Supersymmetric Boundary Conditions in Three Dimensional N = 2 Theories
CCALT 68-2916OU-HET 780
Supersymmetric boundary conditionsinthree-dimensional N = 2 theories Tadashi Okazaki a,b and Satoshi Yamaguchi a a Department of Physics, Graduate School of Science, Osaka University, Toyonaka,Osaka 560-0043, Japan b California Institute of Technology, Pasadena, California 91125, USA
Abstract
We study supersymmetric boundary conditions in three-dimensional N = 2Landau-Ginzburg models and Abelian gauge theories. In the Landau-Ginzburgmodel the boundary conditions that preserve (1 ,
1) supersymmetry (A-type)and (2 ,
0) supersymmetry (B-type) on the boundary are classified in terms ofsubspaces of the target space (“brane”). An A-type brane is a Lagrangiansubmanifold on which the imaginary part of the superpotential is constant, whilea B-type brane is a holomorphic submanifold on which the superpotential isconstant. We also consider the N = 2 Maxwell theory with boundary andthe Abelian duality. Finally we make some comments on N = 2 SQED withboundary condition and the mirror symmetry. tadashi[at]theory.caltech.edu yamaguch[at]het.phys.sci.osaka-u.ac.jp a r X i v : . [ h e p - t h ] J un ontents γ (cid:15) = ¯ (cid:15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 B-type γ (cid:15) = (cid:15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 C.1 Chiral superfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22C.2 Vector superfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Quantum field theories with boundaries are worth investigating since they often playimportant roles in various fields of physics. One of the most interesting ones is aboundary CFT description of D-branes. In particular supersymmetric boundaries oftwo-dimensional N = (2 ,
2) quantum field theories have been well studied because theyare good probes in the study of mirror symmetry [1, 2]. More recently supersymmetricboundary conditions in four-dimensional N = 4 super Yang-Mills theories and theirrelation to S-duality have been examined by [3, 4, 5].It is also interesting to study supersymmetric boundary conditions in three-dimensionalsupersymmetric field theories. One of the most attractive motivations is that it willprovide a description of M5-branes in terms of the boundary condition of M2-branetheories [6, 7, 8, 9]. It will also be a useful tool to investigate various dualities in three2 Boundary
Figure 1: The theory defined in x ≥ x = 0 gives the boundary and weconsider the supersymmetric boundary conditions on this.dimensions such as mirror symmetry [10, 11, 12, 13, 14] and 3d-3d correspondence[15, 16, 17, 18].In this paper, we focus on 1/2 BPS boundary conditions of N = 2 supersymmetrictheories in three dimensions. There are two possibilities of the preserved supersymme-try (SUSY), N = (1 ,
1) and N = (2 , N = (2 ,
2) theories. We consider supersym-metric field theories on the flat half spacetime R , × R + for simplicity. Here R , isparametrized by the time coordinate x and the spatial coordinate x . R + is an infinitehalf line x ≥ J vanishes,as done in [3].Let us summarize the results in this paper. We first consider the N = 2 Landau-Ginzburg theory. We employ the brane picture in the same way as in the two-dimensional case. In some sense, our brane is understood as the extended objectsupon which membranes can end . An A-type brane (A-brane) is a Lagrangian sub-manifold on which the imaginary part of the superpotential is constant, while a B-typebrane (B-brane) is a holomorphic submanifold on which the superpotential is constant.Second we explore 1/2 BPS boundary conditions in N = 2 pure Maxwell theory.This theory is dual to a free field theory which contains a chiral multiplet. We find afew classes of A-type and B-type boundary conditions and interpret them in the freechiral multiplet theory. The results are completely consistent with the analysis of theLandau-Ginzburg theory. Here we use the term “membrane” in a broad meaning. We do not claim that our membrane isthe same as that considered in M-theory. N = 2 supersymmetricquantum electrodynamics (SQED). This theory is supposed to be dual to a Landau-Ginzburg model called “the XYZ model” [11, 12]. We conjecture an example of mirrorsymmetry with boundary and give an evidence in the picture of the moduli space.Although we find the nontrivial solutions of supersymmetric boundary conditionsin three-dimensional N = 2 theories, these are not complete. In the two-dimensionalcontexts, it was discussed that we may also couple bulk theories to boundary degrees offreedom by introducing massless vector bosons, Chan-Paton spaces and so on [19]. Tocomplete our analysis, we should take three-dimensional analogs into account. Theyare extremely interesting, but will be deferred to future work.The organization of this paper is as follows. In section 2 we determine the super-symmetric boundary conditions for three-dimensional N = 2 Landau-Ginzburg model.In section 3 we derive the supersymmetric boundary conditions for three-dimensional N = 2 pure Maxwell theory. Then we consider the duality between pure Maxwelltheory and chiral matter theory. In section 4 we also present the supersymmetricboundary conditions in 3-dimensional N = 2 SQED. Finally section 5 concludes witha discussion of the relating problems and future works. The Appendixes contain ournotations and some useful formulae in three-dimensional N = 2 field theories. In this section we discuss Landau-Ginzburg models in three dimensions. We findsupersymmetric boundary conditions that preserve half of the supersymmetry andshow that the subspace of the sigma model arises as Lagrangian submanifolds orholomorphic submanifolds.Let us consider the Landau-Ginzburg model which has n chiral superfields Φ i ( i =1 , · · · , n ). See the Appendixes for the detail of the convention. The Lagrangian is L = K (Φ , ¯Φ) | − θθ ¯ θ ¯ θ + W (Φ) | θθ + ¯ W ( ¯Φ) | − ¯ θ ¯ θ . (2.1)Here K (Φ , ¯Φ) is the K¨ahler potential and W (Φ) is the superpotential.The K¨ahler potential term is expressed in component fields as K | − θθ ¯ θ ¯ θ = K i ¯ j F i ¯ F ¯ j + 12 K i ¯ j ¯ k F i ( ¯ ψ ¯ j ¯ ψ ¯ k ) − K ij ¯ k ¯ F ¯ k ( ψ i ψ j ) − K ij ¯ k ¯ l ( ψ i ψ j ¯ ψ ¯ k ¯ ψ ¯ l ) − K i ¯ j ∂ µ φ i ∂ µ ¯ φ ¯ j − i K i ¯ j ¯ ψ ¯ j σ µ ∂ µ ψ i − i K i ¯ j ψ i σ µ ∂ µ ¯ ψ ¯ j − i K ij ¯ k ( ∂ µ φ i )( ¯ ψ ¯ k σ µ ψ j ) + i K ij ¯ k ( ∂ µ ¯ φ ¯ j )( ¯ ψ ¯ k σ µ ψ i ) , (2.2)where we use the abbreviation K i ¯ j := ∂ K∂φ i ∂ ¯ φ ¯ j and so on.4n the other hand, the contribution from superpotential is W | θθ + ¯ W | − ¯ θ ¯ θ = F i W i −
12 ( ψ i ψ j ) W ij + ( c.c. ) , (2.3)where ( c.c. ) denotes the complex conjugation. We also use the abbreviations W i := ∂W∂φ i , W ij := ∂ W∂φ i ∂φ j .The supersymmetry transformation of this system is expressed as δφ i = √ (cid:15)ψ i , (2.4) δψ i = √ iγ µ ¯ (cid:15)∂ µ φ i + √ (cid:15)F i , (2.5) δF i = √ i ¯ (cid:15)σ µ ∂ µ ψ i , (2.6) δ ¯ φ ¯ i = −√ (cid:15) ¯ ψ ¯ i , (2.7) δ ¯ ψ ¯ i = −√ iγ µ (cid:15)∂ µ ¯ φ ¯ i + √ (cid:15) ¯ F ¯ i , (2.8) δ ¯ F ¯ i = √ i(cid:15)σ µ ∂ µ ¯ ψ ¯ i . (2.9)We can calculate the supercurrents J µ = −√ K i ¯ j ( ∂ µ ¯ φ ¯ j ) ψ i + √ K i ¯ j ( ∂ ν ¯ φ ¯ j ) γ µν ψ i − √ iγ µ ¯ ψ ¯ i ¯ W ¯ i , ¯ J µ = −√ K i ¯ j ( ∂ µ φ i ) ¯ ψ ¯ j + √ K i ¯ j ( ∂ ν φ i ) γ µν ¯ ψ ¯ j + √ iγ µ ψ i W i . (2.10)Here we investigate this system in the half-space x ≥
0. We restrict ourselves tothe case without boundary terms or boundary degrees of freedom almost throughoutthis paper. Then the equations of motion give nontrivial constraints on the boundaryconditions. Let us use the capital label I which takes values of both i and ¯ i . Thetarget space metric g IJ is defined as g i ¯ j = g ¯ ji = K i ¯ j , g ij = g ¯ i ¯ j = 0 . (2.11)The boundary term coming from the bosonic term becomes δS B,bdy = (cid:90) d xg IJ δφ I ∂ φ J . (2.12)We can use the target space brane picture in the same way as the string theory. SeeFigures. 2 and 3. The target space vector δφ I is tangent to the brane by definition.We should impose the boundary condition in which the boundary term (2.12) vanishes An improvement transformation may give rise to some ambiguity to determine thesupercurrents[20]. Understanding their effects may create an interesting problem. We thank YuNakayama for discussions on these points. δφ I . Thus the target space vector ∂ φ I is normal tothe brane.On the other hand the fermionic boundary term becomes δS F,bdy = (cid:90) d x i g IJ δψ I σ ψ J . (2.13)We impose the boundary condition γ ψ I = S I J ψ J (2.14)with a φ dependent matrix S I J . ( γ ) = 1 leads to the constraint S I J S J K = δ IK . (2.15)We require the boundary term (2.13) to vanish. Then another constraint on S I J isobtained g IJ S I K S J L = g KL . (2.16)Let us turn to the supersymmetry of the boundary condition. A boundary con-dition preserves supersymmetry if and only if the component of the SUSY currentnormal to the boundary vanishes. Thus supersymmetric boundary condition satisfies0 = (cid:15)J − ¯ (cid:15) ¯ J , (2.17)for a certain class of (cid:15) . There are two kinds of choices of (cid:15) for 1/2 BPS boundary asconsidered in [8] ( A ) γ (cid:15) = ¯ (cid:15), N = (1 ,
1) type , ( B ) γ (cid:15) = (cid:15), N = (2 ,
0) type . (2.18)We call them A-type and B-type, respectively, in this paper. They are actually anal-ogous to the A-type and B-type boundary conditions in two-dimensional N = (2 , − K i ¯ j ( ∂ ¯ φ ¯ j )( (cid:15)ψ i ) + K i ¯ j ( ∂ ν ¯ φ ¯ j )( (cid:15)Cγ ν ψ i ) − i ( (cid:15)σ ¯ ψ ¯ i ) ¯ W ¯ i + K i ¯ j ( ∂ φ i )(¯ (cid:15) ¯ ψ ¯ j ) − K i ¯ j ( ∂ ν φ i )(¯ (cid:15)Cγ ν ¯ ψ ¯ j ) − i (¯ (cid:15)σ ψ i ) W i . (2.19)Let us see the geometric meaning of this condition for A-type and B-type.6 .1 A-type γ (cid:15) = ¯ (cid:15) Now we want to discuss the brane of an A-type boundary condition. We call it “A-brane.” Here we will show that an A-brane is a Lagrangian submanifold on whichIm W is constant. This result is similar to the two-dimensional case [2].It is natural to employ the ansatz for the boundary condition for the fermions γ ψ i = S i ¯ j ¯ ψ ¯ j . (2.20)Here S i ¯ j is a φ dependent matrix. Then the matrix S I J in Eq. (2.14) becomes S I J = (cid:32) S ∗ ¯ ij S i ¯ j (cid:33) . (2.21)Before going, we introduce some useful expressions. We introduce the target spacevectors v I and w Ia , a = 0 , v I := (cid:15)ψ i ( I = i ) − ¯ (cid:15) ¯ ψ ¯ i ( I = ¯ i ) w Ia := ¯ (cid:15)σ a ψ i ( I = i ) − (cid:15)σ a ¯ ψ ¯ i ( I = ¯ i ) . (2.22)Then the condition (2.19) is rewritten as0 = − g IJ ∂ φ I v J − g IJ ∂ a φ I w Ja + iW i v i − i ¯ W ¯ i ¯ v ¯ i , where a = 0 , . (2.23)The above condition is satisfied by the ansatz : g IJ ∂ φ I v J = 0 , (2.24) g IJ ∂ a φ I w Ja = 0 , (2.25) iW i v i − i ¯ W ¯ i ¯ v ¯ i = 0 , where a = 0 , . (2.26)Since ∂ φ I is normal to the brane and ∂ a φ I is tangent to the brane, first two con-ditions imply that v I is tangent to the brane and that w Ia is normal to the branerespectively as shown in Fig. 2. The third condition implies that the imaginary partof the superpotential Im W is constant on the brane.Let us define the K¨ahler form of the target space ω IJ ω IJ := ω i ¯ j = iK i ¯ j ( I = i, J = ¯ j ) ω ¯ ij = − iK ¯ ij ( I = ¯ i, J = j )0 (otherwise) (2.27)In order to show that the brane is a Lagrangian submanifold, we should check that We could not find any other solutions without boundary terms, although we could not prove thisis the only solution. If some boundary terms are included, we may have other solutions. -brane membrane target space Figure 2: Membrane, shown in blue, ending on the A-brane, shown in green. Theylive in the target space. v I is parallel to the tangent direction and w Ja is in the normaldirection of the A-brane.1. The real dimension of the submanifold is n in the complex n -dimensional targetspace.2. For two arbitrary tangent vectors v I and v (cid:48) I ω IJ v I v (cid:48) J = 0 , (2.28)are satisfied.Let us check these propositions one by one.First, notice that from the definition (2.22), S I J v J = v J ,S I J w Ja = − w Ja (2.29)are satisfied. In other words tangent vectors and normal vectors are eigen vectors of S with eigenvalues 1 and − S = 1 and Tr( S ) = 0are satisfied because of Eqs. (2.15), (2.21) respectively. Thus the (2 n ) × (2 n ) matrix S has real n -dimensional eigenspace with eigenvalue +1. In other words the A-braneis real n dimensions.The second one is shown as follows. Eq. (2.16) is written in this case as K i ¯ j S i ¯ k S ∗ ¯ jl = K l ¯ k . (2.30)By using this relation the left-hand side of the relation (2.28) is rewritten as ω IJ v I v (cid:48) J = i (cid:16) K i ¯ j v i ¯ v (cid:48) ¯ j − K ¯ ij ( S ∗ ¯ ik v k )( S j ¯ l ¯ v (cid:48) ¯ l ) (cid:17) = 0 . (2.31)This is the relation (2.28). As a result we have shown that an A-brane is a Lagrangiansubmanifold. 8 .2 B-type γ (cid:15) = (cid:15) Next let us turn to the brane of B-type boundary condition. We call it “B-brane”(see Figure 3). We will show that a B-brane is a holomorphic submanifold on whichsuperpotential W is constant. This is also similar to the two-dimensional B-typeboundary condition[2].It is natural to put the ansatz for the boundary condition for the fermions γ ψ i = R ij ψ j , (2.32)where R ij is a φ dependent matrix. Then S I J in Eq. (2.14) becomes S I J = (cid:32) R ij R ∗ ¯ i ¯ j (cid:33) . (2.33)Here we define u I , z Ia , ( a = 0 ,
1) as u I := i ¯ (cid:15)ψ i ( I = i ) i(cid:15) ¯ ψ ¯ i ( I = ¯ i ) z Ia := (cid:15)σ a ψ i ( I = i ) − ¯ (cid:15)σ a ¯ ψ ¯ i ( I = ¯ i ) . (2.34)Then the supersymmetric boundary condition is rewritten as0 = − g IJ ( ∂ φ I ) v J − g IJ ( ∂ a φ I ) z Ja − i ( (cid:15) ¯ ψ ¯ i )( R ∗ ) ¯ i ¯ j ¯ W ¯ i − i (¯ (cid:15)ψ j ) R ij W i , (2.35)which is satisfied by the ansatz g IJ ∂ φ I v J = 0 , (2.36) g IJ ∂ a φ I z Ja = 0 , where a = 0 , , (2.37) u i W i + ¯ u ¯ i ¯ W ¯ i = 0 . (2.38)The first condition and second one imply that the target space vector v I is tangent tothe brane and that z Ia is normal to the brane respectively.Now we would like to show that a B-brane is a holomorphic submanifold. It isnecessary and sufficient to show ω IJ v I z Ja = 0 (2.39)for an arbitrary tangent vector v I and an arbitrary normal vector z Ja . From thedefinitions (2.22) and (2.34), the relations v I = − S I J v J , (2.40) z Ia = S I J z Ja (2.41)9 -brane membrane target space Figure 3: Membrane, shown in blue ending on the B-brane, shown in orange. Theylive in the target space. v I is in the tangent direction and z Ja is in the normal directionof B-brane.are obtained. In other words, the tangent space is the eigenspace of S with eigenvalue −
1, and the normal space is that with eigenvalue +1. The relation (2.16) reads in thisB-type ansatz K i ¯ j R ∗ ¯ j ¯ k R il = K ¯ kl . (2.42)By using this equation and Eq. (2.41) the left-hand side of Eq. (2.39) is rewritten as i ( − K i ¯ j R ik v k R ∗ ¯ j ¯ l ¯ z ¯ la − K ¯ ij R ∗ ¯ i ¯ k ¯ v ¯ k R jl z la ) = 0 . (2.43)Thus the relation (2.39) is satisfied and we can conclude that the B-brane is a holo-morphic submanifold.Next let us turn to show that the superpotential is constant on the B-brane. Thetarget space vector u I , defined in Eq. (2.34) satisfies S I J u J = − u I . (2.44)In other words u I is a eigenvector with eigenvalue − u I is a tangentvector. As a result Eq. (2.38) implies W is constant on the B-brane.10 Pure Maxwell theory
In this section we study three-dimensional N = 2 pure Abelian gauge theory. Thesupersymmetric Lagrangian of the vector multiplet is given by L gauge = − e Σ | − θθ ¯ θ ¯ θ = 1 e (cid:18) − F µν F µν − i ¯ λσ µ ∂ µ λ − ∂ µ σ∂ µ σ + 12 D (cid:19) , (3.1)where we denote the gauge coupling constant by e . Σ is the linear multiplet definedby Σ = − i ¯ DDV . For the detail of the convention, see the Appendixes.Let us first review the duality between pure Abelian gauge theory and masslessfree theory. As discussed in [12], let us start with the action S = (cid:90) d x (cid:90) d θ (cid:18) − e Σ + Σ(Φ + ¯Φ) (cid:19) , (3.2)where Σ is a general real superfield and Φ is a chiral superfield. The fermion integral (cid:82) d θ picks up the coefficient of ( − θθ ¯ θ ¯ θ ). Two dual theories may be thought of as twochoices of variables in path integral. In other words, we can interpret such dualitiesas Legendre transformations.If we integrate out Φ and ¯Φ, we obtain constraints ¯ D Σ = 0 and D Σ = 0, whichmean that Σ is a linear multiplet. Thus this leads to pure Maxwell theory if weintegrate out Φ and ¯Φ.On the other hand, we can integrate out Σ first. This integral can be performedby solving the equation of motion for Σ and substituting Σ in the action (3.2) withthis classical solution. The classical equation of motion is given byΣ = e . (3.3)By substituting this in the action, we can rewrite the action as a functional of Φ and¯Φ: S = e (cid:90) d x (cid:90) d θ ¯ΦΦ (3.4)Now this gives chiral matter theory characterized by the K¨ahler potential K = e ¯ΦΦ.Let us see the duality transformation in components. By expanding (3.3), we11btain the following dictionary: σ = e (Re φ ) (3.5) G µ := 12 (cid:15) µνρ F νρ = e ∂ µ (Im φ ) =: ∂ µ ρ (3.6) λ = e √ ψ (3.7)¯ λ = e √ ψ. (3.8)The relation (3.6) shows that Im φ is the dual photon. It is also convenient to define ρ := e Im φ . Then σ + iρ is the holomorphic coordinate. From charge quantization,we see that ρ is periodic: ρ ∼ ρ + e . (3.9) In Wess-Zumino gauge, the SUSY transformation for the component fields of the vectormultiplet are δA µ = i(cid:15)σ µ λ + i(cid:15)σ µ ¯ λ,δσ = (cid:15) ¯ λ − ¯ (cid:15)λ,δ ¯ λ = − i ¯ (cid:15)D − γ µν ¯ (cid:15)F µν + iγ µ ¯ (cid:15)∂ µ σ,δλ = i(cid:15)D − γ µν (cid:15)F µν − iγ µ (cid:15)∂ µ σ,δD = − (cid:15)σ µ ∂ µ ¯ λ + ¯ (cid:15)σ µ ∂ µ λ. (3.10)Given the above Lagrangian and the SUSY transformation, we can calculate su-percurrents as J µ = − iF µν γ ν ¯ λ + i (cid:15) µρσ ¯ λF ρσ + γ µν ¯ λ∂ ν σ − ¯ λ∂ µ σ, ¯ J µ = + iF µν γ ν λ − i (cid:15) µρσ λF ρσ + γ µν λ∂ ν σ − λ∂ µ σ. (3.11)Then the supersymmetric boundary condition for vector multiplet is given by0 = (cid:15)J − ¯ (cid:15) ¯ J = − iF a ( (cid:15)σ a ¯ λ ) + i ( (cid:15) ¯ λ ) F + ( (cid:15)Cγ a ¯ λ ) ∂ a σ − ( (cid:15) ¯ λ ) ∂ σ − iF a (¯ (cid:15)σ a λ ) + i (¯ (cid:15)λ ) F − (¯ (cid:15)Cγ a λ ) ∂ a σ + (¯ (cid:15)λ ) ∂ σ, where a = 0 , . (3.12)12e focus on the case without boundary terms or boundary degrees of freedom inthis paper, except the boundary theta term: S ϑ = ϑ π (cid:90) x =0 dx dx F . (3.13)This theta term corresponds to the shift for the value of dual photon at the boundaryin the dual picture as pointed out in [21]. In order to see this we begin by defining thereference point ρ as ρ := (cid:104) ρ ( x ) (cid:105) = (cid:90) Dρρ ( x ) e iS , (3.14)where S = − e (cid:90) d x∂ µ ρ∂ µ ρ, (3.15)which is the kinetic term of ρ determined by (3.3). This boundary term (3.13) iswritten in terms of the dual photon ρ : S ϑ = ϑ π (cid:90) dx dx ∂ ρ. (3.16)Using S + S ϑ instead of S , we obtain ρ ϑ := (cid:104) ρ ( x ) (cid:105) ϑ = ρ + e π ϑ. (3.17)From this result we can also see the periodicity of ρ by ϑ whose domain is 0 ≤ ϑ < π .This correspondence is also explained by the usual dualization procedure includingappropriate boundary terms. In two dimensions with boundary this have been donein [22] and it is straightforward to extend it to three dimensions. This Abelian duality is an analog of T-duality in two-dimensions. For example theboundary theta term is the analog of the Wilson line in two-dimensions whose dual isthe position of the D-brane. One may think that this three-dimensional abelian dualityexchanges an A-brane and a B-brane from the analogy of two-dimensions. Howeverit is not true. An A-brane and a B-brane in three-dimensions preserve different typeof supersymmetry. They are N = (1 ,
1) and N = (2 ,
0) on the boundary respectively.Thus an A-brane and a B-brane cannot be dual to each other. The dual of an A-braneis an A-brane, and that of a B-brane is a B-brane.We consider in this paper several examples of boundary conditions with simpleansatze and see the duality, instead of the general classification of the boundary con-ditions. Let us examine the A-type ( N = (1 ,
1) type) and the B-type ( N = (2 , We would like to thank Kentaro Hori for explanation. .2.1 A-type γ (cid:15) = ¯ (cid:15) We put the ansatz γ λ = e iα ¯ λ (3.18)with a real parameter α . Then the condition (3.12) is rewritten as0 = 2 ie − iα (cid:15)σ a λ (cid:16) cos α G a + sin α ∂ a σ (cid:17) + 2 e − iα (cid:15)σ λ (cid:16) sin α G − cos α ∂ σ (cid:17) , where a = 0 , , (3.19)where G µ is the dual strength of the gauge field defined by G µ := (cid:15) µρη F ρη . To satisfythe above condition, we should require thatcos α G a + sin α ∂ a σ = 0 , sin α G − cos α ∂ σ = 0 , where a = 0 , . (3.20)Note that we have theta term contribution (3.13) except when α = π .Let us see this boundary condition in the dual picture. (3.20) become ∂ a (cid:16) cos α ρ + sin α σ (cid:17) = 0 ,∂ (cid:16) sin α ρ − cos α σ (cid:17) = 0 , where a = 0 , . (3.21)Since the first condition is interpreted as Dirichlet-type condition, these conditions canbe shown as in Figure 4. These branes are actually Lagrangian submanifolds on thecylinder and consistent with the analysis in the section 2. (a) α = 0 (b) α = π (c) 0 < α < π (d) Expansion of the σ - ρ plane. Figure 4: A-type boundary conditions for ρ and σ . The angle α in a σ - ρ plane can beidentified with the phase appearing in the action of γ on λ .14 .2.2 B-type γ (cid:15) = (cid:15) Here we consider two ansatze γ λ = + λ, (3.22)and γ λ = − λ. (3.23)We call these ansatze (BI) and (BII) respectively.(BI) γ λ = + λ Noting that (cid:15)λ = 0, (3.12) becomes0 = − (cid:15)σ a ( iF a ¯ λ + ∂ a σγ λ ) − ¯ (cid:15)σ a ( iF a λ − ∂ a σγ λ ) . (3.24)Therefore we obtain the conditions: G a = 0 ,∂ a σ = 0 , where a = 0 , A a because it leadsto F a = 0 , a = 0 ,
1. The second one is Dirichlet one for scalar σ . This brane isa point on the cylinder and a holomorphic submanifold. This is consistent withthe analysis of section 2. We can also introduce the theta term contribution(3.13) which expresses the position in the ρ direction.Let us see this boundary condition in the dual picture. (3.25) is expressed as ∂ a ρ = 0 ,∂ a σ = 0 , where a = 0 , . (3.26)Now both conditions can be understood as Dirichlet-type conditions for σ and ρ . The configurations are illustrated in Figure 5.Figure 5: BI-type boundary conditions for σ and ρ . One imposes Dirichlet boundarycondition on both σ and ρ . The configuration can be expressed as a point in a σ - ρ plane. 15BII) γ λ = − λ Since we have the relation (cid:15)σ a λ = 0 , a = 0 ,
1, the condition (3.12) is simplifiedas 0 = (cid:15) ¯ λ ( iF − ∂ σ ) + ¯ (cid:15)λ ( iF + ∂ σ ) . (3.27)This leads to the condition G = 0 ,∂ σ = 0 . (3.28)In this case the first condition is Dirichlet boundary condition for gauge field A a , a = 0 , F = 0. On the other hand, the second one isNeumann condition for scalar σ . Notice that in this case we have no theta termbecause F = 0.Let us see this brane in the dual picture. (3.28) is rewritten as ∂ ρ = 0 ∂ σ = 0 . (3.29)In this case both of these are Neumann conditions. These are explained in Figure6. This brane is also a holomorphic submanifold and consistent with the analysisof section 2. This brane extends to the ρ direction and thus consistent with thefact that the boundary does not admit the theta term.Figure 6: BII-type boundary conditions for σ and ρ . One imposes Neumann conditionon both σ and ρ . The configuration can be expressed as the entire σ - ρ plane. Now we want to discuss three-dimensional N = 2 supersymmetric electrodynamics(SQED). In this case, in addition to the vector multiplet V , we need to introduce16harged chiral superfields Φ + and Φ − , whose charges are +1 and − L QED = (cid:20) e Σ − ¯Φ + e − V Φ + − ¯Φ − e V Φ − (cid:21) (cid:12)(cid:12)(cid:12) − θθ ¯ θ ¯ θ = 1 e (cid:18) − F µν F µν − i ¯ λσ µ ∂ µ λ − ∂ µ σ∂ µ σ + 12 D (cid:19) − D µ ¯ φ + D µ φ + − i ¯ ψ + σ µ D µ ψ + + ¯ F + F + − iσ ( ¯ ψ + ψ + ) − √ iφ + ( ¯ ψ + ¯ λ ) − √ i ¯ φ + ( ψ + λ ) − ¯ φ + φ + D − ¯ φ + φ + σ − D µ ¯ φ − D µ φ − − i ¯ ψ − σ µ D µ ψ − + ¯ F − F − + iσ ( ¯ ψ − ψ − ) + √ iφ − ( ¯ ψ − ¯ λ ) + √ i ¯ φ − ( ψ − λ ) + ¯ φ − φ − D − ¯ φ − φ − σ . (4.1)Here we define the covariant derivatives as D µ φ ± := ∂ µ φ ± ∓ iA µ φ ± , D µ ψ ± := ∂ µ ψ ± ∓ iA µ ψ ± D µ ¯ φ ± := ∂ µ ¯ φ ± ± iA µ ¯ φ ± , D µ ¯ ψ ± := ∂ µ ¯ ψ ± ± iA µ ¯ ψ ± . (4.2)In Wess-Zumino gauge, the supersymmetric transformation for the chiral multipletis given by δφ ± = √ (cid:15)ψ ± ,δψ ± = √ iγ µ (cid:15)D µ φ ± + √ (cid:15)F ± ∓ √ i ¯ (cid:15)σφ ± ,δF ± = √ i ¯ (cid:15)σ µ D µ ψ ± ± i (¯ (cid:15) ¯ λ ) φ ± ± √ i (¯ (cid:15)ψ ± ) σ. (4.3)Then supercurrents for SQED are calculated as J QED µ = − iF µν γ ν ¯ λ + i (cid:15) ρσµ ¯ λF ρσ + γ µν ¯ λ∂ ν σ − ¯ λ∂ µ σ − √ D µ ¯ φ + ψ + − ¯ φ + φ + γ µ ¯ λ + √ D ν ¯ φ + γ µν ψ + − √ σ ¯ φ + γ µ ψ + − √ D µ ¯ φ − ψ − + ¯ φ − φ − γ µ ¯ λ + √ D ν ¯ φ − γ µν ψ − + √ σ ¯ φ − γ µ ψ − , ¯ J QED µ = iF µν γ ν λ − i (cid:15) ρσµ λF ρσ + γ µν λ∂ ν σ − λ∂ µ σ − √ D µ ¯ φ + ¯ ψ + − ¯ φ + φ + γ µ λ + √ D ν ¯ φ + γ µν ¯ ψ + − √ σ ¯ φ + γ µ ¯ ψ + − √ D µ φ − ¯ ψ − + ¯ φ − φ − γ µ λ + √ D ν φ − γ µν ψ − + √ σφ − γ µ ¯ ψ − . (4.4)Thus we obtain the supersymmetric boundary condition for SQED:0 = (cid:15)J − ¯ (cid:15) ¯ J = − iF a ( (cid:15)σ a ¯ λ ) + i ( (cid:15) ¯ λ ) F + ( (cid:15)Cγ a ¯ λ ) ∂ a σ − ( (cid:15) ¯ λ ) ∂ σ − √ D ¯ φ + ( (cid:15)ψ + ) − ¯ φ + φ + ( (cid:15)σ ¯ λ ) + √ D a ¯ φ + ( (cid:15)Cγ a ψ + ) − √ σ ¯ φ + ( (cid:15)σ ψ + ) − √ D ¯ φ − ( (cid:15)ψ − ) + ¯ φ − φ − ( (cid:15)σ ¯ λ ) + √ D a ¯ φ − ( (cid:15)Cγ a ψ − ) + √ σ ¯ φ − ( (cid:15)σ ψ − )+ ( c.c. ) where a = 0 , . (4.5)17 Figure 7: The moduli space of SQED and the XYZ model. It contains three branches.They are the Higgs branch, the Coulomb branch with σ > σ < X = Y = 0, Z = X = 0 and Y = Z = 0 in termsof the XYZ model. The example of the brane (4.6) is indicated by the red region. Itfills the Coulomb branch. It is the same as Z = 0 region in the moduli space of theXYZ model.Here is an example of B-type boundary condition ( γ (cid:15) = (cid:15) ): γ ψ ± = ψ ± , γ λ = − λ, φ ± = 0 , ∂ σ = F = 0 . (4.6)We will not pursue the full classification of the boundary conditions in this paper.Instead let us discuss the mirror symmetry for the above example of the boundarycondition. The SQED considered above is conjectured to be equivalent to “the XYZ model” inthe low energy limit. The XYZ model contains three chiral superfields
X, Y, Z withthe superpotential W = XY Z. (4.7)This equivalence is called “mirror symmetry.” One piece of evidence is that the modulispace of vacua of SQED coincides with that of the XYZ model[11, 12]. See Figure 7.Let us consider how a boundary condition of SQED is mapped to the XYZ model.Notice that the mirror symmetry in three-dimensions does not exchange an A-braneand a B-brane. An A-brane and a B-brane preserve N = (1 ,
1) SUSY and N =182 ,
0) SUSY respectively in 3-dimensions. Thus an A-brane and a B-brane cannot bemirror to each other. This is one of the differences between two-dimensions and three-dimensions. In two-dimensional field theories with boundary, both A-type and B-typepreserve N = 2 on the boundary. Thus an A-brane can be mirror to a B-brane.Here we conjecture that the B-type boundary (4.6) in SQED corresponds to theB-brane in the XYZ model described by the hypersurface Z = 0 . (4.8)This brane is an holomorphic submanifold and the superpotential W = 0 =(constant)on its world volume. Thus this boundary condition preserves B-type SUSY from theanalysis of section 2.An evidence for this correspondence is the location of the brane in the modulispace. In both sides the brane fills two branches out of three. In the SQED sidethese two branches are Coulomb branches. Actually the boundary condition (4.6) isNeumann to the Coulomb branch direction spanned by σ and the “dual photon” ρ ,while it does not extend to the Higgs branch as seen from φ ± = 0. In the XYZ modelside Z = 0 brane fills the two branches out of three as seen in Figure 7.For a B-type boundary a superconformal index is also defined in the same way as[23, 24, 25]. It will be an interesting future work to calculate the superconformal indexwith boundary and check whether the SQED result and the XYZ model result agreewith each other. In this paper, we provided supersymmetric boundary conditions in three-dimensional N = 2 Landau-Ginzburg model and Abelian gauge theories. We analyzed the Abelianduality of the boundary conditions between pure Maxwell theory and chiral mattertheory. Our result revealed the exact correspondence in terms of supersymmetricboundary conditions. Furthermore we investigated supersymmetric boundary condi-tions in N = 2 SQED, which is supposed to be dual to XYZ model. We made aconjecture on the mirror dual of an example of B-type boundary condition.One can expect many possible applications and future directions related to ouranalysis. It will be a very interesting future problem to calculate the superconformalindex with a boundary to check the mirror symmetry. As discussed in [3], supersym-metric boundary conditions can be identified with BPS domain walls by using thefolding trick. A related calculation of the index in the presence of a domain wall infour dimensions has been done in [26]. It also seems to be an interesting problem to19alculate the partition functions of the N = 2 theory on other spaces with boundarysuch as hemispheres and hemiellipsoids by localization [27, 28]. In particular it shouldbe fruitful to investigate the boundary c-theorem proposed by [29]Another interesting future work is to investigate the role of boundaries or domainwalls in 3d-3d correspondence[15, 16, 17, 18]. We expect that 2d SUSY and 4d-non-SUSY versions of AGT relations might be found by investigating the dualitydomain walls in three-dimensional N = 2 theories. Recent work in [30] is in the samedirection, in which (2 ,
0) supersymmetry, B-type boundary condition is chosen on thetwo-dimensional boundary.Also we would like to understand these results in string theory. In [31, 32], theydiscuss three-dimensional mirror symmetries by using string theory. It is natural tothink of our problems including boundary in such constructions.Moreover, in our construction, we have (2 ,
0) and (1 ,
1) supersymmetry on the two-dimensional boundary. So far there are few examples of mirror symmetries for suchtheories. Our constructions can be useful to explore them.
Acknowledgments
We would like to thank Tohru Eguchi, Abhijit Gadde, Sergei Gukov, Kentaro Hori,Tetsuji Kimura, Yu Nakayama, Hirosi Ooguri, Pavel Putrov, Mauricio Romo, JohnH. Schwarz, Yuji Tachikawa, and Yutaka Yoshida for discussions and comments. Thework of T.O. was supported in part by JSPS fellowships for Young Scientists. Thework of S.Y. was supported in part by JSPS KAKENHI Grant No. 22740165.
A Spinors
In this appendix, we give our notations and useful formulas in three-dimensional N = 2theories. We use the metric η µν = η µν = diag( − , ,
1) and 2 × γ µ matrices satisfy { γ µ , γ ν } = 2 η µν . (A.1) γ is taken as anti-Hermitian and γ and γ as Hermitian.We introduce C matrix C , which has the following properties: C † = C − , C T = − C, ( Cγ µ ) T = Cγ µ (A.2)Two-component spinors ψ α with upper or lower indices transform under C : ψ α := C αβ ψ β , ψ α = ( C − ) αβ ψ β . (A.3)20e use the following summation convention:( χψ ) := χ α ψ α = χ α C αβ ψ β , ( γ µ ψ ) α = γ µαβ ψ β , ( Cγ µ ψ ) α = ( Cγ µ ) αβ ψ β . (A.4)We define σ -matrices as σ µ := Cγ µ , (A.5)and use the summation expression ξσ µ ψ := ξ α ( Cγ µ ) αβ ψ β .We define charge conjugation by¯ ψ α := ( C ( γ ) T ) αβ ( ψ β ) ∗ . (A.6)Here are useful spinor formulas: ξψ = ψξ, ξσ µ ψ = − ψσ µ ξ,ψσ µ ψ = 0 , ψCγ µν χ = − χCγ µν ψ, (A.7)( ξψ ) † = − ¯ ψ ¯ ξ, ( ξσ µ ψ ) † = ¯ ψσ µ ¯ ξ = − ¯ ξσ µ ¯ ψ, (A.8) θ α θ β = 12 C αβ θθ, θ α θ β = −
12 ( C − ) αβ θθ, (A.9)( θψ )( θχ ) = −
12 ( θθ )( ψχ ) , (A.10)( θσ µ χ )( θψ ) = − θθψσ µ χ, (A.11) θσ µ ¯ θθσ ν ¯ θ = 12 θθ ¯ θ ¯ θη µν , (A.12) −
12 ( χλ )( ψξ ) −
12 ( χσ µ λ )( ψσ µ ξ ) = ( χξ )( ψλ ) , (A.13) Cγ µ C − = − γ µT , Cγ µT C − = − γ µ . (A.14)where ψ, ξ, θ, λ are two-component spinors.21 Superspace
We introduce three-dimensional N = 2 superspace coordinates ( x µ , θ α , ¯ θ α ), transform-ing as x µ → x µ − i(cid:15)σ µ ¯ θ − i ¯ (cid:15)σ µ θ , θ → θ + (cid:15) and ¯ θ → ¯ θ + ¯ (cid:15) under the supersymmetrytransformations. We also define the following supersymmetric derivatives: Q α := ∂∂θ α − i ( σ µ ¯ θ ) α ∂ µ , (B.1) Q α := − ∂∂ ¯ θ α + i ( σ µ θ ) α ∂ µ , (B.2) D α := ∂∂θ α + i ( σ µ ¯ θ ) α ∂ µ , (B.3) D α := − ∂∂ ¯ θ α − i ( σ µ θ ) α ∂ µ . (B.4)They have the anticommutation relations { Q α , Q β } = 2 iσ µαβ ∂ µ , { D α , D β } = − iσ µαβ ∂ µ , (B.5)with all the other anticommutators vanishing. The supersymmetry transformation ofa superfield Φ( x, θ, ¯ θ ) is expressed as δ Φ( x, θ, ¯ θ ) = ( (cid:15)Q − ¯ (cid:15)Q )Φ . (B.6) C Superfield
C.1 Chiral superfield
Chiral superfield Φ( x, θ, ¯ θ ) is defined as D α Φ = 0 . (C.1)Using y µ := x µ + iθσ µ ¯ θ , we obtain the component field representations:Φ = Φ( y, θ )= φ ( y ) + √ θψ ( y ) + θθF ( y )= φ ( x ) + iθσ µ ¯ θ∂ µ φ ( x ) − θθ ¯ θ ¯ θ∂ φ ( x ) + √ θψ ( x ) + i √ θθ )(¯ θσ µ ∂ µ ψ ( x )) + θθF ( x ) . (C.2)Antichiral superfield ¯Φ( x, θ, ¯ θ ) with the constraint D α ¯Φ = 0 can be obtained from(C.2) by conjugation:¯Φ = ¯ φ ( x ) − iθσ µ ¯ θ∂ µ ¯ φ ( x ) − θθ ¯ θ ¯ θ∂ ¯ φ ( x ) − √ θ ¯ ψ ( x ) − i √ θ ¯ θ )( θσ µ ∂ µ ¯ ψ ( x )) − ¯ θ ¯ θ ¯ F ( x ) . (C.3)22 .2 Vector superfield Vector superfields satisfy the relation V = ¯ V . (C.4)Choosing Wess-Zumino gauge we obtain the simple expression: V = − θσ µ ¯ θA µ + iθ ¯ θσ − iθθ ¯ θ ¯ λ + i ¯ θ ¯ θθλ + 12 θθ ¯ θ ¯ θD ( x ) . (C.5)We can express field strength as a linear multiplet:Σ := − i DDV. (C.6)In the component description, it is written asΣ = σ + θ ¯ λ − λ ¯ θ − i (¯ θθ ) D + 12 (¯ θCγ µν θ ) F µν − i θθ (¯ θσ µ ∂ µ ¯ λ ) + i θ ¯ θ ( θσ µ ∂ µ λ ) + 14 θθ ¯ θ ¯ θ∂ µ ∂ µ σ. (C.7) References [1] H. Ooguri, Y. Oz, and Z. Yin, “D-branes on Calabi-Yau spaces and theirmirrors,”
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