NUPHB14500S0550-3213(18)30329-810.1016/j.nuclphysb.2018.11.013The Author(s)High Energy Physics – TheoryLinearizing extended nonlinear supersymmetry in two dimensional spacetimeMotomuTsudamotomu.tsuda@gmail.comAizu Hokurei High School, Aizuwakamatsu, Fukushima 965-0031, JapanAizu Hokurei High SchoolAizuwakamatsuFukushima965-0031JapanEditor: Leonardo RastelliAbstractWe linearize nonlinear supersymmetry in the Volkov–Akulov (VA) theory for extended SUSY in two dimensional spacetime (d=2) based on the commutator algebra. Linear SUSY transformations of basic component fields for general vector supermultiplets are uniquely determined from variations of functionals (composites) of Nambu–Goldstone (NG) fermions, which are represented as simple products of powers of the NG fermions and a fundamental determinant in the VA theory. The structure of basic component fields with general auxiliary fields in the vector supermultiplets and transitions to U(1) gauge supermultiplets through recombinations of the functionals of the NG fermions are explicitly shown both in N=2 and N=3 theories as the simplest and general examples for extended SUSY theories in d=2.1IntroductionRelations between Volkov–Akulov (VA) nonlinear supersymmetric (NLSUSY) theory [1] and linear SUSY (LSUSY) ones [2,3] are shown explicitly for N=1 and N=2 SUSY theories [4–8]. In superspace formalism, superfields on specific superspace coordinates which depend on Nambu–Goldstone (NG) fermions [4] give systematically the general relation between the VA NLSUSY and LSUSY theories (NL/LSUSY relation) as demonstrated in N=1 SUSY theories [4–7]. In the component expression, it is possible to construct heuristically functionals (composites) of the NG fermions, which reproduce LSUSY transformations of basic component fields under their NLSUSY transformations, and its heuristic method was used to study the NL/LSUSY relation for N=2 minimal U(1) gauge supermultiplet [8] (and also for a N=3 LSUSY theory in two dimensional spacetime (d=2) [9]), though the constructions of the functionals in all orders of the NG fermions are complicated problems.On the other hand, we have recently proposed a linearization procedure of NLSUSY based on a commutator algebra in the VA NLSUSY theory [10] by introducing a set of fermionic and bosonic functionals which are represented as products of powers of the NG fermions and a fundamental determinant indicating a spontaneous SUSY breaking in the VA NLSUSY action. In this linearization method, variations of basic components defined from the above set of the functionals under the NLSUSY transformations uniquely determine LSUSY transformations for (massless) vector supermultiplets. This is based on a fact that every functional of the NG fermions and their derivative terms satisfies the same commutation relation in the VA NLSUSY theory. Moreover, we have shown in N=1 SUSY theories that U(1) gauge and scalar supermultiplets in addition to a vector one with general auxiliary fields are derived from the same set of the functionals and their appropriate recombinations [11].Because the all-order functional (composite) structure of the NG fermions for vector supermultiplets is manifest in the commutator-based linearization of NLSUSY, its procedure would be useful to understand the NL/LSUSY relation for N≥2 extended SUSY in more detail and furthermore to know low-energy physics of a NLSUSY general relativistic (GR) theory [12]. The Einstein–Hilbert-type (global) NLSUSY fundamental action in the NLSUSY-GR theory possesses rich symmetries [13], which are isomorphic to SO(N) super-Poincaré group and contains the VA NLSUSY action in the cosmological term. Therefore, it is important for the N-extended NLSUSY-GR theory and its composite model interpretation (superon-quintet model (SQM)) [14] in the low energy to study more extensively and explicitly the NL/LSUSY relations for N-extended SUSY.In this paper, we focus on the VA NLSUSY theory for extended SUSY in d=2 and apply the commutator-based linearization procedure to it since explicit calculations for the d=2 NL/LSUSY relation give many suggestive and significant results to d=4 SUSY theories as for the structure of the functionals of the NG fermions for vector supermultiplets [15] and SUSY models with interaction terms [16–18] etc. In Section 2, as in the case of the linearization in d=4 [10], we introduce a set of bosonic and fermionic functionals for vector supermultiplets in the d=2 VA NLSUSY theory, which are represented as products of powers of the NG fermions and the fundamental determinant, and explain some properties of those functionals. In Section 3, we show LSUSY transformations with general auxiliary-field structure for vector supermultiplets, which are uniquely determined from basic components defined from the set of the functionals of the NG fermions by evaluating their variations under NLSUSY transformations based on the commutator algebra.In the remaining section of this paper, we construct N=2 and N=3 LSUSY multiplets from the LSUSY transformations shown in Section 3 as typical and comprehensive examples for the extended SUSY theories. In Section 4, we derive a d=2, N=2 vector supermultiplet [15] by means of the reduction from those general basic components and LSUSY transformations. We also discuss in Section 5 the construction of N=2 minimal U(1) gauge and scalar supermultiplets by using appropriate recombinations of the functionals of the NG fermions in the N=2 vector supermultiplet. Those arguments in the linearization of N=2 NLSUSY give us valuable lessons for N≥3 SUSY theories.In Section 6, as the simplest but a general extended LSUSY model in d=2, we study the structure of the basic components with general auxiliary fields in a N=3 vector supermultiplet by focusing procedures for counting degrees of freedom (d.o.f.) of bosonic and fermionic components. In Section 7, we show a transition from the N=3 vector supermultiplet to a minimal U(1) gauge one by means of general recombinations of the functionals of the NG fermions, which correspond to a generalization of the Wess–Zumino gauge to the NL/LSUSY relation for extended SUSY. Summary and discussions are given in Section 8.2Functionals of Nambu–Goldstone fermions for vector supermultipletsA fundamental action in the VA NLSUSY theory [1] for extended SUSY in d=2 is given in terms of Majorana NG fermions ψi as11The indices i,j,⋯=1,2,⋯,N and Minkowski spacetime indices are denoted by a,b,⋯=0,1. Gamma matrices satisfy {γa,γb}=2ηab with the Minkowski spacetime metric ηab=diag(+,−) and σab=i2[γa,γb] is defined.(2.1)SNLSUSY=−12κ2∫d2x|w|, where κ is a dimensional constant whose dimension is (mass)−1 and a fundamental determinant |w| is defined by means of(2.2)|w|=detwab=det(δba+tab) with tab=−iκ2ψ¯γa∂bψ. NLSUSY transformations of ψi are(2.3)δζψi=1κζi+ξa∂aψi, where ζi are constant (Majorana) spinor parameters and ξa=iκψ¯jγaζj. The NLSUSY transformations (2.3) satisfy a commutator algebra,(2.4)[δζ1,δζ2]=δP(Ξa), with δP(Ξa) (Ξa=2iζ¯1iγaζ2i) meaning a translation. Since the determinant (2.2) also transforms as(2.5)δζ|w|=∂a(ξa|w|), the NLSUSY action (2.1) is invariant under NLSUSY transformations (2.3) except for total derivative terms.As a property of the commutator algebra (2.4), every bosonic or fermionic Lorentz-tensor (or scalar) functional of ψi and their derivative terms (∂ψi, ∂2ψi, ⋯, ∂nψi) satisfies the commutator algebra (2.4) under the NLSUSY transformations (2.3); namely,(2.6)[δζ1,δζ2]FAI=Ξa∂aFAI, where FAI=FAI(ψi,∂ψi,∂2ψi,⋯,∂nψi) are the functionals of the NG fermions with Lorentz index A=(a,ab,⋯,etc.) and the internal one I=(i,ij,⋯,etc.).22The commutation relation (2.6) is proved from Eq. (2.4) and from the fact that the derivative terms (∂ψi, ∂2ψi, ⋯, ∂nψi) and products of two kinds of the functionals FAI and GBJ which are respectively defined in Eq. (2.6) satisfy the same commutation relation (for example, see [19]).From the NLSUSY transformations (2.3) (and (2.5)), a set of bosonic and fermionic functionals of the NG fermions, whose variation under the NLSUSY transformations shows linear exchanges among those functionals, can be constructed by means of products of powers of ψi (with γ-matrices) and the determinant |w|. Let us express the set of bosonic and fermionic functionals for each N NLSUSY as(2.7)biAjkBl⋯mCn((ψi)2(n−1)|w|)=κ2n−3ψ¯iγAψjψ¯kγBψl⋯ψ¯mγCψn|w|,(2.8)fijAklBm⋯nCp((ψi)2n−1|w|)=κ2(n−1)ψiψ¯jγAψkψ¯lγBψm⋯ψ¯nγCψp|w|, which mean(2.9)b=κ−1|w|,biAj=κψ¯iγAψj|w|,biAjkBl=κ3ψ¯iγAψjψ¯kγBψl|w|,⋯,(2.10)fi=ψi|w|,fijAk=κ2ψiψ¯jγAψk|w|,⋯, for n=1,2,⋯, respectively. In the functionals (2.7) and (2.8), (Lorentz) indices A,B,⋯ are used as ones for a basis of γ matrices in d=2, i.e., γA=1,γ5oriγa (γA=1,γ5or−iγa). Note that vector components appear in the functionals biAj of Eq. (2.9) for N≥2 SUSY and the definitions of the functionals (2.7) and (2.8) (or (2.9) and (2.10)) terminate with n=N+1 and n=N, because (ψi)n=0 for n≥2N+1. In the case for N=2 SUSY in d=2, we have already shown that the functionals (2.7) and (2.8) lead to the NL/LSUSY relations for a (massless) vector linear supermultiplet with general auxiliary fields prior to transforming to U(1) gauge supermultiplets by using the superspace formalism [15].Then, the variations of the functionals (2.7) and (2.8) under the NLSUSY transformations (2.3) and (2.5) become(2.11)δζbiAjkBl⋯mCn=κ2(n−1)[{(ζ¯iγAψj+ψ¯iγAζj)ψ¯kγBψl⋯ψ¯mγCψn+⋯}|w|+κ∂a(ξaψ¯iγAψjψ¯kγBψl⋯ψ¯mγCψn|w|)],(2.12)δζfijAklBml⋯nCp=κ2n−1[{ζiψ¯jγAψkψ¯lγBψm⋯ψ¯nγCψp+ψi(ζ¯jγAψk+ψ¯jγAζk)ψ¯lγBψm⋯ψ¯nγCψp+⋯}|w|+κ∂a(ξaψiψ¯jγAψkψ¯lγBψm⋯ψ¯nγCψp|w|)], which indicate that the bosonic and fermionic functionals in Eqs. (2.7) and (2.8) are linearly transformed to each other.3Commutator-based linearization of NLSUSY in d=2In this section, we show general forms of LSUSY transformations of basic components for vector supermultiplets in the d=2 case, which are obtained from the variations (2.11) and (2.12) and the commutation relation (2.6). We also explain their derivations by reviewing the procedures of the commutator-based linearization [10].By using the set of the functionals (2.7) and (2.8), let us define basic bosonic components as(3.1)D=b(ψ),MiAj=α1AbiAj(ψ),CiAjkBl=α3ABbiAjkBl(ψ),EiAjkBlmCn=α5ABCbiAjkBlmCn(ψ),⋯ and fermionic ones as(3.2)λi=fi(ψ),ΛijAk=α2AfijAk(ψ),ΨijAklBm=α4ABfijAklBm(ψ),⋯ where αmA(m=1,2,⋯,5) mean constants whose values are determined from definitions of fundamental actions in d=2 LSUSY theories and the invariances of the actions under LSUSY transformations of the component fields.For the components (3.1) and (3.2), LSUSY transformations which satisfy the commutator algebra (2.4) can be uniquely expressed as follows;(3.3)δζD=−iζ¯i∂̸λi,(3.4)δζλi=Dζi−i2α1AεM1γA∂̸MiAjζj,(3.5)δζMiAj=α1A(ζ¯iγAλj+λ¯iγAζj+i2α2BεΛ1ζ¯k∂̸γBγAΛijBk),(3.6)δζΛijAk=α2A{1α1AMjAkζi−12α1BγB(εM2γAεM2′MiBkζj+γAεM2′MiBjζk)+i4α3ACBεC1εC1′γB∂̸CiCjkACBlζl},(3.7)δζCiAjkBl=α3AB{1α2B(ζ¯iγAΛjkBl+εΛ2ζ¯jγAΛikBl)−12α2CεΛ2(ζ¯kγBγCγAΛijCl+εΛ2′ζ¯lγBγCγAΛijCk)},−iα3AB4α4ABεΨ1εΨ2ζ¯m∂̸γCγBγDγAΨijDklCm, ⋯, etc. In Eqs. from (3.3) to (3.7), we use a sign factor ε (or ε′) which appears from the relation ψ¯jγAψi=εψ¯iγAψj33The sign factor ε is ε=+1 for γA=1 and ε=−1 for γA=γ5,iγa. and we define CiCjkACBl=α3ACBbiCjkACBl (=α3ACBκ3ψ¯iγCψj ψ¯kγAγCγBψl|w|) with constants α3ACB in Eq. (3.6) for convenience of the expression, which can be expanded by means of the components CiAjkBl under the Clifford algebra for γ matrices. Note that the determination of LSUSY transformations terminates with those of bosonic components for the functionals at O{(ψi)2N} in Eq. (2.7).Let us explain below the derivations of the LSUSY transformations from (3.3) to (3.7): First, starting with the variation of D(ψ) in the bosonic components (3.1), the LSUSY transformations of D and λi are unambiguously determined as Eqs. (3.3) and (3.4) by using a Fierz transformation. Then, the closure of the commutator algebra on D under the LSUSY transformations (3.3) and (3.4) is guaranteed by means of the commutation relation (2.6), though its straightforward calculation is easy.Next, as for the LSUSY transformations (3.5) and (3.6), variations of the functionals MiAj(ψ) and ΛijAk(ψ) in Eqs. (3.1) and (3.2) under the NLSUSY transformations (2.3) are calculated by following Eqs. (2.11) and (2.12) as(3.8)δζMiAj=α1A{ζ¯iγAλj+λ¯iγAζj−iκ2∂a(ζ¯kγaψkψ¯iγAψj|w|)},(3.9)δζΛijAk=α2A{1α1AMjAkζi−12α1BγB(εM2γAεM2′MiBkζj+γAεM2′MiBjζk)−i2κ3∂a(γBγaζlψ¯lγBψiψ¯jγAψk|w|)}, respectively. In the variations (3.8) and (3.9), deformations of the functionals ψkψ¯iγAψj and ψ¯lγBψiψ¯jγAψk in the last terms are problems for the definition of the LSUSY transformations. These are solved by examining two supertransformations of λi(ψ) and MiAj(ψ) which satisfy the commutation relation (2.6): Indeed, the two supertransformations of λi(ψ) are obtained from Eqs. (3.3), (3.4) and (3.8) as(3.10)δζ1δζ2λi=δζ1Dζ2i−i4α1AεM1γA∂̸δζ1MiAjζ2j=iζ¯1jγaζ2j∂aλi+[(1↔2)symmetric terms of∂aλ]+14εM1κ2γA∂̸∂a(ζ¯1kγBζ2jγBγaψkψ¯iγAψj|w|). Since the last terms in Eq. (3.10) are symmetric under exchanging the indices 1 and 2 of the spinor transformation parameters (ζ1k, ζ2j) and they vanish in the commutation relation (2.6), the ψk and ψj have to go into bilinear forms ψ¯jγAψk in the last terms of Eq. (3.8) (and Eq. (3.10)) in order to realize straightforwardly the symmetries of the indices of the spinor transformation parameters in the components ΛijAk of LSUSY theories.Thus the LSUSY transformations of MiAj are uniquely determined from Eq. (3.8) as Eq. (3.5) by using a Fierz transformation. The LSUSY transformations (3.4) of λi satisfy the commutator algebra (2.4) under Eqs. (3.3) and (3.5).In the same way, according to the two supertransformations of MiAj(ψ) which are given from Eqs. (3.4), (3.5) and (3.9) as(3.11)δζ1δζ2MiAj=iζ¯1kγaζ2k∂aMiAj+[(1↔2)symmetric terms ofDand∂aMiAj]+14εΛ1κ3∂a∂b(ζ¯2kγaγBγAγCγbζ1lψ¯lγCψiψ¯jγBψk|w|), the ψl and ψk have to take bilinear forms ψ¯kγAψl in the last terms of Eqs. (3.9) and (3.11) in order to realize the symmetries of the indices of the spinor transformation parameters (ζ1l, ζ2k) in the components CiAjkBl of LSUSY theories.Therefore, the LSUSY transformations of ΛijAk are defined from Eq. (3.9) as Eq. (3.6) by using a Fierz transformation and then the LSUSY transformations (3.5) of MiAj satisfy the commutator algebra (2.4) under Eqs. (3.4) and (3.6).Furthermore, as an example of the LSUSY transformations of the components for the higher-order functionals of ψi than ΛijAk(ψ), the LSUSY transformations (3.7) are determined by estimating the variations of CiAjkBl(ψ),(3.12)δζCiAjkBl=α3ABκ2[{(ζ¯iγAψj+ψ¯iγAζj)ψ¯kγBψl+ψ¯iγAψj(ζ¯kγBψl+ψ¯kγBζl)}|w|−iκ2∂a(ζ¯mγaψmψ¯iγAψjψ¯kγBψl|w|)]. In the variations (3.12), we have to consider deformations of the functionals both at O(ψ3) and at O(ψ5).The LSUSY transformations of CiAjkBl to the fermionic components ΛijAk in Eq. (3.7) are determined from the functionals at O(ψ3) in Eq. (3.12) by taking into account derivative terms of ΛijAk (∂Λ-terms) in two supertransformations of ΛijAk(ψ). In fact, the ∂Λ-terms which are given from the LSUSY transformations (3.5) and (3.6) become(3.13)δζ1δζ2ΛijAk[∂Λterms obtained throughδζM]=i2α2A[1α2BεΛ1∂a(ζ¯1lγaγBγAΛjkBlζ2i)−12α2CεM2′εΛ1γB{εM2γA∂a(ζ¯1lγaγCγBΛikClζ2j)+γA∂a(ζ¯1lγaγCγBΛijClζ2k)}], where the ∂a(ΛjkAl,ΛikAl,ΛijAl)-type terms appear. Since these terms in Eq. (3.13) cancel with ones which are obtained from the LSUSY transformations of CiAjkBl in the commutation relation (2.6) on ΛijAk, the variations (3.12) at O(ψ3) have to give Λ-terms with the same arrangement of the internal indices as in Eq. (3.13). Thus the LSUSY transformations of CiAjkBl are defined with respect to the fermionic components ΛijAk as in Eq. (3.7) from the O(ψ3)-terms of Eq. (3.12) by using Fierz transformations. Note that the terms for ΛijCk in Eq. (3.7) give the translations of ΛijAk (i.e., Ξa∂aΛijAk) in a commutator algebra for the LSUSY transformations (3.6).As for the functionals at O(ψ5) in the variations (3.12), if we consider the two supertransformations of ΛijAk(ψ), then we can straightforwardly confirm that second-order derivative terms of those functionals vanish in the commutation relation (2.6) on ΛijAk, provided that the last terms at O(ψ5) in Eq. (3.12) are expressed as(3.14)δζCiAjkBl[∂afkiAjlBmterms]=i2εΨ1α3ABκ4∂a(ζ¯mγaγCγBψkψ¯iγAψjψ¯lγCψm|w|). by means of a Fierz transformation. This is because the form of the variation (3.14) contain the bilinear terms ψ¯lγCψm with symmetries which correspond to the indices of spinor transformation parameters (ζ1m,ζ2l) in the two supertransformations of ΛijAk obtained from the LSUSY transformations (3.6).However, in order to determine the LSUSY transformations of CiAjkBl (as Eq. (3.7)), a deformation of ψkψ¯iγAψj in the functionals of Eq. (3.14) is a remaining problem. Therefore, we further examine two supertransformations of CiAjkBl(ψ) together with the definition of LSUSY transformations of the basic components ΨijAklBm in the fermionic components (3.2): Derivative terms of CiAjkBl (∂C-terms) in two supertransformations of CiAjkBl(ψ), which are given through the LSUSY transformations (3.6) and Λ-terms in Eq. (3.7) are(3.15)δζ1δζ2CiAjkBl[∂Cterms obtained throughδζΛ]=iα3AB16α3ABCεC1εC1′{(ζ¯2iγAγC∂̸CjDklBDCmζ1m+εΛ2ζ¯2jγAγC∂̸CiDklBDCmζ1m)−12εΛ2(ζ¯2kγBγCγAγD∂̸CiEjlCEDmζ1m+εΛ2′ζ¯2lγBγCγAγD∂̸CiEjkCEDmζ1m)}, where the ∂a(CjAklBm,CiAklBm,CiAjlBm,CiAjkBm)-type terms appear.In accordance with the internal indices of ∂C-terms in Eq. (3.15), LSUSY transformations of ΨijAklBm are determined with respect to CiAjkBl by using Fierz transformations in the variations of ΨijAklBm(ψ) as44Note that Λ-terms in the commutation relation (2.6) on ΨijAklBm vanish only if the LSUSY transformations (3.16) are defined, since the LSUSY ones of CiAjkBl to Λ-terms have already been determined as Eq. (3.7).(3.16)δζΨijAklBm[Cterms]=α4AB{1α3ABζiCjAklBm−12α3CB(εC2γCγAζjεC2′CiCklBm+γCγAζkεC2′CiCjlBm)+14α3ADC(εC3γCγBζlεC3′εC3″CiDjkADCm+γCγBζmεC3′εC3″CiDjkADCl)}. Then, in the commutation relation (2.6) on CiAjkBl, the ∂C-terms of Eq. (3.15) cancel with ones obtained through the LSUSY transformations (3.16), provided that the LSUSY transformations of CiAjkBl are defined with respect to ΨijAklBm as in Eq. (3.7). Here we also note that the last terms with respect to CiDjkADCl in the LSUSY transformations (3.16) give the translations of CiAjkBl (i.e., Ξa∂aCiAjkBl) in a commutator algebra for the LSUSY transformations of CiAjkBl through Eq. (3.7).Thus the LSUSY transformations of CiAjkBl are uniquely determined as Eq. (3.7). The LSUSY transformations (3.6) of ΛijAk satisfy the commutator algebra (2.4) under Eqs. (3.5) and (3.7). As for LSUSY transformations of the basic components for higher-order functionals than CiAjkBl(ψ), they can be determined in accordance with the arguments for the definition of LSUSY transformations in this section.4Reduction to a N=2 vector supermultipletIn this section, we reduce the basic components (3.1) and (3.2) to the ones for N=2 SUSY based on the functional structure of the NG fermions and derive LSUSY transformations for a N=2 vector supermultiplets from Eqs. (3.3) to (3.7) as an instructive example in order to discuss the NL/LSUSY relations for N≥3 SUSY in d=2. For N=2 SUSY, the basic components for the vector supermultiplet are constituted from (D, λi, MiAj, ΛijAk, CiAjkBl) in Eqs. (3.1) and (3.2), which are defined from the functionals of ψi(i=1,2) up to O(ψ4). The bosonic and fermionic d.o.f. of the components for the lower-order functionals of ψi (except for |w|), i.e. the d.o.f of (D, λi, MiAj) are (1, 4, 6), respectively, where the components MiAj are decomposed as(4.1)Mij=M(ij)=α11κψ¯iψj|w|,ϕij=ϕ[ij]=α12κψ¯iγ5ψj|w|,vaij=va[ij]=iα13κψ¯iγaψj|w|.As for the components (ΛijAk, CiAjkBl), their d.o.f. have to be counted based on the NG-fermion functional structure under identities which are connected with Fierz transformations: First, the components ΛijAk are decomposed as(4.2)Λijk=Λi(jk)=α21κ2ψiψ¯jψk|w|,Λ5ijk=Λ5i[jk]=α22κ2ψiψ¯jγ5ψk|w|,Λaijk=Λai[jk]=iα23κ2ψiψ¯jγaψk|w|. The d.o.f. of Λijk=Λi(jk) for i=1,2 are apparently 12. However, from the viewpoint of the NG-fermion functional structure, 4 components vanish as(4.3)Λ111(ψ)=0,Λ222(ψ)=0, and 4 components in Λijk are related to each other by means of Fierz transformations as(4.4)Λ112(ψ)=−12Λ211(ψ),Λ221(ψ)=−12Λ122(ψ). Therefore, the d.o.f. of Λijk for N=2 SUSY are effectively 12−8=4.Furthermore, the components Λ5i[jk](ψ) and Λai[jk](ψ) are expressed by using the components (Λ122,Λ211)(ψ) in Eq. (4.4), i.e.(4.5)Λ5112(ψ)=−α222α21γ5Λ211(ψ),Λ5221(ψ)=−α222α21γ5Λ122(ψ),Λa112(ψ)=−iα232α21γaΛ211(ψ),Λa221(ψ)=−iα232α21γaΛ122(ψ), so that fermionic auxiliary fields in the N=2 vector supermultiplet are defined from the remaining components (Λ122,Λ211) in ΛijAk as(4.6)Λi=Λijj.Second, we count the effective d.o.f. of the components CiAjkBl by decomposing them as(4.7)Cijkl=C(ij)(kl)=α31κ3ψ¯iψjψ¯kψl|w|,C5ijkl=C5[ij](kl)=α32κ3ψ¯iγ5ψjψ¯kψl|w|,C˜5ijkl=C˜5(ij)[kl]=α32′κ3ψ¯iψjψ¯kγ5ψl|w|,Caijkl=Ca[ij](kl)=iα33κ3ψ¯iγaψjψ¯kψl|w|,C˜aijkl=C˜a(ij)[kl]=iα33′κ3ψ¯iψjψ¯kγaψl|w|,C55ijkl=C55[ij][kl]=α34κ3ψ¯iγ5ψjψ¯kγ5ψl|w|,Ca5ijkl=Ca5[ij][kl]=iα35κ3ψ¯iγaψjψ¯kγ5ψl|w|,C˜5aijkl=C˜5a[ij][kl]=iα35′κ3ψ¯iγ5ψjψ¯kγaψl|w|,Cabijkl=Cab[ij][kl]=α36κ3ψ¯iγaψjψ¯kγbψl|w|, but C1122 is the only remaining component in Eq. (4.7) for N=2 SUSY based on the NG-fermion functional structure since(4.8)C1212(ψ)=−12C1122(ψ),C551212(ψ)=α332α31C1122(ψ),Cab1212(ψ)=α362α31ηabC1122(ψ), and all other components vanish by means of Fierz transformations. Then, a bosonic auxiliary field in the N=2 vector supermultiplet is defined from C1122 as(4.9)C=Ciijj, where we identify C1122 with C2211 from the NG-fermion functional structure and adopt the SO(2) invariant definition.Thus the d.o.f. of the reduced bosonic and fermionic components for the d=2, N=2 vector supermultiplet are balanced as 8=8 in Eqs. (4.1), (4.6) and (4.9). Then, the LSUSY transformations (3.3) to (3.7) for N=2 SUSY become(4.10)δζD=−iζ¯i∂̸λi,(4.11)δζλi=Dζi−i2α11∂̸Mijζj+i2α12ϵijγ5∂̸ϕζj+12α13ϵijγa∂̸vaζj,(4.12)δζM11=α11(2ζ¯1λ1−iα21ζ¯2∂̸Λ2),(4.13)δζM22=α11(2ζ¯2λ2−iα21ζ¯1∂̸Λ1),(4.14)δζMij=α11{ζ¯iλj+ζ¯jλi+i2α21(ζ¯i∂̸Λj+ζ¯j∂̸Λi)}(i≠j),(4.15)δζϕ=α12ϵij(ζ¯iγ5λj−i2α21ζ¯iγ5∂̸Λj),(4.16)δζva=α13ϵij(iζ¯iγaλj−12α21ζ¯i∂̸γaΛj),(4.17)δζΛi=α21{1α11(Mjjζi−Mijζj)+1α12ϵijϕγ5ζj−iα13ϵijvaγaζj−i4α31∂̸Cζi},(4.18)δζC=4α31α21ζ¯iΛi, where we define ϕ and va by means of(4.19)ϕ=12ϵijϕij=12α12κϵijψ¯iγ5ψj|w|,va=12ϵijvaij=i2α13κϵijψ¯iγaψj|w|. The LSUSY transformations (4.10) to (4.18) just correspond to the ones obtained from a general N=2 superfield in d=2 [20,21].5Transforming to U(1) gauge and scalar supermultiplets for N=2 SUSYThe N=2 vector supermultiplet obtained in Section 4 is transformed into minimal U(1) gauge or scalar supermultiplet by means of appropriate recombinations of the functionals of the NG fermions, which are defined from the basic component fields (D, λi, Mij, ϕ, va, Λi, C) in Eqs. (3.1), (3.2), (4.1), (4.6), (4.9) and (4.19):55We can multiply the functionals of ψi for (D, λi, MiAj, ΛijAk, CiAjkBl) by a overall constant ξ which determine the magnitude of a vacuum expectation value of the D-term, but we take the value ξ=1 for simplicity of the discussions. The U(1) gauge supermultiplet in the N=2 LSUSY theory is derived from component fields which are defined by using the same set of the functionals of ψi as in the N=2 vector supermultiplet as follows;(5.1)λ˜i(ψ)=(λi−i2α21∂̸Λi)(ψ),(5.2)D˜(ψ)=(D−18α31□C)(ψ),(5.3)M(ψ)=Mii(ψ), in addition to ϕ(ψ) and va(ψ) in Eq. (4.19).Indeed, the definition (5.1) of spinor fields λ˜i induce LSUSY transformations with a U(1) gauge field strength Fab=∂avb−∂bva from Eqs. (4.11) and (4.17) (under the NLSUSY transformations (2.3)) as(5.4)δζλ˜i=D˜ζi−i2α11∂̸Mζi+iα12ϵijγ5∂̸ϕζj−12α13ϵijϵabFabζj. Then, LSUSY transformations of an auxiliary scalar field D˜ in the definition (5.2) and the other bosonic component fields (M, ϕ, va) become(5.5)δζD˜=−iζ¯i∂̸λ˜i,(5.6)δζM=2α11ζ¯iλ˜i,(5.7)δζϕ=α12ϵijζ¯iγ5λ˜j,(5.8)δζva=iα13ϵijζ¯iγaλ˜j+∂aWζ, so that the component fields (D˜, λ˜i, M, ϕ, va) constitute the U(1) gauge supermultiplet, in which the components (M11−M22,M12) do not appear.Note that a U(1) gauge transformation parameter Wζ in the LSUSY transformations (5.8) is(5.9)Wζ=−α13α21ϵijζ¯iΛj, which leads to a relation in a commutator algebra for va as(5.10)δζ1Wζ2−δζ2Wζ1=α13(1α11ϵijζ¯1iζ2jM−2α12ζ¯1iγ5ζ2iϕ+2iα13ζ¯1iγaζ2iva). Therefore, the commutator algebra for va under Eqs. (5.4), (5.8) and (5.10) does not induce a U(1) gauge transformation term in contrast to the Wess–Zumino gauge [2,3] as a result of the linearization of NLSUSY based on the commutator algebra (2.4) in the VA NLSUSY theory.When a N=2 LSUSY (free) action for the U(1) gauge supermultiplet,(5.11)SN=2gauge=∫d2x{−14(Fab)2+i2λ˜¯i∂̸λ˜i+12(∂aM)2+12(∂aϕ)2+12D˜2−1κD˜}, is defined, the values of the constants (α11, α12, α13) are determined as α112=14 and α122=α132=1 from the invariance of the action (5.11) under the LSUSY transformations from (5.4) to (5.8). In the action (5.11), the relative scales of the terms for the auxiliary fields Λ and C to the kinetic terms of the physical fields are fixed by taking the values of (α21, α31) in the recombinations (5.1) and (5.2), e.g. as α21=−12 and α31=−18.The above results for the U(1) gauge supermultiplet in d=2, N=2 LSUSY theory, which are obtained from the commutator-based linearization of NLSUSY, coincide with the ones in Ref. [15] based on the superspace formalism. In addition, the relation between the LSUSY action (5.11) and the VA NLSUSY action (2.1) for N=2 SUSY,(5.12)SN=2gauge(ψ)+[a surface term]=SN=2NLSUSY can be shown, e.g. by means of the superspace formalism (for example, see [15,22]).On the other hand, the scalar supermultiplet in the N=2 LSUSY theory is also derived by defining component fields from the same set of the functionals of ψi in the N=2 vector supermultiplet as(5.13)χi(ψ)=(λi+i2α21∂̸Λi)(ψ),(5.14)F(ψ)=(D+18α31□C)(ψ),G(ψ)=∂ava,(5.15)A(ψ)=(M11−M22)(ψ),B(ψ)=2M12(ψ), where the components (M11−M22, M12), which are the redundant ones in the LSUSY transformations from (5.4) to (5.8) for the U(1) gauge supermultiplet, appear in the definition (5.15) of the components (A, B). In fact, the variations of χi in Eq. (5.13) under the NLSUSY transformations (2.3) induce their LSUSY transformations,(5.16)δζχ1=Fζ1+1α13Gζ2−i2α11∂̸(Aζ1+Bζ2),δζχ2=Fζ2−1α13Gζ1+i2α11∂̸(Aζ2−Bζ1), which are also obtained from Eqs. (4.11) and (4.17). Then, LSUSY transformations of the bosonic fields (F,G,A,B) in Eqs. (5.14) and (5.15) become(5.17)δζF=−iζ¯i∂̸χi,δζG=−iα13ϵijζ¯i∂̸χj,(5.18)δζA=2α11(ζ¯1χ1−ζ¯2χ2),(5.19)δζB=2α11(ζ¯1χ2+ζ¯2χ1). Thus the component fields (F, G, χi, A, B) constitute the scalar supermultiplet.When we define a N=2 LSUSY (free) action for the scalar supermultiplet as(5.20)SN=2scalar=∫d2x{12(∂aA)2+12(∂aB)2+i2χ¯i∂̸χi+12(F2+G2)−1κF}, the values of the constants (α11, α13) in the LSUSY transformations (5.16) to (5.19) are determined as α112=14 and α132=1 from the LSUSY invariance of the action (5.11). In the action (5.20), the relative scales of the terms for the auxiliary fields Λ and C to the kinetic terms of the physical fields are fixed by taking the values of (α21, α31) in the recombinations (5.13) and (5.14), e.g. as α21=12 and α31=18. It is expected that the LSUSY action (5.20) and the VA NLSUSY action (2.1) for N=2 SUSY are related to each other from the NL/LSUSY relation in the d=4, N=1 SUSY theory [4,5].We note that the recombinations of the functionals of the NG fermions in Eqs. (5.1) and (5.2) for the U(1) gauge supermultiplet and Eqs. (5.13) and (5.14) for the scalar supermultiplet by using the auxiliary component fields Λi(ψ) and C(ψ) (and va(ψ)) have the same form as the ones for N=1 NL/LSUSY relations in d=4 [11].6Structure of basic components for a N=3 vector supermultipletIn this section, we focus on a N=3 vector supermultiplet derived from the VA NLSUSY theory as the simplest but a general example of the NL/LSUSY relations in d=2 and explain structures of the basic components (3.1) and (3.2) for N=3 SUSY by counting their bosonic and fermionic d.o.f. based on the NG-fermion functionals in Eqs. (2.7) and (2.8). From the consideration of the reduction of the basic components (3.1) and (3.2) to the ones for the N=2 vector supermultiplet in Section 4, let us give two procedures for counting the d.o.f. of the components for vector supermultiplets in N-extended SUSY based on the NLSUSY phase as follows:(a) The d.o.f. of the basic components are counted based on identities for the NG fermions ψi, which are obtained from Fierz transformations, e.g. as in the relations from (4.3) to (4.5) and (4.8) for N=2 SUSY.(b) In order to give the second procedure, we define the basic components expressed as the following bosonic or fermionic functionals of ψi,(6.1)biijj⋯kk((ψi)2(n−1)|w|)=κ2n−3ψ¯iψiψ¯jψj⋯ψ¯kψk|w|,(6.2)fliijjkk((ψi)2n−1|w|)=κ2(n−1)ψlψ¯iψiψ¯jψj⋯ψ¯kψk|w| with i≠j, j≠k, k≠i. In Eqs. (6.1) and (6.2), we consider n≥3 and use a notation without tensor-contraction rule for simplicity of the equations, i.e. they mean(6.3)b1122=κ3ψ¯1ψ1ψ¯2ψ2|w|,b2211=κ3ψ¯1ψ1ψ¯2ψ2|w|,b112233=κ3ψ¯1ψ1ψ¯2ψ2ψ¯3ψ3|w|,b113322=κ3ψ¯1ψ1ψ¯3ψ3ψ¯2ψ2|w|,⋯,(6.4)f12233=κ4ψ1ψ¯2ψ2ψ¯3ψ3|w|,f13322=κ4ψ1ψ¯3ψ3ψ¯2ψ2|w|,⋯, etc. Then, from the NG-fermion functional structure, we identify the each component in the functionals (6.1) or (6.2) with the other ones, i.e. b1122=b2211, b112233=b113322=⋯ and f12233=f13322, ⋯ etc. as the second procedure. This is adopted in the manifest SO(N) covariant (invariant) definition of basic component fields, which is same as that of the component C for N=2 SUSY in Eq. (4.9).By using the above procedures (a) and (b), let us count the d.o.f. of the basic components for the N=3 vector supermultiplet, which are constituted from (D, λi, MiAj, ΛijAk, CiAjkBl, ΨijAklBm, EiAjkBlmCn)(ψ) (i,j,⋯=1,2,3) in Eqs. (3.1) and (3.2): First, the d.o.f. of the components (D, λi, MiAj) are counted (1, 6, 15), straightforwardly, where MiAj are decomposed in Eq. (4.1).Next, as for the fermionic components ΛijAk which are decomposed as Eq. (4.2), for example, the d.o.f. of Λijk=Λi(jk) are apparently 36. However, according to the procedure (a), 6 components in Λijk(ψ) vanish as(6.5)Λiii(ψ)=(Λ111,Λ222,Λ333)(ψ)=0, where we use the same component notation as in Eqs. (6.1) and (6.2) and this notation's rule is also used below in this section. In addition, 12 components in Λijk(ψ) are related to each other by means of Fierz transformations as(6.6)Λiij(ψ)=−12Λjii(ψ)(i≠j). When we subtract the d.o.f. in Eqs. (6.5) and (6.6), the d.o.f. of Λijk are 36−18=18.In the same way, the d.o.f. of Λ5ijk=Λ5i[jk](ψ) and Λaijk=Λai[jk](ψ) are counted as 18−12=6 and 36−24=12, respectively, where the following relations,(6.7)Λ5iij(ψ)=−α222α21γ5Λjii(ψ),Λaiij(ψ)=−iα232α21γaΛjii(ψ)(i≠j), are used for the subtraction of the d.o.f.Furthermore, we notice that there are relations of linear combinations for the remaining components ΛijAk(ψ) (i≠j, j≠k, k≠i) from Fierz transformations as follows;(6.8)(1α21aΛijk+1α22bγ5Λ5ijk+iα23cγaΛaijk)(ψ)=(1α21a′Λkij+1α22b′γ5Λ5kij+iα23c′γaΛakij)(ψ), where coefficients a′, b′, c′ are written in terms of a, b, c as(6.9)a′=−12(a+b−2c),b′=12(a+b+2c),c′=−12(a−b). The relations (6.8) connect the components Λ12A3 with Λ31A2 and Λ23A1 in the NLSUSY phase, so that the 4 d.o.f. for the two independent relations in Eq. (6.8) are subtracted from those of ΛijAk. Therefore, the total d.o.f. of ΛijAk become (18+6+12)−4=32 based on the NG-fermion functional structure of the components.As for the bosonic components CiAjkBl which are decomposed as Eq. (4.7), for example, the d.o.f. of Cijkl=C(ij)(kl) are apparently 36. However, according to the procedure (a) based on the NLSUSY phase, we have to subtract appropriate d.o.f. from the apparent ones as in the case of the components ΛijAk. Indeed, 15 components in Cijkl(ψ) are vanished as(6.10)Ciiii(ψ)=0,Ciiij(ψ)=Cijii(ψ)=0(i≠j). In addition, 12 components in Cijkl(ψ) are related to each other by means of Fierz transformations as(6.11)Cijij(ψ)=−12Ciijj(ψ),Cijik(ψ)=−12(dCiijk+eCjkii)(ψ)(i≠j,j≠k,k≠i). with coefficients d and e. Note that we regard Cjjii as Ciijj in the first relation of Eq. (6.11) because of the procedure (b) for counting the d.o.f. of the components. Therefore, the effective d.o.f. of Cijkl become 36−(15+12)=9.In the same way, if we use relations for the components C5ijkl=C5[ij](kl) and C˜5ijkl=C˜5(ij)[kl],(6.12)C5ijii(ψ)=C˜5iiij(ψ)=0,C5ijij(ψ)=C˜5ijij(ψ)=0,C5ijik(ψ)=−α322α31C5jkii(ψ),C˜5ijik(ψ)=−α32′2α31C5iijk(ψ)(i≠j,j≠k,k≠i), and ones for the components Caijkl=Ca[ij](kl) and C˜aijkl=C˜a(ij)[kl],(6.13)Caijii(ψ)=C˜aiiij(ψ)=0,Caijij(ψ)=C˜aijij(ψ)=0,Caijik(ψ)=−α322α31Cajkii(ψ),C˜aijik(ψ)=−α32′2α31Caiijk(ψ)(i≠j,j≠k,k≠i), then the remaining components in C5ijkl, C˜5ijkl, Caijkl and C˜aijkl are only(6.14)(C51233,C52311,C53122),(C˜53312,C˜51123,C˜52231),(Ca1233,Ca2311,Ca3122),(C˜a3312,C˜a1123,C˜a2231) whose d.o.f. are totally 18.The d.o.f. of the other components in Eq. (4.7), i.e. those of the latter ones (C55ijkl, Ca5ijkl, C˜5aijkl, Cabijkl)(ψ) are effectively 0 since all nonvanishing components in them are expressed in terms of the former ones (Cijkl, C5ijkl, C˜5ijkl, Caijkl, C˜aijkl)(ψ) by meas of Fierz transformations in the case of the N=3 SUSY theory. Thus, the total d.o.f. of CiAjkBl become 9+18=27.As for the components ΨijAklBm and EiAjkBlmCn, in accordance with the procedure (a) and (b) for counting the d.o.f., they are expressed by means of the following components,(6.15)Ψi(ψ)=Ψijjkk(ψ),E(ψ)=Eiijjkk(ψ), respectively, whose d.o.f. are 6 and 1.Therefore, the d.o.f. of the basic components (D, λi, MiAj, ΛijAk, CiAjkBl, ΨijAklBm, EiAjkBlmCn) for N=3 SUSY are summarized as (1, 6, 15, 32, 27, 6, 1), respectively, in which the d.o.f. of the general bosonic and fermionic components are balanced as 44=44. Namely, their functional structure of the NG fermions ψi also determines the effective and general d.o.f. of the component fields in the N=3 vector supermultiplet.7Transforming to a U(1) gauge supermultiplet for N=3 SUSYHelicity states for physical fields in the N=3 LSUSY theory are(7.1)[1_(1),3_(12),3_(0),1_(−12)]+[CPT conjugate], where n_(λ) means the dimension and the helicity in the irreducible representation of SO(3) super-Poincaré algebra. Basic component fields for the helicity states (7.1) are obtained by multiplying the components (3.1) and (3.2) for N=3 SUSY by overall constants ξi which correspond to vacuum expectation values of D-terms in the minimal U(1) gauge supermultiplet [9] as follows;66The d.o.f. of the bosonic and fermionic components in Eq. (7.2) are balanced as 132=132 from the arguments in Section 6.(7.2)Di=ξiD(ψ),λij=ξiλj(ψ),Mijk=Mi(jk)=ξiMjk(ψ),ϕijk=ϕi[jk]=ξiϕjk(ψ),vaijk=vai[jk]=ξivajk(ψ)ΛijkAl=ξiΛjkAl(ψ),CijAklBm=ξiCjAklBm(ψ),ΨijkAlmBn=ξiΨjkAlmBn(ψ),EijAklBmnCp=ξiEjAklBmnCp(ψ), where we use the decomposition of the components MiAj in Eq. (4.1) and (ΛijAk, CiAjkBl) are also decomposed as Eqs. (4.2) and (4.7). In this section, we argue a transition from the general component fields (7.2) to minimal ones for the U(1) gauge supermultiplet.Let us start with a consideration of the U(1) gauge invariance in LSUSY transformations of the triplet spinor component fields in Eq. (7.1) by defining them as(7.3)λi=ϵijkλjk(ψ) and their appropriate functional-recombinations in terms of the auxiliary spinor components ΛijkAl(ψ) in Eq. (7.2): Namely, we consider the most general form for the recombinations of λi(ψ) and ΛijkAl(ψ) as(7.4)λ˜i(ψ)=[λi+ϵijk{iα21∂̸(aΛjkll+bΛljkl)+icα22γ5∂̸Λ5ljkl+1α23(d∂aΛaljkl+eϵabγ5∂aΛbljkl)}](ψ)+O(ψ5), with constants (a,b,c,d,e), and we determine relations among the constants from the U(1) gauge invariance in LSUSY transformations of the spinor fields (7.4).By using the LSUSY transformations of λij and ΛijkAl,(7.5)δζλij=Diζj−i2α11∂̸Mijkζk+i2α12γ5∂̸ϕijkζk+12α13γa∂̸vaijkζk,(7.6)δζΛijkl=α21{12α11(2Miklζj−Mijkζl−Mijlζk)+12α12(ϕijkγ5ζl+ϕijlγ5ζk)−i2α13(vaijkγaζl+vaijlγaζk)}+O(C(ψ)),(7.7)δζΛ5ijkl=α22{−12α11(Mijkγ5ζl−Mijlγ5ζk)+12α12(2ϕiklζj+ϕijkζl−ϕijlζk)+i2α13(vaijkγ5γaζl−vaijlγ5γaζk)}+O(C(ψ)),(7.8)δζΛaijkl=α23{−i2α11(Mijkγaζl−Mijlγaζk)+i2α12(ϕijkγ5γaζl−ϕijlγ5γaζk)+12α13(2vaiklζj+vbijkγbγaζl−vbijlγbγaζk)}+O(C(ψ)), we can estimate va-terms in the variations of λ˜i in Eq. (7.4), which are written as(7.9)δλ˜i[vaterms]=12α13[ϵjkl{(−12+a)ϵab∂avbjklγ5ζi+(12+a)∂avajklζi}+ϵijk{(12−a+b+c−d+e)ϵab∂avbljkγ5ζl+(−12−a+b+c+d−e)∂avaljkζl+(b−c+d−3e)ϵab∂avbljlγ5ζk+(b−c−3d+e)∂avaljlγ5ζk}]. In order to make this variations U(1) gauge invariant, the vanishments of the terms for ∂avajkl, ∂avaljk, ϵijkϵab∂avbljk and ϵijkϵab∂avbljl in Eq. (7.9) are required, i.e.(7.10)12+a=0,−12−a+b+c+d−e=0,12−a+b+c−d+e=0,b−c+d−3e=0, which mean a=−12, c=−b−12, d=b+1 and e=b+12. Then, the variations (7.9) become(7.11)δλ˜i[vaterms]=−1α13(14ϵabFabγ5ζi+ϵijk∂avaljlγ5ζk), where Fab=∂avb−∂bva with a vector field,(7.12)va=12ϵijkvaijk(ψ). Note that the terms for ∂avaljl(ψ) in the variations (7.11), whose functional forms correspond to the auxiliary fields G=G(ψ) in Eq. (5.14), are absorbed into recombinations with respect to the auxiliary components Di as shown below.Under the conditions (7.10), the variations of λ˜i are expressed at least up to O(ψ2) as(7.13)δζλ˜i=ϵijk(Dˆj−iα11∂̸Aj)ζk+iα12γ5∂̸ϕζi−12α13ϵabFabγ5ζi, where bosonic components Dˆi, Ai and ϕ are defined by means of(7.14)Dˆi=(Di−1α13∂avajij)(ψ),Ai=(12Mijj−Mjij)(ψ),ϕ=12ϵijkϕijk(ψ).The values of the constants (a,b,c,d,e) in Eq. (7.4), which satisfy the relations (7.10), will be determined from considerations for LSUSY transformations of other component fields in the minimal U(1) gauge supermultiplet. Moreover, as for higher-order terms of ψi in the bosonic components (7.12) and (7.14) and the singlet spinor field in the helicity states (7.1), whose leading term of ψi is defined as(7.15)χ=ξiψi, it is expected that those higher-order terms are determined from general arguments as in Eqs. from (7.4) to (7.10), though most of the calculations are complicated.With the help of the study in the relation between the VA NLSUSY theory and the minimal U(1) gauge supermultiplet for d=2, N=3 SUSY [9], we find that the following recombinations of ψi,(7.16)D˜i={Di−1α13∂avajij−18α31□(Cijjkk−4Cjijkk)}(ψ)+O(ψ6),(7.17)λ˜i=[λi−i2ϵijk{1α21∂̸(Λjkll+Λljkl)+iα23∂aΛaljkl}](ψ)+O(ψ5),(7.18)χ˜=(χ+i2α21∂̸Λiijj)(ψ)+O(ψ5),(7.19)A˜i=(Ai−α112α34∂aCajijkk)(ψ)+O(ψ6),(7.20)ϕ˜=(ϕ+α124α34ϵabϵijk∂aCbijkll)(ψ)+O(ψ6),(7.21)v˜a=(va−α134α32ϵabϵijk∂bC5ijkll)(ψ)+O(ψ6), give LSUSY transformations in the minimal U(1) gauge supermultiplet at least up to O(ψ2) or O(ψ3). Note that the recombinations (7.17) are obtained from Eq. (7.4) by taking the values of the constants (a,b,c,d,e) as (−12,−12,0,12,0), which satisfy the conditions (7.10).Then, U(1) gauge invariant and minimal expressions of LSUSY transformations for (D˜, λ˜i, χ˜, A˜i, ϕ˜, v˜a) are confirmed at least up to O(ψ2) or O(ψ3) as follows;(7.22)δζD˜i=−i(ϵijkζ¯j∂̸λ˜k+ζ¯i∂̸χ˜),δζλ˜i=ϵijk(D˜j−iα11∂̸A˜j)ζk+iα12γ5∂̸ϕ˜ζi−12α13ϵabF˜abγ5ζi,δζχ˜=(D˜i+iα11∂̸A˜i)ζi,δζA˜i=α11(ϵijkζ¯jλ˜k−ζ¯iχ˜),δζϕ˜=−α12ζ¯iγ5λ˜i,δζv˜a=−α13(iζ¯iγaλ˜i+∂aWζ), where a gauge transformation parameter X in δζv˜a is(7.23)Wζ=12ϵijk{1α21ζ¯iΛjkll−12(3α21ζ¯lΛijkl−1α22ζ¯lγ5Λ5ijkl+iα23ζ¯lγaΛaijkl)}. We notice that the relations for the auxiliary fields ΛijAk in Eqs. from (6.6) to (6.8) are used in the derivation of the LSUSY transformations (7.22), in particular, for the bosonic components (D˜, A˜i, ϕ˜, v˜a). Namely, those relations among the general auxiliary fields play the role not only in counting the d.o.f. for the component fields but also in transforming to the minimal U(1) gauge supermultiplet from the general one.As is the case with the LSUSY transformations (5.8) in the N=2 U(1) gauge supermultiplet, the U(1) gauge transformation parameter Wζ for v˜a in Eq. (7.22) leads to a relation (up to O(ψ2)), δζ1Wζ2−δζ2Wζ1=−θ(ζ1i,ζ2i;A˜i,ϕ˜,v˜a), in the commutator algebra for v˜a. Thanks to this relation, a U(1) gauge transformation term with the parameter θ is not induced in the commutation relation.The relation between a N=3 LSUSY (free) action for the U(1) gauge supermultiplet, which is denoted as SN=3gauge, and the VA NLSUSY action (2.1) for N=3 SUSY, i.e.,(7.24)SN=3gauge(ψ)+[a surface term]=SN=3NLSUSY has been shown in the Ref. [9] at least up to O(ψ3). In the relation (7.24), we define the LSUSY action SN=3gauge as(7.25)SN=3gauge=∫d2x{−14(F˜ab)2+i2λ˜¯i∂̸λ˜i+i2χ˜¯∂̸χ˜+12(∂aA˜i)2+12(∂aϕ˜)2+12(D˜i)2−ξiκD˜i}, where F˜ab=∂av˜b−∂bv˜a and the values of the constants (α11, α12, α13) are determined as α112=α122=α132=1 from the invariance of the action (7.25) under the LSUSY transformations from (7.22). Note that the relative scales of the terms for the general auxiliary fields to the kinetic terms of the physical fields in the action (7.25) are fixed by taking the values of (α21, α23, α31, α32, α34, etc.) appropriately in the recombinations from (7.16) to (7.21).8Summary and discussionsWe have discussed the linearization of NLSUSY based on the commutator algebra (2.4) and derived the general structure of the vector supermultiplets for extended LSUSY theories in d=2. By defining the bosonic and fermionic component fields (3.1) and (3.2) from the set of the functionals (2.7) and (2.8), we have obtained the LSUSY transformations from (3.3) to (3.7) for vector supermultiplets with general auxiliary fields. Those LSUSY transformations are uniquely determined from the variations (2.11) and (2.12) under the commutation relation (2.6) as is the case with the linearization of NLSUSY in d=4 [10]. In the definition of the functionals (2.7) and (2.8), the fundamental determinant |w| in the VA NLSUSY theory play the important role.The reduction from the general vector supermultiplets obtained in Section 3 to the N=2 one gives some instructions for counting the d.o.f. of the bosonic and fermionic components up to the general auxiliary fields: Indeed, in Section 6, the relations from (4.3) to (4.5) and (4.8) lead the counting procedure (a), while the definition (4.9) of the auxiliary component C gives its procedure (b). The minimal U(1) gauge or scalar supermultiplet for N=2 SUSY is obtained by means of appropriate recombinations of the functionals of the NG fermions as in Eqs. (5.1) and (5.2) or Eqs. from (5.13) to (5.15), which are similar to the case of the d=4, N=1 SUSY theory [11] corresponding to the Wess–Zumino gauge. In d=4, the Komargodski–Seiberg (KS) Goldstino theory [23] is also constructed from the nilpotent chiral superfield X with X2=0. It relates to the VA NLSUSY theory as was shown explicitly in Ref. [24], in which an independent auxiliary scalar field in the KS theory is expressed as a functional (composite) of the NG fermion in the VA one. It is interesting to consider the description of spontaneous SUSY breaking through the nilpotency of superfields in d=2.We have found in Section 6 that the d.o.f. of the general bosonic and fermionic components for the N=3 vector supermultiplet are balanced as 44=44 by using the procedure for counting the d.o.f. of the components, which is the simplest but a general example for the extended LSUSY theory with the general auxiliary-field structure. In particular, the relations (constraints) for each auxiliary fields (ΛijAk, CiAjkBl, ΨijAklBm, EiAjkBlmCn) are determined from their functional (composite) structure of the NG fermions.Moreover, by introducing the basic component fields (7.2) for the helicity states (7.1), we have derived the U(1) gauge supermultiplet for N=3 SUSY in Section 7. The component fields (D˜, λ˜i, χ˜, A˜i, ϕ˜, v˜a) in the U(1) gauge supermultiplet are determined from the general recombinations of the functionals of the NG fermions in Eqs. from (7.16) to (7.21), which are a generalization of the Wess–Zumino gauge to the extended SUSY theories.The LSUSY transformations (7.22) for the d=2, N=3 minimal U(1) gauge supermultiplet are obtained from the variations of the functionals from (7.16) to (7.21) under the NLSUSY transformations (2.3), where we have used the relations from (6.6) to (6.8) for the auxiliary fields ΛijAk. That is, from the viewpoint of the commutator-based linearization of NLSUSY, the relations (constraints) for each auxiliary fields, which are determined from the behavior of the NG fermions, have the crucial role not only in counting the d.o.f. for the general component fields in LSUSY theories but also in transforming to the U(1) gauge supermultiplet.Moreover, the (p,q) SUSY [25] is defined in d=2, where non-negative integers p and q are considered for positive and negative-chirality charges. The twisted version of the (p,q) SUSY is also considered, e.g. in Ref. [26]. It is an important and open problem to extend the arguments of the commutator-based linearization in this paper to the case where p and q are different. 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