A 6D nonabelian (1, 0) theory

Journal of High Energy Physics, May 2018

Abstract We construct a 6D nonabelian \( \mathcal{N}=\left(1,\ 0\right) \) theory by coupling an \( \mathcal{N}=\left(1,\ 0\right) \) tensor multiplet to an \( \mathcal{N}=\left(1,\ 0\right) \) hypermultiplet. While the \( \mathcal{N}=\left(1,\ 0\right) \) 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 \( \mathcal{N}=\left(2,\ 0\right) \), 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 \( \mathcal{N}=1 \) supersymmetric Yang-Mills theory in 5D. We discuss briefly the possible applications of the theories to multi M5-branes.

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A 6D nonabelian (1, 0) theory

HJE nonabelian (1, 0) theory Fa-Min Chen 0 1 2 0 Conformal Field Models in String Theory , Extended Supersymmetry 1 No. 3 Shang Yuan Cun , Hai Dian District, Beijing, 100044 P.R. China 2 Department of Physics, Beijing Jiaotong University We construct a 6D nonabelian N = (1; 0) theory by coupling an N = (1; 0) tensor multiplet to an N = (1; 0) hypermultiplet. While the N = (1; 0) 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), 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 supersymmetric Yang-Mills theory in 5D. We discuss brie y the possible applications of the theories to - A M-Theory 1 Introduction and summary 2 Review of the minimal (1; 0) tensor multiplet 2.1 2.2 (1; 0) tensor multiplet with manifest SU( 2 ) R-symmetry 3 Nonabelian (1; 0) theory 3.1 3.2 Summary of the nonabelian (1; 0) stheory 4 Enhancing to (2; 0) LP theory 5 Relating to 5D SYM A Closure of the (1; 0) tensor multiplet superalgebra B Super-variations of the equations of motion of the (1; 0) theory C Supercurrents C.1 Supercurrent of N = (1; 0) theory C.2 Supercurrent of N = (2; 0) LP theory 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 di cult to construct the gauge theory of multi M5-branes. One particular reason is that it is di cult to construct an action: the theory contains a self-dual three-form eld strength H the kinetic term H H 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 C4=Z2. Thus the (2; 0) LS theory may = 31! " H , implying that { 1 { 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 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 eld 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 di erent 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 2 space-like vector vev hC i = gYM 5 . Here C is an auxiliary eld, and gYM the coupling constant of the supersymmetric Yang-Mills theory. In section 5, we discuss some other cases with hC i 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 hC i = 2 gYM 5 in the (1; 0) theory; we also brie y 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. 2 Review of the minimal (1; 0) tensor multiplet In this section, we rst review the 6D nonabelian (1; 0) tensor multiplet theory2 constructed in section 2 of [10]. We then recast it such that the SU( 2 ) R-symmetry is manifest. 1In 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 2We 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. { 2 { Following the convention of [10], we will rst work with 32-component Majorana fermions. (More precisely, we will work with SO(9; 1) Majorana fermions.) The gamma matrices satisfy the anti-commutation relations f ; g = 2 f s; tg = 2 st; ; f s; is anti-chiral with respect to 012345, but chiral with respect (2.2) (2.3) (2.4) (2.5) (2.6) is a Weyl spinor. Recall that we assumed that is an SO(9; 1) Majorana spinor, so is an SO(9; 1) Majorana-Weyl The supersymmetry transformations are 012345 6789 = = : 0123456789 ; = = = i ; + is chiral with respect to 012345 as well as 6789, i.e., The super-poincare algebra is closed by imposing the equations of motion (EOM) ] = 0: However, due to the self-duality nature of H , it is di cult to construct a Lagrangian. The reason is as follows: the kinetic term H is proportional to 012345 = ; 6789 = : " H ; H H { 3 { which vanishes by the self-duality conditions. Here "012345 = 3In ref. [10], the supersymmetry parameter is denoted as +, which is a 10D Majorana-Weyl spinor. H m = 1 3! " Hm : m = m = Am = i C = 0; i m; D m + mC ; where C is an abelian auxiliary eld, and f npm the structure constants of the Lie algebra of the gauge group. The covariant derivative is de ned as follows D The equations of the nonabelian (1; 0) theory are given by [10] One can generalize the above free (1; 0) tensor multiplet to be the nonabelian one [10], ( m; H m; m): (In ref. [10], the fermionic eld is denoted as m+.) Here m is an adjoint index of the Lie algebra of gauge group, and kmn is an invariant form on the Lie algebra. If the Lie algebra is semi-simple, then kmn is nothing but the Killing-Cartan metric, whose inverse will be denoted as kmn. We will use kmn to lower indices, and use its inverse kmn to raise indices; for instance, m = kmn n. The the components of the eld strength H m also obey the self-dual conditions After introducing the nonabelian gauge symmetry, the law of supersymmetry reads [10]: The eld strength F m is de ned as The supersymmetry transformations (2.9) are closed, provided that the equations (2.11) are obeyed. { 4 { where s = (~ ; i12 2) and sy = (~ ; i12 2), with ~ the pauli matrices. And where A; B = 1; 2 and A_ ; B_ = 1_; 2_ are the undotted and dotted indices of SU( 2 ) SU( 2 ), respectively. The 8 8 gamma matrices ( )6D are de ned as Using equations (2.14) (2.18), we see that (2.13) indeed satisfy the commutation relations (2.1). Equations (2.13) are essentially the decomposition: SO(9; 1) ) SO(5; 1) SU( 2 ) SU( 2 ). Equations (2.1), (2.2), and (2.13) suggest that the SO(9; 1) Majorana-Weyl fermion can be converted into an SU( 2 ) symplectic-Majorana chiral spinor A: where labels the Majorana-Weyl representation of SO(9; 1), and labels the Weyl representation of SO(5; 1), more precisely, and A = 1; 2 is a fundamental index of the SU( 2 ) R-symmetry group. In the basis (2.13), the reality condition (Majorana condition) reads Using (2.13), (2.19), (2.23), and (2.23), the law of supersymmetry transformation (2.9) can be recast into the form4 ( Am) = ABB6D Bm: and the equations (2.11) can be recast into (2.22) 4In appendix B, the super-variation \ " in (2.25) will be replaced by \ ", while the super-variation in (3.39) will be still denoted as \ ". { 6 { where the gamma matrices in (2.25) and (2.26) are de ned by (2.16), and we have dropped the subscript \6D", i.e. ( )6D ! we rederive equations (2.26) by requiring the closure of the super Poincare algebra. It is well know that the gauge eld of the N = 6 ABJM theory [4] is non-dynamical. Here the gauge eld Am is also non-dynamical. If it were a dynamical eld, its super-partner (gaugino) would be also an independent dynamical eld. However, the third equation of (2.25) indicates that the gaugino can be expressed in term of the fermionic eld the tensor multiplet and the auxiliary eld C . So the gaugino is just an auxiliary eld. m of In other words, the gaugino is non-dynamical. 3 Nonabelian (1; 0) theory In this section, we will construct the nonabelian N = (1; 0) theory by coupling the N = (1; 0) tensor multiplet theory to an N = (1; 0) hypermultiplet theory. 3.1 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) (3.2) (3.3) Here A satis es the reality and chirality conditions (2.24), and A = 1; 2 is a fundamental index of the R-symmetry group SU( 2 ). The fermionic eld 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 and A = 0 are satis ed. 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, Instead, we assume that the hypermultiplet can be in the arbitrary representation of the A = i A ; = A : 012345 = : { 7 { gauge group; in particular, it can be in the fundamental representation of the gauge group.5 With this understanding, the component elds of the nonabelian hypermultiplet can be written as is de ned as where I labels an arbitrary representation of the Lie algebra of gauge symmetry. The mJ I are a set of representation matrices of the generators of the gauge group, and A m = kmnAn. To ensure the positivity of the theory, we assume that mJ I obeys the reality condition: We postulate the law of supersymmetry transformations as follows ( IA; I ); D A I mJ I A m J ; A ( mJ I ) = mI J : AHm + a1 AC mJ A I m J ; BC ( JA IB + BJ AB IB, and a1, b1, d1, and d2 are real constants, to be determined later. We now check the closure of the super-Poincare algebra. The supersymmetry transformation of the scalar eld m is m; (3.8) 5We emphasize this point because the matter elds 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 elds matter elds 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. { 8 { The transformation on the scalar eld IA is v 2i 2A 1A: [ 1; 2] IA = v D IA + m with v de ned 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 eld: We see that the second term is a gauge transformation by the parameter m. Requiring the second term of (3.10) to be a gauge transformation determines the constant b1: Also, since [ ; ]m = 0, eq. (3.10) can be written in the desired form: To close the super-poincare algebra on the gauge eld, we must require the last two terms of (3.12) to vanish separately. This determines the equations of motion for the gauge elds where where (3.9) (3.10) (3.11) and the constraint equation on the scalar elds Taking a super-variation on the above equation gives The supersymmetry transformation of the fermionic eld I is given by I + C mJ I . The rst line of (3.18) is the translation and the gauge transformation. So the second line must be the equations of motion 0 = D I + C In deriving (3.18), we have used the Fierz identity 1A 2B = 1 4 B 2 Here = 012345 is the chirality matrix of SO(5; 1); and ; 1A + : obeying the duality conditions The above two equations are special cases of the identity The transformation on the fermionic elds Am is given by where with 2A = AB 2B; and 1A 1 2 1 4 v 1 8 1 8 1 8 1 8 i 24 (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) J = ( J ) ; and B 1 is the inverse of B, de ned 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(AB). We are thus led to v(AB) 2A 1B + 2B 1A ; d2 = a1 and d1 = a1: D Am + C [ ; A]m + a1 C B 1 J The rst 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, 0 = Substituting (3.27) into (3.24), a short calculation gives [ 1; 2] Am = v D which are the equations of motion of Am. m = v D H After some algebraic steps, we obtain the super-variation of the self-dual eld strengths: (3.28) (3.30) (3.31) (3.33) (3.34) and the equations of motion of H m: ]m 1 4 " i 8 " Taking super-variations on (3.31), one obtains The second line vanishes by the equations of motion of the gauge elds (3.15); in order to close the superalgebra on H m, the last four lines must also vanish. This gives the constraint equations on the scalar elds 0 = C D IB = C D I ; B The Bianchi identity D[ F ]m = 0 and eqs. (3.15) and (3.32) imply that 0 = C D The equations of motion of IA and 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 0 = D2 A I One can of course keep this continuous free parameter a1 in the (1; 0) theory. If a1 = 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 a1. We see that a1 cannot be xed by the closure of superalgebra. However, if a1 6= 0, it can be absorbed into the rede nitions of the hypermultiplet elds: In summary, the equations of the (1; 0) theory are given by 0 = D2 A I 0 = 0 = Here a1 has been absorbed into the rede nitions of the elds (see (3.37)). And the law of supersymmetry transformations are as follows m + I = 2 AD A I AC The details are presented in appendix B. 4 Enhancing to (2; 0) LP theory We have veri ed 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 rst equation of (3.38) (the EOM of IA), we will obtain the equations of motion of the I . In other words, under supersymmetry It would be interesting to re-construct the theory using a superspace approach [11]. 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). let us de ne Now we are ready to enhance the SU( 2 ) R-symmetry to USp(4) = SO(5). To do so, I ! n; A I ! A n ; mI J ! ( m n ) p f nmp; A_n = i p 2 B 1 n n ! ; A_n = A_B_ B A_n: B 1 is the inverse of B, and B is de ned by the second equation of (2.22), with \6D" omitted. It is not di cult to check that A_n obeys the reality conditions (3.39) (3.40) (4.1) (4.2) (4.3) (4.4) where !AB is the invariant antisymmetric tensor of USp(4). form a 4 of SO(4): The set of scalar elds of the hypermultiplet can be re-arranged such that they transsn = i p 2 sy A _ 1 An sy2_ A AB nB ; where sy _ A = (~ ; i12 2), with ~ the pauli matrices. And sn and n can be combined A to form a 5 of SO(5): an = ( sn; n): a B = ( s; 5); A Here a = 1; : : : ; 5 is a fundamental index of SO(5). Similarly, we use the matrices (2.14) and (2.15) to de ne the set of SO(5) gamma matrices where we have dropped the subscript \4D". Using equations (4.1) (4.9), the equations of motion (3.38) can be recast into An = !AB = An A_n ! ; AB 0 0 A_B_ ! ; (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) to form a 4 of USp(4) = SO(5):6 Here A_B_ is de ned by the last equation of (2.22). It can be seen that the dotted representation of SU( 2 ) SU( 2 ). Now it is possible to combine A_n transforms in A_n and A_n where in left hand side, A = 1; : : : ; 4 is a fundamental index of USp(4); and in the right hand side, A = 1; 2 and A_ = 1_; 2_ are un-dotted and dotted index of SU( 2 ) SU(2), respectively; the reality conditions become HJEP05(218) ( An) = !ABB Bn; 0 = D2 a p amD anC f mnp; 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 6To avoid introducing too many indices, we still use the capital letters A; B; : : : , to label the USp(4) indices. We hope this will not cause any confusion. 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 + Later we will see that the above discrete symmetry allows us to enhance the N = (1; 0) For convenience, we de ne two sets of parameters of supersymmetry transformations as follows: where 1 2 A = (1 5 )AB B; (A; B = 1; : : : ; 4) A = A _ A ! : In the right hand side, A = 1; : : : ; 4 is a fundamental index of USp(4), and the right hand side, A = 1; 2 and A_ = 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 H m = 3i A+ [ We see that in (4.15), if we replace A+ by A , while switch + we will obtain another independent N R-symmetry is another SU( 2 ): am = Bm; 1 2 1 2 A+Hm + abB B+C A + The equations of motion for closing the poincare supersymmetry algebra (4.17) can be simply obtained by applying the discrete transformation + Am $ Am to (4.10). However, since + Am $ 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 de ned by (4.10) are invariant under the supersymmetry transformations (4.15) and (4.17). Eqs. (4.15) and (4.17) can be uni ed to give the N = (2; 0) supersymmetry transformations: am = Am = where A is de ned 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]. 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 as follows hC i = g(0; : : : ; 0; 1) = g 5 ; where the constant g has dimension 1. Later we will see that it should be identi ed 2 with gYM [5], i.e. g = gYM, where gYM is the coupling constant of the 5D SYM theory. Using (5.1), the equations of motion of gauge elds (the third equation of (3.38)) are decomposed into F m = gH 5m; F5 m = gH5 5m = 0; where ; = 0; 1; : : : ; 4: The second equation says that (5.1) (5.2) (5.3) So A5 is a at connection. We may set A5 = 0 at least locally, leading to = 0: Namely, the gauge connection A is independent of the fth coordinate x5. Also, substituting (5.1) into the last line of (3.38), we nd that all other elds are also independent of the fth coordinate x5: For convenience, we de ne the SO(4; 1) gamma matrices as follows HJEP05(218) 4 matrices de ned by (2.17). Using (2.17), one can check that the set of gamma matrices obeys the Cli ord 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 elds (i 5 Am)6D with the spinor elds ( Am)5D. (We have used \6D" and \5D" to indicate the dimensions of the corresponding spacetimes.) i( 5 Am)6D = ( Am)5D; i( 5 I )6D = ( I )5D: ( Am) = ABB Bm; B = i 3 2 : (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) Speci cally, We have used i( 5 Am)6D = 0 )4 4 is the gamma matrix de ned in (2.17). Similarly, applying (5.1) and (5.5) to (3.38), it is possible to identify the 4-component Weyl spinor elds i( 5 I )6D with the spinor elds ( 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 elds as we formulate the 5D SYM theory in the following paragraphs. The 5D spinor elds Am obey the reality conditions where the 4 4 matrix B is de ned as Using (5.1) (5.10), we are able to reduce the 6D equations (3.38) into the set of 5D equations of motion: D D 0 = 0 = 0 = D D 0 = D D I Am m A I 1 2 g Am J mJ I + g2 A K m n nK J mJ I +g2( BK LA + AC BD CK LD) JB mLK p + (A )m nf mnp: To formulate an action, we set g = gYM; and re-scale the elds as follows LYM = F m Fm 1 2 gYM + J 1 4 Am g IA ! I ; A g I ! I ; g Am ! A m ; g m ! m ; while leave the gauge eld A unchanged, i.e. A with 8 supersymmetries is given by ! A . The action of the 5D SYM theory i Am 2 D Am 1 2 i I 2 1 Am 2 mI J IA + A m J mJ I JA Ap nf npm m = IA = Am = I = Am = i A i A I ; Am; AD m + i 2 2 A AD m A : A I 2i A (5.12) (5.13) (5.14) (5.16) 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 a1 by using (3.37). Using (5.1){(5.10), one can reduce the law of supersymmetry transformations (3.39) into 1 2 D I + D mD m + D IAD A I 2 1 J I m mI J AFm + i B( JA IB + AC BD J C D I ) mI J ; The action (5.15) is invariant under the above supersymmetry transformations. If (4.1) and (4.2) are satis ed, i.e., if the scalar elds IA and fermion fermionic elds 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 hC i is a light-like vector. In refs. [5, 8], it was argued that if one uses the null reduction i.e. hC i 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 eld 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. Using the three equation of (3.38), one can solve Hm in terms of the eld strength of the gauge eld: hC i = g(1; 0; : : : ; 0; 1); hC ihC i = 0; (5.17) HJEP05(218) Substituting (5.17) into (5.18), we nd that the eld strength obeys the duality condition: which can be decomposed into C2Hm = 3F[m C ] + F m C : 1 2 " 0 = 3F[m C ] + 1 2 " F m C ; F m5 = F m0; F m = 1 2 " F m; ( ; ; ; = 1; : : : ; 4) where "1234 = "1234 = 1. We see that the eld strength F m is anti-selfdual. Let hC i be a time-like vector, namely, hC i = g(1; 0; : : : ; 0): sional spacetime. theories, see [13{22]. Then the elds are covariantly static, that is 0 = D0 m = D0 m = D0 IA = D0 Am = D0 I = D0H According to [5], this theory may be a dual gauge theory for static 5-branes in 11 dimenFor more discussions on M5-branes and 6D (1; 0) and (2; 0) theories and 5D SYM (5.18) (5.19) (5.20) (5.21) (5.22) (5.23) Acknowledgments We are grateful to Jun-Bao Wu and Zhi-Guang Xiao for useful discussions. This work is supported in part by the National Science Foundation of China (NSFC) under Grant No. 11475016, and supported partially by the Ren-Cai Foundation of Beijing Jiaotong University through Grant No. 2013RC029, and supported partially by the Scienti c Research Foundation for Returned Scholars, Ministry of Education of China. A Closure of the (1; 0) tensor multiplet superalgebra In this section, we verify the closure of the poincare superalgebra of the (1; 0) theory of section 2.2, using manifest SU( 2 )-notations. For convenience, we cite the supersymmetry transformations (2.25) here HJEP05(218) m = Am = The variation of the scalar elds reads where where It can be seen that the right-hand side of (A.2) is a covariant transformation. Let us now consider the gauge elds. After some algebraic steps, one obtains The rst 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 D m gives (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) By the de nition of (see (A.5)), we see that [ ; ] = 0. So equation (A.2) can be recast into the expected form We now check the closure on the fermionic elds. 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 H obey the self-dual conditions As for the auxiliary eld C , we have [ 1; 2]C = 0. On the other hand, we expect However, since C is not \charged" by the gauge group, we must have [ ; C ] = 0, leadH m = 1 3! H : ing to i.e. C is a constant eld. (A.11) m to (A.12) (A.13) (A.14) (A.15) (A.17) Finally, we compute the super-variations of the tensor elds: m = v D H The second line vanishes by equation (A.6); the third line turns out to be the equations of motions for the tensor elds: Combining the Bianchi identity D[ F m] = 0 and the equations of motion F m = Hm C (see (A.6)), we learn that D[ Hm] C = 0, which is equivalent to 1 4 1 4 i 8 4 3 i 8 C D[ Hm However, the rst term vanishes by the equations of motion (A.16). We therefore have the constraint equation: The above equation implies that C D Hm = 0: C D F m = 0; ( D A m C [ A; ]m) = 0; i 2 ( nA one obtains the equations of motion (A.6) and (A.16), and the equations of motion for the scalar elds: In summary, the equations of motion (A.6), (A.11), (A.16), and (A.21) are in agreement with (2.11); and the constraint equations (A.7), (A.8), (A.14), (A.18), and (A.19) are exactly the same as the last line of (2.11). B Super-variations of the equations of motion of the (1; 0) theory In this section, we will check that the set of equations of motion (3.38) in section 3 are closed under supersymmetry transformations (3.39). First of all, we have already learned that the super-variation which can be also derived by using the Bianchi identity D[ F m] = 0 and the equations of motion F m = Hm C (see (A.6)). Taking a super-variation on the equations of motion for the fermions (A.18) (A.19) (A.20) (A.21) (B.1) (B.2) (B.3) (B.4) (B.5) gives one obtains 0 = (C D m ) 0 = C D 0 = (C D m A : Am); (See also (A.7) and (A.8).) Taking a super-variation on the above equation, 0 = A[[C F ; ]m + D (C D m)] + 1 12 AC D Hm + BC [C D ( JA IB + BJ AI )] mIJ : C (see the third equation of (3.38)), the rst term of the rst line vanishes; the rest terms are the following constraint equations 0 = C D As for the self-dual eld strength, we have 0 = (C D H = 0, the rst term of the rst line of (B.6) vanishes; using 0 = C D C F , the second term of the rst line vanishes; using 0 = C D Am = C D p, the third term of the rst line vanishes; the terms of the second line are the constraint equations: HJEP05(218) We now calculate the super-variation of the constraint equation for the scalar elds IA : Since also satis ed. = 0, the rst term of (B.8) vanishes; the second term is nothing but I : By the reality condition IA = ( IA) , the equation 0 = (C D A I ) must be We now turn to the constraint equation for fermionic elds I , 0 = C D m)J I JA + C D (i A I ) : (B.6) Am = (B.7) (B.8) (B.11) AmC )( m)J I J + C D ( 2 A D A I 2 AC mJ I m JA) : (B.9) The rst term of (B.9) vanishes due to that C C = 0; by 0 = C D C D m, we see that the rest terms of (B.9) also vanishes. Because of the reality condition IA = C F = I = ( I ) , the equation 0 = (C D I ) must also hold. Let us consider the super-variation of the equations of motion for the fermionic elds Am: 0 = ( D Am + A straightforward calculation gives 0 = A D2 m nD " " C ; ]m: C i 2 ( nA JBD It can be seen that the second and third lines are the equations of motion for the tensor elds, while the last line is the equations of motion for the gauge elds. So the rst line must be the equations of motion for the scalar elds m. In deriving (B.11), we have used the constraint equations 0 = C D We now study the supersymmetry transformation of the equations of motion for the fermionic elds I : 0 = ( D I + C The rst line is the equations of motion for the gauge elds. So the second and third lines must be the equations of motion for scalar elds IA. In deriving (B.13), we have also used The super-variation of the equations of motion for the gauge elds is given by which is equivalent to i.e. the constraint equations for the fermionic elds 0 = (F m Hm C ); 0 = i A (C D Under supersymmetry transformations (3.39), the 6th equation of (3.38) becomes 0 = D[ H ( nA Ap)f npm + ( J pf npm + JBD IB mI J JBD I J B mI : (B.16) 1 4 ]m i 8 C C nD i 8 Am); m. A 1 4 " 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) rst: without coupling to the matter elds, the equations of motion of H are given by fourth equation of (2.26). Under the supersymmetry transformations (2.25), the fourth equation of (2.26) should obey7 0 = 7To distinguish the supersymmetry transformations (2.25) and (3.39), in this appendix, we replace the super-variation \ " in (2.25) by \ ", while the super-variation in (3.39) is still denoted as \ ". 1 4 i A + " 2 + i A rewritten as vanishes. 4 ( nA After verifying the above equation, it will be much easier to verify (B.16), since the proof of the above equation can be used to verify (B.16). Under (2.25), eq. (B.17) reads 3i A [ [F ]; A]m + i A i A HJEP05(218) A; ]m ( nA 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 0 = 3i A [ [F ]; A]m i A 48 C ( nA A)H f npm [ [D ] A; ]mC + " Using the self-dual conditions (2.8), the last term of the rst line of (B.19) can be 2 3i A [ [F ]; A]m + i AC The above two terms cancel the rst term of the rst line of (B.19), so the rst line of (B.19) Am = 0, the rst term of the second line of (B.19) can be written as 0 = D Am C An pf npm); (B.19) (B.20) (B.21) (B.22) Am (see the (B.23) 4 5i A [ [D ] A; ]mC : So the second line of (B.19) becomes The above expression is zero by the equations of motion for the fermions third equation of (2.26)). To see this, we multiply the third equation of (2.26) by D ] AmC + " C D Am Multiplying (B.25) by A, and then taking commutator with , the right hand side turns out to be exactly the same as (B.22), so (B.22) must vanish. p = 0, one can show that the last line of (B.19) also vanishes. This nishes 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 fth equation of (3.38) by 4i " C ; the result is 0 = D Am + D Am + C [ ; A]m + C [ ; A]m; and ), and multiplying both sides by 4i " i 4 (B 1 Jy AI I JA) pf npm nI J : Notice that the rst 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 I or the fourth equation of (3.38) by (C 0 = The conjugate equation of (B.29) is 0 = ( A A D D I + C nK I n K 2 C An nK I AK ; , we are able to derive eq. (B.24) becomes 0 = 5i + " + " JA mIJ ) 4i " i 4 ( A i 4 +( A (B.24) C , (B.25) (B.26) (B.27) (B.28) (B.29) 0 = D ] AmC + " C D Am As a check, if we set I = 0, then (B.26) is reduced to (B.25). Multiplying (B.26) by A and then taking commutator with , we obtain 0 = C 0 = (D I A)C A J We now try to calculate (B.16). Taking account of the relation of (3.39) and (2.25), we nd that under supersymmetry transformations (3.39), (B.16) becomes + " IBD C J B 1 4 C ([D Apf npm C C ( C A J ( 0 Ap)f npm J I mJ ( I ) mIJ ; JBD IB + JB An( nK I ) BK + JBD B I IB An( n)J K BK IBD JB ( m)I J where \ " and \ " refer to the super-variations in (2.25) and (3.39), respectively, and 0 Ap BC ( AK LB + BK AL) pLK : Notice that the rst 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 rst three lines of (B.31) turn out to be Plugging (B.33) into (B.31), and using (3.39) and (B.32), we obtain 4 0 = i 4 + " + " i 4 4 i 4 4 + " C C A I D A)C J I (D (B.31) (B.32) (B.33) HJEP05(218) " + " 4 i 4 ( A ( J ( A ( J I A)C J mI B)C C ( AK LB + BK AL) pLK f npm mK I nI J ) : Substituting (B.29) and (B.30) into (B.34), one obtains HJEP05(218) 0 = C C A I JA) pf npm nI J i AD2 Am Apf npm : i A 2 (B.34) (B.35) (B.36) (B.37) In the rst line, one can use the reality condition (2.24) to write A B 1 Jy AI as J A IA; then, using the commutator mI nI K J n I mIK = fmnp pJ K , it is easy to J prove that the right hand side of (B.35) vanishes. This nishes the proof of (B.16). We now consider the super-variation on the second equation of (3.38): We shall use the same trick for verifying (B.16): without coupling to the matter elds I and IA, eq. (B.36) is reduced to 0 = D2 m 2 2 2 0 = D2 m C ( nB Bp)f npm iC Ay y 0D n + 12 1 Ay y 0Hn A n Apf npm In the second line of the above equation, \ " refers to the supersymmetry transformation (2.25). Using the constraint equations C D n = 0 and F = H C (see (2.26)), one can simplify (B.37) to give i A To prove that (B.38) vanishes, let us look at the EOM of Am (see (2.26)), 0 = D Am + Multiplying the above equation by D , a short calculation gives 0 = D2 Am + Multiplying the above equation by i A, the right-hand side turns out to be exactly the same as (B.38), so (B.38) must vanish. This nishes the proof of (B.37). After coupling with the scalar multiplets, eq. (B.39) becomes (see also the fth equation of (3.38)): 0 = D Am + In exactly the same way for deriving (B.40), we can show that i 2 Using the relation between the supersymmetry transformations (2.25) and (3.39), we can write (B.43) as 0 = [ A ; D ]m + D ([ A ; ]m) + D2( m) A Apf npm iC 0 nA Apf npm where \ " and \ " refer to the super-variations in (2.25) and (3.39), respectively, and in the second line 0 An = BC ( JA IB + BJ AI ) nI J : (B.44) (B.45) 0 = D2 Am + C [(B 1D Jy) AI + B 1 JyD AI D I JA I D JA] mIJ : (B.42) We now begin to calculate (B.36); it can be written as 0 = [ A ; D ]m + D ([ A ; ]m) + D2( m) iC A Apf npm + C [( J I ) + ( J I )] mI J +2C2[( JA) IA n + JA( IA) n + JA IA( n)]( (m n))I J : (B.43) Using the results for proving (B.37), and using eq. (B.42), (B.44) can be converted into 0 = i A C [(B 1D Jy) AI + B 1 JyD AI (D I ) JA I D JA] mIJ iC 2 i 2 A BC ( JA IB + BJ AI ) pI J f npm + C [ J ( 2 + C [( 2 AD AD A I 2 A I 2 AC AC mJ I m JA)] mI J mJ I m JA)y 0 +2C2[(i J A) IA n + JA(i A I ) n + JA IA( i B nB)]( (m n))I J : (B.46) To prove above equation, we consider the following EOM (see the fourth equation of (3.38)): 0 = D I + C Multiplying the above equation by C , and using C D I = 0 (see the last line of (3.38)), we obtain 0 = C D I + C2 mJ I m J 2C2 Am mJ I JA : (B.47) (B.48) Substituting (B.48) into the rst line of (B.46), a straightforward calculation shows that HJEP05(218) the right-hand side of (B.46) vanishes. This nishes the calculation of (B.36). We now try to calculate the super-variation of the rst equation of (3.38): 0 = [D2 A I nJ I )D A iC [( +C2[( I AK BL)( JB) mLK mJ I BK ) LA + BK ( LA) + ( AK ) BL + AK ( BL)] JB mLK mJ I : (B.49) Substituting the supersymmetry transformations (3.39) into (B.49), and after some work, (B.49) reads i( B D nB)C A J J mI +i AD2 I +i( A 2 ( A J )C D m J mI J )F m J mI 2i( B (Bn) m)C2 A K K n J mJI + 2i( Am B)C2 n BK ( n m)K I the fth equation of (3.38)) by ( JA nJ I C )i B , we have The rst line of (B.50) is related to the EOM of nB. Multiplying the EOM of nB (see 0 = ( JA nJ I C )i B nB + n + C B 1 K BL m K C L BK mLK : (B.51) D L Using the reality condition (2.24), we obtain ( BB 1 Ky) BL = ( K B) LB; on the other hand, we have C D nB = 0 (see the last line of (3.38)). Using these two equation, one can convert equation (B.51) into the form (i B D nB)C A J J n I = (B.52) i B BmC2 p JAf pmn nJ I [i( K B) LB JA i( B L) BK JA]C2 L n K nJ I : The second line of (B.50) can be taken care of by using the EOM of I . Multiplying the EOM of I (see the fourth equation of (3.38)) by i A Simplifying the above equation gives i AD2 I = i A 2 2i( A J F m D i A Bm)C B J J C D 2i A m + i( A D J )C BmC D B J mJI : (B.54) One can also take care of the third term of the right-hand side of (B.54) using the EOM of I . Multiplying the EOM of I (see the fourth equation of (3.38)) by i A C , which can be written as i( A D J )C m J mI = 2i A nBC2 B m K i A K C2 n m ( n m)K I : (B.56) We have used C D 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. C C.1 Supercurrents Supercurrent of N = (1; 0) theory The supercurrent of the N = (1; 0) theory of section 3.2 can be de ned as follows, A j A = ic Am Am + ic 2 I I ic I 2 I ; where A = 1; 2 is an SU( 2 ) R-symmetry index, and c is an overall constant. (In the current of the LP theory [18], c = i.) A short calculation gives D , C (C.1) (C.2) j A = ic +ic[ +ic[ AmD I (D B 1 I m I A AB(D C 1 2 AmH m BmC ( JA IB + BJ AI ) mI J m A J mI J )] IB + C m J B mJ I )]: In fact, one can easily verify that 10 3) 5 2 + 5 1) 5 3 + 5 1) If we set hypermultiplet elds s A = x~jA; A = ~jA + B 1 I C m AJ mJ I I C m JA mI J : I = IA = 0 A short calculation gives the \ -trace" of ~j A, j A = (4ic + 5 1) AmD 5 2) B 1 I D AI 5 3) I D BmC J B mI J A I 10 2) BmC BJ AI mI J 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 de ne the following modi ed current ~j A = j A + 1 (C.3) traceless", i.e., current [23] Here 1 and the total derivative terms do not contribute to the supercharges. If ~j A were \ A = 0, it would be possible to de ne the conserved superconformal HJEP05(218) (C.4) (C.5) I A (C.6) (C.7) (C.8) (C.9) (C.10) (C.11) by setting a1 = 0 in (3.37), the right-hand side of (C.6) vanishes, i.e., ~ A = 0; and it is possible to construct a superconformal current s , de ned by (C.4); as a result, the minimal (1; 0) tensor multiplet theory of section 2 may have a superconformal symmetry. However, if a1 6= 0 in (3.37), the right-hand side of (C.6) fails to vanish without imposing additional constraints on the elds, though one can make either the rst line or the last three lines to vanish by choosing the values of 1 , 2, and 3 properly. If we set 1 = 2 = 3 = 2ic=5; the last three lines of (C.6) vanish, but the rst line remains: ~ A = 2ic AmD 2ic B 1 I D AI 2ic the rst line of (C.6) vanishes, and the remaining part is 0 = 4ic + 5 1 = 4ic 5 2 = 4ic 5 3; ~ A = 4ic +4ic BmC B 1 I C J B mI J + 4ic A I m AJ mJ I BmC 4ic AI mI J m JA mI J : In either case, one cannot construct the conserved superconformal current s , meaning that the general N = (1; 0) theory does not have a superconformal symmetry. However, if we impose the additional constraint A = 0 or at least ~jAjphyi = 0, with jphyi the physical states, it is possible to construct a conserved superconformal s , and the (1; 0) theory may admit a superconformal symmetry. (See also (C.19) and the discussion below (C.19).) C.2 Supercurrent of N = (2; 0) LP theory If the hypermultiplet is also in the adjoint representation of the gauge group (see section 4), the supercurrent (C.1) becomes A j A = ic mA m A ; (C.12) where A = 1; : : : ; 4 is a USp(4) = SO(5) R-symmetry index, and and Am is de ned by the second equation of (4.18). (The de nitions of all elds of the (2; 0) theory can be found in section 4.) We see that supercurrent is indeed enhanced from Am is de ned by (4.5), N = (1; 0) to (2; 0). The expression of the (2; 0) supercurrent is jA = ic a B BmD A a AmHm abB BmC A a b f npm : n p (C.13) Using the 32-component spinor formalism (see section 2), it can be written as 1 2!3! 1 2!3! 1 2 j = ic ab mC a b f npm : (C.14) n p Here = 0; 1; : : : ; 5 and a = 6; : : : ; 10, and 10 = The three-algebra counterpart of (C.14) was constructed in [18]; its expression is j3alg = ab mCo n p a b f onpm; (C.15) where f onpm, being totally antisymmetric in four indices, are the structure constants of three-algebra.8 If we set c = i in (C.14), and make the replacement in (C.15), we see that (C.15) is exactly the same as (C.14). We now try to calculate the \ -trace" of the modi ed current Co f onpm ! C f npm ~ j3alg = j3alg + where is a constant. A short computation gives ~ j3alg = ( 4 + 5 ) a mD am + (2 5 ) ab mCo n p a b f onpm : Again, no matter how we choose the value of , the right-hand cannot vanish. So the general N = (2; 0) LP theory does not have a superconformal symmetry. 8For convenience, we have converted the convention of [18] into our convention. (C.16) (C.17) (C.18) We now consider the possibility of constructing a superconformal current s3alg by imposing an additional constraint on the elds. In (C.18), if we set = 4=5, and assume that the 3-bracket m[C ; a; b]m = mCo n p a b f onpm = 0; (C.19) or at least that the 3-bracket annihilates the physical states, i.e., then we have j3alg = 0 or j3algjphyi = 0. As a result, it is possible to construct the conserved superconformal current s3alg, and the N = (2; 0) LP theory may have a superconformal symmetry. It would be interesting to investigate the physical signi cance m[C ; a; b]mjphyi = 0, of the additional constraint (C.19). Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. HJEP05(218) [1] J.M. 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Fa-Min Chen. A 6D nonabelian (1, 0) theory, Journal of High Energy Physics, 2018, 185, DOI: 10.1007/JHEP05(2018)185