The energy-momentum multiplet of supersymmetric defect field theories
PPreprint typeset in JHEP style - HYPER VERSION
KCL-MTH-17-01
The energy-momentum multiplet of supersymmetric defect field theories
Nadav Drukker, Dario Martelli and Itamar Shamir
Department of Mathematics, King’s College London,The Strand, WC2R 2LS, London, United Kingdom [email protected] , [email protected] , [email protected] Abstract:
Defects in field theories break translation invariance, resulting in thenon-conservation of the energy-momentum tensor in the directions normal to thedefect. This violation is known as the displacement operator . We study 4d N =1 theories with 3d defects preserving 3d N = 1 supersymmetry by analyzing theembedding of the 3d superspace in the 4d superspace. We use this to construct theenergy-momentum multiplet of such defect field theories, which we call the defectmultiplet and show how it incorporates the displacement operator. We also derivethe defect multiplet by using a superspace Noether procedure. a r X i v : . [ h e p - t h ] A ug ontents
1. Introduction 22. N = 1 supersymmetry in 3d 5
3. Energy-momentum multiplets in 3d 7
4. Coupling of 3d and 4d theories 10
5. Warm-up – global conserved currents 16
6. Energy-momentum multiplet in 4d 19
7. Superspace Noether approach to energy-momentum multiplets 27
8. Concluding remarks 32A. The displacement operator in scalar and gauge field theory 33B. Superspace conventions and useful formulas 35
B.1 4d superspace 35B.2 3d superspace 36
C. The S -multiplet 37 – 1 – . Introduction In this paper we consider co-dimension one defects in 4d theories with N = 1 su-persymmetry. By this we mean 4d theories coupled to 3d theories living on a 3dsubmanifold. We focus on planar submanifolds, specified by a constant and space-like normal vector n µ . The submanifold can be taken to be x n = x µ n µ = 0. Thepresence of the defect leads to an explicit breaking of translation symmetry in thedirection orthogonal to it. This manifests itself as a violation of the conservation ofthe energy-momentum tensor T µν by an operator local to the defect, which reads ∂ µ T νµ = n ν δ ( x n ) f d . (1.1)Here f d is called the displacement operator. The presence of the delta function meansthat away from the defect the energy-momentum tensor is conserved. Equation (1.1)can be easily generalized to defects with co-dimension greater than one. We presenttwo explicit examples of the displacement operator in bosonic theories in appendix A.The displacement operator appears in several applications. The Bremsstrahlungfunction describing the radiation of an accelerating charge can be extracted as thecoefficient of the two-point function of the displacement operator of a Wilson line[1, 2, 3]. More recently, the displacement operator was used to study the dependenceof entanglement entropy on the shape of the entangling surface [4, 5, 6, 7, 8]. Addi-tionally, conformal methods were used to constrain the form of correlation functionsof the energy-momentum tensor and the displacement operator and to obtain con-straints on the flow of defect field theories [9, 10, 11]. For co-dimension 2 defects intheories with N = 2 supersymmetry in 4d, the displacement operator was discussedin [12].Focusing on N = 1 supersymmetry in 4d, the main goal of this paper is to con-struct the supersymmetric multiplet of the displacement operator. When there areno defects, it was shown that any N = 1 theory in 4d admits a so-called S -multiplet[13]. It generalizes the Ferrara-Zumino (FZ), R and superconformal multiplets whichexist only under additional assumptions (see for instance [14, 15]). The S -multipletmay be defined as a real vector superfield S α ˙ α satisfying ¯ D ˙ α S α ˙ α = 2( χ α − Y α ) . (1.2)Here χ α is a chiral superfield satisfying D α χ α = ¯ D ˙ α ¯ χ ˙ α and Y α is constrained by¯ D Y α = 0 and D ( α Y β ) = 0. These conditions mean that we can locally solve Y α = D α X with X chiral. An explicit computation shows that the components of S µ include a symmetric and conserved energy-momentum tensor T νµ and a conservedsupercurrent S αµ . Schematically, the component expansion of S µ takes the form S µ = − iθ ( S µ + . . . ) + θσ ν ¯ θ (2 T νµ + . . . ) + . . . (1.3) Note that we use different conventions from [13]. In particular, bi-spinors are (cid:96) α ˙ α = σ µα ˙ α (cid:96) µ ,where we are using the notation of Wess and Bagger [16]. – 2 –he main result of this paper is a modification of (1.2) by terms arising fromthe presence of a defect. Since the defect necessarily breaks some of the translationand Lorentz symmetries it can at most preserve a subalgebra of supersymmetry. For N = 1 in 4d and n µ space-like the interesting cases are: • co-dimension one defects preserving N = 1 in 3d. • co-dimension two defects preserving N = (0 ,
2) in 2d.Both these subalgebras preserve half of the original supersymmetries. In this pa-per we consider the first case. We choose coordinates x µ = ( x i , x n ) where x i arespace-time coordinates, used along the world-volume of the defect. The preservedsupercharges take the form ˆ Q α = √ (cid:0) Q α + ( σ n ¯ Q ) α (cid:1) with { ˆ Q α , ˆ Q β } = 2(Γ i ) αβ P i . (1.4)Here the 3d gamma matrices are defined by Γ i ≡ σ n ¯ σ i . Notice that only momentaorthogonal to n µ appear in this algebra.We propose the following modification of (1.2):¯ D ˙ α S α ˙ α = 2( χ α − Y α ) + 2 δ (˜ y n ) Z α , (1.5)which we take as the definition of the defect multiplet . Let us explain the ingredientswhich enter in the new term. The argument of the delta function is ˜ y n ≡ x n + iθσ n ¯ θ − iθ . It has two virtues: (1) it is chiral (annihilated by ¯ D ˙ α ) and (2) it is invariantunder the subalgebra (1.4). This means that it breaks the symmetry in the correctway. We demand ¯ D Z α = 0 and the reality conditions Z α + ( σ n ¯ Z ) α → −−→ , ¯ D ˙ α Z α + D α ¯ Z ˙ α → −−→ − iσ nα ˙ α D . (1.6)The arrows imply a projection of the 4d superspace to the 3d N = 1 superspace and D is a real scalar superfield of the 3d superspace.We show that (1.5) implies the existence of an energy-momentum tensor satis-fying (1.1) where f d is now the top component of D . The energy-momentum tensoris conserved in the other directions, i.e. ∂ µ T jµ = 0, but it is generally not symmet-ric. Moreover, unlike the S -multiplet (1.3) in which S αµ is a conserved supercurrent,in (1.5) only the combination S αµ + σ nα ˙ α ¯ S ˙ αµ is conserved. This is the combinationassociated with the subalgebra (1.4).In a purely 3d theory, the energy-momentum sits in a 3d N = 1 multipletanalogous to the S -multiplet (1.2). Such multiplets were discussed in the literaturein the superconformal case [17, 18, 19] (and for N = 2 in 3d [20]). Using the3d N = 1 superspace coordinates ( x i , Θ α ), where the Grassmannian coordinate isMajorana, satisfying the reality conditions (Θ α ) † = Θ α σ nα ˙ α , we define a 3d N = 1energy-momentum multiplet by D α J αj = − ∂ j Σ , (Γ j ) αβ J βj = i D α ( H − Σ) , (1.7)– 3 –here D α is the covariant derivative in the 3d superspace. Σ and H are real scalarsuperfields and J αj is Majorana and for Σ = H = 0 this multiplet reduces to thesuperconformal case. We show that (1.7) leads to a component expansion, whichincludes J αj = − S (3) αj − i (Γ i Θ) α (2 T (3) ij + . . . ) + . . . , (1.8)where S (3) αj is a conserved Majorana supercurrent and T (3) ij is a conserved and sym-metric energy-momentum tensor. We study the structure of improvements of thismultiplet and discuss two examples.When a defect field theory is constructed as a coupling of a 4d theory with a 3dtheory, the total energy-momentum tensor of the system has a contribution localizedon the defect T µν = T (4) µν + δ ( x n ) P µi P ν j T (3) ij , (1.9)where P ni = 0 and P ki = δ ki is an embedding. The superspace analog of thisstatement, which is another result of this note, is that the 3d energy-momentummultiplet (1.7) can be written as the S -multiplet in the 4d superspace. This isachieved by studying the embedding of the 3d superspace in the 4d one. We definea change of variablesΘ α = 1 √ θ + σ n ¯ θ ) α , (cid:101) Θ α = i √ θ − σ n ¯ θ ) α (1.10)in the 4d superspace and identify Θ α with the 3d Grassmannian coordinate. Thisallows us to embed (1.7) in the S -multiplet as S (3) α ˙ α = δ (˜ x n ) (cid:101) Θ β J βj (Γ j σ n ) α ˙ α , (1.11)where ˜ x n = x n − Θ α (cid:101) Θ α is an invariant of the subalgebra. This gives rise to thestructure in (1.9). When the 3d theory interacts with the 4d one, the S -multipletmust be modified to include the new term in (1.5) which leads to the displacementoperator.The outline of this paper is as follows. In section 2 we review the 3d N = 1superspace. In section 3 we construct energy-momentum multiplets in 3d and discussexamples. In section 4 we study the embedding of the 3d superspace in the 4dsuperspace as a tool for coupling 4d theories with 3d defect theories. We consider tworepresentative examples: 4d chirals coupled to 3d scalars via a scalar potential anda bulk gauge multiplet coupled to a global symmetry on the defect. In section 5 weconsider global conserved currents as a simple application of the formalism developed.In section 6 we construct the defect multiplet. In section 7 we show how to obtain theenergy-momentum multiplets in 3 and 4 dimensions as well as the defect multipletusing a superspace Noether procedure. In section 8 we discuss some applications and– 4 –uture directions. We include 3 appendices. In appendix A we review a computationof the displacement operator in two simple bosonic theories. Appendix B includestwo parts: In the first we review some necessary material on the 4d superspace, andin the second we collect some useful formulas corresponding to the embedding of the3d superspace in 4d. Finally, in appendix C we review the S -multiplet as well as theexample of chiral superfields which is used in the paper. N = 1 supersymmetry in 3d In this section we review some basic facts about N = 1 supersymmetry in 3d. Mostof our presentation in this section is close in spirit to [15] although our conventionsare different. The need to juggle two superspaces at the same time inevitably putspressure on the available resources of letters and indices. We have chosen a mini-malistic approach, whereby the reader is trusted with understanding from contextwhich object lives in which universe. We hope this does not lead to much confusion.Let us begin by specifying our conventions for 3d, and their relation to 4d. Itis important to emphasize that the constructions discussed in this section as wellas the next one are strictly 3d. The invocation of the 4d embedding in our choiceof conventions here is meant to facilitate the discussion of section 4, in which weconsider the coupling of 3d and 4d theories. As described in the introduction, theembedding is specified by a constant space-like vector n µ . This leads to a split x µ = ( x n , x i ), where as before x n = n µ x µ and x i are coordinates of a 3d Minkowskispace. Similarly, the 4d Pauli matrices split according to σ µα ˙ α = ( σ nα ˙ α , σ iα ˙ α ). Thebasic spinor in 3d is a Majorana doublet with a reality condition¯ χ ˙ α = ( χ α ) † = ( χσ n ) ˙ α . (2.1)Even though, contrary to 4d, χ α and its conjugate transform in equivalent repre-sentations, it is convenient to keep track of dotted and undotted indices, which areconverted by the use of σ nα ˙ α . In this way, spinor contraction as well as spinor in-dices lowering and raising follow straightforwardly from the 4d conventions. The 3dgamma matrices are (Γ i ) αβ = 2( σ ni ) αβ and satisfy Γ i Γ j = − η ij − i(cid:15) ijk Γ k . Here the3d and 4d epsilon tensors are related by (cid:15) ijk = (cid:15) ijkn .The superspace coordinates are ( x i , Θ α ) where Θ α are Grassmannian coordinatessubject to the reality condition of eq. (2.1). The supersymmetry generators insuperspace are defined by Q α = ∂∂ Θ α − i (Γ j Θ) α ∂ j , {Q α , Q β } = 2 i (Γ j ) αβ ∂ j . (2.2)We also define covariant derivatives by D α = ∂∂ Θ α + i (Γ j Θ) α ∂ j , {D α , D β } = − i (Γ j ) αβ ∂ j . (2.3) Our conventions for 4d are of course based on Wess and Bagger [16]. In particular we have therelation σ µ ¯ σ ν = − η µν + 2 σ µν , where σ µν = − σ νµ . See appendix B.1 for more useful 4d formulas. – 5 –s usual, the covariant derivatives are defined so that {Q α , D β } = 0. Let us quote afew useful identities for the covariant derivatives D α D β = − i∂ αβ + 12 (cid:15) αβ D , D β D α D β = 0 , (2.4) D D α = 2 i∂ αβ D β , D α D = − i∂ αβ D β . (2.5)Here the bi-spinor is defined as ∂ αβ = (Γ j ) αβ ∂ j . We also use (Γ j ) αβ = (cid:15) βγ (Γ j ) αγ ,which is symmetric in the spinor indices. The simplest multiplet contains a real scalar, a Majorana fermion and a real auxiliaryfield. In superspace it is described by a scalar multiplet with the following componentexpansion A = a + Θ χ + 12 Θ f a . (2.6)It is immediate to derive the supersymmetry variation δA = ζ Q A by using (2.2).We find δa = ζχ,δχ α = ζ α f a + i (Γ i ζ ) α ∂ i a,δf a = iζ Γ i ∂ i χ. (2.7) Vector multiplet
Vector fields sit in a spinor multiplet V α with gauge symmetry acting by δ V α = D α ω .The gauge symmetry can be used to fix the Wess-Zumino gauge, in which V α takesthe form V α = i (Γ i Θ) α v i − Θ λ α . (2.8)A gauge invariant field strength is defined by W α = 12 D β D α V β = λ α − i (cid:15) kij (Γ k Θ) α F ij + i (Γ i ∂ i λ ) α , (2.9)where F ij = ∂ i v j − ∂ j v i . It follows immediately from the identity D β D α D β = 0 that D α W α = 0. In fact, this gives the Bianchi identity.Fully covariant derivatives are defined by D α = D α + i V α and D αβ = ∂ αβ + i V αβ ,satisfying the following algebra { D α , D β } = − i D αβ , [ D α , D βγ ] = − ( (cid:15) αβ W γ + (cid:15) αγ W β ) . (2.10)In particular, we have the relation V αβ = i D ( α V β ) with V αβ | = v αβ .– 6 – urrent multiplet Conserved currents ∂ i j i = 0 sit in a spinor multiplet J α satisfying D α J α = 0. Incomponents this is solved as J α = χ α + i (Γ i Θ) α j i + i (Γ i ∂ i χ ) α . (2.11)Clearly the field strength W α defined above is a current multiplet, with the dual fieldstrength conserved by the Bianchi identity. A supersymmetric Lagrangian is a top component of a real scalar multiplet (2.7). Asimple example is V M = D α A D α A which corresponds to a canonical kinetic termfor A V M | Θ = −
12 ( ∂ j a ) − i χ Γ j ∂ j χ + 12 f a . (2.12)This can be generalized to include multiple fields A I with a non-standard kineticterm V M = G IJ ( A ) D α A I D α A J . A scalar potential is constructed as a real function P ( A I ) and the equations of motion for such a model are D A I + Γ IJK D α A J D α A K = G IJ ∂ J P, (2.13)where Γ IJK is the usual Levi-Civita connection of G IJ .A gauge invariant interaction can be succinctly written by promoting D α to D α (see (2.10) for the definition of D α ). For simplicity we shall stick to Abelian gaugefields and take V MG = D α ¯ A D α A , where A is here a complexified scalar multiplet.The equation of motion is D A = 0 and the gauge invariant current J α = i ( A D α ¯ A − ¯ A D α A ) (2.14)is obtained by the variation δV MG = δ V α J α . The Lagrangian for the gauge field isderived from the multiplet V G = − W α W α . More explicitly it is given by V G | Θ = − i λ Γ j ∂ j λ − F ij F ij . (2.15)The equation of motion is D β D α W β = J α . It is also possible to include a Chern-Simons term V CS = κ π W α V α , but being topological, it does not matter for anythingwe do in the sequel.
3. Energy-momentum multiplets in 3d
Any local supersymmetric field theory contains a conserved and symmetric energy-momentum tensor T (3) ij and a conserved supercurrent S (3) αi . It follows from the algebra– 7 –f supersymmetry that these two operators sit in the same multiplet. When a super-space realization of the algebra is available, then these operators can be incorporatedin a superfield. In this section we define such superfields. All 3d N = 1 theoriesadmit a maximal multiplet with 6 + 6 components, but some theories admit shortermultiplets with 4 + 4 or 2 + 2 components (in the superconformal case). The su-perconformal multiplet was described by several groups before [17, 18, 19], whichalso discussed extended supersymmetry. However, to the best of our knowledge thenon-conformal energy-momentum multiplets were not considered elsewhere in theliterature. (See [20] for 3d N = 2.) We discuss the structure of improvements ofthese multiplets and review some examples.We define a real multiplet J αi by D α J αi = − ∂ i Σ , (Γ i ) αβ J βi = i D α ( H − Σ) , (3.1)with Σ and H both real multiplets. The component expansion of Σ and H isΣ = σ + Θ ψ + 12 Θ f σ , H = η + Θ κ + 12 Θ f η . (3.2)Let us emphasize that only the derivatives of σ and η are guaranteed to be well-defined. More generally we can write D α J αi = − i and J βαβ = i H α which werequire to satisfy ∂ [ i Σ j ] = 0 and D α D β H α = 0. This means that locally we can solveΣ i = ∂ i Σ and H α = D α ( H − Σ). To simplify the notation we will not make thisexplicit.Solving (3.1) for the components of J αi we obtain J αj = − S (3) αj + i (Γ j ψ ) α + Θ α ∂ j σ − i (Γ i Θ) α (cid:18) T (3) ij − η ij f σ + 12 (cid:15) ijk ∂ k η (cid:19) −
12 Θ (cid:0) i (Γ i ∂ i S j ) α − (Γ j Γ i ∂ i ψ ) α (cid:1) . (3.3)Here T (3) ij is symmetric and conserved, S (3) αi is conserved, and the following relationshold T (3) ii = f η + 2 f σ , S (3) αβ α = i ( κ β + 2 ψ β ) . (3.4)The combination H + 2Σ is the ‘trace multiplet’. Before continuing, let us mentionthat a simple generalization of (3.1) is obtained by changing the second equation to J βαβ = i D α ( H − Σ) + iJ α , where J α is a conserved current. This is a multiplet whichencompasses a non-symmetric energy-momentum tensor. Let us examine how this multiplet can be modified. For any real multiplet U = u + Θ ρ + Θ f u , we can act on J αi by the following transformation δ J αi = i (Γ i ) αβ D β U, δ
Σ =
U, δH = − U, (3.5)– 8 –nder which the energy-momentum tensor and the supercurrent do not change. Inparticular, it is easy to see that the ‘trace multiplet’ remains unmodified. Anotherway to transform the multiplet is by δ J αi = − ∂ i D α U, δ
Σ = D U, δH = 0 , (3.6)which is a bona fide improvement. The resulting transformation is δS (3) α k = 2(Γ [ k Γ j ] ∂ j ρ ) α , δT (3) ij = ( ∂ i ∂ j − η ij ∂ ) u. (3.7)The general J αi multiplet we obtained, has 6 + 6 components, but is not minimal(similarly to the S -multiplet is 4d). A submultiplet with 4 + 4 components can beachieved by using equation (3.5) to set some linear relation between Σ and H . Theform of the improvement (3.6) suggests that a natural choice is H = 0. This isachieved by taking U = H in (3.5), which results in the multiplet D α J αi = − ∂ i Σ , (Γ i ) αβ J βi = − i D α Σ . (3.8)As explained by Komargodski and Seiberg [13] such improvements only make senseif H is a well-defined operator ( e.g. gauge invariant). It is interesting that in theexamples we study below, H indeed turns out to be well-defined, perhaps suggestingthe (3.8) always exists. We will see in section 6 that this multiplet is closely relatedto the 4d FZ multiplet.In some theories we may be able to further shorten the multiplet using theimprovement (3.6). This is possible if and only if there is a well-defined U such thatΣ = D U . When this is the case, we can set to zero the ‘trace multiplet’ leading toa multiplet satisfying D α J αi = 0 and (Γ i ) αβ J βi = 0 which is a 2 + 2 superconformalmultiplet (see [17, 18, 19]). As a first example we consider the sigma model of scalar multiplets A I described inthe previous section. The energy-momentum multiplet is then given by J αi = − G IJ ∂ i A I D α A J , Σ = V M + P, H = − V M . (3.9)The bottom component of P is not necessarily a well-defined operator. (For example,as in [20] we can take A I ∼ A I + 1 and a linear potential.) However, we note that H is well-defined, and we can define the shorter multiplet (3.8), in which Σ = V M + P .If the theory is free and massless, i.e. G IJ = δ IJ and P = 0, we can use theequations of motion to write Σ = D ( A I A I ) which means that the theory admits asuperconformal multiplet.Next, consider an Abelian gauge field coupled to a complex scalar multiplet. Thematter and gauge contributions to the energy-momentum multiplet are given by( J MG ) αβγ = − D βγ ¯ A D α A + D βγ A D α ¯ A ) , (3.10)( J G ) αβγ = − i W α D ( β W γ ) , (3.11)– 9 –nd satisfy D α ( J MG ) αβγ = − ∂ βγ ( D α ¯ A D α A ) − i W ( β J γ ) , (3.12) D α ( J G ) αβγ = − ∂ βγ (cid:18) W α W α (cid:19) + 4 i W ( β J γ ) , (3.13)( J MG ) αβα = − iD β ( D α ¯ A D α A ) , ( J G ) αβα = − iD β ( W α W α ) , (3.14)where J α is the gauge current (2.14). We can identify that Σ MG = − H MG = V MG and Σ G = − H G = − V G , where V MG and V G are the matter and gauge kinetic termsrespectively and are defined in the previous section.Let us note the cross-term 4 i W ( β J γ ) in (3.12), which clearly cancels in the sumwith (3.13). We have separated here the contributions of the matter part and thegauge part deliberately to exhibit this term. The reason is that, unlike the case here,when we couple the 3d matter fields to 4d gauge fields below, then the term in (3.12)will not be sufficient to completely cancel the 4d contribution. The remainder willbe identified as the displacement operator.
4. Coupling of 3d and 4d theories
In the previous two sections we have discussed various aspects of theories with N = 1supersymmetry in 3d. We are now ready to begin our exploration of the main themeof this paper: the supersymmetric coupling of 3d and 4d theories, and the structureof their combined energy-momentum multiplet. In this section we will explain howthis supersymmetric coupling can be performed in a manifest fashion, via superspace.Some of the discussion parallels a previous work by Bilal [21], who considered4d N = 1 theories with a boundary preserving a 3d N = 1 subalgebra (see also[22, 23] and [24, 25] in 4d N = 2). But there are also important differences, to bepointed out below, which are crucial to the main goals of this paper. For example,we demonstrate how to write superspace equations of motion for the coupled system.The basis for constructing supersymmetric coupling of 3d and 4d theories is tostudy the 3d superspace embedding in the 4d superspace. A simple approach utilizesthe pattern of symmetry breaking. As discussed in the introduction, the embeddingcan preserve at most two supersymmetries. Clearly, the broken Poincar´e symmetriescan be used to fix the normal vector n µ to some specified direction and translate the3d subspace to a point in the x n direction, say the origin. Supersymmetry acts onthe 4d superspace coordinates ( x µ , θ α , ¯ θ ˙ α ) by δx µ = iθσ µ ¯ ζ − iζσ µ ¯ θ, δθ α = ζ α , δ ¯ θ ˙ α = ¯ ζ ˙ α . (4.1)The subalgebra we consider is determined by the relation ζ α = ( σ n ¯ ζ ) α . Equivalently,we consider a supersymmetry generator which is a linear combination of supercharges The broken R -symmetry corresponds to a possible phase ζ α = e iη ( σ n ¯ ζ ) α . If the 4d field theorythat we consider has an R -symmetry, we can use it to dial η = 0. Otherwise, it is a genuineparameter of the embedding. In any event, we shall keep using η = 0 to simplify the notation. – 10 –ith opposite chirality ˆ Q α = 1 √ (cid:0) Q α + ( σ n ¯ Q ) α (cid:1) (4.2)As can be seen from (1.4) this generates an algebra isomorphic to N = 1 in 3d. Theaction leaves the following combinations of superspace coordinates invariant˜ x n ≡ x n − i θ − ¯ θ ) , (cid:101) Θ α = i √ θ − σ n ¯ θ ) α . (4.3)We can use these two coordinates to generate other invariants. For example,˜ y n ≡ ˜ x n + i (cid:101) Θ (4.4)is a chiral combination. It is illuminating to write ˜ y n in terms of the chiral coordinateof superspace y µ = x µ + iθσ µ ¯ θ . We find ˜ y n = y n − iθ , which is clearly a chiralcombination. We can complete ˜ x n and (cid:101) Θ α to a basis of the 4d superspace byincluding x i and another Grassmann coordinateΘ α = 1 √ θ + σ n ¯ θ ) α . (4.5)In total, we have the change of basis( x µ , θ α , ¯ θ ˙ α ) ←→ ( x i , ˜ x n , Θ α , (cid:101) Θ α ) . (4.6)These coordinates are natural from the point of view of the preserved subalgebra.Clearly, ( x i , Θ α ) can be identified with the coordinates in the 3d superspace, andare acted upon by the subalgebra in the expected way. Unlike [21, 22] where the3d superspace is identified via the relation θ α = ( σ n ¯ θ ) α (or simply (cid:101) Θ α = 0 in ourlanguage), we here find it very useful to keep track of all the coordinates including (cid:101) Θ α . It will become apparent below why this is advantageous.Consider now a general superfield F ( x µ , θ, ¯ θ ). We would like to understand howto decompose it into representations of the subalgebra. This is obtained by writingthe superfield in the coordinates system introduced above and expanding in (cid:101) Θ α F ( x i , ˜ x n , Θ , (cid:101) Θ) = F ( x i , ˜ x n , Θ) + (cid:101) Θ α F α ( x i , ˜ x n , Θ) + 12 (cid:101) Θ F ( x i , ˜ x n , Θ) . (4.7)It is obvious from the discussion above that the component superfields transformindependently under the subalgebra. However, for practical reasons it is usually For example, a chiral superfield Φ = ( φ, ψ α , F ) invariant under ˆ Q α must be a function of ˜ y n ,leading to the component expansion Φ(˜ y n ) = φ ( y n ) − iθ ∂ n φ ( y n ). This is explained as follows.Observing the variations of Φ and demanding invariance implies ψ α = 0 and setting δψ α = √ ζ α ( F + i∂ n φ ) + 2 i ( σ nµ ζ ) α ∂ µ φ ]to zero means that φ = φ ( x n ) and F = − i∂ n φ . This is precisely the component expansion above. – 11 –ore convenient to work with fields which are functions of x n instead of ˜ x n (notethat ˜ x n = x n − (cid:101) ΘΘ). Namely, in the coordinate system ( x i , x n , Θ α , (cid:101) Θ α ). However,this brings about a small complication, as one observes by writing explicitly thepreserved supercharge in these coordinates ˆ Q α = Q α + (cid:101) Θ α ∂ n . Here Q α is the 3dexpression in (2.2). In other words, this means that component superfields in a (cid:101) Θ α expansion mix under ˆ Q α . As usual, the problem is solved by introducing covariantderivatives. We define∆ α ≡ √ (cid:0) D α + ( σ n ¯ D ) α (cid:1) = ∂∂ Θ α + i (Γ i Θ) α ∂ i − (cid:101) Θ α ∂ n , (cid:101) ∆ α ≡ − i √ (cid:0) D α − ( σ n ¯ D ) α (cid:1) = ∂∂ (cid:101) Θ α + i (Γ i (cid:101) Θ) α ∂ i + Θ α ∂ n . (4.8)By construction we have that { ˆ Q α , ∆ α } = { ˆ Q α , (cid:101) ∆ α } = 0. The component superfieldsof (4.7) can thus be obtained by taking (cid:101) ∆ α derivatives and projecting to the 3dsuperspace by setting (cid:101) Θ = 0. Strictly speaking, a projection should also include x n = 0 (or some other point). Nevertheless, it is convenient to keep the location ofthe defect unspecified, i.e. keep explicit dependence on x n .The simplest example is that of a chiral superfield Φ = ( φ, ψ, F ), for which weobtain Φ( y i , y n , θ ) | (cid:101) Θ=0 = Φ( x i , x n + i Θ , √ Θ) = φ + Θ ψ + 12 Θ ( F + i∂ n φ ) . (4.9)A similar expression for this projection was obtained in [21, 22]. For a chiral superfield (cid:101) ∆ α does not give a new superfield since from (4.8) it has the same effect as ∆ α .Similarly, the anti-chiral superfield projects to¯Φ(¯ y i , ¯ y n , ¯ θ ) | (cid:101) Θ=0 = ¯Φ( x i , x n − i Θ , √ Θ σ n ) = ¯ φ + Θ σ n ¯ ψ + 12 Θ ( ¯ F − i∂ n ¯ φ ) . (4.10)Conversely, given a 3d superfield (with or without x n dependence) we can embed itinto the 4d superspace. As demonstrated below, this is required in order to writeequations of motion for the coupled system in the 4d superspace. As a simple ex-ample, a 3d scalar multiplet A = ( a, χ, f a ) can be embedded as a chiral multipletby A ( x i , x n , Θ) → −−→ A ( y, θ ) ≡ A ( y i , ˜ y n , √ θ )= a + √ θχ + θ ( f a − i∂ n a ) . (4.11)Let us remark that this is a “real chiral superfield”. Its existence is a by-product of thecoupling to 3d and will be important in the sequel. It is useful for later computations Let us note the relations θσ n ¯ θ = 12 (Θ + (cid:101) Θ ) , θσ i ¯ θ = − i (cid:101) ΘΓ i Θ . We refer the reader to appendix B.2 for more superspace relations. – 12 –o show more explicitly the relation between A and A . This is achieved by expandingaround ( x i , Θ) in the following way A = A ( y, √ θ ) = A ( x i + (cid:101) ΘΓ i Θ , ˜ x n + i (cid:101) Θ , Θ − i (cid:101) Θ)= A ( x i , ˜ x n , Θ) − i (cid:101) Θ α ∆ α A + 14 (cid:101) Θ ∆ A (4.12)This relation shows that A is the unique chiral superfield whose projection (cid:101) Θ = 0 is A . It is also useful as a trick to simplify certain computations below. In a similarway, we can embed A in an anti-chiral superfield ¯ A ≡ A (¯ y i , ˜¯ y n , √ σ n ¯ θ ).We note that projecting a chiral (anti-chiral) superfield to 3d and then liftingit to a chiral (anti-chiral) returns the original field. However, if we start with ananti-chiral ¯Φ, project to 3d ¯Φ | (cid:101) Θ=0 and then lift to a chiral we get (cid:101) ¯Φ( y, θ ) = ¯ φ + √ θσ n ¯ ψ + θ ( ¯ F − i∂ n ¯ φ ) . (4.13)It is easy to check that under the subalgebra (1.4) the multiplet (cid:101) ¯Φ = ( ¯ φ, σ n ¯ ψ, ¯ F − i∂ n ¯ φ ) transforms as a chiral. In the same sense (cid:101) Φ = ( φ, − ¯ σ n ψ, F + 2 i∂ n φ ) is ananti-chiral superfield.The embedding of a 3d superfield in 4d superspace can be written in anotherway, that is more useful for computations. Starting with a superfield A ( x i , x n , Θ) wedefine 4d chiral and anti-chiral superfields by A → −−→ A ≡
12 ¯ D ( (cid:101) Θ A ) , A → −−→ ¯ A ≡ D ( (cid:101) Θ A ) . (4.14)Clearly A and ¯ A are chiral and anti-chiral superfield respectively. With some labourthis can be computed explicitly and shown to be equivalent to the expansion (4.12)(and similarly for the anti-chiral). However, a simple trick renders this computationtrivial. Because of the (cid:101) Θ factor we can change the arguments of A in the chiralembedding by (cid:101) Θ α terms without changing the expression. There is a unique way ofdoing it which makes A chiral, namely A ( y i , ˜ y n , √ θ ). Then ¯ D acts only on (cid:101) Θ andthe result follows.Another useful relation allows us to rewrite 3d Lagrangians in 4d superspace.Recall that a 3d Lagrangian is a top component of a real scalar multiplet. Multiplyingby (cid:101) Θ allows us to write this as a D -term of a 4d real multiplet. Specifically, let P = p + Θ χ + Θ f p , then ( − (cid:101) Θ P = . . . + θ ¯ θ f p . We therefore have theprescription (cid:90) d x (cid:90) d Θ P = (cid:90) d x (cid:90) d θ ( − δ (˜ x n ) (cid:101) Θ P. (4.15)Notice that (cid:101) Θ can be thought of as a Grassmannian delta function. More gener-ally, we can replace − δ (˜ x n ) (cid:101) Θ by any function f = f (˜ x n , (cid:101) Θ) without breaking thesymmetry further. This can be interpreted as a smeared defect.– 13 –o do the same for gauge fields, consider first a real multiplet V with components( C, χ, M, v µ , λ, D ) which we decompose following the procedure given above. Mostinteresting is the component containing the vector. It is given by (cid:101) ∆ α V | (cid:101) Θ=0 = 1 √ χ + σ n ¯ χ ) α + Θ α (cid:18)
12 ( M + ¯ M ) + ∂ n C (cid:19) + i (Γ i Θ) α v i −
12 Θ (cid:18) √ λ + σ n ¯ λ ) + i √ i ∂ i ( χ + σ n ¯ χ ) (cid:19) . (4.16)This multiplet can be identified with the 3d vector multiplet V α mentioned in theprevious sections. In particular the gauge symmetry of the real multiplet δV = i (Ω − ¯Ω) translates into (cid:101) ∆ α δV | (cid:101) Θ=0 = 12 √ D − σ n ¯ D ) α (Ω − ¯Ω) | (cid:101) Θ=0 = 12 ∆ α (Ω + ¯Ω) | (cid:101) Θ=0 , (4.17)where Ω is a chiral superfield. We can identify the 3d gauge parameter multiplet ω with the real scalar multiplet ω = (Ω + ¯Ω) | (cid:101) Θ=0 as above equation (2.8).Next consider the field strength W α = − ¯ D D α V which satisfies by construction D α W α = ¯ D ˙ α ¯ W ˙ α . It is decomposed as W α ≡ i √ W − σ n ¯ W ) α | (cid:101) Θ=0 , (cid:102) W α ≡ √ W + σ n ¯ W ) α | (cid:101) Θ=0 . (4.18)Expanding in components they give W α = 1 √ λ + σ n ¯ λ ) α − i (cid:15) kijn (Γ k Θ) α F ij + i √ Γ i ∂ i ( λ + σ n ¯ λ ) α , (4.19) (cid:102) W α = − i √ λ − σ n ¯ λ ) α + Θ α D − i (Γ i Θ) α F ni + 12 Θ (cid:18) − √ i ∂ i ( λ − σ n ¯ λ ) α + √ ∂ n ( λ + σ n ¯ λ ) α (cid:19) . (4.20)Clearly W α can be identified with the field strength defined in 3d (2.9). In particularwe have ∆ α W α = i ( D α W α − ¯ D ˙ α ¯ W ˙ α ) = 0. Consider 4d chiral superfields Φ a with a K¨ahler potential K (Φ a , ¯Φ ¯ a ) and a super-potential W (Φ a ). On the defect we consider real scalars A I with a kinetic term V M = G ij ∆ α A I ∆ α A J as described around (2.13). The 3d and the 4d theoriesinteract through a potential P (Φ a , ¯Φ ¯ a , A I ) | (cid:101) Θ=0 localized on the defect. It is clearthat the 3d equations of motion stay the same as in (2.13) with P (Φ a , ¯Φ ¯ a , A I ) | (cid:101) Θ=0 substituting for the purely 3d potential. We would also like to obtain the equations Note that since A I are 3d fields (independent of x n ), D α and ∆ α can be used interchangeably. A similar potential was considered in [21] as a boundary interaction. – 14 –f motion of Φ a with the defect interaction. Following the discussion above we canlift the potential to the 4d superspace by (cid:90) d x (cid:90) d Θ P | (cid:101) Θ=0 = (cid:90) d x (cid:90) d θd ¯ θ (cid:16) ( − δ (˜ x n ) (cid:101) Θ P (cid:17) . (4.21)Here we switched ˜ x n for x n in the delta function so that it manifestly preserves thedesired symmetries. The difference is proportional to (cid:101) Θ α and does not change theexpression. To compute the equations of motion we change to integration over halfsuperspace (cid:90) d θd ¯ θ (cid:16) ( − δ (˜ x n ) (cid:101) Θ P (cid:17) = (cid:90) d θ ¯ D (cid:18) δ (˜ x n ) (cid:101) Θ P (cid:19) , (4.22)and use the relation12 ¯ D (cid:16) δ (˜ x n ) (cid:101) Θ P (Φ , ¯Φ , A ) (cid:17) = δ (˜ y n ) P (Φ , (cid:101) ¯Φ , A ) . (4.23)Here A and P are the chiral lifts of A and P and (cid:101) ¯Φ is the chiral associated with theanti-chiral ¯Φ as per (4.13). This leads to the equation of motion¯ D K a = 4 W a + 2 δ (˜ y n ) P a . (4.24)Clearly for this equation to make sense the delta function must be a chiral superfield. Consider a 4d U (1) gauge theory. As demonstrated above, V α ≡ (cid:101) ∆ α V | (cid:101) Θ=0 is equiva-lent to a 3d gauge multiplet. Therefore we can take a 3d theory with a global U (1)symmetry and gauge it by coupling to the U (1) gauge field coming from 4d. Thecoupling is identical to the minimal coupling for a complexified scalar multiplet A considered above (2.14) and so are the resulting 3d equations of motion for A . Herewe obtain the 4d gauge field equation of motion coupled to the 3d matter current(2.14).As usual the 4d gauge part is given by (cid:82) d θW W + (cid:82) d ¯ θ ¯ W ¯ W . The uncon-strained variable which we must vary to obtain the equation of motion is V . Bystandard superspace maneuvers we obtain (cid:90) d θ δV (cid:18) −
12 ( D α W α + ¯ D ˙ α ¯ W ˙ α ) (cid:19) . (4.25)For comparison, consider a charged 4d chiral field Φ. The Lagrangian is (cid:82) d θ ¯Φ e V Φ.Identifying the 4d current as J = ¯Φ e V Φ, the contribution to the equations of motionis −
12 ( D α W α + ¯ D ˙ α ¯ W ˙ α ) = 2 J. (4.26)– 15 –imilarly, from (2.14), the variation of the 3d coupling gives (cid:82) d Θ δ V α J α , which uponlifting to 4d becomes − (cid:90) d θδ (˜ x n ) (cid:101) Θ δ V α J α = − (cid:90) d θδV δ (˜ x n ) (cid:101) Θ α J α . (4.27)Here we have used the relation δ V α ≡ (cid:101) ∆ α δV | (cid:101) Θ=0 and the (cid:101) Θ factor to change the x n dependence of J α to ˜ x n since (cid:101) ∆ α ˜ x n = 0. The equations of motion we obtain are −
12 ( D α W α + ¯ D ˙ α ¯ W ˙ α ) = 2 δ (˜ x n ) (cid:101) Θ α J α ( x i , ˜ x n , Θ) (4.28)with the identification of δ (˜ x n ) (cid:101) Θ α J α as the embedding of the current in the 4d su-perspace. We discuss this embedding in more details in the next section.
5. Warm-up – global conserved currents
In this section we study multiplets of global conserved currents. We reviewed insection 2 the structure of such multiplets in the case of N = 1 in 3d. They aregiven by a spinor superfield J α satisfying D α J α = 0. We shall shortly remind thereader of its 4d counterpart. Our goal in this section is to formulate the mostgeneral conservation equation which is consistent with the symmetries of a 4d theoryinteracting with a 3d defect. This is a useful preliminary to our study of currentmultiplets pertaining to spacetime (superspace) symmetries to which we turn in thenext section.Let us begin by recalling that the conserved current multiplet in 4d is defined asa real multiplet J satisfying ¯ D J = 0. To see what relations this constraint implieson the components of J let us consider the standard superspace expansion of a realmultiplet, given in appendix B.1. In terms of these components we find (B.6)¯ D J = 2 i ¯ M + 4 θ ( iλ − σ µ ∂ µ ¯ χ ) − θ ( D + ∂ C − i∂ µ v µ ) , (5.1)and imposing the constraint implies ∂ µ v µ = 0 and the following component expansionof J , in which we renamed the fields for later convenience, J = f + iθρ − i ¯ θ ¯ ρ − θσ µ ¯ θj µ + θ ¯ θ ¯ σ µ ∂ µ ρ − ¯ θ θσ µ ∂ µ ¯ ρ − θ ¯ θ ∂ f. (5.2)Let us note that the constraint ¯ D J = 0 sets to zero a chiral submultiplet (5.1) of J . Next we show that a 3d current multiplet can be embedded in a 4d real multipletsatisfying the same constraint. As discussed previously, a 3d current resides in aspinor multiplet J α satisfying ∆ α J α = 0. Since the fields are so far 3d with no x n J α has x n dependence since the gauge invariant current depends on the 4d V α . – 16 –ependence we might as well use ∆ α instead of D α . A natural guess for the 4dembedding is (cid:101) J = δ (˜ x n ) (cid:101) Θ α J α , (5.3)and a simple computation confirms that ¯ D (cid:101) J = − i δ (˜ y n ) ¯ D ( (cid:101) Θ ∆ α J α ). This showsthat the 3d constraint for J α is exactly equivalent to the 4d one for (cid:101) J . Another wayto understand the expression for (cid:101) J is to consider the decomposition of J followingthe prescription given above. We find J | (cid:101) Θ=0 = f + Θ κ −
12 Θ j n , (5.4) (cid:101) ∆ α J | (cid:101) Θ=0 = χ α + Θ α ∂ n f + i (Γ i Θ) α j i + i (cid:0) Γ i ∂ i χ + 2 i∂ n κ (cid:1) α , (5.5)where κ α = i √ ( ρ − σ n ¯ ρ ) α and χ α = √ ( ρ + σ n ¯ ρ ) α . Setting J | (cid:101) Θ=0 = 0 means that j i is conserved in the 3d sense. Moreover, we find that (cid:101) ∆ α J | (cid:101) Θ=0 is identical to theexpression for the 3d conserved current multiplet (2.11).Let us now consider the case where the 3d and 4d theories are coupled. It turnsout that in this case the two terms above J and (cid:101) J are not sufficient for the constraintto hold. (We illustrate this below in an explicit example.) It is not far-fetched tospeculate that what we are missing is a (cid:101) Θ term, however there is a more elegantway of discovering this term.Looking back at (5.1), we see that to guarantee a conserved current it is suffi-cient to constrain the imaginary part of the θ component of the chiral superfield.Normally, a 4d chiral superfield must be complex and hence the constraint above isthe minimal possible. However as discussed in the previous section, owing to thecoupling with 3d, we have a natural construction of “real chiral superfields”. Wetherefore relax the constraint to¯ D J = δ (˜ y n ) B = δ (˜ y n ) (cid:16) b + √ θχ b + θ ( f b − i∂ n b ) (cid:17) , (5.6)where b and f b are real and χ b is Majorana. The argument of the delta functionis again crucial. Expanding ˜ y n = y n − iθ , we see that the imaginary part of the θ component is a total derivative. This implies ∂ µ v µ = − ∂ n ( δ ( x n ) b ) and lets usdefine a conserved current by j µ = v µ + δ µn δ ( x n ) b . The new term in the current isunderstood in light of the form of the projection in (4.9)-(4.10). The normal deriva-tives in Φ | (cid:101) Θ=0 and ¯Φ | (cid:101) Θ=0 mean that the potential involves derivative interactions andtherefore contributes to the current, as follows from Noether’s formula.As a final comment, let us show that the new term on the right hand sideof (5.6) can be written as a (cid:101) Θ contribution to (cid:101) J , as remarked above. For thispurpose, define the projection B = B| (cid:101) Θ=0 . Then using (4.14) we have the equality δ (˜ y n ) B = ¯ D ( δ (˜ x n ) (cid:101) Θ B ), demonstrating our claim.– 17 – .1 A derivation using superspace Noether procedure We now show how the equation for the current can be obtained from a variationalapproach. Let us start from a global U (1) symmetry. It acts on the matter fieldsby δ Φ a = iωq a Φ a and δA I = iωq I A I , where ω is the parameter of transformationand q a and q I are the charges. To obtain the current, the symmetry is gauged bygiving a space time dependence to the symmetry parameter. This is implementedin superspace in the following way. In the case of 4d chirals, ω is lifted to a chiralsuperfield Ω by defining δ Φ a = i Ω q a Φ a . The global limit is obtained by equatingΩ = ¯Ω. Chiral and anti-chiral fields are equal if and only if all fields vanish exceptfor the real part of the bottom component which has to be constant. This meansthat the variation of the 4d Lagrangian takes the form [26] δ L (4) = (cid:90) d θi (Ω − ¯Ω) J (5.7)for some J . The variation must vanish on the equations of motion for any Ω andtherefore we can obtain ¯ D J = D J = 0. Similarly, we introduce in 3d a realmultiplet ω which gauges the symmetry, and the global limit is obtained by ∆ α ω = 0.The Lagrangian hence transforms as δ L (3) = (cid:90) d Θ∆ α ωJ α (5.8)for some J α . This leads to the conservation equation ∆ α J α = 0.Let us now assume that the theories are coupled in a supersymmetric way. Wemake the identification ω = (Ω + ¯Ω) | (cid:101) Θ=0 and define ω (cid:48) = − i (Ω − ¯Ω) | (cid:101) Θ=0 . Varyingthe Lagrangian as above we get a new term since ω (cid:48) vanishes in the global limit, i.e. δ L tot = (cid:90) d θi (Ω − ¯Ω) J + (cid:90) d Θ δ ( x n ) (∆ α ωJ α − ω (cid:48) B ) . (5.9)The 4d part can be written as − i (cid:82) d θ Ω ¯ D J + c.c. and then projected into the 3dsuperspace − i (cid:90) d Θ( ω + iω (cid:48) ) ¯ D J | (cid:101) Θ=0 + i (cid:90) d Θ( ω − iω (cid:48) ) D J | (cid:101) Θ=0 . (5.10)We obtain the conservation equations − i (cid:0) ¯ D J − D J (cid:1) | (cid:101) Θ=0 = δ ( x n )∆ α J α , (cid:0) ¯ D J + D J (cid:1) | (cid:101) Θ=0 = δ ( x n ) B, (5.11)or more conveniently ¯ D J | (cid:101) Θ=0 = δ ( x n )( i ∆ α J α + B ). Using the same methods asabove this can be lifted to the 4d superspace expression we found in the previoussection.As an example, consider 4d chirals Φ a coupled to 3d scalar multiplets A I trans-forming as indicated above. A potential P (Φ a , ¯Φ ¯ a , A I , ¯ A ¯ I ) | (cid:101) Θ=0 is invariant if δP = i (cid:88) a q a ( P a Φ a − P ¯ a ¯Φ ¯ a ) + i (cid:88) I q I ( P I A I − P ¯ I ¯ A ¯ I ) = 0 . (5.12)– 18 –fter gauging, the fields transform by δ Φ a | (cid:101) Θ=0 = iq a ( ω + iω (cid:48) )Φ a | (cid:101) Θ=0 and δA I = iq I ωA I . This leads to δP | (cid:101) Θ=0 = − ω (cid:48) (cid:88) a q a ( P a Φ a − P ¯ a ¯Φ ¯ a ) | (cid:101) Θ=0 = − ω (cid:48) B. (5.13)It is a trivial exercise to compute J and J α assuming some U (1) invariant kineticterms for Φ a and A I and to show that the conservation equation (5.11) is satisfied.
6. Energy-momentum multiplet in 4d
The purpose of this section is to suggest a modification of the S -multiplet that comesfrom the interaction with a 3d defect preserving N = 1 supersymmetry. We do it intwo stages. First, we show how to embed the energy-momentum multiplet of a purely3d theory, i.e. equation (3.1), in the 4d S -multiplet. This is important since the totalenergy-momentum tensor should be of the form T (4) νµ + δ ( x n ) P ν i P µj T (3) ij , where P µj is the embedding defined in the introduction, with P nj = 0 and P kj = δ kj . However,the structure that is obtained is not sufficient to describe the coupling of 4d theorywith a 3d theory. We therefore study in section 6.2 what terms can appear on theright hand side of the S -multiplet which are consistent with an energy-momentumtensor conserved in the 3 directions tangent to the defect and a conserved Majoranasupercurrent. Finally, we elaborate on two examples and compute the resultingdisplacement operators.Let us reiterate here, for convenience, the definitions of the 3 and 4 dimensionalenergy-momentum multiplets. First, the 4d S -multiplet [13] is given by¯ D ˙ α S α ˙ α = 2( χ α − Y α ) , (6.1)with χ α chiral and D α χ α = ¯ D ˙ α ¯ χ ˙ α and Y α satisfying ¯ D Y α = 0 and D ( α Y β ) = 0. Thecondition on χ α means that it can be solved locally as − ¯ D D α V where V is a realmultiplet. Similarly, the condition on Y α means that it can be solved locally as D α X with X chiral. In appendix C we review the S -multiplet in more detail, includingits component expansion, improvements and some examples. In 3d we found themultiplet (3.1) ∆ α J αi = − ∂ i Σ , (Γ i ) αβ J βi = i ∆ α ( H − Σ) , (6.2)where Σ and H are real 3d multiplets. Note that we have replaced D α with ∆ α . Asremarked before, on 3d fields their action is identical. To determine the way the 3d energy-momentum multiplet sits in the 4d S -multipletit is most illuminating to consider its component expansion. Keeping in mind ourdiscussion of global conserved currents (see (5.3) and below), it is natural to guess– 19 –hat the 3d multiplet should appear as the (cid:101) Θ α component of the S -multiplet. Indeed,this is verified by computing the (cid:101) ∆ α derivative of the component expansion appearingin equation (C.3). In a purely 3d theory the normal component (such as j n ) andnormal derivatives are null and we find (cid:101) ∆ α S j | (cid:101) Θ=0 = − √ S j + σ n ¯ S j ) α + 2 i (cid:0) Γ j ( ψ + σ n ¯ ψ ) (cid:1) α + 2Θ α ∂ j ( x + ¯ x ) − i (Γ i Θ) α (cid:0) T ij − η ij A + (cid:15) ijkn ∂ k v n (cid:1) −
12 Θ (cid:18) i √ (cid:0) Γ i ∂ i ( S j + σ n ¯ S j ) (cid:1) α − (cid:0) Γ j Γ i ∂ i ( ψ + σ n ¯ ψ ) (cid:1) α (cid:19) . (6.3)This expression matches the component expansion of the 3d energy-momentum mul-tiplet (3.3), by identifying Σ = 2( X + ¯ X ) | (cid:101) Θ=0 and V = − (cid:101) Θ H = − θσ n ¯ θ η + . . . (recall χ α = − ¯ D D α V ). The latter is implied by identifying v n = η and using therelation V | θσ ν ¯ θ = − v ν . What we have shown is that we can embed J αj as S α ˙ α = (cid:101) Θ β J βαγ σ nγ ˙ α ,χ α = 18 ¯ D D α (cid:16) (cid:101) Θ H (cid:17) , Y α = 18 D α ¯ D (cid:16) (cid:101) Θ Σ (cid:17) (6.4)which solves the 4d S -multiplet equations (6.1).It is also useful to show this by an explicit computation. One quickly finds ¯ D ˙ α (cid:16) (cid:101) Θ β J βαγ σ nγ ˙ α (cid:17) = − i √ D (cid:16) (cid:101) Θ J βαβ (cid:17) − i σ n ¯ D ) γ (cid:16) (cid:101) Θ ∆ β J βαγ (cid:17) . (6.5)Notice that the terms in the two parentheses on the right hand side exactly corre-spond to the two terms in (6.2) and since both multiply (cid:101) Θ we can interchange D α with ∆ α as in (6.2), without restricting the dependence of the operators on x n . Wenow use this relation to obtain¯ D ˙ α (cid:16) (cid:101) Θ β J βαγ σ nγ ˙ α (cid:17) = 14 ¯ D D α (cid:16) (cid:101) Θ ( H − Σ) (cid:17) + i ( σ i ¯ D ) α ∂ i (cid:16) (cid:101) Θ Σ (cid:17) = 2( χ α − Y α ) − i ( σ n ¯ D ) α ∂ n (cid:16) (cid:101) Θ Σ (cid:17) . (6.6) Actually, that is somewhat of a lie. The computation is quite tedious if one attempts to carry itout by brute force. Therefore, out of consideration for the reader we show how it can be trivializedby a simple trick. The idea is to use (4.12), which here gives J βi ( x i , ˜ x n , Θ) = J βi ( y i , ˜ y n , √ θ ) + i (cid:101) Θ α ∆ α J βi ( x i , ˜ x n , Θ) + O ( (cid:101) Θ ) . From this we get (cid:101) Θ β J βi ( x i , ˜ x n , Θ) = 12 (cid:101) Θ β ¯ D (cid:16) (cid:101) Θ J βi (cid:17) − i (cid:101) Θ ∆ β J βi Here relation (4.14) was used. Applying ¯ D ˙ α now leads to (6.5). – 20 –f Σ has no x n dependence, then the last term in the second line drops out andwe obtain the result from before. Roughly speaking, this term is responsible forcancelling the ∂ n in Y α . We define2 Y (cid:48) α = 14 D α ¯ D (cid:16) (cid:101) Θ Σ (cid:17) + i ( σ n ¯ D ) α ∂ n (cid:16) (cid:101) Θ Σ (cid:17) , (6.7)which satisfies 2 ¯ D ˙ α Y (cid:48) α = − iσ iα ˙ α ∂ i Σ. We note the absence of the normal derivative inthis expression. Compare this with the corresponding 4d term in (6.1), which gives2 ¯ D ˙ α Y α = − iσ µα ˙ α ∂ µ X .For the application we want to consider, the 3d contribution should come witha delta function, and hence is defined as S (3) α ˙ α = δ (˜ x n ) (cid:101) Θ β J βαγ σ nγ ˙ α . (6.8)Note that we can change δ (˜ x n ) → δ (˜ y n ) because of the (cid:101) Θ β factor. It satisfies¯ D ˙ α S (3) α ˙ α = 2 δ (˜ y n )( χ α − Y (cid:48) α ) . (6.9)In fact, we can swallow the delta function in Σ and H , which is most clearly observedin the first line of (6.6). Since δ (˜ y n ) is chiral it goes through ¯ D ˙ α with no effect. But wecan pull the delta through D α as well since the commutation relation is proportionalto (cid:101) Θ α .Lastly, let us note that when considering also the 4d theory, we can form thecombination S µ ≡ S (4) µ + S (3) µ = 2 θσ ν ¯ θ (cid:16) T (4) νµ + δ ( x n ) P ν i P µj T (3) ij (cid:17) + · · · . (6.10)Since we showed that the 3d terms can be swallowed in the S -multiplet terms (thedifference between Y α and Y (cid:48) α is immaterial), it should be obvious that this can notlead to a displacement operator. In other words, the resulting energy-momentumtensor will be fully conserved, which follows straightforwardly by acting with ¯ D ˙ α on(6.9) and noting that the right hand side is a total derivative (see also the discussionaround (6.12) below). This means that the structure we have described can notaccommodate 4d theories coupled to 3d theories, which requires the appearance of anew term. In the next section we investigate the form such terms can take. Note thatalso in the purely bosonic cases reviewed in appendix A, the displacement operatorvanishes when the 4d and 3d degrees of freedom are not coupled. We would now like to find new terms that can appear on the right hand side of the S -multiplet and are consistent with the existence of a conserved energy-momentumtensor in the 3 directions parallel to the defect and a conserved Majorana super-current ( i.e. a supercurrent which is a linear combination of 4d supercurrents of– 21 –pposite chirality). Since the equation for the S -multiplet is linear and the solutionfor the 4d terms χ α and Y α is well known, we might as well discard them and focuson the term of interest to us. We therefore take the following starting point¯ D ˙ α V α ˙ α = 2 δ (˜ y n ) Z α ≡ Z (cid:48) α . (6.11)Here V µ = ( C µ , χ µ , M µ , v νµ , λ µ , D µ ) is a real vector multiplet. It is obtained simplyby adding a vector index to the usual real multiplet V discussed in appendix B.Our goal is to find constraints on Z α that lead to a multiplet with the requirementsspecified above. Consistency requires ¯ D Z α = 0, and we can define a chiral superfieldΠ α ˙ α = − i ¯ D ˙ α Z α .A quick way to show what conditions ensure the existence of a conserved energy-momentum tensor in the directions parallel to the defect and to see how the dis-placement operator emerges is to follow a similar argument to that which appearedin section 5. From (6.11) we can derive ¯ D V µ = 2 iδ (˜ y n )Π µ . This is compared to anexpression similar to (5.1)¯ D V µ = · · · − θ ( D µ + ∂ C µ − i∂ ν v νµ ) . (6.12)We learn from this that the existence of a conserved energy-momentum tensor im-poses that the real part of Π i is a total derivative while the real part of Π n givesthe displacement operator. It should be noted that v µν is not symmetric here so weought to be a little careful. In particular the current index in S µ is the free vectorindex.To argue more systematically, we obtain the following equation ∂ µ V µ = − i D α Z (cid:48) α − ¯ D ˙ α ¯ Z (cid:48) ˙ α ) , (6.13)which is derived from (6.11). It is useful to look closer at the components of ∂ µ V µ .In particular the interesting sub-multiplet is given by (cid:101) ∆ α ∂ µ V µ | (cid:101) Θ=0 = 1 √ ∂ µ ( χ µ + σ n ¯ χ µ ) α + Θ α ∂ µ (cid:18)
12 ( M µ + ¯ M µ ) + ∂ n C µ (cid:19) + i (Γ j Θ) α ∂ µ v jµ −
12 Θ ∂ µ (cid:18) κ µα + i √ (cid:0) Γ j ∂ j ( χ µ + σ n ¯ χ µ ) (cid:1) α (cid:19) . (6.14)Here we have introduced κ µα ≡ √ λ + σ n ¯ λ ) µα only for the sake of keeping the lengthof the expression in check. We recognize that this sub-multiplet contains the com-ponents that we want to keep conserved, namely χ µ + σ n ¯ χ µ and v jµ .Projecting to the same sub-multiplet on the right hand side of (6.13) we find √ (cid:101) ∆ α ∂ µ V µ | (cid:101) Θ=0 = ∆ β ∆ α ( Z (cid:48) + σ n ¯ Z (cid:48) ) β + i √ β (Π (cid:48) αβ + ¯Π (cid:48) αβ )+2 i∂ n ( Z (cid:48) − σ n ¯ Z (cid:48) ) α . (6.15)– 22 –ere Π (cid:48) αβ = δ (˜ y n )Π i (Γ i ) αβ . Since δ (˜ y n ) | (cid:101) Θ=0 = δ ( x n ) and the ∂ n term in ∆ α is oforder (cid:101) Θ α we can simplify to= δ ( x n ) (cid:18) ∆ β ∆ α ( Z + σ n ¯ Z ) β + i √ β (Π αβ + ¯Π αβ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:101) Θ=0 + 2 i∂ n (cid:0) δ ( x n )( Z − σ n ¯ Z ) α (cid:1) | (cid:101) Θ=0 . (6.16)What are the conditions which guarantee the existence of a conserved energy-momen-tum tensor and supercurrent? For the first term we can demand that the 3d projec-tion of Z α + ( σ n ¯ Z ) α is either a total covariant derivative ∆ α ( . . . ) or it is a current, i.e. annihilated by ∆ α . Likewise we demand that the 3d projection of Π i + ¯Π i is atotal derivative ∂ i ( . . . ). The obvious solutions are the ones we already encounteredabove, namely Z α = χ α , Y α and Y (cid:48) α . Let us focus here on a different solution givenby imposing (cid:0) Z α + ( σ n ¯ Z ) α (cid:1) | (cid:101) Θ=0 = 0 , (Π µ + ¯Π µ ) | (cid:101) Θ=0 = − n µ D . (6.17)Here D is a real scalar multiplet (of the 3d superspace), which we now show containsthe displacement operator.To see more explicitly the conservation equation for the energy-momentum tensorwe proceed as follows. The θσ ν ¯ θ component of ∂ µ V µ (where the energy-momentumsits) is obtained as the bottom component of[ D α , ¯ D ˙ α ] ∂ µ V µ = 14 D ( δ (˜ y n )Π α ˙ α ) + 14 ¯ D (cid:0) δ (˜¯ y n ) ¯Π α ˙ α (cid:1) + ∂ β ˙ α (cid:0) D α Z (cid:48) β + D β Z (cid:48) α (cid:1) − ∂ α ˙ β (cid:16) ¯ D ˙ β ¯ Z (cid:48) ˙ α + ¯ D ˙ α ¯ Z (cid:48) ˙ β (cid:17) (6.18)It is immediate to evaluate the bottom component of the top line by using (B.14) ∂ µ v νµ = − i ∂ n (cid:0) δ ( x n )(Π ν − ¯Π ν ) (cid:1) − δ ( x n )∆ (Π ν + ¯Π ν ) + · · · , (6.19)where the ellipses represent the contributions from the second line of (6.18), which aretotal derivatives, as in fact is also the first term here, so the interesting contributionis the second term ∂ µ v nµ = − δ ( x n ) f d + · · · , (6.20)where ∆ D | = − f d . As we see below, − T νµ = v νµ + · · · which leads us the formof the displacement operator given in (1.1). To find the components of V µ we first need to solve the constraints (6.17) on Z α moreexplicitly. Using a chiral superfield expansion we can write Z α = i Λ α − i ( σ µ ¯ θ ) α Π µ ,– 23 –here Λ α and Π µ are chirals (although Λ α does not transform standardly ). In fact,it is even more convenient to redefine Λ α → Λ α + ( σ µ ¯ σ n θ ) α Π µ , which leaves Λ α chiral. The advantage is that now Z α = i Λ α − √ σ µ ¯ σ n (cid:101) Θ) α Π µ (6.21)and Z α | (cid:101) Θ=0 = i Λ α | (cid:101) Θ=0 (hence Λ α | (cid:101) Θ=0 does transform standardly under the preservedsubalgebra), while still maintaining the relation − i ¯ D ˙ α Z α = Π α ˙ α . The componentsare given by Λ α = ρ α + θ α B − i ( σ µν θ ) α Λ µν + θ κ α , Π µ = g µ + √ θψ µ + θ F µ . (6.22)Here Λ µν may be taken to be real. We also expand the 3d multiplet D as D = d + Θ χ d + 12 Θ f d . (6.23)The first constraint in (6.17) implies ρ = σ n ¯ ρ, Im( B ) = 0 , Λ ij = 0 , κ + i∂ n ρ = σ n (¯ κ − i∂ n ¯ ρ ) . (6.24)We define (cid:96) µ = Λ nµ . The second constraint in (6.17) givesRe( g µ ) = − n µ d, ψ µ + σ n ¯ ψ µ = − n µ χ d , Re( F µ ) − ∂ n Im( g µ ) = − n µ f d . (6.25)We are now ready to solve (6.11), with Z α subject to the constraints (6.17), byexpressing the components of V µ ( v µν and χ µ ) in terms of the conserved quantities T νµ and S µ + σ n ¯ S µ . As an example, consider taking the bottom component of (6.15).Recalling that V µ = C µ + iθχ µ − i ¯ θ ¯ χ µ + . . . as in (B.2), this leads to the relation ∂ µ ( χ µ + σ n ¯ χ µ ) = − ∂ n ( ρ (cid:48) + σ n ¯ ρ (cid:48) )= − ∂ µ ( σ µ ¯ ρ (cid:48) − σ n ¯ σ µ ρ (cid:48) ) . (6.26)Here we are again using the shorthand ρ (cid:48) = δ ( x n ) ρ , and in the second line we haveused that ρ is a Majorana spinor, namely ρ = σ n ¯ ρ . This allows us to define aconserved supercurrent by √ S µ = − ( χ µ + σ n ¯ χ µ ) − δ ( x n )( σ µ ¯ ρ − σ n ¯ σ µ ρ ) . (6.27)We can now write √ S µ = S µ + σ n ¯ S µ and decompose the relation above to χ µ = − S µ − δ ( x n ) σ µ ¯ ρ, ¯ χ µ = − ¯ S µ + 2 δ ( x n )¯ σ µ ρ, (6.28) By this we mean, that the supersymmetry variation of Λ α contains Π µ terms. This is a conse-quence of the explicit use of ¯ θ ˙ α in this definition. The same is not true of Π µ since it has a naturalsuperspace definition as the covariant derivative of Z α . – 24 –oting that S µ is here determined only up to a shift by imaginary spinors ( i.e. ζ † = − ζσ n ), such that ˆ S µ remains unchanged. With a similar analysis of v µν we getthe expansion V µ = C µ − iθ ( S µ + 2 δ ( x n ) σ µ ¯ ρ ) + i ¯ θ (cid:0) ¯ S µ − δ ( x n )¯ σ µ ρ (cid:1) + i θ δ ( x n )¯ g µ − i θ δ ( x n ) g µ + θσ ν ¯ θ (cid:18) T νµ − (cid:15) νµρκ ∂ ρ C κ − δ ( x n ) (cid:0) n ν Im( g µ ) − n [ ν (cid:96) µ ] (cid:1)(cid:19) + · · · , (6.29)where the different fields satisfy the following conservation equations ∂ µ T νµ = n ν δ ( x n ) f d , ∂ µ ( S µ + σ n ¯ S µ ) = 0 . (6.30)The violation of conservation of momentum in the normal direction, i.e. , the displace-ment operator, is accompanied by a similar statement for the supercurrent, whichtakes the form − i∂ µ ( S µ − σ n ¯ S µ ) = ( κ (cid:48) + σ n ¯ κ (cid:48) ) + i∂ µ ( σ µ ¯ ρ (cid:48) + σ n ¯ σ µ ρ (cid:48) ) + 4 √ χ (cid:48) d . (6.31)This term, which like the displacement operator is localized on the defect, is afflictedby the ambiguity in S µ mentioned below (6.28), which implies that we can shift thisby a total derivative. In addition, we have the relations T µµ = 0 , ¯ σ µ S µ = 6 δ ( x n ) ¯ ρ, ∂ µ C µ = 2 δ ( x n )( d − B ) , (6.32)and lastly the antisymmetric part of the energy-momentum tensor is given by T [ ni ] = 14 δ ( x n ) (Im( g i ) − (cid:96) i ) , T [ ij ] = 0 . (6.33)Let us note that the new term Z α does not contribute to the trace of the energy-momentum tensor. Since the trace of the supercurrent ¯ σ µ S µ is a Majorana spinor,we can also define a conserved superconformal current by x ν ( σ ν ¯ S µ + σ n ¯ σ ν S µ ). Ofcourse, generically the traces receive contributions from χ α and Y α in (6.1) (as wellas the analogous terms coming from 3d) so the conformal currents are not conserved.Here we are only considering the contribution from the new term Z α . In this example there are 4d chiral superfields Φ a with K¨ahler potential K andsuperpotential W and 3d real scalar multiplets A I with target space metric G IJ . Thetwo theories are coupled through a potential P (Φ a , ¯Φ ¯ a , A I ) | (cid:101) Θ=0 . As before, we use P to denote the chiral embedding of P . The equations of motion are¯ D K a = 4 W a + 2 δ (˜ y n ) P a , D α ( G IJ D α A J ) = 12 ∂ I G JK D α A J D α A K + P I . (6.34)– 25 –e define the 4d and 3d parts of the energy-momentum multiplet by S (4) α ˙ α = K a ¯ a ¯ D ˙ α ¯Φ ¯ a D α Φ a , (6.35) S (3) α ˙ α = δ (˜ x n ) (cid:101) Θ β J βαγ σ nγ ˙ α = δ (˜ x n ) (cid:101) Θ β (cid:0) − G IJ ∂ i A I ∆ β A J (cid:1) (Γ i σ n ) α ˙ α . (6.36)We find for the 4d part¯ D ˙ α S (4) α ˙ α = 2( χ α − Y α ) − δ (˜ y n ) P a D α Φ a , (6.37)where χ α = − ¯ D D α K and Y α = D α W . For the 3d part, using identity (6.5)¯ D ˙ α S (3) α ˙ α = 2( χ α − Y (cid:48) α ) − δ (˜ y n ) P I D α A I . (6.38)We can write the new terms as 2( Z α − δ Y (cid:48) α ) with2 Z α = − (cid:16) P a D α Φ a − P ¯ a D α (cid:101) ¯Φ ¯ a (cid:17) − √ (cid:101) Θ α ∂ n P , δ Y (cid:48) α = 14 D α P + i ( σ n ¯ D ) α ( (cid:101) Θ P ) . (6.39)The second term has the form of Y (cid:48) α in (6.7) and can be absorbed in it. Z α satisfies (cid:0) Z α + ( σ n ¯ Z ) α (cid:1) | (cid:101) Θ=0 = 0 , (cid:0) Π µ + ¯Π µ (cid:1) | (cid:101) Θ=0 = − n µ ∂ n P. (6.40)In particular 2 D = ∂ n P . This is the obvious supersymmetric generalization of thescalar expressions in (A.6). In this model we have a 4d Abelian gauge field W α coupled to a 3d matter field A .The equations of motion are D α W α = ¯ D ˙ α ¯ W ˙ α = 2 δ (˜ x n ) (cid:101) Θ α J α , (6.41) D A = D ¯ A = 0 . (6.42)The two parts of S µ are S (4) α ˙ α = − W ˙ α W α , (6.43) S (3) α ˙ α = δ (˜ x n ) (cid:101) Θ β (cid:0) − D i A D β ¯ A − D i ¯ A D β A (cid:1) (Γ i σ n ) α ˙ α , (6.44)satisfying ¯ D ˙ α S (4) α ˙ α = 4 δ (˜ y n ) (cid:101) Θ β J β W α , ¯ D ˙ α S (3) α ˙ α = 2( χ α − Y (cid:48) α ) + √ iδ (˜ y n )(Γ i (cid:101) Θ) α J Γ i W . (6.45)It should be noted that J α and W α both represent the chiral embedding of thecorresponding 3d fields. We identify Z (4) α = 2 (cid:101) Θ β J β W α and Z (3) α = i √ (Γ i (cid:101) Θ) α J Γ i W . For the first equation it follows from the fact that J α is a current (satisfying ∆ α J α = 0).Then reasoning similar to those in footnote 9 show that in (cid:101) Θ α J α we can take J α to be the chiralembedding. – 26 –bviously Z α | (cid:101) Θ=0 = 0 so our first condition for Z α is trivially satisfied. We can thenfind Π (4) µ + ¯Π (4) µ = − n µ J (cid:102) W + P µi iJ Γ i W , Π (3) µ + ¯Π (3) µ = −P µi iJ Γ i W . (6.46)The displacement multiplet is therefore 2 D = J (cid:102) W and ( c.f. (A.10)) f d = j i F in + fermions . (6.47)
7. Superspace Noether approach to energy-momentum mul-tiplets
We now implement the Noether procedure as an alternative method of deriving theenergy-momentum multiplet. This was considered by several authors. Our discussionhere is mostly based on [26, 27] (see also [28, 29]). To do this, we must promotesupersymmetry to a local symmetry, so we consider the set of chirality preservingdiffeomorphisms of superspace δy µ = v µ ( y, θ ) , δ ¯ y µ = ¯ v µ (¯ y, ¯ θ ) ,δθ α = λ α ( y, θ ) , δ ¯ θ ˙ α = ¯ λ ˙ α (¯ y, ¯ θ ) , (7.1)On chiral functions of superspace this corresponds to the differential operator L + = v µ ∂ µ + λ α ∂∂θ α = h µ ∂ µ + λ α D α ,h µ ≡ v µ ( y, θ ) + 2 i ¯ θ ¯ σ µ λ ( y, θ ) . (7.2)Similarly, for anti-chiral functions L − = ¯ h µ ∂ µ + ¯ λ ˙ α ¯ D ˙ α . By definition, the action L + preserves chirality [ ¯ D ˙ α , L + ] = 0 and [ D α , L − ] = 0. We have the relation¯ D ˙ α h µ = − i ( λσ µ ) ˙ α , D α ¯ h µ = 2 i ( σ µ ¯ λ ) α . (7.3)Hence λ and ¯ λ are determined by h µ and ¯ h µ , which are free except for the constraint¯ D ( ˙ β h ˙ α ) α = 0 , D ( β ¯ h α ) ˙ α = 0 . (7.4)This in particular means that we can write h ˙ αα = − i ¯ D ˙ α L α and ¯ h ˙ αα = − iD α ¯ L ˙ α for an unconstrained superfield L α L α = (cid:96) α − i σ µ ¯ θ ) α v µ + ¯ θ λ α , (7.5)where (cid:96) α is an irrelevant chiral superfield, since the gauge transformation is given interms of ¯ D ˙ α L α . – 27 –ecall how in the case of global symmetries in section 5.1, the gauging involvespromoting the global (Abelian) transformation δ Φ = i Φ to an action by a chiralsuperfield Ω given by δ Φ = i ΩΦ. The global limit is then obtained by taking Ω = ¯Ω,and implies that the current J should appear in the variation of the Lagrangian as i (Ω − ¯Ω) J . In the same spirit, the basic assertion is that the global limit is given by h µ = ¯ h µ . (7.6)More precisely, this equation is equivalent to the superconformal Killing equations[30]. For example, letting v µ | = (cid:15) µ + ib µ and − D α λ α | = Λ (1) + i Λ (2) , one can verifythat (7.6) implies ∂ µ (cid:15) ν + ∂ ν (cid:15) µ = 4 η µν Λ (1) . (7.7)In other words, (7.6) imposes the conformal Killing vector equation on (cid:15) µ . For asuperconformal theory the variation of the Lagrangian must assume the form δ L (4) = − i (cid:90) d θ ( h µ − ¯ h µ ) S µ = 12 (cid:90) d θ ( ¯ D ˙ α L α − D α ¯ L ˙ α ) S α ˙ α . (7.8)Indeed, expanding in components (and remembering that S µ = 2 θσ ν ¯ θT νµ + . . . ) onefinds terms such as δ L (4) = − ∂ ν (cid:15) µ T νµ . Since L α is not constrained, we obtain thesuperconformal multiplet ¯ D ˙ α S α ˙ α = 0.To obtain the other multiplets, we need to constrain the gauge parameters L α .The R -constraint is given by further imposing D α ¯ D L α + ¯ D ˙ α D ¯ L ˙ α = 0, which byusing (7.5) gives D α λ α + ¯ D ˙ α ¯ λ ˙ α = 0. This implies Λ (1) = 0 so (7.7) now reduces to theKilling vector equation of flat space. The FZ-constraint is given by imposing thatthe chiral superfield σ ≡ − ¯ D D α L α vanishes. In terms of fields in (7.5) this reads σ = ∂ µ h µ + D α λ α . The bottom component of σ gives ∂ µ (cid:15) µ = 2Λ (1) , which togetherwith (7.7) once more means that Λ (1) = 0. Finally, the S -constraint is obtained byimposing both conditions. We can use two strategies for applying the constraints to the Noether procedure.The first, as presented above, is to think of the constraints as applying to the gaugesymmetry itself. Then, varying the action we again obtain (7.8) but this time since L α is constrained not all the component of ¯ D ˙ α S α ˙ α vanish. A slightly more convenientapproach is to take L α unconstrained and think of the R - and FZ-constraints as partof the global limit, so on the same footing as (7.6). Using this point of view forthe Noether procedure, we expect additional terms to appear in the variation of theLagrangian. The most general variation with the R -constraint is δ L (4) = − i (cid:90) d θ ( h µ − ¯ h µ ) S µ − (cid:90) d θ (cid:0) D α ¯ D L α + ¯ D ˙ α D ¯ L ˙ α (cid:1) V, (7.9) It follows from [13] that supersymmetric field theories are generally consistent only with the S -constraint, i.e. the smallest gauge symmetry. Additional assumptions are needed to consider themore general gauge symmetries. For example, only superconformal theories can accommodate thegauge symmetry with L α unconstrained, otherwise we are gauging a broken symmetry. – 28 –here V is a real multiplet. Since L α is not constrained, this leads to the R -multiplet¯ D ˙ α S α ˙ α = 2 χ α with χ α = − ¯ D D α V a chiral satisfying D α χ α = ¯ D ˙ α ¯ χ ˙ α . For thegauged FZ-symmetry the most general variation of the Lagrangian is δ L (4) = − i (cid:90) d θ ( h µ − ¯ h µ ) S µ − (cid:90) d θσX − (cid:90) d ¯ θ ¯ σ ¯ X, (7.10)where X is a chiral superfield. This leads to the FZ-multiplet ¯ D ˙ α S α ˙ α = − Y α with Y α = D α X . For the S -multiplet we simply impose both constraints to get¯ D ˙ α S α ˙ α = 2( χ α − Y α ).Let us consider the example of chiral superfields Φ a with a K¨ahler potential K (Φ a , ¯Φ ¯ a ) and superpotential W (Φ a ). The action of the gauge symmetry on a chiralsuperfield is given by [26] δ Φ a = L + Φ a . (7.11)For the FZ-constraint we find up to total derivatives δK = i h − ¯ h ) ˙ αα (cid:18) K a ¯ a ¯ D ˙ α ¯Φ ¯ a D α Φ a + 13 [ D α , ¯ D ˙ α ] K (cid:19) −
13 ( σ + ¯ σ ) KδW = − σW. (7.12)Comparing this with (7.10) leads to the FZ-multiplet, which is reviewed in (C.7).If the theory has an R -symmetry we can also consider the R -constraint. Let R a be the R -charges of Φ a . R -invariance implies the relations (cid:88) i ( R a K a Φ a − R a K ¯ a ¯Φ ¯ a ) = 0 , (cid:88) R a W a Φ a = 2 W. (7.13)It follows from the first relation that U R = (cid:80) R a K a Φ a is a real multiplet. Thegauge transformation of a chiral superfield is now [26, 27, 30] δ Φ a = L + Φ a + 12 σR a Φ a . (7.14)Here there is no sum over a . It is easy to check that δW = 0 up to a total derivativeunder this gauge symmetry. For the K¨ahler potential we find δK = i h ˙ αα − ¯ h ˙ αα ) (cid:0) K n ¯ n ¯ D ˙ α ¯Φ ¯ n D α Φ n + [ D α , ¯ D ˙ α ] U R (cid:1) + ( D α λ α + ¯ D ˙ α ¯ λ ˙ α )( K − U R ) . (7.15)Comparing with (7.9) one can derive the R -multiplet which agrees with (C.8). Fi-nally, for the S -constraint with (7.11) we find δK = i h ˙ αα − ¯ h ˙ αα ) (cid:0) K n ¯ n ¯ D ˙ α ¯Φ ¯ n D α Φ n (cid:1) + ( D α λ α + ¯ D ˙ α ¯ λ ˙ α ) K,δW = − σW, (7.16)which gives the S -multiplet in (C.6). The following identities, derived from the definitions of h µ and λ α , are useful ∂ µ h µ = − i
12 [ D α , ¯ D ˙ α ] h ˙ αα + 43 σ, ∂ µ ¯ h µ = i
12 [ D α , ¯ D ˙ α ]¯ h ˙ αα + 43 ¯ σ. – 29 – .2 3d multiplets In 3d we consider diffeomorphisms of superspace (see [31] for a related discussion inthe superconformal case) δx i = ˇ v i , δ Θ α = K α . (7.17)The action on (scalar) superfields is given byˆ L = ˇ v i ∂ i + K α ∂∂ Θ α = K i ∂ i + K α D α ,K i = ˇ v i + i ΘΓ i K. (7.18)Contrary to the 4d case, this does not imply any relation between K i and K α , inother words the two superfields are unconstrained. It is not difficult to check thatthe equation D α K i = 2 i (Γ i K ) α (7.19)corresponds to the superconformal Killing equation in 3d. To obtain the superPoincar´e Killing equation we can constrain the gauge symmetry by ∂ i K i + D α K α = 0.Together with (7.19) this implies also D α K α = 0.A general variation of the Lagrangian therefore takes the form δ L (3) = − (cid:90) d Θ( D α K i + 2 i ( K Γ i ) α ) J αi + (cid:90) d Θ( − ( ∂ i K i + D α K α )Σ + D α K α H ) . (7.20)Since K i and K α are unconstrained we readily get the 3d multiplet (3.1). As anexample we consider a sigma model of real scalar multiplets A I with kinetic term V M = G IJ D α A I D α A J and potential P ( A I ). The action of the gauge symmetry on A I is δA I = ˆ L A I = K i ∂ i A I + K α D α A I . (7.21)A simple computation gives δV M = − (cid:0) D α K i + 2 i ( K Γ i ) α (cid:1) ( − G IJ ∂ i A I D α A J ) − ( ∂ i K i + D α K α ) V M − D α K α V M ,δP = − ( ∂ i K i + D α K α ) P. (7.22)Comparing with (7.20) leads to the desired form (3.9).Let us identify the 3d parameters with the 4d ones. We haveˇ v i = 12 ( δy i + δ ¯ y i ) | (cid:101) Θ=0 = 12 ( v i + ¯ v i ) | (cid:101) Θ=0 ,K = 1 √ δθ + σ n δ ¯ θ ) | (cid:101) Θ=0 = 1 √ λ + σ n ¯ λ ) | (cid:101) Θ=0 . (7.23)– 30 –dditionally, − i h i − ¯ h i ) = (cid:101) K i − (cid:101) Θ α (cid:0) ∆ α K i − i (Γ i K ) α (cid:1) + O ( (cid:101) Θ ) , (7.24) (cid:101) K i = − i v i − ¯ v i ) + i ΘΓ i (cid:101) K, (cid:101) K α = − i √ λ − σ n ¯ λ ) α . (7.25)Clearly δ (cid:101) Θ α = − (cid:101) K α . We can see that the 4d global limit h i = ¯ h i corresponds to the3d equation (7.19) but includes an additional condition (cid:101) K i = 0. Similarly, for thenormal component we find − i h n − ¯ h n ) = (cid:101) K n − (cid:101) Θ α (cid:16) ∆ α K n − i (cid:101) K α (cid:17) + O ( (cid:101) Θ ) , (7.26) (cid:101) K n = − i v n − ¯ v n ) − Θ K, K n = 12 ( v n + ¯ v n ) + Θ (cid:101) K. (7.27)The constraints on the gauge symmetry also match12 (cid:0) ∂ µ h µ + ∂ µ ¯ h µ + D α λ α + ¯ D ˙ α ¯ λ ˙ α (cid:1) | (cid:101) Θ=0 = ∂ µ K µ + ∆ α K α . (7.28)We now use (7.24) to rewrite the first term of (7.20) in the 4d superspace as − (cid:90) d θ (cid:101) Θ β (cid:0) D β K i − i (Γ i K ) β (cid:1) δ (˜ x n ) (cid:101) Θ α J αi = − i (cid:90) d θ ( h i − ¯ h i ) δ (˜ x n ) (cid:101) Θ α J αi . (7.29)Here we have discarded the term in h − ¯ h which is of zeroth order in (cid:101) Θ (see (7.24))since it does not contribute to the integral. Evidently, this represents a contributionto S µ in the form δ (˜ x n ) (cid:101) Θ α J αi in agreement with (6.8). Similarly, the terms in thesecond line of (7.20) are rewritten as − (cid:90) d Θ( ∂ i K i + D α K α )Σ = (cid:90) d θL α (cid:18) D α ¯ D + i σ n ¯ D ) α ∂ n (cid:19) (cid:16) δ (˜ x n ) (cid:101) Θ Σ (cid:17) + c.c., (cid:90) d Θ D α K α H = 18 (cid:90) d θL α ¯ D D α (cid:16) δ (˜ x n ) (cid:101) Θ H (cid:17) + c.c., (7.30)where in the first line we used ∂ i K i + D α K α = ( σ + ¯ σ ) − ∂ n K n . This clearly confirmsthe structure we have found for embedding the 3d energy-momentum multiplet inthe S -multiplet. Finally, let us see how to obtain the defect multiplet from a variation approach. Weproceed by arguments similar to those appearing in the discussion of global currents,see (5.9). It follows from (7.24)-(7.26) that in the global limit (cid:101) K µ and ∆ α K n − i (cid:101) K α vanish. Moreover, we must demand the vanishing of K n as well. This guarantees thatthe solutions to the Killing equations will not include the translation corresponding– 31 –o the normal direction (and associated transformations). This discussion leads tothe following additional terms in the variation of the Lagrangian (cid:90) d Θ (cid:16) − i (cid:101) K µ Π µ + 2 √ (cid:101) K α Λ α + 2 K n D (cid:17) . (7.31)Clearly Π µ must be imaginary while Λ α and D are real. The dependence on the tildefields and K n implies that such terms come from interactions of 4d fields localizedon the defect. There terms can be rewritten as (cid:90) d θL α δ (˜ y n ) (cid:18) i Λ α − √ σ µ ¯ σ n (cid:101) Θ) α Π µ (cid:19) + c.c. , (7.32)where we have redefined (Π n − D ) → Π n . After this redefinition Π n has a real partwhich gives the displacement multiplet. Note that, as one can verify by followingthe derivation, in this equation Λ α and Π µ are the chiral embeddings of the fieldsintroduced in (7.31).As an example consider 4d chiral superfields Φ a coupled to real 3d scalar mul-tiplet A I through a potential P (Φ a , ¯Φ ¯ a , A I ) | (cid:101) Θ=0 . Projecting the transformations ofthe 4d chiral to 3d gives δ Φ = ( K µ + i (cid:101) K µ ) ∂ µ Φ + ( K α + i (cid:101) K α )∆ α Φ . (7.33)Applying this to the interaction potential leads to δP = − ( ∂ i K i + ∆ α K α ) P + K n ∂ n P + i (cid:101) K µ ( P a ∂ µ Φ a − P ¯ a ∂ µ ¯Φ ¯ a )+ i (cid:101) K α ( P a ∆ α Φ a − P ¯ a ∆ α ¯Φ ¯ a ) (7.34)up to a total derivative. The first term clearly gives rise to a Y (cid:48) α term. We canfurther obtain Z α = − √ D (cid:16) (cid:101) Θ ( P a ∆ α Φ a − P ¯ a ∆ α (cid:101) ¯Φ ¯ a ) (cid:17) − √ σ µ ¯ σ n (cid:101) Θ) α (cid:16) P a ∂ µ Φ a − P ¯ a ∂ µ (cid:101) ¯Φ ¯ a + n µ ∂ n P (cid:17) . (7.35)Recall that (cid:101) ¯Φ is the chiral lift of ¯Φ | (cid:101) Θ=0 . This gives D = ∂ n P and matches theresults obtained previously.
8. Concluding remarks
In this note we discussed 4d N = 1 supersymmetric field theories in the presence ofa 3d planar defect, preserving half of the supersymmetry. In particular, we describedhow the displacement operator in these theories is contained in a modified energy-momentum multiplet, which we named the defect multiplet. Our main motivation– 32 –or this work is to understand systematically how to place defects on curved mani-folds in a supersymmetric fashion. It will be interesting to develop a formalism thataddresses this issue using ideas similar to [32]. A related problem concerns the studyof manifolds with boundaries, where one would like to find all possible supersym-metric boundary geometries arising from the rigid limit of background supergravitystudied on a manifold with boundaries [33, 34, 35, 36, 37, 38].It would be nice to understand the moduli space of all supersymmetric defectsand the geometry that characterizes such embeddings. This can then be appliedto localization computations and can shed light on the problem of mapping defectsand boundaries under dualities. In particular, it would be interesting to follow thedependence of the partition function on the moduli space (as in [43, 44]). It is possiblethat these methods may also help the study of configurations defined on manifoldswith (conformal) boundaries [45, 46], or be useful for developing a supersymmetricformulation of holographic renormalization [47].It will also be interesting to generalize our results to other defects in variousdimensions and extended supersymmetry. These include co-dimension two defectsin 4d N = 1 field theories, preserving (0 ,
2) supersymmetry in two dimensions,as well as starting from N = 2 in 4d (see [12] for early work in this direction).The representation of the displacement multiplet for 3d defects preserving N = 4supersymmetry was studied in [48]. Acknowledgments
We are grateful to Benjamin Assel, Stefano Cremonesi, Cristian Vergu and DaisukeYokoyama for useful discussions, and especially to Cyril Closset, Lorenzo Di Pietro,and Zohar Komargodski for comments on the manuscript. The work of D.M. and I.S.is supported by the ERC Starting Grant N. 304806, “The gauge/gravity duality andgeometry in string theory”. The work of N.D. is supported by Science & TechnologyFacilities Council via the consolidated grant number ST/J002798/1.
A. The displacement operator in scalar and gauge field theory
Consider a 4d scalar φ and a 3d scalar a , confined to a planar submanifold Σ. The4d and 3d actions are (cid:90) L (4) = (cid:90) (cid:18) − ∂ µ φ∂ µ φ − V ( φ ) (cid:19) , (cid:90) Σ L (3) = (cid:90) Σ (cid:18) − ∂ i a∂ i a − V ( a ) (cid:19) . (A.1) Some example of exact results in supersymmetric field theories in the presence of defects include[39, 40, 41, 42]. – 33 –o make the system interesting, we need to couple the 3d and 4d fields. The simplestway to do that is (cid:90) Σ L ( I ) = − (cid:90) Σ V I ( φ, a ) , (A.2)with an arbitrary coupling potential V I .There are 4d and 3d terms in the energy-moment tensor T (4) µν = ∂ µ φ∂ ν φ + η µν L (4) T (3) ij = ∂ i a∂ j a + η ij ( L (3) + L ( I ) ) . (A.3)The full energy-momentum tensor will include both parts, which requires the em-bedding P µi on the directions tangent to Σ T µν = T (4) µν + δ ( x n ) P µi P ν j T (3) ji . (A.4)Using the classical equations of motion we find ∂ µ T µν = n ν δ ( x n ) ∂ φ V I ( φ, a ) ∂ n φ, (A.5)where x n is the coordinate normal to Σ. The displacement operator, defined in (1.1),is therefore given by f d = ∂ n V I ( φ, a ) . (A.6)In the presence of a 4d Abelian gauge field, the 4d action contains the term (cid:90) L (4) = − (cid:90) F µν F µν . (A.7)This can couple to a 3d theory on the defect, by gauging a global U (1) symmetry,with current j k (3) , via the coupling (cid:90) L ( I ) = (cid:90) Σ v µ P µk j k (3) . (A.8)The bulk energy-momentum is T (4) µν = F µρ F ρν + η µν L (4) , (A.9)and the 3d energy-momentum tensor T (3) µν will depend on the details of the 3d theory,which we do not specify. We need T (3) µν to establish the conservation in directionstangent to the defect but not in order to compute the displacement as T µn = T (4) µn from (A.4). This leads to ∂ µ T µn = δ ( x n ) F nk j k (3) . (A.10)– 34 – . Superspace conventions and useful formulas B.1 4d superspace
Our conventions follow quite closely Wess and Bagger. For convenience we mentionhere a few formulas which are used in the paper. The superspace coordinates are( x µ , θ, ¯ θ ) and the chiral combination is y µ = x µ + iθσ µ ¯ θ . A chiral superfield is afunction of ( y µ , θ ) Φ( y µ , θ ) = φ + √ θψ + θ F. (B.1)On several occasions we use a general real multiplet given by the following θ expansion V = C + iθχ − i ¯ θ ¯ χ + i θ M − i θ ¯ M − θσ µ ¯ θv µ + iθ ¯ θ (cid:18) ¯ λ + i σ µ ∂ µ χ (cid:19) − i ¯ θ θ (cid:18) λ + i σ µ ∂ µ ¯ χ (cid:19) + 12 θ ¯ θ (cid:18) D + 12 ∂ C (cid:19) . (B.2)In fact, it will be much more convenient for us to define the component fields bytaking bottom component of V acted upon by covariant derivative. That is V | = C, D α V | = iχ α , ¯ D ˙ α V | = − i ¯ χ ˙ α ,D V | = − iM, ¯ D V | = 2 i ¯ M , [ D α , ¯ D ˙ α ] V | = − v α ˙ α , ¯ D D α V | = 4 iλ α , D ¯ D ˙ α V | = − i ¯ λ ˙ α , D α ¯ D D α V | = 8 D. (B.3)Also useful: D α ¯ D ˙ α V | = − i∂ α ˙ α C − v α ˙ α , ¯ D ˙ α D α V | = − i∂ α ˙ α C + v α ˙ α . (B.4)To analyse the energy-momentum multiplets we also consider a vector real multiplet V µ = C µ + iθχ µ + · · · . All the formulas above are applied by adding a vector indexin an obvious way. For example [ D α , ¯ D ˙ α ] V µ | = − v α ˙ αµ . The following covariantderivatives identities are useful:[ ¯ D ˙ α , D ] = 4 iD α ∂ α ˙ α , [ D α , ¯ D ] = 4 i ¯ D ˙ α ∂ ˙ αα . (B.5)This form is far superior than the θ expansion in terms of the efficiency of computa-tions.We use two other chiral superfields which are derived from V . We write themhere as reference. The first, ¯ D V , is in components (in ( y, θ ) coordinates)¯ D V = 2 i ¯ M + 4 iθ ( λ + iσ µ ∂ µ ¯ χ ) − θ ( D + ∂ C − i∂ µ v µ ) . (B.6)This arises in the context of the current multiplet, which is a real multiplet satisfying¯ D V = 0. The second chiral superfield is the field strength associated with V viewedas an Abelian gauge multiplet W α = −
14 ¯ D D α V = − iλ α + θ α D − i ( σ µν θ ) α F µν + θ ( σ µ ∂ µ ¯ λ ) α . (B.7)– 35 – .2 3d superspace The 3d superspace has coordinates ( x (cid:48) i , Θ (cid:48) α ). To embed it in the 4d superspace wedefine new fermionic coordinatesΘ α = 1 √ θ + σ n ¯ θ ) α , (cid:101) Θ α = i √ θ − σ n ¯ θ ) α , (B.8)and ˜ x n = x n − i ( θ − ¯ θ ). The embedding is given by ( x i , ˜ x n , Θ α , (cid:101) Θ α ) = ( x (cid:48) i , , Θ (cid:48) α , x i = x (cid:48) i , Θ α = Θ (cid:48) α and forget about the tilded coordinates.We have also described in the paper an embedding in the chiral superspace ( y µ , θ α )which is similarly defined. As explained, the motivation for this definition is that thesubspace is invariant under the super-algebra preserved by the defect. The followingrelations are easily derived θ = 12 (Θ − (cid:101) Θ ) − i Θ (cid:101) Θ , ¯ θ = 12 (Θ − (cid:101) Θ ) + i Θ (cid:101) Θ , (B.9) θσ µ ¯ θ = 12 (Θ + (cid:101) Θ ) , θσ i ¯ θ = i ΘΓ i (cid:101) Θ , θ ¯ θ = − Θ (cid:101) Θ . (B.10)The change of basis in the 4d superspace is accompanied with the associated covariantderivatives ∆ α = 1 √ (cid:0) D α + ( σ n ¯ D ) α (cid:1) = ∂∂ Θ α + i (Γ i Θ) α ∂ i − (cid:101) Θ α ∂ n , (B.11) (cid:101) ∆ α = − i √ (cid:0) D α − ( σ n ¯ D ) α (cid:1) = ∂∂ (cid:101) Θ α + i (Γ i (cid:101) Θ) α ∂ i + Θ α ∂ n , (B.12)which satisfy { ∆ α , ∆ β } = − i (Γ i ) αβ ∂ i , ∆ α ∆ β ∆ α = 0 , ∆ ∆ = 4 ∂ i ∂ i , ∆ α ∆ β = − i∂ αβ + (cid:15) αβ ∆ , ∆ ∆ α = − ∆ α ∆ = − i∂ αβ ∆ β . (B.13)(Similarly for (cid:101) ∆.) For bookkeeping, we present the following relations for convertingcovariant derivatives in the different bases∆ = 12 ( D + ¯ D ) + Dσ n ¯ D − i∂ n (B.14)= 12 ( D + ¯ D ) − ¯ D ¯ σ n D + 2 i∂ n , (B.15) i ∆ α (cid:101) ∆ α = 12 ( D − ¯ D ) − i∂ n , (B.16)∆ ( β (cid:101) ∆ α ) = i (cid:0) ( ¯ D ¯ σ n ) ( β D α ) − ( σ n ¯ D ) ( α D β ) (cid:1) , (B.17) √ i ∆ (cid:101) ∆ α = ¯ D D α − D ( σ n ¯ D ) α + 2 i Γ j ( D − σ n ¯ D ) α ∂ j . (B.18)– 36 –hese are useful for computing the 3d components of 4d superfields. As an example,consider the decomposition of the 4d real multiplet V (B.2). We find V | (cid:101) Θ=0 = C + i √ χ − σ n ¯ χ ) + 12 Θ (cid:18) i M − ¯ M ) − v n (cid:19) , (B.19) (cid:101) ∆ α V | (cid:101) Θ=0 = 1 √ χ + σ n ¯ χ ) α + Θ α (cid:18)
12 ( M + ¯ M ) + ∂ n C (cid:19) + i (Γ j Θ) α v j −
12 Θ (cid:18) √ λ + σ n ¯ λ ) α + i √ (cid:0) Γ j ∂ j ( χ + σ n ¯ χ ) (cid:1) α (cid:19) . (cid:101) ∆ V | (cid:101) Θ=0 can be computed similarly but we shall not need it. To demonstrate thiscomputation let consider the Θ α component of (cid:101) ∆ α V | (cid:101) Θ=0 . It is obtained by applyingthe covariant derivative and using (B.16). This leads to −
12 ∆ α (cid:101) ∆ α V | = i D − ¯ D ) V | + ∂ n V | , (B.20)which together with (B.3) can be expressed in terms of the components of V . C. The S -multiplet In our conventions the S -multiplet [13] S α ˙ α = σ µα ˙ α S µ is given by¯ D ˙ α S α ˙ α = 2( χ α − Y α ) . (C.1)Here χ α satisfies D α χ α = ¯ D ˙ α ¯ χ ˙ α . In components this is solved by χ α = − iλ α + θ α D − i ( σ µν θ ) α F µν + θ ( σ µ ∂ µ ¯ λ ) α , (C.2)with D real and F µν = − F νµ satisfying the Bianchi identity, that is it can locally bewritten as F µν = ∂ µ v ν − ∂ ν v µ . In addition, we can locally define a chiral superfield X = x + √ θψ + θ F such that Y α = D α X . Solving for the components of S µ gives S µ = j µ − iθ (cid:16) S µ − √ iσ µ ¯ ψ (cid:17) + i ¯ θ (cid:16) ¯ S µ − √ i ¯ σ µ ψ (cid:17) + 2 iθ ∂ µ ¯ x − i ¯ θ ∂ µ x + θσ ν ¯ θ (cid:18) T νµ − η νµ A − (cid:15) νµρσ ( ∂ ρ j σ − F ρσ ) (cid:19) − θ ¯ θ (cid:16) ¯ σ ν ∂ ν S µ + 2 √ i ¯ σ µ σ ν ∂ ν ¯ ψ (cid:17) + 12 ¯ θ θ (cid:16) σ ν ∂ ν ¯ S µ + 2 √ iσ µ ¯ σ ν ∂ ν ψ (cid:17) + 12 θ ¯ θ (cid:18) ∂ µ ∂ ν j ν − ∂ j µ (cid:19) . (C.3)In this expression S αµ is conserved, T µν is symmetric and conserved, and T µµ = 6 A + D, ( σ µ ¯ S µ ) α = − λ α − √ iψ α , ∂ µ j µ = 4 B, (C.4)where F = A + iB . – 37 –mprovements by a real multiplet U take the form S α ˙ α → S α ˙ α − [ D α , ¯ D ˙ α ] U,χ α → χ α − ¯ D D α U, Y α → Y α + D α ¯ D U. (C.5)For a sigma model with K¨ahler potential K ( ¯Φ ¯ a , Φ a ) and superpotential W (Φ a )the S -multiplet is given by S α ˙ α = K a ¯ a ¯ D ˙ α ¯Φ ¯ a D α Φ a ,χ α = − ¯ D D α K, Y α = D α W. (C.6)The FZ-multiplet exists if the improvement U FZ = − K is well defined in which case χ α = 0 and J α ˙ α = K a ¯ a ¯ D ˙ α ¯Φ ¯ a D α Φ a + [ D α , ¯ D ˙ α ] K, Y α = D α W − D α ¯ D K. (C.7)If there is an R -symmetry, with R [Φ a ] = R a , we may define U R = (cid:80) R a Φ a K a .Using the equations of motion ¯ D K n = 4 W n this leads to Y α = 0 and R α ˙ α = K a ¯ a ¯ D ˙ α ¯Φ ¯ a D α Φ a − [ D α , ¯ D ˙ α ] U R ,χ α = − ¯ D D α ( K + 3 U R ) . (C.8) References [1] A. M. Polyakov and V. S. Rychkov,
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