#### A nonabelian (1, 0) tensor multiplet theory in 6D

Fa-Min Chen
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Open Access
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c The Authors. Article funded by SCOAP
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Department of Physics, Beijing Jiaotong University
, No.3 Shang Yuan Cun, Hai Dian District,
Beijing 100044, P.R. China
We construct a general nonabelian (1, 0) tensor multiplet theory in six dimensions. The gauge field of this (1, 0) theory is non-dynamical, and the theory contains a continuous parameter b. When b = 1/2, the (1, 0) theory possesses an extra discrete symmetry enhancing the supersymmetry to (2, 0), and the theory turns out to be identical to the (2, 0) theory of Lambert and Papageorgakis (LP). Upon dimension reduction, we obtain a general N = 1 supersymmetric Yang-Mills theory in five dimensions. The applications of the theories to D4 and M5-branes are briefly discussed.
Contents
1 Introduction and summary
2 (1, 0) tensor multiplets without coupling to hypermultiplets
2.1 Closure of the superalgebra
2.2 Summary of the theory
3 (1, 0) tensor multiplets coupling to hypermultiplets
3.1 (1, 0) tensor multiplets coupling to hypermultiplets
3.2 Summary of the theory
3.3 Promoting to (2, 0) LP theory
3.4 Relating to N = 1 SYM in 5D
A Conventions and useful identities
1 Introduction and summary
In recent years, the construction of nonabelian (1, 0) and (2, 0) superconformal theories in
6D has attracted a lot of attention. Using a Nambu 3-algebra, Lambert and Papageorgakis
have been able to build up a nonabelian (2, 0) tensor multiplet theory in 6D [1]; in [2, 3], the
same theory has been constructed in term of ordinary Lie algebra. One particular feature
of the (2, 0) LP theory is that the gauge field is nondynamical. The (2, 0) LP theory may be
a candidate of dual gauge description of interacting multiple M5-branes (a general review
of superconformal field theories and multi M-branes can be found in ref. [4]; see also [5]).
On the other hand, upon a dimension reduction, the (2, 0) LP theory can be reduced to
a maximally supersymmetric Yang-Mills in 5D [1], which can used to describe interacting
multiple D4-branes.
A little later, a nonabelian (1, 0) theory with the same field content of a (2, 0) theory,
has been constructed in a series of papers [68]; in this theory, the gauge field is dynamical.
In this paper, we consider another possibility: we construct a nonabelian (1, 0) theory
with the same field content of a (2, 0) theory, but its gauge field is nondynamical.
Specifically, the R-symmetry of this (1, 0) theory is SU(2), and the theory contains 5 bosonic
fields, a non-dynamical gauge field, two anti-chiral spinor fields (transforming in the 2 of
SU(2)L and 2 of SU(2)R of the global symmetry group SU(2)L SU(2)R, respectively),
and a selfdual field strength H. The fields are in the adjoint representation of the Lie
algebra of gauge group; and the Lie algebra of gauge symmetry can be chosen as the Lie
algebra of ADE type.
One important feature of our (1, 0) theory is that it contains a continuous
(dimensionless) parameter b. However, in the special case of b = 1/2, the theory possesses an extra
(1, 0) tensor multiplets without coupling to hypermultiplets
Closure of the superalgebra
For simplicity, in this section we will try to construct a 6D nonabelian (1, 0) tensor multiplet
theory with SU(2) R-symmetry. This theory does not contain hypermultiplets, and its
gauge field is non-dynamical. Another motivation is that this theory is interesting in its
own right: in [6], the (1, 0) tensor multiplet theory, with a dynamical gauge field, does not
contain hypermultiplets as well; it is natural to construct a similar (1, 0) theory with a
non-dynamical gauge field, and to compare these two types of theories. Of course, in the
next section, we will add the hypermultiplets into this theory such that there are non-trivial
interactions between the tensor multiplets and hypermultiplets, and the field content of the
final (1, 0) theory becomes the same as that of the (2, 0) theory of LP [1].
The component fields of the (1, 0) tensor multiplets in 6D are given by
where m are the scalar fields, with m an adjoint index of the Lie algebra of gauge
symmetry. The fermionic fields m+ are defined as m+ = 21 (1 + 6789)m, with m anti-chiral
fields with respect to 012345, i.e. 012345m = m (our conventions are summarized in
In ref. [1], it was proved that there are no suitable fields Bm such that Hm = 3D[Bm]
(except that the gauge group is abelian), where the covariant derivative is defined by (2.4).
So in this paper, we do not try to define Hm in terms of 3D[Bm]. But later we will see
that Hm can be solved in terms of the gauge field strength Fm (see (2.30)). We will make
further comments on Hm below eq. (2.30).
We postulate the law of supersymmetry transformations as follows,
where C is an abelian auxiliary field1 with scaling dimension 1, and f npm the structure
constants of the Lie algebra. c1 and d1 are constants; they will be determined by the closure
of superalgebra. The supersymmetry generators are defined as + = 21 (1 + 6789), where
are chiral with respect to 012345, i.e. 012345 = . The covariant derivative is defined
as follows
Notice that in (2.3), it is fine to replace m+ by m = 12 (1 6789)m, if we replace +
by = 21 (1 6789) at the same time; but the resulted theory is a new theory.
Now we would like to examine the closure of super-Poincare algebra. Let us begin by
considering the scalar fields. A short computation gives
In the case of gauge fields, we obtain
Fm = Am Am + [A, A ]m,
1In a 3-algebra approach, a similar auxiliary field Ca is introduced, where a is a 3-algebra index [1]; our
reason for introducing C into the theory is similar to that of ref. [1] (see also [2, 3]).
with [A, A]m = AnApfnpm. In order to close the Poincare superalgebra (up to the gauge
transformation Dm), we must require the third and fourth terms of (2.7) to vanish.
This gives the equations of motion of gauge fields and the constraint equations on the
scalar fields:
Taking a super-variation on (2.11), we obtain the constraint equations on the fermionic
fields
Note that using (2.8), one can re-write (2.5) as the expected form
Let us now look at the auxiliary field:
[1, 2]C = 0 = vDC + [, C].
[1, 2]m+ = vDm+ + [, +]m
+ 3 (1 c1 + d1)vijijCn+pf npm
8
14 vDm+ + 81 (7c1 5d1 3)vCn+pf npm
The second line of (2.15) must vanish; this can be achieved by setting
Using (2.17), the third and fourth lines of (2.15) become
1 c1 + d1 = 0.
14 vDm+ + 81 (2d1 + 4)vCn+pf npm
+ 1 vDm+ 1 (2d1 + 4)vCn+pf npm,
4 8
= 1 v(Dm+ (d1 + 2)Cn+pf npm),
4
which leads us to impose the equations of motion
0 = Dm+ (d1 + 2)Cn+pf npm.
On the other hand, taking a super-variation on (2.2) gives
By the last equation of (2.3), a short computation converts the above equation into
Comparing the above equation and eq. (2.19) determines d1:
0 = Dm+ + d1Cn+pf npm.
d1 = 1.
c1 = 0.
The second line vanishes on account of eqs. (2.10) and (2.23), while the third line must be
the equations of motion
Using eqs. (2.10) and (2.25), and the Bianchi identity D[F]p = 0, one obtains the
constraint equations
In summary, the equations of the (1, 0) theory of this section are given by
Notice that the second equation is equivalent to the equation
C2Hm = 3F[m C] + 12 FmC .
We emphasize again that the gauge field of the (1, 0) theory of this section is
nondynamical; it seems that due to this reason, one cannot define the nonabelian selfdual field
strength in the following way: Hm = 3D[B]m, as analyzed by the authors of [1].
However, in the (1, 0) theory of [6], the gauge field is dynamical, and it is possible to
construct a field strength Hm associated with the nonabelian covariant derivative D[B]m
(see also [1522]). So these two types of theories are not the same; but still, there may
be a connection between them. It would be interesting to investigate these two types of
theories further.
The supersymmetry transformations are
m+ = +Dm + 31! 21! +Hm,
Hm = 3i+[ D]m+ i+Cn+pf npm.
If we make the following replacements
+ ,
in (2.28) and (2.31), we will obtain a new (1, 0) theory.
(1, 0) tensor multiplets coupling to hypermultiplets
(1, 0) tensor multiplets coupling to hypermultiplets
Having constructed the (1, 0) tensor multiplet theory, our next challenge is to couple this
theory to a (1, 0) hypermultiplet theory. We begin by considering the free (1, 0)
hypermultiplet (Xi, ). Here Xi (i = 6, 7, 8, 9) are a set of bosonic fields; and the ferminoic field
is anti-chiral with respect to 6789 as well as 012345, that is, = 12 (1 6789) and
012345 = (our conventions are summarized in section A). The (1, 0) supersymmetry
transformations are
The superalgebra is closed by imposing the equations of motion
Taking a super-variation on (3.2) gives the equations of motion of the free bosonic fields:
Xi = 0. Clearly, this free (1, 0) hypermultiplet theory has an SU(2) R-symmetry.
To couple tensor multiplets and hypermultiplets, it is natural to assume that they
share the same gauge symmetry, and the hypermultilets furnish the adjoint representation
of the algebra of gauge symmetry, as the tensor multiplets do. Under these assumptions,
we propose the supersymmetry transformations
Xmi = i+im,
m = i+DXmi + ai+CXnipf npm,
m+ = +Dm + 31! 21! +Hm + bij +CXniXpj f npm,
Hm = 3i+[ D]m+ i+Cn+pf npm
where a, b, and d are constants, to be determined later. We see that after coupling to the
hypermultiplets, the field content of this (1, 0) theory is the same as that of the (2, 0) theory
of LP [1]. However, the R-symmetry of the (1, 0) theory is only SU(2), while the
Rsymmetry of the (2, 0) theory is SO(5).
Let us now examine the closure of the superalgebra. Using the results of the last
section, the task of examining the closure of the algebra becomes much simpler. The
transformation of scalar fields m remains the same form as that of the last section:
[1, 2]Xmi = v D Xmi + a[, Xi]m.
The last two terms must vanish separately. In this way, we obtain the equations of motion
of gauge fields and the constraint equations on the scalar fields m:
a = 1.
It can be seen that we must set
d = 2b,
Using the identities in appendix A, one obtains
[1, 2]m = v D m + [, ]m
0 = Dm + Cnpf npm + iCn+Xpi f npm.
[1, 2]m+ = v D m+ + [, +]m
where vij is defined by (2.16). The first line and the first two terms of second line are
the results of section 2 (see (2.15)). The rest terms are due to that we have coupled the
(1, 0) theory of section 2 to the (1, 0) hypermultiplet theory. It can be seen immediately
that if
the last two lines vanish simultaneously. Thus the second line of (3.12) must be the
equations of motion
The above equation can be also derived by combining the selfdual conditions Hm =
31! Hm and eq. (3.13).
Let us now compute the super-variation of the H-fields. After some algebraic steps,
we obtain
i
+ 4 b (m n)Cf mnp +
48bvij (CDXmi)Xnjf mnp.
The first three lines are adopted from (2.24), while the last two lines are due to the coupling
of (1, 0) tensor multiplet and hypermultiplet theories. To close the algebra, we must require
the last line to vanish. This can be done by either setting b = 0 or CDXmi = 0.
However, if b = 0, there wouldnt be nontrivial interactions between the tensor multiplets
and hypermultiplets. We are therefore led to
The second line of (3.15) goes away by the equations of motion for the gauge fields, while
the third and fourth lines must be the equations of motion of the H-fields
Also, in exactly the same way of deriving eq. (2.26), one can obtain the constraint equations
Finally, one can derive the equations of motion of the bosonic fields Xpi and p by
taking super-variations on eqs. (3.11) and (3.14), respectively; they are given by
In summary, the equations of the (1, 0) theory of this section are
0 = Dm + Cnpf npm + iCn+Xpi f npm,
The supersymmetry transformations are
Xmi = i+im,
m = i+DXmi + i+CXnipf npm,
m+ = +Dm + 31! 21! +Hm + bij +CXniXpj f npm,
Hm = 3i+[ D]m+ i+Cn+pf npm
In constructing the (1, 0) theory, we have used + as the set of generators of
supersymmetry. The generators + transform as 2 of SU(2)L of the global symmetry group
SO(4) = SU(2)L SU(2)R. We can of course choose , transforming as 2 of SU(2)R, as
the set of supersymmetry generators. In fact, if we make the replacement
we will obtain a new (1, 0) theory, provided that b 6= 12 , because the discrete
transformation (3.24) is not a symmetry of the theory (see (3.27) and the comments below (3.27)).
The gauge groups can be classified by specifying the structure constants f mnp and the
invariant symmetric tensor kmn on the Lie algebras. (In the case of simple or semi-simple
Lie algebra, kmn is the Killing-Cartan metric.) For instance, the Lie algebras can be chosen
as the Lie algebras of type ADE.
+
Finally, we expect that the (1, 0) theory has a full OSp(8|2) suerconformal symmetry.
One should be able to verify this symmetry explicitly. The ideas for proving the OSp(N |4)
(N = 4, 5, 6, 8) superconformal symmetries associated with the 3D N 4 Chern-Simons
matter theories may be useful [2326].
Promoting to (2, 0) LP theory
Since a theory with fewer supersymmetries must be more general than those with higher
supersymmetries, we should be able to obtain the (2, 0) theory as a special case of the (1, 0)
theory. To enhance the supersymmetry is equivalent to promote the SU(2) R-symmetry to
SO(5). The first step is to require the bosonic fields to transform as the 5 of SO(5); this
leads us to define
where i = 6, 7, 8, 9 and I = 6, . . . , 10. Similarly, we can combine i and 10 = 0123456789
to form
With these notations and the properties of +, m+, and m (see (A.15)), the fourth and
fifth equations of (3.21) can be unified into the equation
where m = m+ + m. It can be seen that under the discrete transformation m+
m, the first two terms of (3.27) are invariant, but the last term becomes
Hence the transformation is not a symmetry of the theory in general. However, in the
special case of
the last term of (3.27) drops, and equation (3.27) is invariant under the transformation
m+ m; more importantly, this equation becomes manifestly SO(5) covariant.
Similarly, using b = 1/2, the first two equations of (3.21) can be combined into a single equation
with manifest SO(5) symmetry. The rest equations of (3.21) can be taken care of as well.
In summary, if b = 1/2, we have
0 = CDXmI = CDm = CDHm = C .
These are essentially the same equations of the (2, 0) LP theory, constructed in terms of
3-algebra [1]. Similarly, if b = 1/2, the supersymmetry transformations (3.22) can be recast
into the forms
Hm = 3i+[ D]m + i+I nCXpI f npm.
+ ,
to eqs. (3.30) and (3.31), we find that (1) the equations of motion (3.30) do not change
at all, meaning that m+ m is just a discrete symmetry of the theory; (2) the
supersymmetry transformations (3.31) become
Hm = 3i[ D]m + iI nCXpI f npm,
which must be considered as another independent set of supersymmetry transformations
of the theory. The two sets of supersymmetry transformations (3.31) and (3.33) can be
unified into
m = I DXmI + 31! 21! Hm + 21 IJ CXnI XpJ f npm,
Hm = 3i[ D]m + iI nCXpI f npm,
where = + + . These are essentially the same supersymmetry transformations of
the (2, 0) LP theory, expressed in terms of Lorentian 3-algebra [1]. In this way, we have
recovered the whole (2, 0) LP theory.
Relating to N = 1 SYM in 5D
In this section, we will show our (1,0) theory can be reduced to an N = 1 super Yang-Mills
theory in 5D by specifying the auxiliary field C. Since our C plays the same role as that
of Ca of the LP theory [1] (see also [2, 3]), it is natural adopt the method in [1] and to
choose the space-like vector vev
where the constant g has dimension 1 and obeys the equation g = 0. The equations of
motion of gauge fields (the 3rd equation of (3.21)) now become
hCi = g(0, . . . , 0, 1) = g5,
where we have decomposed into = (, 5), with = 0, 1, . . . , 4. On the other hand,
since F5m = gH55m = 0, we have
0 = Dm+ g5n+pf npm + 2bg5inXpif npm,
Here the covariant derivative is defined as
m = i+DXmi + g5i+Xnipf npm,
m+ = +Dm + 21g 5+Fm + bg5ij+XniXpjf npm,
These are the N = 1 supersymmetry transformations associated with the 5D SYM theory.
As observed by LP [1], the coupling constant of the SYM theory is related to the constant
g as follows
g = gY2 M .
Notice that the continuous parameter b still survives in the N = 1 SYM theory.
The N = 1 SYM theory can be the dual gauge theory of multi D4-branes. It would
be interesting to study their large N limit and to construct their gravity duals.
Recall that in the special case of b = 12 , the supersymmetry of the 6D (1, 0) theory is
promoted to (2, 0), and theory becomes the (2, 0) theory (see section 3.3). Substituting the
vev (3.35) into the equations (3.30) and the supersymmetry transformations (3.34) of the
(2, 0) theory, one can obtain the maximally supersymmetric (N = 2) Yang-Mills theory in
5D. For details, see ref. [1].
We end this section by commenting on the other possibilities: hCi is a light-like vector
or a time-like vector. In ref. [1], it was argued that if one uses the null reduction
hCi = g(1, 0, . . . , 0, 1), hCihCi = 0,
i.e. hCi is a light-like vector, the (2, 0) theory can be used to describe a system consisting
of both M2 and M5-branes. Since the (2, 0) theory is a special case of our (1, 0) theory,
we expect that the (1, 0) theory can be also a gauge description of some system containing
M2 and M5-branes by choosing the null reduction. On the other hand, like its (2, 0)
counterpart [27], this (1, 0) theory may be also a light-cone description of multiple M5-branes.
However, we leave the work of exploring this particular 6D (1, 0) theory to the future.
Finally, if we choose hCi as the time-like vector
all fields are static:
and the theory may be used to describe static 5-branes in 11D [1]. It would be nice to
investigate this theory further.
Acknowledgments
We are grateful to Bin Chen for useful discussions. This work is supported by the
RenCai Foundation of Beijing Jiaotong University through Grant No. 2013RC029, and
supported by the Scientific Research Foundation for Returned Scholars, Ministry of Education
of China.
Conventions and useful identities
Following the idea of ref. [1], we will also work with 32-component Majorana fermions.
However, in our case, these are SO(9, 1) Majorana fermions. The gamma matrices are real.
Under the decomposition SO(9, 1) SO(5, 1) SO(4), one can define the chirality matrix
012345 of SO(5, 1) and the chirality matrix 6789 of SO(4). The fermionic fields are
anti-chiral with respect to 012345, i.e.
while the parameters of supersymmetry transformations are chiral,
We also define
The product of two spinors is defined as
where and have opposite chiralities with respect to 012345. We have chosen 0 = 0T
as the charge conjugation matrix C. It obeys the equations
where = 0, . . . , 5 and i = 6, . . . , 9.
To work out the Fierz identities, it is convenient to introduce the eleventh gamma
matrix
Now the gamma matrices satisfy the Clifford algebra in eleven spacetime dimensions
The antisymmetric part of the above equation is
(1) 12 (p1)p(2m1...mp 1)m1...mp .
12 21 = 116 p=X1,2,5 p1!
(1) 12 (p1)p(2m1...mp 1)m1...mp .
Using eqs. (A.1), (A.2), (A.4), and (A.7), one can reduce equation (A.10) to
1 1
(2++)1+ (1++)2+ = 4 (2+1+)+ 192
(2+1+)ij +, (A.11)
ij
ij = 1 (ij 1 ijklkl) (6789 = 1),
2 2
and the identity
Also, with the definition of (A.7), we find that
Finally, we find that the following identity
(1) 12 (p1)p
(6 p)!
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