A 6D nonabelian (1, 0) theory
HJE
nonabelian (1, 0) theory
FaMin 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 YangMills theory in 5D. We discuss brie y the possible applications of the theories to

A
MTheory
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
) Rsymmetry
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 Supervariations 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 M2branes 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 ChernSimons matter theories
with variety gauge groups. However, it seems more di cult to construct the gauge theory
of multi M5branes. One particular reason is that it is di cult to construct an action: the
theory contains a selfdual threeform
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 threealgebra
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
M5branes (for reviews on gauge theories of Mbranes, see [6] and [7]). More recently,
using the Nambu threealgebra, Lambert and Sacco (LS) have constructed a more general
(2; 0) theory by introducing an additional nondynamical abelian threeform into the LP
theory [8]. Remarkably, upon a dimension reduction, the LS theory is reduced to the 3D
N = 8 BLG theory, describing two M2branes in C4=Z2. Thus the (2; 0) LS theory may
= 31! "
H
, implying that
{ 1 {
be a dual gauge theory for two M5branes or two M2branes. 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 Rsymmetry is
only SU(
2
). If the (1; 0) hypermultiplet also takes value in the adjoint representation,
then the SU(
2
) Rsymmetry 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
) Rsymmetry 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 YangMills (SYM) theory with 8 supersymmetries, by choosing the
2
spacelike vector vev hC i = gYM 5
. Here C is an auxiliary
eld, and gYM the coupling
constant of the supersymmetric YangMills theory. In section 5, we discuss some other cases
with hC i being a lightlike or a timelike 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
M5branes. 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
) Rsymmetry 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 32component Majorana fermions.
(More precisely, we will work with SO(9; 1) Majorana fermions.) The gamma matrices
satisfy the anticommutation relations
f
;
g = 2
f s; tg = 2 st;
;
f s;
is antichiral 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) MajoranaWeyl
The supersymmetry transformations are
012345
6789
=
= :
0123456789
;
=
=
=
i ;
+
is chiral with respect to 012345 as well as 6789, i.e.,
The superpoincare algebra is closed by imposing the equations of motion (EOM)
] = 0:
However, due to the selfduality 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 selfduality conditions. Here "012345 =
3In ref. [10], the supersymmetry parameter is denoted as +, which is a 10D MajoranaWeyl 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 semisimple, then kmn is nothing but the KillingCartan 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 selfdual 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) MajoranaWeyl fermion
can be converted into an SU(
2
) symplecticMajorana chiral spinor
A:
where
labels the MajoranaWeyl 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
) Rsymmetry 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 supervariation \ " in (2.25) will be replaced by \ ", while the supervariation
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 nondynamical.
Here the gauge eld Am is also nondynamical. If it were a dynamical eld, its superpartner
(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 nondynamical.
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 Rsymmetry group SU(
2
). The fermionic eld
is a 6D Weyl spinor, and it
is antichiral with respect the 6D chirality matrix, i.e.
The superPoincare 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 superPoincare 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 ChernSimons 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 righthand 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 superpoincare 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 supervariation 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 superPoincare 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 supervariation of the selfdual 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 supervariations 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 supervariations 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 supervariation on any equation of (3.38) can transform
it into some other equations of (3.38). For instance, if we take a supervariation 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 reconstruct 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
) Rsymmetry 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 rearranged 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 undotted 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 Rsymmetry group USp(4). We now
see that the USp(4) = SO(5) Rsymmetry 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
3algebra [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
Rsymmetry 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 recasting 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 ChernSimons 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) Rsymmetry, are
the ordinary Lie 2algebra 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 spacelike 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 4component 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 4component 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 rescale 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 EulerLagrange 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 lightlike vector. In refs. [5, 8], it was
argued that if one uses the null reduction
i.e. hC i is a lightlike 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 M5branes [12]. It would be interesting to explore this special (1; 0) theory further.
In particular, it would be interesting to introduce an additional abelian 3form
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 ChernSimons 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 antiselfdual.
Let hC i be a timelike 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 5branes in 11
dimenFor more discussions on M5branes 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 JunBao Wu and ZhiGuang 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 RenCai 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 righthand 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 supervariation 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 selfdual 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 supervariations 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
Supervariations 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 supervariation
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 supervariation 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 supervariation 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 selfdual 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 supervariation 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 supervariation 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 supervariation 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
supervariation \ " in (2.25) by \ ", while the supervariation 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 selfdual 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 supervariations 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 supervariation 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 righthand 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 supervariations 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 righthand side of (B.46) vanishes. This nishes the calculation of (B.36).
We now try to calculate the supervariation 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 righthand 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 supervariation 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
) Rsymmetry 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 righthand 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 righthand 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) Rsymmetry 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 32component 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 threealgebra 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
threealgebra.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 righthand 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 3bracket
m[C ; a; b]m =
mCo n p
a b f onpm = 0;
(C.19)
or at least that the 3bracket 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.
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