On the linearization of nonlinear supersymmetry based on the commutator algebra

We discuss a linearization procedure of nonlinear supersymmetry (NLSUSY) based on the closure of the commutator algebra for variations of functionals of Nambu–Goldstone fermions and their derivative terms under NLSUSY transformations in Volkov–Akulov NLSUSY theory. In the case of a set of bosonic and fermionic functionals, which leads to (massless) vector linear supermultiplets, we explicitly show that general linear SUSY transformations of basic components deﬁned from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory.

On the other hand, the NLSUSY general relativistic (GR) theory has been constructed [7,8] as a generalization of the NLSUSY theory to curved spacetime, where a NLSUSY Einstein-Hilbert type action is defined in terms of a vierbein and the NG fermions. The fundamental action in the NLSUSY-GR theory has symmetries which are isomorphic to S O (N) super-Poincaré group and it contains the VA NLSUSY action in the cosmological term, where a dimensional constant in the NLSUSY theory is expressed by using the cosmological and gravitational constants. Therefore, the NL/LSUSY E-mail address: tsuda@sit.ac.jp. relations in flat spacetime contribute to the understanding of the low-energy physics in the NLSUSY GR theory and it is an interesting and important problem to know more explicitly the NL/LSUSY relations for N ≥ 2 SUSY including up to general auxiliary fields in linear supermultiplets.
In order to investigate the NL/LSUSY relations in more detail, it is a useful method to identify basic components for general linear supermultiplets by considering variations of functionals of the NG fermions based on a commutator algebra for the NLSUSY transformations in the VA NLSUSY theory and by constructing LSUSY transformations of bosonic and fermionic components, which satisfy the same commutation relation. In this letter, we discuss the linearization procedure of NLSUSY in extended SUSY based on the fact in flat spacetime that every functional of the NG fermions and their derivative terms satisfies the commutator algebra in the VA NLSUSY theory under the NLSUSY transformations. By introducing a set of bosonic and fermionic functionals, which leads to (massless) vector supermultiplets, we show that general LSUSY transformations of basic components defined from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory.
The fundamental action in the VA NLSUSY theory [1] is defined in terms of (Majorana) NG fermions ψ i as where κ is a dimensional constant whose dimension is (mass) −2 and the determinant |w| is 1 where ξ a = iκψ j γ a ζ j with constant (Majorana) spinor parameters ζ i . The commutator algebra for the NLSUSY transforma- where δ P ( a ) means a translation with parameters a = 2iζ i It is also shown that every Lorentz-tensor (or scalar) functionals of ψ i and their derivative terms satisfy the commutator algebra (4) under the NLSUSY transformations (3): If we define bosonic or fermionic functionals of ψ i and their derivative terms (∂ψ i , ∂ 2 ψ i , · · · , ∂ n ψ i ), in which γ -matrices are also used, as with A meaning the Lorentz indices of (a, ab, · · · , etc.) and I being the (internal) ones of (i, ij, · · · , etc.), then the commutator algebra for variations of the functionals F I A (or F A , F I and F ) under the NLSUSY transformations (3) closes as This relation (6) is proved from the fact that the derivative terms (∂ψ i , ∂ 2 ψ i , · · · , ∂ n ψ i ) and products of two kinds of the functionals F I A and G J B which are respectively defined as Eqs. (5) and (6) satisfy the same commutation relation (for example, see [9]).
Then, the variations of the functionals (7) and (8) under the NLSUSY transformations (3) become 1 The indices i, j, · · · = 1, 2, · · · , N and Minkowski spacetime indices are denoted by a, b, · · · = 0, 1, 2, 3. Gamma matrices satisfy {γ a , γ b } = 2η ab with the Minkowski by using the variations of |w|, i.e. δ ζ |w| = ∂ a (ξ a |w|). The variations (9) and (10) indicate that the bosonic and fermionic functionals in Eqs. (7) and (8) are linearly exchanged with each other through those variations. In fact, the functionals (7) and (8)  From now on, studying the variations (9) and (10) by starting with a bosonic component which is defined as D = b for the lowest-order functional in Eq. (7) and by examining the commutation relation (6) on the functionals (7) and (8), we show that LSUSY transformations of basic bosonic and fermionic components defined from those functionals are uniquely determined, which satisfy the same commutator algebra as Eq. (4). First, the variation of D is where spinor components λ i are defined as λ i = f i . Then, the variations of λ i become by using a Fierz transformation. In the variation (12), we introduce bosonic components M i mean the following kinds of components, we use γ A = 1, −iγ 5 , −iγ a , −γ 5 γ a or − √ 2iσ ab and a sign factor ε M1 which appear from the relation ψ j γ A ψ i = εψ i γ A ψ j . 2 The constants α and the sign factor ε are also used in the same meanings below. 2 The sign factor ε is ε = +1 Up to the variations of D and λ i , their LSUSY transformations are unambiguously determined as Eqs. (11) and (12), in which the LSUSY transformation (11) of D satisfies the commutator algebra (4) under Eq. (12) (because of the commutation relation (6) on D = D(ψ i )).
In the variations (14), in order to define LSUSY transformations of  12) and (14) as follows; Since these two supertransformations satisfy Eq. (6)   which satisfy the commutation relation (6) and which are obtained from Eqs. (12), (16) and (17) as by using a Fierz transformation in the last terms of Eq. (17). In Eq.
Here we also note that the last terms with respect to C i l ), they can be determined in accordance with the above arguments for the definition of LSUSY transformations, which terminates with those of bosonic components for the functionals at O{(ψ i ) 4N } in Eq. (7).
We summarize our results as follows. In this letter, we have discussed a linearization procedure of NLSUSY based on the commutation relation (6) on the functionals (5) in terms of the NG fermions ψ i and their derivative terms under the NLSUSY transformations (3). In the case of the bosonic and fermionic functionals (7) and (8), we have shown that the general LSUSY transformations of the basic components defined from those functionals are uniquely determined in the linearization procedure; indeed, the variations (9) and (10)  These results for the basic components which are defined straightforwardly from the functionals (7) and (8) show that their LSUSY transformations are uniquely determined in the linearization of NLSUSY by examining the two supertransformations of them, i.e. the commutation relation (6) from on the lowest-order functional to on the same-order ones, including up to the vanishing terms in the commutation relation.
Finally, we mention transitions from the basic components in the general LSUSY transformations to component fields in (massless) vector supermultiplets prior to transforming to gauge supermultiplets. For N = 1 SUSY, the bosonic and fermionic functionals (7) and (8) with the constants α m (m = 1, 2, · · · , 5), though the "vector" field v a defined in Eq. (28) is expressed in terms of the axial-vector functional b 5a [5]. Note that the degrees of freedom of the above bosonic and fermionic components for N = 1 SUSY are balanced as 8 = 8 and the basic component fields in N = 1 SUSY theories are defined, in general, by further multiplying those components by an overall constant ξ which gives a vacuum expectation value of the

D-term.
From the functional representation of the basic component fields (28), we have recently shown [12] that LSUSY transformations for a N = 1 vector supermultiplet with the general auxiliary fields ( , C ) [13,14] are derived by using the commutator-based linearization procedure in this letter. In addition, both U (1) gauge and scalar supermultiplets in N = 1 SUSY theories are also constructed from the same set of the functionals (27) by means of appropriate recombinations of the basic components (28) as follows [12]; namely, recombinations, lead to the U (1) gauge supermultiplet with the component fields (D, λ , v a ), whereas ones, give the scalar supermultiplet with the component fields Moreover, in N = 1 SUSY theories, a constrained-superfield NLSUSY action of a goldstino for low-energy effective theories was constructed in Ref. [15], in which a spinor field g identified as the goldstino and the (auxiliary) scalar F -term constitute a nonlinear supermultiplet through a chiral superfield X with a constraint 3 In the reduction of the functionals (7) and (8)  X 2 = 0. In the constrained chiral superfield, a scalar component (i.e., a superpartner of the goldstino) is represented in terms of g (which corresponds to χ = χ (ψ)) and the auxiliary scalar field F c (= F (ψ) + iG(ψ)) as a composite g 2 2F c (in the two-component spinor notation). From the viewpoint of the NG-fermion functional expression (30) for the scalar supermultiplet, the composite scalar field in X with X 2 = 0 relates to a complex scalar field φ defined from the scalar components (A, B) in Eq. (28); its relation is confirmed easily at least at a leading order for the composite scalar as The all-order NG-fermion functional correspondence for Eq. (31) is also expected from the equivalence of the VA and Komargodski-Seiberg NLSUSY actions, which has been shown explicitly in Ref. [16], in addition to the NL/LSUSY relation for the N = 1 scalar supermultiplet.