations. For simplicity we focus mostly on gauge group U (N c ) and N f fundamental chiral multiplets. For a vector multiplet and one fundamental chiral multiplet this gives where we have defined the 2d chirality projectors and all spinor contractions are of the form λγ µ = λ α γ µ β α β . The generalization to N f fundamental chiral multiplets is straightforward, we just With implicit contraction of the flavor indices, the supercurrent is given by the same expression (2.40). It is also straightforward to generalize to chiral multiplets in other representation of the gauge group.
For the bulk equations of motion to be consistent the boundary terms must vanish when deriving the Euler-Lagrange equations. This leads to the following Euler-Lagrange boundary conditions on a boundary at fixed x 1 = 0 There are two ways to make the fermionic terms vanish while preserving the maximum number of half the independent components. We can impose either • A-type: a linear relation between the spinor and its complex conjugate, i.e. a type of reality condition • B-type: a projection condition on the spinor.
For λ we need to satisfy δλγ 1λ + δλγ 1 λ = 0 . (2.45) For gauge group U (1), taking A-type the most general solution is while for B-type the most general solution is where α is an arbitrary real constant. Similar conditions must be imposed on ψ. Note that these boundary conditions can be generalized in the case of larger gauge group (for λ) or multiple flavors (for ψ). This is discussed below in sections 2.1.5 and 2.2.8.
We will now see that these boundary conditions are compatible with supersymmetry, in particular allowing preservation of half of the supersymmetry, i.e. preserving two supercharges. For this we need to take either A-type for all spinors or B-type for all spinors.
Note that we must also impose suitable Euler-Lagrange boundary conditions on the bosonic fields in order to satisfy (2.43). The most obvious is to impose either Neumann or Dirichlet boundary conditions for each field, e.g. for φ we could have Neumann D 1 φ = D 1 φ † = 0 or Dirichlet δφ = δφ † = 0 so that φ takes a constant boundary value. For the gauge field the conditions are F 01 = θ/(4π) for Neumann or δA 0 = 0 for Dirichlet, which we may as well write as A 0 = 0. However, more involved conditions are also possible such as a Dirichlet condition on a linear combination of A 0 , σ and σ † without any of the individual fields having to satisfy Dirichlet or Neumann conditions. We will see an example of this in section 2.2.6.

A-type boundary conditions
We can impose a condition on the spinors as follows = e iϑ γ 1 ,λ = e iα γ 1 λ ,ψ = e 2iβ γ 1 ψ , (2.48) for some real α, β and ϑ. This leads to identifications of spinor bilinears such as So defining the boundary conditions for supersymmetry are Clearly we can set ϑ = 0 without loss of generality by redefining α and β, so we now do that. We then have two special choices of α = 0, π giving rise to Neumann (for c − = 0) or Dirichlet (for c + = 0) boundary conditions for the gauge multiplet complex scalar σ. For the Neumann case we also have a Neumann boundary condition for the gauge field, F 01 = 0 while for Dirichlet we see that the constant boundary values of σ and σ † must commute. If we impose gauge field boundary conditions A 0 = 0 for Dirichlet, these two options satisfy the boundary conditions (2.43) required for the variational principle, except that for the Neumann case the supersymmetry and Euler-Lagrange boundary conditions are consistent only for vanishing theta angle. For the Neumann case with θ = 0 we would also need to impose the Dirichlet boundary condition δ (Tr A 0 ) = 0 which would completely fix the U (1) part of the gauge field on the boundary. Of course, for gauge group U (N c ) = SU (N C ) × U (1) we could also impose different Dirichlet or Neumann boundary conditions for the SU (N c ) and U (1) gauge fields.
In general we need not choose α = 0 or α = π but let us first consider the basic boundary conditions, i.e. Neumann and Dirichlet boundary conditions which neatly project 2d N = (2, 2) supermultiplets onto 1d N = 2 A supermultiplets.

A-type Neumann b.c. for the vector multiplet
We start with the basic boundary conditions for the vector multiplet. For exp(iα) = 1 we find the Neumann boundary conditions for the vector multiplet: The condition (2.62) would be interpreted as the Neumann boundary condition for the gauge field when θ = 0 whereas (2.64) and (2.65) impose the Neumann boundary condition on σ and σ † , consistent with the Euler-Lagrange Neumann conditions. These A-type Neumann boundary conditions for the vector multiplet admit the 1d N = 2 A gauge multiplet which will be obtained from the 3d N = 1 gauge multiplet [53] or 2d N = (1, 1) gauge multiplet [54] and described by the (2, 2, 0) supermultiplet whose complex bosonic scalar fields are compatible with the remaining σ and σ † obeying the Neumann boundary conditions (2.64) and (2.65). While in the absence of the chiral multiplets, under the conditions (2.62), (2.64) and (2.65), the gauge symmetry could be completely preserved at the boundary, when the chiral multiplets are coupled to the vector multiplet, the UV boundary conditions get more complicated as additional conditions, including (2.63), are required. Of course, without chiral multiplets, or with chiral multiplets with boundary conditions φ = 0, we must take vanishing FI parameter in order to satisfy (2.63).

A-type Dirichlet b.c. for the vector multiplet
For exp(iα) = −1 we have the Dirichlet boundary conditions for the vector multiplet: The equations (2.67) and (2.68) fix the values of σ and σ † at the boundary. We also must impose the Dirichlet boundary condition for the gauge field. The A-type Dirichlet boundary conditions for the vector multiplet admit the 1d N = 2 A scalar multiplet described by the (1, 2, 1) supermultiplets whose bosonic physical scalar field is identified with the surviving A 1 component of the gauge field. Together with the condition (2.66), these boundary conditions can be solved by The Dirichlet boundary condition for the vector multiplet breaks the gauge symmetry G at the boundary and the global gauge transformation at the boundary would lead to the global symmetry G ∂ . We can also deform the condition (2.70) to where s = 0 is some constant satisfying [s, s † ] = 0. In this case, the flavor symmetry G ∂ is broken to the stabilizer of s, that is the subgroup whose action on s is trivial.

A-type b.c. for the chiral multiplet
Consider the boundary conditions for the chiral multiplets. Similarly to the choice of α = 0, π for the vector multiplet, we can take the special choices 2β = 0, π. For e 2iβ = 1 we have D 0 (Im(φ)) = 0 (2.72) while for e 2iβ = −1 we have D 0 (Re(φ)) = 0 (2.76) Note however that the second set of boundary conditions is equivalent to the first up to a redefinition φ → iφ. The boundary conditions (2.72) and (2.73) (or (2.76) and (2.75)) for the chiral multiplet are the Dirichlet and Neumann boundary conditions on the imaginary and real (or real and imaginary) part of φ, which are compatible with the Lagrangian splitting of the chiral multiplet to the 1d N = 2 A scalar multiplets described by the (1, 2, 1) supermultiplets whose bosonic physical scalar field is identified with the real (or imaginary) part of φ.
When the chiral multiplets are coupled to a vector multiplet, we have additional conditions (2.74) and (2.75) (or (2.78) and (2.79)). As they involve both the vector multiplet scalar field σ and the chiral multiplet scalar φ, the Coulomb branch and Higgs branch vevs obstruct one another.
Let us consider the case where the vector multiplet obeys the Dirichlet boundary conditions where σ is frozen according to the Dirichlet boundary conditions (2.67) and (2.68). The boundary condition (2.72) is solved by requiring that Im φ obeys the Dirichlet boundary conditions and that A 0 · Im φ vanishes. They can be solved by simply setting Imφ to zero. On the other hand, (2.73) can be solved by requiring that Re φ is subject to the Neumann boundary condition. Accordingly, the additional conditions (2.74) and (2.75) become Noting that φ = Re φ and defining σ S = (σ + σ T )/2 these equations are equivalent to There are two obvious solutions to this condition: σ S = 0 or φ = 0. When the vector multiplet satisfies Dirichlet boundary conditions, we can have solutions with constant non-zero σ at the boundary and this can allow non-zero values for both σ S and φ if σ S and σ † S have common eigenstates with eigenvalue 0.
For example, for G = SU (4) we could have for arbitrary real φ 3 and φ 4 . Instead, when the vector multiplet is subject to the Neumann boundary condition, the conditions (2.74) and (2.75) (or (2.78) and (2.79)) would require the modification of the basic boundary conditions. In particular when the U (1) vector multiplet satisfies the Neumann boundary conditions (2.64) and (2.65), we cannot find a solution to the constraints (2.74) and (2.75) (or (2.78) and (2.79)). This is because the Lagrangian splitting of a charged chiral multiplet leads to scalar multiplets which cannot be charged under the U (1) gauge symmetry.
While it is not clear whether or how one can obtain the Neumann boundary conditions, we note that our boundary conditions on fermions (2.48) may be generalized when the theory has multiple matter fields or/and higher rank gauge group, as we will see in section 2.1.5. In such a case, one may modify the additional conditions or ordinary Neumann boundary conditions to those which break G down to H and find some consistent Lagrangian splitting of chiral multiplets so that the surviving scalar multiplets are coupled to the H gauge field. In other words, it may be possible to obtain boundary conditions in which the gauge group G is broken to its subgroup H so that G decomposes as where g and h are the Lie algebras of G and H and h ⊥ is the orthocomplement that is not Lie algebra. Correspondingly, we split the adjoint-valued fields as where σ ∈ h and σ ⊥ ∈ h ⊥ . Then we would get the boundary conditions which reduce the gauge group G to its subgroup H at the boundary. In addition, one can introduce extra boundary terms which may modify or remove the constraints encountered above. We leave the issue of the Neumann boundary condition for the vector multiplet coupled to chiral multiplets to future work.

Generic A-type boundary conditions
Above we commented on the specific cases where either c + = 0 or c − = 0 but in general we will have the following equations If we define Note that (2.92) and (2.93) are mixed boundary conditions which may be problematic as they both contain derivatives with respect to x 0 and x 1 . Hence whether we impose Dirichlet or Neumann conditions on σ to satisfy (2.43), we will also have a constraint on the other derivative. Hence we should look for singular solutions for the generic A-type boundary conditions, otherwise we would only expect solutions with σ completely fixed.
However, although it could be interesting to explore singular solutions for the generic boundary conditions (2.87)-(2.97), we cannot find them for physical theories. If fact, as we will discuss in section 4.2.3, the brane configuration for this case does not admit such singular boundary behavior. As we show in appendix B it is possible to have singular boundary conditions, but only with a non-compact non-Abelian gauge group. In particular, the specific signs in (2.90), given (2.92) and (2.93) mean that we have Nahm equations which admit singular solutions but based on the Lie algebra su(1, 1) rather than su (2). It is not clear if such singular solutions can appear in any physically relevant cases. Also they are only possible for non-Abelian groups. If we have an Abelian theory, in the gauge A 1 = 0 we would have a boundary condition 0 = F 01 = −∂ 1 A 0 which clearly does not admit singular behavior of the gauge potential. Then the boundary conditions forσ also only admit regular solutions.

General gauge group projections and multiple matter multiplets
More generally, when we have a gauge group other than U (1) we can impose matrix projection conditions on λ a , and similarly for multiple matter multiplets ψ i . In other words, there are more general A-type boundary conditions by taking such matrix projection conditions on fermions. Specifically, we can impose the boundary conditions Together these conditions mean that R and S are both unitary and symmetric.
This leads to the following identifications of spinor bilinears Clearly we can absorb the factor e iϑ into R. So the boundary conditions for supersymmetry are Similarly for multiple chiral multiplets we have so again we can absorb e iϑ by redefining S ī j , giving supersymmetric boundary condi-tions¯ Note that R and S may depend on bosonic fields, which would lead to more general boundary conditions. Although we do not complete the analysis above in this paper, it would be interesting to explore such general boundary conditions.

B-type boundary conditions
The B-type conditions arise from imposing a projection condition on the spinors. The most general such boundary condition is The boundary conditions for supersymmetry then become Note that if we simultaneously shift α and β by ϑ and redefine σ by a phase factor e iϑ we can absorb all ϑ dependence in these boundary conditions. From now on we assume this has been done, which is equivalent to setting ϑ = 0, leaving us with If we choose e iα = ±1 then we can have Dirichlet type boundary conditions for σ + and Neumann type for σ − or vice-versa, where we define We also have additional simplification as some terms vanish in this case. Similarly, we can get either Dirichlet or Neumann type conditions for both φ and φ † for the choices e iβ = ±1. These can all happen simultaneously for the four special cases of e iα = ±1 and e iβ = ±1.
As in the A-type boundary conditions, let us first consider the basic boundary conditions, i.e. Neumann and Dirichlet boundary conditions (labelled here by the type of boundary condition obeyed by the field strength F 01 ) which project 2d N = (2, 2) supermultiplets onto 1d N = 2 B supermultiplets. Some basic boundary conditions were already discussed in [43,44].

B-type Neumann b.c. for the vector multiplet
Let us begin with the basic B-type boundary conditions for the vector multiplet. For exp(iα) = 1, we find the Neumann boundary conditions for the vector multiplet: The boundary conditions (2.140) and (2.141) are the Neumann boundary conditions for σ + and the Dirichlet boundary conditions for σ − respectively, consistent with the Euler-Lagrange boundary conditions if we also impose A 0 σ − = 0 at the boundary. This is analogous to the Lagrangian splitting of the A-type boundary conditions for the chiral multiplets in section 2.1.3. It allows for a Lagrangian submanifold L of the space g C of the Kähler manifold labeled by σ. The B-type Neumann boundary conditions for the vector multiplet are compatible with the 1d N = 2 B gauge multiplet described by the (1, 2, 1) supermultiplets [55] whose real bosonic scalar field is identified with σ + . As σ − obeys the Dirichlet boundary condition, the additional condition (2.142) can be simply solved by setting In this case, the gauge symmetry G can be preserved at the boundary as we need not impose any constraints on the boundary value of A 0 . Such Neumann boundary conditions for the vector multiplet were studied in the context of gauged linear sigma models [42,43,44]. We note that the Euler-Lagrange boundary conditions are satisfied we could have the Neumann boundary conditions F 01 = 0 for the SU (N c ) gauge fields, but if we impose (2.139) for the U (1) gauge field we would need to impose Dirichlet boundary conditions for the Abelian gauge field, fixing A 0 . When instead we choose where t − is some nonzero valued constant, the gauge symmetry G may be explicitly broken to H in such a way that σ + valued in H commutes with σ − according to the condition (2.142).

B-type Dirichlet b.c. for the vector multiplet
On the other hand, the Dirichlet boundary conditions for the vector multiplet can be obtained for exp(iα) = −1: The boundary conditions (2.145) and (2.146) are the Neumann boundary condition for σ − and Dirichlet boundary condition for σ + respectively, satisfying the Euler-Lagrange boundary conditions provided also A 0 σ + = 0. The B-type Dirichlet boundary conditions admit the 1d N = 2 B chiral multiplet described by the (2, 2, 0) supermultiplet whose complex scalar field corresponds to the A 1 component of gauge field and σ − . Similarly to the A-type, a simple solution would lead to the boundary flavor symmetry G ∂ as the global gauge transformation of G. Its deformation where t + is some nonzero real valued constant would break the flavor symmetry G ∂ to the stabilizer of t + . For example, for generic t + , the flavor symmetry is broken to its maximal torus.
In the presence of the chiral multiplets coupled to the vector multiplet, the additional condition (2.147) is imposed. For the Dirichlet boundary conditions (2.154) and (2.155) of the chiral multiplets with gauge group U (1) it can be simply solved by fixing the squared norm of φ as ζ/g. However, the Neumann boundary conditions (2.150) and (2.151), together with (2.145) imply that the boundary values of σ − and φ should not be fixed, but this is incompatible with (2.147), (2.152) and (2.153) which prevent the chiral multiplets from freely fluctuating at the boundary. This indicates that the pairing of the Dirichlet boundary condition for the vector multiplet and the Neumann boundary condition for the chiral multiplet should be generalized, as we will see below.

B-type Neumann b.c. for the chiral multiplet
For exp(iβ) = −1 we get Neumann boundary conditions for the chiral multiplet: The boundary conditions (2.150) and (2.151) are Neumann boundary conditions for the chiral multiplet φ. The B-type Neumann boundary conditions for the chiral multiplet admit the 1d N = 2 B chiral multiplet described by the (2, 2, 0) supermultiplets [55] whose complex bosonic scalar field is identified with φ. The additional conditions (2.152) and (2.153) involving the vector multiplet scalar σ − can be satisfied for the Neumann boundary condition of vector multiplet with σ − = 0 as in (2.143). But when σ − = 0 as in (2.144), the gauge symmetry G may be explicitly broken to H and (2.152) and (2.153) would consistently project out the scalar φ so that φ takes values in H. However, for the Dirichlet boundary condition of vector multiplet with (2.145), they cannot be simply solved. Again it indicates that we need to generalize the pairing of the Dirichlet boundary condition for the vector multiplet and the Neumann boundary condition for the chiral multiplet.

B-type Dirichlet b.c. for the chiral multiplet
For exp(iβ) = 1 we find Dirichlet boundary conditions for the chiral multiplet: The boundary conditions (2.154) and (2.155) are Dirichlet boundary conditions for the chiral multiplet φ provided we also impose A 0 φ = 0. The B-type Dirichlet boundary conditions for the chiral multiplet admit the 1d N = 2 B Fermi multiplet described by the (0, 2, 2) supermultiplets [55]. The additional constraints (2.156) and (2.157) can be solved by simply setting the boundary value of φ to zero. So a simple solution to (2.154)-(2.157) is This can lead to the maximal flavor symmetry G ∂ from the global transformation of the gauge symmetry when the vector multiplet obeys the Dirichlet boundary condition.

Singular B-type boundary conditions
We can find more general B-type boundary conditions. For e iα = 1 we find which generalize the Neumann boundary conditions for the vector multiplet. For e iα = −1, we get which contain the Dirichlet boundary conditions for the vector multiplet.
Similarly for e iβ = 1 we find which contain the Dirichlet boundary conditions for the chiral multiplet. For e iβ = −1 which generalize the Neumann boundary conditions for the chiral multiplet. It is convenient to define The chiral multiplet boundary conditions (2.163) and (2.165) can then be written as D 0 φ = 0 and D 1 φ = 0 respectively. This also allows us to rewrite the boundary conditions (2.159) and (2.160) for the vector multiplet as a single equation To get solutions of the general B-type boundary conditions, we first consider the boundary conditions (2.165) and (2.166) for the chiral multiplets, which generalize the Neumann boundary conditions (2.150) and (2.151). They play a key role to find singular solutions. In fact, when we consider the chiral multiplet φ = φ 1 + iφ 2 transforming in the adjoint representation, the equations (2.165) and (2.166) together with (2.162), which generalizes the Dirichlet boundary condition for the vector multiplet, are lifted to the well-known Nahm pole boundary conditions which admit singularity after identifying φ 1 , φ 2 and σ − with three real scalar fields. In terms of the generalized differential operators (2.168), the boundary conditions (2.165) and (2.166) for the chiral multiplets can be expressed as This would imply that the chiral multiplets are covariantly constant and therefore a gauge invariant polynomial in φ should take the same values at any value of x 1 . The equation (2.170) generalizes the Neumann boundary conditions for the chiral multiplet. In the axial gauge we find a singular solution to the boundary conditions (2.165) and (2.166) with the form where t is some constant element of the Lie algebra of the gauge group, which we take hermitian. Given the singular configuration (2.171) of σ − near the boundary, we further consider the boundary conditions for other fields in the vector multiplet. To find the solutions, we set A 1 = 0 by gauge transformation in the following.

Neumann b.c. for the vector multiplet with singularity
Consider the singular solution (2.171) along with the set of boundary conditions (2.159) and (2.160), which generalize the Neumann boundary conditions for the vector multiplet. For the basic Neumann boundary conditions for the vector multiplet that freezes σ − , the regular value (2.144) of σ − may break the gauge symmetry. But for the singular solution (2.171) of σ − , it turns out that the scaling behavior of the scalar φ is affected.
For the U (1) vector multiplet, we find a solution we find a solution where φ 0 is some constant value. When t is positive, φ vanishes at the boundary, which essentially realizes the Dirichlet boundary condition. When t is negative, φ has poles at the boundary of order |t|. Therefore the singularity (2.171) in σ − dramatically alters the boundary conditions for φ in such a way that the scaling behavior of φ is controlled by t.
For the vector multiplet of non-Abelian gauge group G, the commutator in the equation (2.159) allows the A 0 component of the gauge field to have non-zero value. Now, we still need to satisfy (2.43), and the contribution from the vector multiplet now requires One option is to take Dirichlet boundary conditions for σ − , i.e. in (2.175) we have δσ − = 0. Then the boundary condition (2.159) requires that A 0 and σ + take values in the same element of the Lie algebra, up to a term which commutes with σ − , which modifies the ordinary Neumann boundary condition (2.160). Specifically, where we require [σ − , Σ − ] = 0. Then (2.160) reduces to and (2.175) becomes simply This is solved by imposing Dirichlet boundary conditions on Σ − so it is fixed like σ − . As in the case of U (1), the boundary condition (2.170) for the chiral multiplet leads to a non-trivial profile as a solution to a matrix differential equation (2.179) For example, for G = SU (4) when we choose where α 1 , α 2 , α 3 and α 4 are some constant values. In this case, the gauge symmetry is completely broken and the Neumann boundary condition for φ changes so that all the components depend on x 1 . Alternatively, when we pick the gauge symmetry is broken to SU (2) and we find a solution where φ 3 and φ 4 obey the Neumann boundary condition. We see that the non-zero values in t modify the Dirichlet boundary condition of σ − (or equivalently the Neumann boundary condition for the vector multiplet) and the Neumann boundary condition on φ changes in such a way that the two components have nontrivial dependence on x 1 . As can easily be seen in the above example, in general the solutions are linear combinations of the eigenvectors of t. In particular if t has eigenvectors v i with eigenvalues t i , the general solution takes the form (2.184) Consequently, the gauge symmetry G is generically broken to some H depending on t.
In particular, when the chiral multiplet scalar field has a simple pole where a is the eigenvector of t with eigenvalue −1 according to the equation ( where c ∈ C is some constant. Plugging (2.188) into the condition (2.165), we find that σ − = ∓i/(gx 1 ), so we see that (2.43), or more specifically (2.175), is satisfied with σ − obeying Dirichlet boundary conditions. Together with the condition (2.162), this gives rise to |c| 2 = 1, i.e. c is an arbitrary phase. Recall that the ordinary Dirichlet boundary conditions for the U (1) vector multiplet together with the ordinary Neumann boundary conditions for the chiral multiplet have no simple solution. In the absence of boundary terms, the only consistent condition seems to be the singular configurations (2.171) and (2.188) which resolve this obstruction. For the non-Abelian case, we also find solutions (2.187) and the gauge symmetry is completely broken. However, c is a vector c rather than a constant: Then (2.165) requires it to be an eigenvector of t with eigenvalue −1, where σ − = ±it/(gx 1 ), and consequently (2.162) requires that Clearly c is automatically an eigenvector of t, and it has eigenvalue −1 provided c is normalized so that c † c = 1. Note that this means that Tr t = 0 so such solutions are possible with gauge group U (N c ) but not SU (N c ). In fact, the above gives the leading order behaviour at the boundary even without the specific assumptions in (2.188) or (2.189) if we simply assume that φ can be expressed as a Taylor series in x 1 , with non-zero constant term, multiplied by an arbitrary non-zero power of x 1 . That is because in this case (2.165) requires the leading behaviour of σ − to be a simple pole in x 1 and consequently (2.162) requires the leading order term for φ to also be a simple pole. We could solve (2.165) by including a more singular term in σ − provided it annihilated φ but that would not be consistent with (2.162). The only other alternative is that φ is a regular Taylor series, in which case (2.165) shows that σ − is also regular at the boundary. Then the boundary values of φ and σ − are arbitrary, with (2.162) and (2.165) determining the boundary values of their normal derivatives.
If we have multiple flavors, we can find more general solutions. In this case each flavor φ i satisfies (2.165) but (2.162) is generalized to However, we can use the flavor symmetry to do a field redefinition so that for each J we use a unitary transformation acting on the φ i with I(i) = J so that only one scalar has a singular part. Hence we will split the N f flavors φ i into N s with a singular boundary condition. Specifically, including the subleading regular terms we have N s scalars with orthonormal {c I } and the remaining N f − N s will have regular boundary conditions φÎ = φÎ 0 + x 1 φÎ 1 (2.193) where we have assumed a power series expansion of the fields near the boundary and included only those terms relevant for the boundary conditions. Similarly, writing the boundary condition  which means that all φ i 0 ∈ V r . Then, noting that ( showing that also all φ i 1 ∈ V r . This simplifies (2.200) to Note that the singular and regular parts of the solutions are completely independent other than the determination of the splitting into orthogonal subspaces V s and V r . In particular, if we choose φ i 0 = 0 and t 0 = 0 we have only singular terms in the solution, except for the FI parameter contribution to ∂ 1 σ − through t 1 .
Now we also consider the symmetries preserved by such boundary conditions. Since we have A 0 = 0 the gauge symmetry will become a global symmetry U (N c ) on the boundary. For N s > 0 this is explicitly broken to U (N s ) × U (N c − N s ) simply by the choice of V s . Without any regular terms (i.e. with φ i 0 = 0 and t 0 = 0) the form of σ − given in (2.194) is invariant, even with an FI parameter. However, the φ I given by (2.192) are not invariant under the U (N s ) transformations. Nevertheless, we can compensate using the explicitly broken U (N s ) subgroup of the flavor symmetry. So, together with the unbroken U (N f − N s ) flavor symmetry we have the global symmetry N s ). Now, introducing non-zero regular terms into the solutions will generically completely break the U (N c −N s ) and U (N f −N s ) symmetries, but since the regular terms are related to V r only, the form of φÎ in (2.193) and of σ − in (2.194) will not break the U (N s ) global symmetry. So, the U (N s ) symmetry is only broken by the regular terms in φ I in (2.192), since these are invariant under the U (N s ) from the gauge symmetry but not under the compensating transformation from the flavor symmetry. Therefore, if N s of the φ I 0 = 0 a U ( N s ) symmetry will be preserved, as from (2.203) also the corresponding φ I 1 = 0. We note that the above analysis is for chiral multiplets in the fundamental representation. It can easily be generalized to other representations. For example if we had a single chiral multiplet in the adjoint representation we would have the Nahm equations (deformed by the FI term) for the three fields σ − and the Hermitian and anti-Hermitian parts of the scalar φ.

General gauge group projections and multiple matter multiplets
Like for the A-type constraints we can generalize the B-type boundary conditions when we have a gauge group other than U (1), or/and multiplet matter multiplets by imposing matrix projection conditions on fermions. In this case we get Although we do not pursue the detail here, we can find certain combinations of the boundary conditions we discussed above for the theory with higher rank gauge group or multiplet flavors by choosing appropriate matrices R and S. It would be also interesting to explore more general boundary conditions by allowing R and S to depend on the scalar fields σ and/or φ.

Quarter-BPS boundary conditions
We can impose the quarter-BPS boundary conditions by writing each spinor and its complex conjugate in terms of a single spinor. In order to satisfy δλγ 1λ + δλγ 1 λ = 0 . the most general condition we can have is and similarly for ψ we have We can also parameterize in terms of a single real Grassmann parameter 0 =¯ 0 as Then the conditions to preserve 1d N = 1 supersymmetry reduce to two equations − Re ae iα 2 + be i(α 2 −α 1 ) F 01 and The first equation (3.9) is the quarter-BPS boundary conditions for the vector multiplet and the second equation (3.10) is that for the chiral multiplet. The parameters a and b fix a choice of the supercharge. There are four basic boundary conditions for each of vector and chiral multiplets so that totally we find sixteen types.

Simple examples
We first consider the case with a = b = 1 where the supersymmetric boundary conditions have fewer terms. Although the following conditions would be required to preserve N = 1 supersymmetry, one needs to impose additional boundary conditions to preserve locality of the bulk fields, which do not show up in the projection of supercurrents. In the following we group the resulting quarter-BPS boundary conditions into four different sets of boundary conditions for the vector multiplet, which we call N N , N D , D N and D D .

N N boundary conditions
There are four distinct boundary conditions which are compatible with A-type Neumann boundary conditions and B-type Neumann boundary conditions for the vector multiplet.

N D boundary conditions
There are four boundary conditions which are compatible with A-type Dirichlet boundary condition and B-type Neumann boundary condition for the vector multiplet. For α 1 = 0, α 2 = π 2 , β 1 = 0 and β 2 = 0 we find

D N boundary conditions
There are four boundary conditions which are compatible with A-type Neumann boundary condition and B-type Dirichlet boundary condition for the vector multiplet. They can be obtained from the N D boundary conditions by replacing the value of α 1 with π.

D D boundary conditions
There are four boundary conditions which are compatible with A-type Dirichlet boundary condition and B-type Dirichlet boundary condition for the vector multiplet. They can be obtained from the N N boundary conditions by replacing the value of α 1 with π.

Singular solutions
We can also find singular solutions of the quarter-BPS boundary conditions. Note that as in the B-type boundary conditions, the two kinds of boundary conditions and for the chiral multiplet can be solved by postulating the simple pole for σ − in the axial gauge: where u is some constant valued in the Lie algebra.
Given the singular profile (3.45), it is now straightforward to get solutions. For the N N , N D and D D boundary conditions one can solve them by following the previous discussion for the generalized B-type Neumann boundary conditions whereas for the D N boundary conditions we can find solutions by following the discussion for the generalized B-type Dirichlet boundary conditions.

Other cases 3.2.1 Mixed cases
Next consider the case with a = 0 and b = 1.
For fixed α 1 and α 2 one finds four different types of supersymmetric boundary conditions of the vector multiplet:

Corner
Another interesting situation with a quarter of supersymmetry is a corner configuration of the 2d N = (2, 2) gauge theory placed on a quadrant R + × R + . It should realize 0d N = 1 supersymmetry. In order to preserve supersymmetry at a corner, say at  129)), one finds that the only consistent conditions consist of the A-type boundary conditions along the x 1 and the B-type boundary conditions along the x 0 or vice versa. The resulting conditions imposed at a corner would be stronger than the quarter-BPS boundary conditions that we discussed in the above. We defer a study of the quarter-BPS corner conditions to future work.

Brane setup in Type IIA string theory
In this section, we construct the BPS-boundary conditions in terms of branes in Type IIA string theory. We start with brane configurations producing 2d N = (2, 2) gauge theories and then add additional branes to give either A-type of B-type boundary conditions. We also show that the M-theory lift of these brane configurations is the same for both A-type and B-type -the difference being simply which direction is compactified to reduce to Type IIA string theory.

2d N = (2, 2) supersymmetric gauge theories
First, let us briefly review the Hanany-Hori construction [48] of 2d N = (2, 2) gauge theory. We start from the brane configuration consisting of the following branes in Type IIA string theory: There are four preserved supercharges obeying the projection conditions (4.3). We consider N c D4-branes suspended between the NS5-brane say at x 6 = x 7 = x 8 = x 9 = 0 and NS5-brane at x 6 = L, x 4 = x 5 = 0 and N f D4-branes which have the same x 6 position as the NS5-brane in the upper-half space x 7 > 0 and N a D4-branes in the lower-half space x 7 < 0. Similarly one can also introduce D4-branes which have the same x 6 position as the NS5-brane.
Since the D2-branes have finite extent along x 6 , the low-energy effective theory on the world-volume of the D2-branes is a two-dimensional field theory along (x 0 , x 1 ) preserving 2d N = (2, 2) supersymmetry.
The D2-D2 strings yield a 2d N = (8,8) U (N c ) vector multiplet. However, six of eight scalar fields are frozen due to the boundary condition required at the end points of NS5-brane and NS5-brane. The surviving two of eight scalar fields describe the motion of the D2-branes along the (x 2 , x 3 ) directions and correspond to the complex scalar field σ in the vector multiplet. 5 The vector multiplet scalar σ has charge +2 under the U (1) 23 .
The open strings stretched between the D2-branes and upper-half D4-branes give rise to the fundamental chiral multiplets Φ i , i = 1, · · · , N f . Similarly, the open strings between the D2-branes and lower-half D4-branes lead to the antifundamental chiral multiplets Φ j , j = 1, · · · , N a . These matter multiplets carry charge +1 under the U (1) 89 .
The U (1) 23  When N f = N a , the D4-branes can combine to form N f infinite D4-branes and the flavor symmetry is broken to SU (N f ). For N f ≥ N c there are N f Nc supersymmetric ground states and supersymmetry is broken for N f < N c as a consequence of the s-rule.
The parameters in the 2d N = (2, 2) gauge theories are realized as the positions of the D4-branes and NS5-branes. The gauge coupling is given by the distance L between the NS5-and NS5-branes along the x 6 direction. 1 where g st is the Type IIA string coupling constant. The FI parameter for the U (1) factor of the U (N c ) gauge symmetry is realized by the x 7 position of NS5-brane The mass parameter is given by the positions in the (x 4 , x 5 ) directions of joined i-th upper-half D4-brane and j-th lower-half D4-brane The twisted mass parameter m The theta parameter is most simply described in the M-theory lift, corresponding to the separation of the M5-brane and M5-brane along the x 10 direction where the M5brane and M5-brane are lifted from the NS5-brane and NS5-brane respectively. The D4 and D4 brane also lift to M5-branes while the D2-branes become M2-branes. These M-theory brane configurations describe M2-branes ending on intersecting M5-branes.
The classical Coulomb branch of the theory corresponds to the (x 2 , x 3 ) positions of the D2-branes suspended between NS5-and NS5-branes. The classical Higgs branch of the theory is parametrized by the (x 7 , x 8 , x 9 ) positions of D2-branes stretched between the D4-branes.
We note that there is another type of D4-brane, which we call D4-brane. When we have both D4-branes and D4-branes, the theory may have superpotential terms [56].
The 2d N = (2, 2) supersymmetric gauge theories with orthogonal and symplectic gauge group studied in [52] can be constructed by using orientifold planes [57]. For simplicity we do not include orientifold planes and so focus only on unitary gauge groups.

A-type boundaries 4.2.1 A-type boundary conditions
Now we would like to find the brane construction of the A-type boundary conditions. We further introduce the NS5 -branes or/and D4 -branes to the configuration (4.1): It follows that the configuration (4.8) preserves two supercharges. We identify this with 1d N = 2 A supersymmetry along the x 0 direction as the continuous R-symmetry is classically broken. We consider the configuration with the NS5-brane at x 6 = x 7 = x 8 = x 9 = 0, the NS5 at x 6 = L, x 4 = x 5 = x 7 = 0 and the D4-branes at x 6 = L, x 2 = x 3 = x 4 = x 5 = 0 so that FI parameters, mass parameters and twisted mass parameters are set to zero.

NS5 -brane
When a D2-brane is stretched between the NS5-brane and NS5-brane along the x 6 direction, it admits the 2d N = (2, 2) U (1) vector multiplet. The boundary condition coming from the NS5 -brane at x 1 = x 5 = x 7 = x 8 = 0 projects out the A 1 component of the gauge field while the complex scalar field σ describing the motion of D2-branes along the (x 2 , x 3 ) directions can fluctuate and the A 0 component of gauge field survives. Thus the NS5 -brane provides the A-type Neumann boundary condition for the vector multiplet F 01 = 0, (4.10) which correspond to (2.62) and (2.64) for α = 0. However, for the bulk supersymmetry to be unbroken, the D4-brane must be introduced.
When an upper-half D4-brane emanating from the NS5-brane at x 6 = L is further added, the effective D2-brane theory has a chiral multiplet of charge +1 which arises from D2-D4 strings. As the D4-brane intersects with the NS5-brane at x 6 = L, the complex scalar field in the chiral multiplet would correspond to the fluctuations of the D2-brane along the (x 8 , x 9 ) directions. The boundary condition arising from the NS5 -brane classically fixes the x 8 position of the D2-brane. This would split the complex scalar fields into real scalar fields obeying the Neumann and Dirichlet boundary conditions Imφ = 0, (4.12) which can be obtained from (2.72) and (2.73) for β = 0 in the A-type boundary condition for the chiral multiplet. Thus the NS5 -like boundary conditions may naturally correspond to the case of α = 0 and β = 0, equipped with the fermionic boundary conditions (4.14) Similarly, the NS5 -brane at x 1 = x 4 = x 7 = x 9 = 0 may naturally lead to the A-type Neumann boundary conditions (4.10) and (4.11) for the U (1) vector multiplet and the A-type boundary conditions for the chiral multiplet opposite to the conditions (4.12) and (4.13), which are realized when α = 0 and β = π.
However, we found from the field theory analysis that the additional conditions (2.74) and (2.75) (or (2.78) and (2.79)) cannot be simply solved together with the Neumann boundary conditions for the vector multiplet. At this stage it is not clear to see such an obstruction associated to the NS5 -brane (or NS5 -brane) from the brane picture.

D4 -brane
On the other hand, the D4 -brane at x 1 = x 2 = x 3 = x 4 = x 9 on which the D2-brane ends projects out the A 0 component of the gauge field and the motion of D2-branes along the (x 2 , x 3 ) directions described by complex scalar field σ whereas A 1 is not frozen. Hence the D4 -brane yields the A-type Dirichlet boundary conditions for the vector multiplet.
which can be found from the conditions (2.67) and (2.68) for α = π. We note that, unlike for the B-type configurations we will discuss later, this boundary condition, fixing σ, excludes any singular boundary behaviour for the D2-branes. So, for A-type boundary conditions we do not get any Nahm pole like behaviour.
When an upper-half D4-brane coinciding with the NS5-brane is introduced, the theory has a charged multiplet coupled to the U (1) vector multiplet and the D4brane then fixes the x 9 position of the D2-brane. This requires a splitting of the complex scalar field into two real scalar fields obeying the boundary conditions Reφ = 0, (4.18) D 1 (Imφ) = 0, (4.19) which are different from the boundary conditions (4.12) and (4.13) imposed by the NS5 -brane. Therefore the D4 -brane leads to the A-type boundary condition with α = π and β = π. Correspondingly the fermionic boundary conditions for the D4brane are 20) Analogously, the D4 -brane at x 1 = x 2 = x 3 = x 5 = x 8 also introduces the A-type Dirichlet boundary condition for the U (1) vector multiplet and the A-type boundary conditions (4.12) and (4.13) for the chiral multiplets. This corresponds to the A-type boundary conditions with α = π and β = 0.
In summary, any of these additional four branes, NS5 , NS5 , D4 , D4 , placed at x 1 = 0 would naturally correspond to the basic A-type boundary conditions in 2d N = (2, 2) gauge theories. The angle α that defines the boundary conditions (2.48) on the gaugino λ controls a ratio of the 5-branes and the 4-branes for each system. The angle β that determines the boundary condition (2.48) on the matter fermion ψ describes the ratio for two types of 5-brane, NS5 -and NS5 -branes and that for two types of 4-branes, D4 -and D4 -branes.

M-theory configuration
Consider the M-theory lift of the D2-NS5-NS5-NS5 configuration. It is again recognized as the M2-M5 system, however, unlike the D2-NS5-NS5 system of the 2d N = (2, 2) gauge theories, all the types of NS5-branes and D4-branes can become a single wrapped M5-brane. This M5-brane fills the (x 0 , x 2 , x 3 ) directions and wraps a special Lagrangian submanifold of a Calabi-Yau three-fold and the M2-brane has a boundary on this special Lagrangian cycle. The M5-brane will lead to a 3d N = 2 field theory in the (x 0 , x 2 , x 3 ) directions while the M2-brane will be understood as a charged particle in the theory.
In fact the complete A-type system can be realized in M-theory as an M5-brane wrapped on the product of a holomorphic curve in C 2 , having complex coordinates x 2 + ix 3 and x 7 + ix 10 , with a special Lagrangian 3-cycle in C 3 , having complex coordinates x 1 + ix 6 , x 4 − ix 8 and x 5 + ix 9 . In particular, reducing to Type IIA, branes wrapping x 7 + ix 10 become D4-branes while those wrapping x 2 + ix 3 become NS5-branes.
In general an M5-brane wrapping a special Lagrangian 3-cycle will preserve one eighth on the supersymmetry, i.e. 4 supercharges. We see the four types of NS5-branes and D4-branes arising from M5-branes wrapping the real 3-cycles in the 145, 189, 469 and 568 directions.  so that overall the system we considered indeed preserves 2 supercharges. Note that because the M5-brane wraps a special Lagrangian 3-cycle where one of the complex coordinates is x 1 + ix 6 , an M2-brane will always have a boundary on (or codimensionone intersection with) the M5-brane.

N = 2 A line operators
Let us further introduce the following additional D2 -branes or/and D2 -branes: The supercharges preserved in the brane configuration (4.31) satisfy the conditions (4.3), (4.9) and D2 : Γ 048 One can check that the configuration (4.31) preserves 1d N = 2 A supersymmetry along the x 0 direction without further breaking supersymmetry. When the D2 -branes or/and D2 -branes at x 1 = 0 are added, they would realize domain walls or line operators compatible with the A-type boundary condition in 2d N = (2, 2) gauge theories. Although we do not pursue here, it would be interesting to figure out the field theory interpretation. We also note that the M-theory lift could in general describe a single M2-brane wrapping a holomorphic cycle in C 3 .

B-type boundaries 4.3.1 B-type boundary conditions
Now let us consider the brane construction of the B-type boundary conditions. We take another set of additional NS5 -branes or/and D4 -branes as follows: It can be checked that the configuration (4.33) preserves two supercharges. In this case the space-time symmetry is classically broken to SO(2) 45 × SO(2) 89 . We identify the corresponding supersymmetry with 1d N = 2 B supersymmetry along x 0 . Again we take the configuration with the NS5-brane at x 6 = x 7 = x 8 = x 9 = 0, the NS5 at x 6 = L, x 4 = x 5 = x 7 = 0 and the D4-branes at x 6 = L, x 2 = x 3 = x 4 = x 5 = 0 so that FI parameters, mass parameters and twisted mass parameters are turned off.

NS5 -brane
For the theory of a 2d N = (2, 2) U (1) vector multiplet arising from a D2-brane suspended between the NS5-brane and NS5-brane along the x 6 direction, the NS5brane at x 1 = x 3 = x 4 = x 5 = 0 on which the D2-brane terminates fixes the motion of the D2-brane along the x 3 direction and the A 1 component of gauge field, while the motion of the D2-brane along x 2 is not fixed. Accordingly, the NS5 -brane realizes the B-type Neumann boundary conditions for the U (1) vector multiplet F 01 = 0, (4.35) where σ + and σ − correspond to the x 2 and x 3 positions of the D2-brane. These can be found from the equations (2.139), (2.140) and (2.141) for α = 0.
In the presence of an upper-half D4-brane, which leads to a charged chiral multiplet, when the D2-brane ends on the NS5 -brane, the motion of the D2-brane along the (x 8 , x 9 ) directions can still fluctuate. This corresponds to the B-type Neumann boundary conditions for the chiral multiplet which are the conditions (2.150) and (2.151) obtained for β = π. One can then identify the NS5 -like boundary conditions with the B-type boundary conditions with α = 0 and β = π. Whereas these basic boundary conditions will correspond to the NS5brane at x 1 = x 3 = x 4 = x 5 = 0, it would be interesting to shift the transverse positions of the NS5 -brane to find the generalized boundary conditions, as we argued for the deformation of the Neumann boundary conditions for the vector multiplet with a singular profile of σ − in the field theory analysis in section 2.2.6. On the other hand, the B-type Neumann boundary conditions (4.35), (4.36) and (4.37) for the U (1) vector multiplet can be also given by the NS5 -brane at x 1 = x 3 = x 8 = x 9 = 0. However, in this case the B-type boundary conditions for the chiral multiplets are the Dirichlet boundary conditions, rather than the Neumann boundary conditions (4.38) and (4.39). Thus the NS5 -brane would lead to the B-type boundary conditions with α = 0 and β = 0.

D4 -brane
When the D2-brane terminates on a D4 -brane at x 1 = x 2 = x 7 = x 8 = x 9 = 0, the motion of the D2-brane along the x 2 direction and the A 0 component of gauge field are frozen, while the D2-brane can still free move along the x 3 direction. Thus the D4 -brane realizes the B-type Dirichlet boundary conditions for the U (1) vector multiplet A 0 = 0, (4.40) σ + = 0, (4.41) which we found in the equations (2.145), (2.146) for α = π.
In the presence of an upper-half D4-brane, the effective theory has a charged chiral multiplet. When the D2-brane ends on the D4 -brane, the fluctuations of the D2-brane along the (x 8 , x 9 ) directions are projected out. This gives rise to the B-type Dirichlet boundary conditions for the chiral multiplet which can be obtained from the equations (2.154) and (2.155) for β = 0. Hence the D4 -like boundary condition is identified with the B-type boundary condition with α = π and β = 0. So far, we have argued that the NS5 -branes, the NS5 and the D4 -branes introduced at x 1 = 0 can produce the basic B-type boundary conditions in 2d N = (2, 2) gauge theories in such a way that the phase α in the boundary conditions (2.128) on the gaugino λ determines a ratio of 5-branes and the 4-branes, while the phase β in the boundary conditions (2.129) on the matter fermion ψ encodes the ratio for two types of 5-branes and that for two types of 4-branes.
Although one may expect that a D4 -brane at x 1 = x 2 = x 4 = x 5 = x 7 = 0 also realizes the B-type Dirichlet boundary conditions (4.40), (4.41) and (4.42) for the U (1) vector multiplet, as well as the B-type Neumann-type boundary conditions for the chiral multiplet, it does not seem to be the case because we found from the field theory analysis that such a combination of the basic boundary conditions for vector and chiral multiplets cannot be simply solved due to the obstruction from the conditions (2.147), (2.152) and (2.153). Instead, we found in section 2.2.7 that the corresponding B-type boundary conditions are generalized so that they allow for the consistent singular solutions where σ − and φ have a simple pole at the boundary. Indeed this might be expected from the brane configuration as the D2-D4 branes are T-dual to the familiar D1-D3 system described by the Nahm equation [49,50].
As the residue t at the pole (2.171) for σ − typically breaks the boundary global symmetry G ∂ = U (N c ) down to U (N c − N s ), this will correspond to the brane configuration where N s of the N c D2-branes end on a single D4 -brane in a similar manner as the regular Nahm pole boundary condition of rank r realized by the D5-brane on which r D3-branes end [1]. In fact, when we rotate the NS5-brane to be parallel to the NS5-brane and T-dualize the system along x 2 , the D2-branes ending on the D4 -brane become the D3-branes ending on the D5-brane. However, in our case the fundamental scalar field φ arising from the D2-D4 strings also contains a pole (2.189) whose residue c satisfies the relation t = −cc † , which may also break the U (N f ) flavor symmetry down to U (N f − N s ) as we discussed in section 2.2.7. In the brane configuration, such a symmetry breaking may occur when some of N f flavor D4-branes terminate on the single D4 -brane at x 1 = 0. On the other hand, we argued for the regular terms appearing in the solutions. However, it is not clear from the brane configuration. We illustrate the case with a maximal rank of the pole with N c = N s where all the D2-branes and N s D4-branes terminate on the D4 -brane in Figure 1.

M-theory configuration
As for the A-type brane configuration, we can lift the IIA configuration to M-theory. Interestingly, we then see that in eleven dimensions the A-type and B-type brane configurations are equivalent, i.e. they can both be described as an M5-brane wrapping the product of a holomorphic curve in C 2 with a special Lagrangian 3-cycle in C 3 . The only difference is our identification of the coordinates, and in particular when reducing to ten dimensions with the M-theory circle in the C 2 we get A-type while if it is in the C 3 we get B-type.
Specifically, for A-type we saw that the complex coordinates in C 2 were x 2 + ix 3 and x 7 + ix 10 , while for B-type we have x 4 + ix 5 and x 8 + ix 9 . For the C 3 we had coordinates x 1 + ix 6 , x 4 − ix 8 and x 5 + ix 9 for A-type while for B-type we have instead x 1 + ix 6 , x 2 + ix 10 and x 3 + ix 7 . In both cases we preserve 2 supercharges from the wrapped M5-brane, the M2-branes spanning 016 always have a boundary on the M5brane since x 1 + ix 6 is a complex coordinate in C 3 in both cases, and the M2-branes do not break any further supersymmetry. For B-type one set of four independent projection conditions is In summary, the A-type and B-type boundary conditions can be realized in Mtheory setup as follows: where the M5-brane is wrapped on the special Lagrangian 3-cycle M 3 in C 3 and the holomorphic 2-cycle Σ in C 2 while the M2-branes wrap the holomorphic 2-cycle C in C 3 whose boundary is in M 3 . It may be interesting to explore the supergravity description of such M-brane configurations. We are not aware of any such solutions but solutions for the M5-brane wrapping Σ × M 3 have been described [58]. It may be possible to understand some aspects of the field theory using M-brane probes in these backgrounds, such as calculating central charges similar to [59]. It would be particularly interesting to investigate supergravity solutions including M2-branes giving rise to AdS 2 geometry as duals of superconformal QM. Examples without M2-branes were found in [60,58].

N = 2 B line operators
We also note that there are other objects preserving N = 2 B supersymmetry along x 0 . Let us further introduce the fundamental strings (F1) and D2 -branes: The configuration (4.50) preserves 1d N = 2 B supersymmetry along the x 0 direction without further breaking supersymmetry.
The fundamental strings along the (x 0 , x 2 ) directions or/and D2 -branes would realize domain walls or line operators supported along the boundary which are compatible with the B-type boundary condition in 2d N = (2, 2) gauge theories. In the M-theory lift, the F1 and D2' become M2-branes, and as for the A-type case, we can have a single M2-brane wrapping a holomorphic cycle in C 3 .

Dualities
As we commented above, the A-type and B-type configurations are equivalent when lifted to M-theory, the only difference being the choice of direction to compactify on to reduce to Type IIA string theory, along with a relabelling of some of the coordinates.

One choice of mapping of coordinates is
A − type 0 1 2 3 4 5 6 7 8 9 10 B − type 0 1 4 5 2 3 6 8 10 7 9 (4.52) which results in the following mapping of branes Now we also have several mappings of the coordinates which preserve the A-type or B-type configurations. We consider only those which map D2-branes to D2-branes and do not result in new orientations of branes we have not considered. The mappings which satisfy this condition and have a non-trivial effect on some branes are for A-type any combination of and for B-type any combination of This gives the following mapping of branes Another possible duality is to T-dualize to Type IIB along x 2 , then perform Sduality before T-dualizing back to Type IIA, again along x 2 . This would result in new orientations of branes, but if we then exchange x 3 ↔ x 7 we get the same type of branes back. However, this is already included in the above mappings via M-theory, e.g. mapping A0 to A1 or B0 to B4. It is worth noting that this TST duality does not exchange A-type with B-type, but as we see above this is possible via M-theory.
There will also be interesting Seiberg-like dualities arising from Hanany-Witten brane rearrangements. It would be interesting to explore some of these dualities and find interpretations in the field theory, but we leave that for future work. In particular we expect to find dualities of boundary conditions related to 2d mirror symmetry and through T-duality this should be closely related to 3d mirror symmetry [61,62,63,64,65]. 4.4

.1 N = 1 quarter BPS boundary conditions
When we consider the configuration in which the NS5 -and D4 -branes and the NS5and D4 -branes exist 0 1 2 3 4 5 6 7 8 9 D2 one can check that there remains N = 1 supersymmetry. In addition, one can also introduce the fundamental strings and three kinds of D2-branes in the configurations (4.31) and (4.50), corresponding to the line operators keeping N = 1 supersymmetry. According to the identification of the parameters in the half-BPS boundary conditions, we can identify the phases α 1 , α 2 , β 1 and β 2 introduced in the field theory analysis with parameters which characterize the different types of boundary branes. The angle parameter α 1 distinguishes two kinds of branes for the B-type so that the case with α 1 = 0 (resp. π) describes the 4-brane (resp. 5-brane). The parameter α 2 −α 1 distinguishes the two kinds of branes for the A-type so that one finds the same kinds of branes in A-type for α 2 −α 1 = 0 mod π and the opposite kinds of branes in A-type for α 2 − α 1 = π/2 mod π. The angle parameters β 1 and β 2 further characterize the rotation of each of the 4-branes (and 5-branes) in B-type and A-type respectively.
As shown in Table 1, sixteen combinations of A-type and B-type boundary branes correspond to the basic quarter-BPS boundary conditions discussed in section 3. We see that the N N boundary conditions are realized for (α 1 , α 2 ) = (0, 0), the N D boundary conditions for (α 1 , α 2 ) = (0, π 2 ), the D D boundary conditions for (α 1 , α 2 ) = (π, 0) and the D N boundary conditions for (α 1 , α 2 ) = (π, π 2 ). In particular, we can again check that σ + which corresponds to the x 2 positions of a pair of branes boundary condition α 1 α 2 β 1 β 2 D2-branes is fixed by a pair of A-type and B-type D4-branes while it can still freely move for a pair of A-type and B-type NS5-branes as we found in the D D boundary conditions and N N boundary conditions. We have not yet understood the brane construction of the mixed quarter-BPS boundary conditions with a = 0, b = 1. It would be intriguing to explore the additional objects such as fluxes which realize them while maintaining the 1d N = 1 supersymmetry.
Finally, we note that the quarter-BPS configurations have a simple M-theory description as a single M5-brane wrapping a special Lagrangian 5-cycle in C 5 with complex coordinates x 1 + ix 6 , x 2 + ix 10 , x 3 + ix 7 , x 4 − ix 8 and x 5 + ix 9 . This specializes to the half-BPS configurations, noting that in that case the holomorphic curve in C 2 can equivalently be described as a special Lagrangian 2-cycle with a different complex structure. As for the half-BPS configurations, we are not aware of any supergravity solutions for the M-brane system, but for the M5-brane alone wrapping a special • We denote the 2d Minkowski metric η µν with η 00 = −1 and in the reduction from 4d we identify Lorentz indices m = 0 with µ = 0 but m = 3 with µ = 1.
• The 4d vector, chiral and anti-chiral multiplets all reduce to the corresponding 2d multiplets. The 4d gauge potential v m becomes the 2d gauge potential A µ with A 0 = v 0 , A 1 = v 3 and a complex scalar σ = 1 √ 2 (v 1 − iv 2 ). • The 2d spinor indices are the same as those in 4d except we no longer distinguisḣ α from α and all index contractions are top-left to bottom-right. This can lead to some changes of sign compared to 4d expressions, e.g.λψ =λαψα →λ αψ α = −λ αψ α = −λψ.

A.2.1 Vector multiplet
We work in WZ gauge where the vector multiplet V is a real superfield with component superfield expansion The supersymmetric transformations of these components are Expanding to write all fields as functions of x m rather than y m we have For gauge theories, replace partial derivatives with gauge covariant derivatives D m = ∂ m + igv m .

A.2.3 Twisted chiral and anti-chiral multiplets
A twisted chiral superfield Φ is defined in 2d byD − Φ = 0 = D + Φ while and its conjugate¯ Φ is automatically a twisted anti-chiral superfield, in general defined by D +¯ Φ = 0 = D −¯ Φ. If we define z 0 = x 0 + iθγ 1θ (A.24) 25) where in 2d notation θγ µθ = θ α γ µ β αθ β the general solution for a twisted chiral superfield takes the form Compared to the chiral superfield we have the following replacements θ + ↔θ + , θ − ↔θ − (A.27) with the other components of θ and unchanged, and other fields trivially gaining a tilde. Therefore we can map all the result for the chiral superfield. In particular, the supersymmetric transformations are

A.3 Supercurrent
In WZ gauge the action for the gauge field and chiral multiplets are given by If we vary the Lagrangians with constant we find a total derivative so the action is invariant with suitable boundary conditions. If we let depend on the spacetime coordinates we find additional terms of the form J m ∂ m +J m ∂ m¯ which define the supercurrents J andJ. Note, in our conventions this definesJ m to be the conjugate of J m . Explicitly we have δL gauge = ∂ m Tr −λσ m D − iλσ n v mn −¯ σ n λṽ mn +i (∂ m¯ )σ n λ −λσ n (∂ m ) v mn +λσ n (∂ m )ṽ mn + (∂ m¯ )σ n λṽ mn (A.44) where we have definedṽ mn = 1 2 mnpq v pq (A.46) Extracting the supercurrents from each part we find for the gauge multiplet

B More on A-type boundary conditions
We note that (2.90), (2.92) and (2.93) take a form similar to Hitchin's equations [69] F A = [Φ, Φ], (B.1) The singular solution of the Hitchin's equation was mathematically studied in [70] and it was used to define the surface operator in 4d N = 4 SYM theory [71,72]. where the Higgs field Φ has a pole at the origin. Now let us go back to the boundary conditions. To find the solutions of the boundary conditions (2.90), (2.92) and (2.93) for the vector multiplet, we first fix the gauge so that A u and A u commute. We then take the ansatz A u = c 1 s 3 x 1 + · · · , A u = c * 1 s 3 x 1 + · · · , (B.8) σ = c 2 s + x 1 + · · · ,σ † = c * 2 s − x 1 + · · · (B.9) where we have only indicated the singular terms at the boundary, s 1 , s 2 and s 3 are constant elements of the Lie algebra, s ± = s 1 ± is 2 and c 1 , c 2 ∈ C are some numerical where s ± = s 1 ± is 2 are raising and lowering operators of su(1, 1).