A 6D nonabelian (1, 0) theory

We construct a 6D nonabelian N=1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=\left(1,\ 0\right) $$\end{document} theory by coupling an N=1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=\left(1,\ 0\right) $$\end{document} tensor multiplet to an N=1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=\left(1,\ 0\right) $$\end{document} hypermultiplet. While the N=1,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=\left(1,\ 0\right) $$\end{document} tensor multiplet is in the adjoint representation of the gauge group, the hypermultiplet can be in the fundamental representation or any other representation. If the hypermultiplet is also in the adjoint representation of the gauge group, the supersymmetry is enhanced to N=2,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=\left(2,\ 0\right) $$\end{document}, and the theory is identical to the (2, 0) theory of Lambert and Papageorgakis (LP). Upon dimension reduction, the (1, 0) theory can be reduced to a general N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=1 $$\end{document} supersymmetric Yang-Mills theory in 5D. We discuss briefly the possible applications of the theories to multi M5-branes.


Introduction and summary
M2-branes and M5-branes are two types of fundamental objects in M-theory. According to the gauge/gravity correspondence, they admit dual gauge descriptions [1]. The gauge theories of multi M2-branes have been constructed successfully: they are the 3D N = 8 BLG theory with gauge group SO(4) [2,3], the N = 6 ABJM theory with gauge group U(N ) × U(N ) [4], and the other extended superconformal Chern-Simons matter theories with variety gauge groups. However, it seems more difficult to construct the gauge theory of multi M5-branes. One particular reason is that it is difficult to construct an action: the theory contains a self-dual three-form field strength H µνρ = 1 3! ε µνρσλτ H σλτ , implying that the kinetic term H µνρ H µνρ vanishies.
Fortunately, it is possible to construct the equations of motions and the laws of supersymmetry transformations of 6D (r, 0) theories. Here r = 1, 2. Using a three-algebra approach, Lambert and Papageorgakis (LP) was able to derive a nonabelian (2, 0) tensor multiplet theory [5], which may be a candidate of the gauge description of multiple M5-branes (for reviews on gauge theories of M-branes, see [6] and [7]). More recently, using the Nambu three-algebra, Lambert and Sacco (LS) have constructed a more general (2, 0) theory by introducing an additional non-dynamical abelian three-form into the LP theory [8]. Remarkably, upon a dimension reduction, the LS theory is reduced to the 3D N = 8 BLG theory, describing two M2-branes in C 4 /Z 2 . Thus the (2, 0) LS theory may JHEP05(2018)185 be a dual gauge theory for two M5-branes or two M2-branes. The LS theory has been investigated in ref. [9], and an intereting solution was found in [9].
In this paper, we generalize the (2, 0) LP theory in another direction. We construct a 6D nonabelian N = (1, 0) theory by coupling a "minimal" N = (1, 0) tensor multiplet to an N = (1, 0) hypermultiplet. The "minimal" (1, 0) tensor multiplet, constructed in our previous work [10], is in the adjoint representation of the gauge group, but the (1, 0) hypermultiplet, can be in the fundamental representation or any other representation. The field content of theory is the same as that of the LP theory, but the R-symmetry is only SU (2). If the (1, 0) hypermultiplet also takes value in the adjoint representation, then the SU(2) R-symmetry can be promoted to SO (5), and the supersymmetry gets enhanced to (2,0), and our theory becomes identical to the (2, 0) LP theory. However, if the hypermultiplet is not in the adjoint representation of the gauge group, our theory is a real (1, 0) theory. In fact, if the tensor multiplet and hypermultiplet are in different representations, it is impossible to promote the SU(2) R-symmetry to SO (5), meaning that one cannot enhance the (1, 0) supersymmetry to (2,0). 1 Following the method of [5], we show that this (1, 0) theory can be reduced to a general 5D supersymmetric Yang-Mills (SYM) theory with 8 supersymmetries, by choosing the space-like vector vev C µ = g 2 YM δ µ 5 . Here C µ is an auxiliary field, and g YM the coupling constant of the supersymmetric Yang-Mills theory. In section 5, we discuss some other cases with C µ being a light-like or a time-like vector. It would be interesting to investigate these SYM theories.
Our paper is organized as follows. In section 2, we review the "minimal" (1, 0) tensor multiplet theory of our previous work [10]; in section 3, we construct the 6D (1, 0) theory by coupling a (1, 0) hypermultiplet theory to this (1, 0) tensor multiplet theory. In section 4, we derive the (2, 0) LP theory by enhancing the supersymmetry from (1, 0) to (2,0). In section 5, we construct the action of the N = 1 SYM theory in 5D, by setting C µ = g 2 YM δ µ 5 in the (1, 0) theory; we also briefly discuss the applications of these theories to M5-branes. In appendix A, we verify the closure of the superalgebra of the minimal (1, 0) tensor multiple theory. In appendix B, we prove that the set of equations of motion of the 6D (1, 0) theory are closed under supersymmetry transformations. In appendix C, we construct the conserved supercurrents and discuss the possibilities for enhancing the Poincare supersymmetries to the full superconformal symmetries.

Review of the minimal (1, 0) tensor multiplet
In this section, we first review the 6D nonabelian (1, 0) tensor multiplet theory 2 constructed in section 2 of [10]. We then recast it such that the SU(2) R-symmetry is manifest. 1 In our previous work [10], only the "minimal" (1, 0) tensor multiplet theory is a genuine (1, 0) theory (see section 2 of [10]). After coupling to the hypermultiplet, which is also in the adjoint representation of the gauge group, the resulted (1, 0) tensor multiplet theory in ref. [10] is not a real (1, 0) theory with SU(2) R-symmetry. A careful analysis can show that the parameter "b" in ref. [10] can be absorbed into the re-definition of the fields, and the theory turns out to be the (2, 0) LP theory; in other words, it is actually a re-derivation of (2, 0) LP theory using a different approach. 2 We also call it a "minimal" (1, 0) tensor multiplet theory. After coupling to the (1, 0) hypermultiplet theory, it will be called a nonabelian (1, 0) theory.
The supersymmetry transformations are 3) The self-dual field strength is defined as H µνρ = 3∂ [µ B νρ] . The supersymmetry parameter 3 is chiral with respect to Γ 012345 as well as Γ 6789 , i.e., Γ 012345 = , The super-poincare algebra is closed by imposing the equations of motion (EOM) However, due to the self-duality nature of H µνρ , it is difficult to construct a Lagrangian. The reason is as follows: the kinetic term H µνρ H µνρ is proportional to which vanishes by the self-duality conditions. Here ε 012345 = −ε 012345 = 1.

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One can generalize the above free (1, 0) tensor multiplet to be the nonabelian one [10], (In ref. [10], the fermionic field is denoted as ψ m+ .) Here m is an adjoint index of the Lie algebra of gauge group, and k mn is an invariant form on the Lie algebra. If the Lie algebra is semi-simple, then k mn is nothing but the Killing-Cartan metric, whose inverse will be denoted as k mn . We will use k mn to lower indices, and use its inverse k mn to raise indices; for instance, φ m = k mn φ n . The the components of the field strength H µνρm also obey the self-dual conditions After introducing the nonabelian gauge symmetry, the law of supersymmetry reads [10]: where C µ is an abelian auxiliary field, and f np m the structure constants of the Lie algebra of the gauge group. The covariant derivative is defined as follows The equations of the nonabelian (1, 0) theory are given by [10] The field strength F m µν is defined as The supersymmetry transformations (2.9) are closed, provided that the equations (2.11) are obeyed.

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In the basis (2.13), the reality condition (Majorana condition) reads We denote the inverses of the anti-symmetric forms ȦḂ and AB as ḂĊ and BC , respectively, satisfying ȦḂ ḂĊ = δȦ C and AB BC = δ A C . Now the reality condition (2.21) is equivalent to SU(2) Majorana condtion (2.23) Similarly, the 10D Majorana-Weyl spinor Σ can be converted into the SU(2) simplectic Majorana spinor αA , i.e. Σ → αA . Here A obeys the reality and chirality conditions: and the equations (2.11) can be recast into It is well know that the gauge field of the N = 6 ABJM theory [4] is non-dynamical. Here the gauge field A m µ is also non-dynamical. If it were a dynamical field, its super-partner (gaugino) would be also an independent dynamical field. However, the third equation of (2.25) indicates that the gaugino can be expressed in term of the fermionic field χ m of the tensor multiplet and the auxiliary field C µ . So the gaugino is just an auxiliary field. In other words, the gaugino is non-dynamical.

Closure of the N = (1, 0) superalgebra
We begin by presenting a quick review of the free theory of hypermultiplet. The supersymmetry transformations are given by (3.1) Here A satisfies the reality and chirality conditions (2.24), and A = 1, 2 is a fundamental index of the R-symmetry group SU (2). The fermionic field ψ is a 6D Weyl spinor, and it is anti-chiral with respect the 6D chirality matrix, i.e.
The super-Poincare algebra is closed provided the equations of motion are satisfied.
To couple the hypermultiplet and the tensor multiplet, it is natural to assume that they share the same gauge symmetry. Recall that the tensor multiplet constructed in the last section is in the adjoint representation of the Lie algebra of gauge group. However, it is not necessary to assume that the hypermultiplet is also in the adjoint representation. Instead, we assume that the hypermultiplet can be in the arbitrary representation of the JHEP05(2018)185 gauge group; in particular, it can be in the fundamental representation of the gauge group. 5 With this understanding, the component fields of the nonabelian hypermultiplet can be written as where I labels an arbitrary representation of the Lie algebra of gauge symmetry. The complex conjugation of φ A I will be denoted asφ I A , i.e.φ I A = (φ A I ) * . The covariant derivative is defined as where τ mJ I are a set of representation matrices of the generators of the gauge group, and A µm = k mn A n µ . To ensure the positivity of the theory, we assume that τ mJ I obeys the reality condition: We postulate the law of supersymmetry transformations as follows whereφ BJ = ε BAφJ A and φ AI = AB φ B I , and a 1 , b 1 , d 1 , and d 2 are real constants, to be determined later.
We now check the closure of the super-Poincare algebra. The supersymmetry transformation of the scalar field φ m is (3.8) 5 We emphasize this point because the matter fields of the N = 6 ABJM theory are also in the bifundamental representation of the gauge group U(N )×U(N ). In fact, to achieve enhanced supersymmetries (N ≥ 4), the Lie algebras of gauge groups of 3D Chern-Simons matter theories must be chosen as the bosonic parts of certain superalgebras, and the matter fields matter fields must be in the fundamental representations of these Lie algebras. However, here the Lie algebra of the gauge group of the (1, 0) theory can be arbitrary, not necessarily restricted to the bosonic part of some superalgebra. It would be interesting to study the Lie algebras of gauge groups and the corresponding representations for both 3D and 6D theories.

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where The transformation on the scalar field φ A I is with v µ defined by (3.9). Later we will see that the second term of the right-hand side of (3.10) is a gauge transformation.
Let us now look at the gauge field: We see that the second term is a gauge transformation by the parameter Λ m . Requiring the second term of (3.10) to be a gauge transformation determines the constant b 1 : Also, since [Λ, φ] m = 0, eq. (3.10) can be written in the desired form: To close the super-poincare algebra on the gauge field, we must require the last two terms of (3.12) to vanish separately. This determines the equations of motion for the gauge fields and the constraint equation on the scalar fields φ m : Taking a super-variation on the above equation gives The supersymmetry transformation of the fermionic field ψ I is given by where Λ J I = Λ m τ mJ I . The first line of (3.18) is the translation and the gauge transformation. So the second line must be the equations of motion

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In deriving (3.18), we have used the Fierz identity Here Γ = Γ 012345 is the chirality matrix of SO(5, 1); and obeying the duality conditions The above two equations are special cases of the identity The transformation on the fermionic fields χ Am is given by and B −1 is the inverse of B, defined by the second equation of (2.22), with the subscript "6D" omitted. The third and fourth lines of (3.24) must vanish separately, since they contain the set of unwanted parameters v µνρ (AB) . We are thus led to

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Substituting (3.27) into (3.24), a short calculation gives The first line of (3.28) is a covariant translation and a gauge transformation. In order to close the super-Poincare algebra, one must require the last line of (3.28) to vanish, which are the equations of motion of χ Am . After some algebraic steps, we obtain the super-variation of the self-dual field strengths: The second line vanishes by the equations of motion of the gauge fields (3.15); in order to close the superalgebra on H µνρm , the last four lines must also vanish. This gives the constraint equations on the scalar fields and the equations of motion of H µνρm : Taking super-variations on (3.31), one obtains The Bianchi identity D [µ F νρ]m = 0 and eqs. (3.15) and (3.32) imply that The equations of motion of φ A I and φ p can be derived by taking super-variations on eqs. (3.19) and (3.29), respectively (for details, see appendix B). They are given by We see that a 1 cannot be fixed by the closure of superalgebra. However, if a 1 = 0, it can be absorbed into the redefinitions of the hypermultiplet fields: One can of course keep this continuous free parameter a 1 in the (1, 0) theory. If a 1 = 0, the theory is reduced to the minimal (1, 0) tensor multiplet theory of section 2. It would be interesting to investigate the physical meaning of this continuous free parameter a 1 .

Summary of the nonabelian (1, 0) stheory
In summary, the equations of the (1, 0) theory are given by Here a 1 has been absorbed into the redefinitions of the fields (see (3.37)). And the law of supersymmetry transformations are as follows We have verified that the set of equations (3.38) are closed under the supersymmetry transformations (3.39): taking a super-variation on any equation of (3.38) can transform it into some other equations of (3.38). For instance, if we take a super-variation on the first equation of (3.38) (the EOM of φ A I ), we will obtain the equations of motion of the gauge fields A m µ , and spinor fields χ Am and ψ I . In other words, under supersymmetry transformations (3.39), The details are presented in appendix B. It would be interesting to re-construct the theory using a superspace approach [11].

Enhancing to (2, 0) LP theory
In this section, we will promote the (1, 0) theory to the (2, 0) LP theory [5]. Recall the (1, 0) tensor multiplet is in the adjoint representation of the gauge group, while the (1, 0) hypermultiplet can be in arbitrary representation. To promote the supersymmetry to (2, 0), it is necessary that the (1, 0) tensor multiplet and hypermultiplet are in the same representation of the gauge group. We are therefore led to require that the hypermultiplet is also in the adjoint representation, i.e.
where n is an adjoint index of the Lie algebra of the gauge group. Accordingly, the representation matrices should be the structure constants, i.e.
which also obey the reality condition (3.6). Now we are ready to enhance the SU(2) R-symmetry to USp(4) ∼ = SO(5). To do so, let us define B −1 is the inverse of B, and B is defined by the second equation of (2.22), with "6D" omitted. It is not difficult to check that ψȦ n obeys the reality conditions

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Here ȦḂ is defined by the last equation of (2.22). It can be seen that ψȦ n transforms in the dotted representation of SU(2) × SU (2). Now it is possible to combine ψȦ n and χȦ n to form a 4 of USp(4) ∼ = SO(5): 6 where in left hand side, A = 1, . . . , 4 is a fundamental index of USp(4); and in the right hand side, A = 1, 2 andȦ =1,2 are un-dotted and dotted index of SU(2) × SU(2), respectively; the reality conditions become where ω AB is the invariant antisymmetric tensor of USp(4). The set of scalar fields of the hypermultiplet can be re-arranged such that they transform a 4 of SO (4): where σ s †ȦA = ( σ, −i1 2×2 ), with σ the pauli matrices. And φ s n and φ n can be combined to form a 5 of SO(5): φ a n = (φ s n , φ n ). where we have dropped the subscript "4D". Using equations (4.1)−(4.9), the equations of motion (3.38) can be recast into where A = 1, . . . , 4 is a fundamental index of the R-symmetry group USp(4). We now see that the USp(4) ∼ = SO(5) R-symmetry is manifest. These equations are essentially the JHEP05(2018)185 same equations of motion of the N = (2, 0) LP theory, constructed in terms of Nambu 3-algebra [5]. If we introduce the notation we see that the equations of motion (4.10) are invariant if we switch ψ + Am and ψ − Am : Later we will see that the above discrete symmetry allows us to enhance the N = (1, 0) supersymmetry to N = (2, 0). For convenience, we define two sets of parameters of supersymmetry transformations as follows: (4.14) In the right hand side, A = 1, . . . , 4 is a fundamental index of USp(4), and the right hand side, A = 1, 2 andȦ =1,2 are undotted and dotted index of SU(2) × SU(2). Using equations (4.1)−(4.9), the supersymmetry transformations (3.39) can be rewritten as We see that in (4.15), if we replace A+ by A− , while switch ψ + Am and ψ − Am , that is, we will obtain another independent N = (1, 0) supersymmetry transformations, whose R-symmetry is another SU (2):

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The equations of motion for closing the poincare supersymmetry algebra (4.17) can be simply obtained by applying the discrete transformation ψ + Am ↔ ψ − Am to (4.10). However, since ψ + Am ↔ ψ − Am is just a discrete symmetry of (4.10). So the equations for closing (4.17) are nothing but (4.10). In other words, the theory defined by (4.10) are invariant under the supersymmetry transformations (4.15) and (4.17). Eqs. (4.15) and (4.17) can be unified to give the N = (2, 0) supersymmetry transformations: where A is defined by (4.14). The above law of supersymmetry transformations is essentially the same as that of the N = (2, 0) LP theory [5]. The above (2, 0) supersymmetry transformations (4.18) can be also obtained by re-casting the (2, 0) supersymmetry transformations of [10], using the gamma matrix decompositions in section 2.2. In enhancing the supersymmetry from (1, 0) to (2, 0), the Lie algebra of the gauge group of the theory can still be arbitrary, unlike the 3D N ≥ 4 superconformal Chern-Simons matter whose Lie algebras must be restricted to the bosonic parts of certain superalgebras. In summary, eqs. (4.10) and (4.18), with manifest USp(4) ∼ = SO(5) R-symmetry, are the ordinary Lie 2-algebra version of the N = (2, 0) theory [5].

Relating to 5D SYM
In this section, we will demonstrate that upon dimension reduction, the 6D N = (1, 0) theory in section 3 can be reduced to a general 5D N = 1 SYM theory. Following the idea of ref. [5], we specify the space-like vector vev of C µ as follows where the constant g has dimension −1. Later we will see that it should be identified with g 2 YM [5], i.e. g = g YM , where g YM is the coupling constant of the 5D SYM theory. Using (5.1), the equations of motion of gauge fields (the third equation of (3.38)) are decomposed into where α, β = 0, 1, . . . , 4. The second equation says that

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So A 5 is a flat connection. We may set A 5 = 0 at least locally, leading to Namely, the gauge connection A β is independent of the fifth coordinate x 5 . Also, substituting (5.1) into the last line of (3.38), we find that all other fields are also independent of the fifth coordinate x 5 : For convenience, we define the SO(4, 1) gamma matrices as follows where the Γ-matrices are the 4 × 4 matrices defined by (2.17). Using (2.17), one can check that the set of gamma matrices γ α obeys the Clifford algebra Applying (5.1) to the rest equations of (3.38), and taking account of (5.5), it is natural to identify the 4-component Weyl spinor fields (iΓ 5 χ Am ) 6D with the spinor fields (χ Am ) 5D .
(We have used "6D" and "5D" to indicate the dimensions of the corresponding spacetimes.) Specifically, We have used where (Γ 5 ) 4×4 is the gamma matrix defined in (2.17). Similarly, applying (5.1) and (5.5) to (3.38), it is possible to identify the 4-component Weyl spinor fields i(Γ 5 ψ I ) 6D with the spinor fields (ψ I ) 5D . In summary, The above equations are also in accordance with (5.6). Without causing confusion, we will drop the subscript "5D" of the spinor fields as we formulate the 5D SYM theory in the following paragraphs. The 5D spinor fields χ Am obey the reality conditions where the 4 × 4 matrix B is defined as

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Using (5.1)−(5.10), we are able to reduce the 6D equations (3.38) into the set of 5D equations of motion: To formulate an action, we set and re-scale the fields as follows while leave the gauge field A α unchanged, i.e. A α → A α . The action of the 5D SYM theory with 8 supersymmetries is given by All equations of motions in (5.12) can be derived as Euler-Lagrange equations from the above action, and one can restore the continuous parameter a 1 by using (3.37). Using (5.1)-(5.10), one can reduce the law of supersymmetry transformations (3.39) into

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The action (5.15) is invariant under the above supersymmetry transformations. If (4.1) and (4.2) are satisfied, i.e., if the scalar fields φ A I and fermion fermionic fields ψ I are also in the adjoint representation of gauge group, we expect that the N = 1 supersymmetry is enhanced to N = 2, and theory is promoted to be the maximum supersymmetric Yang-Mills theory in 5D.
We now consider the possibility that C µ is a light-like vector. In refs. [5,8], it was argued that if one uses the null reduction C µ = g (1, 0, . . . , 0, 1), i.e. C µ is a light-like vector, the (2, 0) theory can be used to describe a system of M5branes. So it is natural to expect that this (1, 0) theory may be also used to describe multiple M5-branes [12]. It would be interesting to explore this special (1, 0) theory further.
In particular, it would be interesting to introduce an additional abelian 3-form field into this (1, 0) theory (like Lambert and Sacco did in their work [8]), and see that whether the theory can be reduced to some 3D superconformal Chern-Simons matter theory or not.
The variation of the scalar fields reads It can be seen that the right-hand side of (A.2) is a covariant transformation. Let us now consider the gauge fields. After some algebraic steps, one obtains The first term of (A.4) is the covariant translation, while the second term is a gauge transformation. The second term and third term of (A.4) must be the equations of motion: A super-variation on 0 = C ν D ν φ m gives

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By the definition of Λ (see (A.5)), we see that [Λ, φ] = 0. So equation (A.2) can be recast into the expected form We now check the closure on the fermionic fields. A lengthy calculation gives Clearly, the last two terms must be the equations of motion for the fermions: In computing (A.10), we have used the Fierz identity (3.20). As observed in [5], the equations of motion of the fermions (A.11) can be also derived by requiring δH µνρm to obey the self-dual conditions As for the auxiliary field C µ , we have [δ 1 , δ 2 ]C µ = 0. On the other hand, we expect However, since C µ is not "charged" by the gauge group, we must have [Λ, C µ ] = 0, leading to 14) i.e. C µ is a constant field. Finally, we compute the super-variations of the tensor fields: The second line vanishes by equation (A.6); the third line turns out to be the equations of motions for the tensor fields: Combining the Bianchi identity D [µ F m νρ] = 0 and the equations of motion F m νρ = H m νρλ C λ (see (A.6)), we learn that D [µ H m νρ]λ C λ = 0, which is equivalent to

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However, the first term vanishes by the equations of motion (A.16). We therefore have the constraint equation: The above equation implies that which can be also derived by using the Bianchi identity D [µ F m νρ] = 0 and the equations of motion F m νρ = H m νρλ C λ (see (A.6)). Taking a super-variation on the equations of motion for the fermions one obtains the equations of motion (A.6) and (A. 16), and the equations of motion for the scalar fields: First of all, we have already learned that the super-variation (See also (A.7) and (A.8).) Taking a super-variation on the above equation, By F ρµ = H ρµν C ν (see the third equation of (3.38)), the first term of the first line vanishes; the rest terms are the following constraint equations

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As for the self-dual field strength, we have Since Γ λσ C λ C σ = 0, the first term of the first line of (B.6) vanishes; using 0 = C λ D λ χ Am = C λ F λρ , the second term of the first line vanishes; using 0 = C λ D λ χ Am = C λ D λ φ p , the third term of the first line vanishes; the terms of the second line are the constraint equations: We now calculate the super-variation of the constraint equation for the scalar fields φ A I : Since Γ λν C λ C ν = 0, the first term of (B.8) vanishes; the second term is nothing but 0 = C λ D λ ψ I . By the reality conditionφ I A = (φ A I ) * , the equation 0 = δ(C λ D λφ I A ) must be also satisfied.
We now turn to the constraint equation for fermionic fields ψ I , The first term of (B.9) vanishes due to that Γ λν C λ C ν = 0; by 0 = C λ D λ φ A I = C λ F λµ = C λ D λ φ m , we see that the rest terms of (B.9) also vanishes. Because of the reality condition ψ I = (ψ I ) * , the equation 0 = δ(C λ D λ ψ I ) must also hold.
Let us consider the super-variation of the equations of motion for the fermionic fields χ Am : It can be seen that the second and third lines are the equations of motion for the tensor fields, while the last line is the equations of motion for the gauge fields. So the first line

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must be the equations of motion for the scalar fields φ m . In deriving (B.11), we have used the constraint equations 0 = C µ D µ φ m = C µ D µφ I A = C µ D µ φ AI . We now study the supersymmetry transformation of the equations of motion for the fermionic fields ψ I : The result is The first line is the equations of motion for the gauge fields. So the second and third lines must be the equations of motion for scalar fields φ A I . In deriving (B.13), we have also used The super-variation of the equations of motion for the gauge fields is given by which is equivalent to i.e. the constraint equations for the fermionic fields χ m A . Under supersymmetry transformations (3.39), the 6th equation of (3.38) becomes The above equation is complicated, hence it is not easy to verify it directly. Our strategy is to take care of a simpler version of (B.16) first: without coupling to the matter fields, the equations of motion of H µνρ are given by fourth equation of (2.26). Under the supersymmetry transformations (2.25), the fourth equation of (2.26) should obey 7 Note that the third term of the third line of (B.18) cancels the second term of the fourth line; we group the rest terms of (B.18) as follows Using the self-dual conditions (2.8), the last term of the first line of (B.19) can be rewritten as The above two terms cancel the first term of the first line of (B.19), so the first line of (B. 19) vanishes.
Using C λ D λ χ Am = 0, the first term of the second line of (B.19) can be written as So the second line of (B.19) becomes (B.22) The above expression is zero by the equations of motion for the fermions χ Am (see the third equation of (2.26)). To see this, we multiply the third equation of (2.26) by Γ ν , Relabeling the indices (ν → λ and µ → κ), and multiplying both sides by i 4 ε µνρσλτ C τ , eq. (B.24) becomes Multiplying (B.25) by¯ A , and then taking commutator with φ, the right hand side turns out to be exactly the same as (B.22), so (B.22) must vanish.
Using C λ D λ φ p = 0, one can show that the last line of (B.19) also vanishes. This finishes the proof of (B.17).
We are ready to verify (B.16). We begin by proving three important equations which are useful in verifying (B.16). In exactly the same way for deriving (B.25), we multiply the fifth equation of (3.38) by i 4 ε µνρσλτ C τ Γ ν ; the result is As a check, if we set ψ I = 0, then (B.26) is reduced to (B.25). Multiplying (B.26) by¯ A , and then taking commutator with φ, we obtain Notice that the first line of (B.27) is exactly the same as (B.22). This is expected, since now the tensor multiplets are coupling with the hypermultiplets. Similarly, multiplying the EOM of ψ I or the fourth equation of (3.38) The conjugate equation of (B.29) is

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We now try to calculate (B.16). Taking account of the relation of (3.39) and (2.25), we find that under supersymmetry transformations (3.39), (B.16) becomes where "δ" and "δ" refer to the super-variations in (2.25) and (3.39), respectively, and Notice that the first three lines of (B.25) are nothing but (B.18). Using the results for proving (B.18) and (B.19), and using eq. (B.27), the first three lines of (B.31) turn out to be Plugging (B.33) into (B.31), and using (3.39) and (B.32), we obtain Substituting (B.29) and (B.30) into (B.34), one obtains In the first line, one can use the reality condition (2.24) to write¯ A Γ λη B −1 ψ J † φ AI as −ψ J Γ λη A φ A I ; then, using the commutator τ J m I τ I n K − τ J n I τ I m K = f mn p τ J p K , it is easy to prove that the right hand side of (B.35) vanishes. This finishes the proof of (B.16).
We now consider the super-variation on the second equation of (3.38): (B.36) We shall use the same trick for verifying (B.16): without coupling to the matter fields ψ I and φ A I , eq. (B.36) is reduced to In the second line of the above equation, "δ" refers to the supersymmetry transformation (2.25). Using the constraint equations C ν D ν φ n = 0 and F µν = H µνρ C ρ (see (2.26)), one can simplify (B.37) to give To prove that (B.38) vanishes, let us look at the EOM of χ Am (see (2.26)),

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To prove above equation, we consider the following EOM (see the fourth equation of (3.38)): Multiplying the above equation by C ν Γ ν , and using C µ D µ ψ I = 0 (see the last line of (3.38)), we obtain Substituting (B.48) into the first line of (B.46), a straightforward calculation shows that the right-hand side of (B.46) vanishes. This finishes the calculation of (B.36).
We now try to calculate the super-variation of the first equation of (3.38): Substituting the supersymmetry transformations (3.39) into (B.49), and after some work, (B.49) reads The first line of (B.50) is related to the EOM of χ n B . Multiplying the EOM of χ n B (see the fifth equation of (3.38)) by (φ A J τ J n I C ν )i¯ B Γ ν , we have

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Using the reality condition (2.24), we obtain (¯ B B −1 ψ K † )φ BL = (ψ K B )φ B L ; on the other hand, we have C µ D µ χ n B = 0 (see the last line of (3.38)). Using these two equation, one can convert equation (B.51) into the form The second line of (B.50) can be taken care of by using the EOM of ψ I . Multiplying the EOM of ψ I (see the fourth equation of (3.38)) by i¯ A Γ ν D ν , Simplifying the above equation gives One can also take care of the third term of the right-hand side of (B.54) using the EOM of ψ I . Multiplying the EOM of ψ I (see the fourth equation of (3.38)) by i¯ A Γ µ C µ , which can be written as We have used C µ D µ ψ I = 0 (see the last line of (3.38)). Substituting (B.52) and (B.54) into (B.50), and using (B.56), a straightforward computation shows that (B.50) does vanishes. This complete the calculation of (B.49).
In summary, the super-variation of every EOM vanishes. In other words, eq. (3.40) is obeyed.

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It is straightforward to verify that the current is conserved, using equations of motion and Fierz identities. By adding three total derivative terms, one can define the following modified current Here α 1 , α 2 , and α 3 are constants. The physics remains the same, since ∂ µj µA = ∂ µ j µA = 0 and the total derivative terms do not contribute to the supercharges. Ifj µA were "Γtraceless", i.e., Γ µj µ A = 0, it would be possible to define the conserved superconformal current [23] s µ A = Γ · xj µ A ; (C.4) In fact, one can easily verify that A short calculation gives the "Γ-trace" ofj µA , If we set hypermultiplet fields ψ I = φ I A = 0 (C.7) by setting a 1 = 0 in (3.37), the right-hand side of (C.6) vanishes, i.e., Γ µj µ A = 0, and it is possible to construct a superconformal current s µ A , defined by (C.4); as a result, the minimal (1, 0) tensor multiplet theory of section 2 may have a superconformal symmetry.

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We now consider the possibility of constructing a superconformal current s µ 3alg by imposing an additional constraint on the fields. In (C.18), if we set α = 4/5, and assume that the 3-bracket ψ m [C ν , φ a , φ b ] m = ψ m C ν o φ a n φ b p f onp m = 0, (C. 19) or at least that the 3-bracket annihilates the physical states, i.e., ψ m [C ν , φ a , φ b ] m |phy = 0, then we have Γ µj µ 3alg = 0 or Γ µj µ 3alg |phy = 0. As a result, it is possible to construct the conserved superconformal current s µ 3alg , and the N = (2, 0) LP theory may have a superconformal symmetry. It would be interesting to investigate the physical significance of the additional constraint (C. 19).
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