Linearizing extended nonlinear supersymmetry in two dimensional spacetime

We 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 $d = 2$.


Introduction
Relations 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 dimesional 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 relativisitic (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 ferimions, 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.

Functionals of Nambu-Goldstone fermions for vector supermultiplets
A fundamental action in the VA NLSUSY theory [1] for extended SUSY in d = 2 is given in terms of Majorana NG fermions ψ i as † where κ is a dimensional constant whose dimension is (mass) −1 and a fundamental determinant |w| is defined by means of where ζ i are constant (Majorana) spinor parameters and ξ a = iκψ j γ a ζ j . The NLSUSY transformations (2.3) satify a commutator algebra, with δ P (Ξ a ) (Ξ a = 2iζ i 1 γ a ζ i 2 ) meaning a translation. Since the determinant (2.2) also transforms as As a property of the commutator algebra (2.4), every bosonic or fermionic Lorentztensor (or scalar) functional of ψ i and their derivative terms (∂ψ i , ∂ 2 ψ i , · · ·, ∂ n ψ i ) satisfies the commutator algebra (2.4) under the NLSUSY transformations (2.3); namely, 6) † The indices i, j, · · · = 1, 2, · · · , N and Minkowski spacetime indices are denoted by a, b, · · · = 0, 1. Gamma matrices satisfy {γ a , γ b } = 2η ab with the Minkowski spacetime metric η ab = diag(+, −) and where F I A = F I A (ψ i , ∂ψ i , ∂ 2 ψ i , · · · , ∂ n ψ i ) are the functionals of the NG fermions with Lorentz index A = (a, ab, · · · , etc.) and the internal one I = (i, ij, · · · , etc.). ‡ 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 ψ i (with γ-matrices) and the determinant |w|. Let us express the set of bosonic and fermionic functionals for each N NLSUSY as 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., γ A = 1, γ 5 or iγ a (γ A = 1, γ 5 or − iγ a ). Note that vector components appear in the functionals b i A j 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 (ψ i ) 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 12) which indicate that the bosonic and fermionic functionals in Eqs.(2.7) and (2.8) are linearly transformed to each other. ‡ The commutation relation (2.6) is proved from Eq.(2.4) and from the fact that the derivative terms (∂ψ i , ∂ 2 ψ i , · · ·, ∂ n ψ i ) and products of two kinds of the fuctionals F I A and G J B which are respectively defined in Eq.(2.6) satisfy the same commutation relation (for example, see [19]).

Commutator-based linearization of NLSUSY in d = 2
In 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 and fermionic ones as where α mA (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;

7)
· · · , etc. In Eqs. from (3.3) to (3.7), we use a sign factor ε (or ε ′ ) which appears from the relationψ j γ A ψ i = εψ i γ A ψ j § and we define C iCjk Since the last terms in Eq.(3.10) are symmetric under exchanging the indices 1 and 2 of the spinor transformation parameters (ζ k 1 , ζ j 2 ) and they vanish in the commutation relation (2.6), the ψ k and ψ j have to go into bilinear formsψ j γ A ψ k 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 Λ ij the ψ l and ψ k have to take bilinear formsψ k γ A ψ l 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 Therefore, the LSUSY transformations of Λ ij (3.14) by means of a Fierz transformation. This is because the form of the variation (3.14) contain the bilinear termsψ l γ C ψ m with symmetries which correspond to the indices of spinor transformation parameters (ζ m 1 , ζ l 2 ) in the two supertransformations of Λ ij = α 34 κ 3ψi γ 5 ψ jψk γ 5 ψ l |w|, but C 1122 is the only remaining component in Eq.(4.7) for N = 2 SUSY based on the NG-fermion functional structure since and all other components vanish by means of Fierz transformations. Then, a bosonic auxiliary field in the N = 2 vector supermultiplet is defined from C 1122 as where we identify C 1122 with C 2211 from the NG-fermion functional structure and adopt the SO (2) ǫ ij γ 5 ∂φζ j + 1 2α 13 ǫ ij γ a ∂v a ζ j , (4.11) 14) where we define φ and v a by means of 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].

5)
We can multiply the functionals of ψ i for (D, l ) 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. Note that a U(1) gauge transformation parameter W ζ in the LSUSY transformations (5.8) is which leads to a relation in a commutator algebra for v a as When a N = 2 LSUSY (free) action for the U(1) gauge supermultiplet, is defined, the values of the constants (α 11 , α 12 , α 13 ) are determined as α 2 11 = 1 4 and α 2 12 = α 2 13 = 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 (α 21 , α 31 ) in the recombinations (5.1) and (5.2), e.g. as α 21 = − 1 2 and α 31 = − 1 8 .
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, S N =2 gauge (ψ) + [ a surface term ] = S N =2 NLSUSY (5.12) can be shown, e.g. by means of the superspace formalism (for example, see [15,22]).
On the other hand, the scalar supermutiplet in the N = 2 LSUSY theory is also derived by defining component fields from the same set of the functionals of ψ i in the N = 2 vector supermultiplet as  (5.16) which are also obtained from Eqs. When we define a N = 2 LSUSY (free) action for the scalar supermultiplet as  [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 Λ i (ψ) and C(ψ) (and v a (ψ)) have the same form as the ones for N = 1 NL/LSUSY relations in d = 4 [11]. (b) In order to give the second procedure, we define the basic components expressed as the following bosonic or fermionic functionals of ψ i , b iijj···kk (ψ i ) 2(n−1) |w| = κ 2n−3ψi ψ iψj ψ j · · ·ψ k ψ k |w|, (6.1) f liijjkk (ψ i ) 2n−1 |w| = κ 2(n−1) ψ lψi ψ iψj ψ j · · ·ψ k ψ k |w| (6.2) 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  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 Λ ijk (ψ) are related to each other by means of Fierz transformations as are used for the subtraction of the d.o.f.
Furthermore, we notice that there are relations of linear combinations for the remaining components Λ ij A k (ψ) (i = j, j = k, k = i) from Fierz transformations as follows; where coefficients a ′ , b ′ , c ′ are written in terms of a, b, c as The relations (6.8) connect the components Λ 12 In addition, 12 components in C ijkl (ψ) are related to each other by means of Fierz transformations as with coefficients d and e. Note that we regard C jjii as C iijj 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 C ijkl become 36 − (15 + 12) = 9.
In the same way, if we use relations for the components C ijkl and ones for the components C ijkl     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 and their appropriate functional-recombinations in terms of the auxiliary spinor components Λ ijk A l (ψ) in Eq.(7.2): Namely, we consider the most general form for the recombinations of λ i (ψ) and Λ ijk A l (ψ) as 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 λ ij and Λ ijk A l , γ a ∂v ijk a ζ k , (7.5) we can estimate v a -terms in the variations ofλ i in Eq. (7.4), which are written as In order to make this variations U(1) gauge invariant, the vanishments of the terms for ∂ a v jkl a , ∂ a v ljk a , ǫ ijk ǫ ab ∂ a v ljk b and ǫ ijk ǫ ab ∂ a v ljl b in Eq.(7.9) are required, i.e. 1 2 + a = 0, Then, the variations (7.9) become Note that the terms for ∂ a v ljl a (ψ) in the variations (7.11), whose functional forms corresonds to the auxiliary fields G = G(ψ) in Eq.(5.14), are absorbed into recombinations with respect to the auxiliary components D i as shown below.
Under the conditions (7.10), the variations ofλ i are expressed at least up to O(ψ 2 ) as where bosonic componentsD i , A i and φ are defined by means of 14) 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 ψ i in the bosonic conponents (7.12) and (7.14) and the singlet spinor field in the helicity states (7.1), whose leading term of ψ i is defined as 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 ψ i , where a gauge transformation parametar X in δ ζṽ a is (7.23) We notice that the relations for the auxiliary fields Λ ij A k 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,Ã i ,φ,ṽ 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 W ζ forṽ a in Eq.(7.22) leads to a relation (up to O(ψ 2 )), , in the commutator algebra forṽ a . Thanks to this relation, a U(1) gauge transformation term with the parameter θ is not induced in the commutation relation.

Summary and discussions
We 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 auxilialy 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.
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 (Λ ij 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,Ã i ,φ,ṽ 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 Λ ij A k . 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.