#### Higher spin currents in the orthogonal coset theory

Eur. Phys. J. C
Higher spin currents in the orthogonal coset theory
Changhyun Ahn 0
0 Department of Physics, Kyungpook National University , Taegu 41566 , Korea
In the coset model (D(N1) ⊕ D(N1), D(N1)) at levels (k1, k2), the higher spin 4 current that contains the quartic WZW currents contracted with a completely symmetric S O(2N ) invariant d tensor of rank 4 is obtained. The threepoint functions with two scalars are obtained for any finite N and k2 with k1 = 1. They are determined also in the large N 't Hooft limit. When one of the levels is the dual Coxeter number of S O(2N ), k1 = 2N − 2, the higher spin 27 current, which contains the septic adjoint fermions contracted with the above d tensor and the triple product of structure constants, is obtained from the operator product expansion (OPE) between the spin 23 current living in the N = 1 superconformal algebra and the above higher spin 4 current. The OPEs between the higher spin 27 , 4 currents are described. For k1 = k2 = 2N − 2 where both levels are equal to the dual Coxeter number of S O(2N ), the higher spin 3 current of U (1) charge 43 , which contains the six products of spin 21 (two) adjoint fermions contracted with the product of the d tensor and two structure constants, is obtained. The corresponding N = 2 higher spin multiplet is determined by calculating the remaining higher spin 27 , 27 , 4 currents with the help of two spin 23 currents in the N = 2 superconformal algebra. The other N = 2 higher spin multiplet, whose U (1) charge is opposite to the one of the above N = 2 higher spin multiplet, is obtained. The OPE between these two N = 2 higher spin multiplets is also discussed.
1 Introduction
The proposal by Gaberdiel and Gopakumar [1], the duality
between the higher spin gauge theory on Ad S3 space [2] and
the large N ’t Hooft limit of a family of WN (≡ W AN −1)
minimal models is the natural analog of the Klebanov and
Polyakov duality [3] relating the O(N ) vector model in
threedimensions to a higher spin theory on Ad S4 space. Then the
obvious generalization of [1] is to consider the Klebanov and
Polyakov duality in one dimension lower. By replacing the
SU (N ) group by S O(2N ), the relevant most general coset
model is described as [4,5]
(1.1)
SˆO(2N )k1 ⊕ SˆO(2N )k2
SˆO(2N )k1+k2
One can also consider the case where the SU (N ) group by
S O(2N + 1) but this is not described in this paper. It is well
known that the conformal weight (or spin) of the primary
state is equal to the quadratic Casimir eigenvalue divided by
the sum of the level and the dual Coxeter number of the finite
Lie algebra [6,7]. For example, for S O(2N ), the quadratic
Casimir eigenvalue for the adjoint representation is given by
2N − 2, while the dual Coxeter number is 2N − 2. Then
we are left with the adjoint fermion of spin 21 at the critical
level which is equal to the dual Coxeter number. One can
apply this critical behavior to the two numerator factors in
(1.1) simultaneously. In the description of these adjoint free
fermions, the central charge grows like N 2 in the large N ’t
Hooft limit: the so-called stringy coset model [8]. See also
the relevant work in [9–13].
Although some constructions on the higher spin currents
in [14] have been done, there are two unknown coefficients
in the expression of higher spin 4 current. Moreover, the spin
1 currents in the numerators of (1.1) are described with the
double index notation. Each index is a vector representation
of S O(2N ) and because of the antisymmetry property of
these spin 1 currents, the number of independent fields is
given by 21 [(2N )2 − 2N ] = N (2N − 1). In order to obtain
the description of the above free adjoint fermions, one should
write down the spin 1 currents with a single adjoint index.
It is well known that the real free fermions transforming in
the adjoint representation of S O(2N ) realize the affine Kac–
Moody algebra for the critical level. It is equivalent to the
theory of 21 2N (2N − 1) = N (2N − 1) free fermions [7].
Before one considers the adjoint free fermion description,
one should obtain the higher spin 4 current from the spin
1 currents living in the numerator factors of (1.1) and
having a single adjoint index. The higher spin 4 current is the
S O (2N ) singlet field [6]. Then one should have a quantity
contracted with the quartic terms in the above spin 1 currents.
This is known as the d symbol; it is a completely symmetric
S O (2N ) invariant tensor of rank 4. In the calculation of any
OPE between the higher spin currents, one should use various
contraction identities between the above d symbol and the
structure constant f . Recall that in the defining OPE between
the spin 1 currents, the structure constant f symbol appears.
As far as I know, there are no known identities between f
symbol and d symbol except of the f f contraction in the
literature. This is one of the reasons why the double index
notation in [14] is used.
In this paper, one starts with the definition of the d symbol
which is given by one half times the trace over six quartic
terms in the S O (2N ) generators. When one meets the
relevant contraction identities in the calculation of any OPE, one
can try to obtain the tensorial structure in the right-hand sides
of these identities. Of course in each term, there should be
present N dependence coefficients explicitly. The tensorial
structure in terms of multiple product of f symbol, d
symbol and the symmetric S O (2N ) invariant tensor δ of rank
2 occurs naturally during the explicit calculation of OPE.
As one applies for N = 2, 3, 4 and 5 cases in the S O (2N )
generators, one can determine the N dependence coefficients
explicitly.
It turns out that the higher spin 4 current is obtained
completely except of overall normalization factor. The eigenvalue
equations of zero mode of the higher spin 4 current acting on
several primary states can be determined explicitly. The
corresponding three-point functions can be obtained. By
choosing the overall factor correctly, one observes the standard
three-point functions in the large N ’t Hooft limit from the
asymptotic symmetry algebra in the Ad S3 bulk theory.
According to the observations in [8, 9], the Gaberdiel and
Gopakumar proposal in the unitary case is still valid for
arbitrary N and k2. One of the main novelties of this paper is the
fact that the (scalar–scalar-higher spin current) three-point
functions for finite N and k2 are obtained. They do depend
on these finite values and in the large N ’t Hooft limit they
contain the ’t Hooft coupling constant which is the ratio of
these two quantities as usual. Of course, the standard
threepoint functions depend on the three complex coordinates
appearing in the above three quantities via two-point
function between the two scalars and some factors which can be
determined by the conformal symmetry. See also [15]. There
are also, in general, spin dependent pieces in the three-point
functions. In our case, the higher spin is fixed by s = 4.
In the large N ’t Hooft limit one can also analyze the next
leading order behavior (for example, N1 ) of the three-point
functions. It is interesting to describe the asymptotic
symmetry algebra in the Ad S3 bulk theory and see whether the
above finite N and k2 behavior in the corresponding
eigenvalues or in the corresponding three-point functions can be
reproduced.
From the description of adjoint fermions living in the
first factor in the numerator of (1.1), one obtains the well
known N = 1 superconformal algebra generated by the spin
2 stress energy tensor and its superpartner, spin 23 current.
It turns out that the higher spin 27 current consists of
septic, quintic, cubic and linear terms in the adjoint fermions
with appropriate derivative terms. The N = 2
superconformal algebra is realized by two adjoint fermions living in the
two numerator factors in (1.1). In this case, the higher spin 3
current with U (
1
) charge 43 is given by the multiple product
of two fermions contracted with d f f or f f tensors without
any derivative terms. Moreover, its three partners, higher spin
27 , 27 , and 4 currents, are determined.
In Sect. 2, the higher spin 4 current is obtained, the
threepoint functions are given and the OPE between the higher
spin 4 current and itself is described under some constraints.
In Sect. 3, the higher spin 27 current is obtained, and the
three OPEs between this higher spin 27 current and the higher
spin 4 current are described using the Jacobi identities.
In Sect. 4, the lowest higher spin 3 current is obtained, and
its three other higher spin 27 , 27 and 4 currents are obtained
which can be denoted as N = 2 lowest higher spin multiplet
with definite U (
1
) charge 43 . Furthermore, another N = 2
lowest higher spin multiplet with definite U (
1
) charge − 43 is
obtained. The OPEs between these higher spin multiplets in
N = 2 superspace are given using the Jacobi identities.
In Sect. 5, we list some future directions related to this
work.
In Appendices A–L, which appear in the arXiv version
only (arXiv:1701.02410), the technical details appearing in
Sects. 2–4 are given.
2 The coset model with arbitrary two levels (k1, k2)
From the spin 1 currents of the coset model, one constructs
the spin 2 stress energy tensor. By generalizing the
Sugawara construction, the higher spin 4 current is obtained from
the quartic terms in the spin 1 currents with the S O (2N )
invariant tensors of ranks 4, 2. The corresponding
threepoint functions of zero mode of the higher spin 4
current with two scalars are described. The OPE between the
higher spin 4 current and itself for particular k1 and N is
obtained.
2.1 Spin 2 current and Virasoro algebra
The standard stress energy tensor satisfies the following OPE
[6]:
1 c 1
T (z) T (w) = (z − w)4 2 + (z − w)2 2T (w)
1
+ (z − w) ∂ T (w) + · · · .
For the coset model in (1.1), the above stress energy tensor
can be obtained by the usual Sugawara construction [6],
1
T (z) = − 2(k1 + 2N − 2)
1
− 2(k2 + 2N − 2)
1
+ 2(k1 + k2 + 2N − 2)
( J a + K a )( J a + K a )(z).
The affine Kac–Moody algebra SˆO(2N )k1 ⊕ SˆO(2N )k2
in (1.1) is described by the following OPEs [6]:
1
J a (z) J b(w) = − (z − w)2 k1δab
1
+ (z − w)
1
K a (z) K b(w) = − (z − w)2 k2δab
The adjoint indices a, b, . . . corresponding to S O(2N ) group
run over a, b = 1, 2, . . . , 21 2N (2N − 1). The Kronecker
delta δab appearing in (2.3) is the second rank S O(2N )
symmetric invariant tensor. The structure constant f abc is
antisymmetric as usual. The diagonal affine Kac–Moody algebra
SˆO(2N )k1+k2 in (1.1) can be obtained by adding the above
two spin 1 currents, J a (z) and K a (z). Of course, we have
J a (z) K b(w) = + · · · .
The central charge appearing in the above OPE (2.1) is
given by [6]
1
c(k1, k2, N ) = 2 2N (2N − 1)
k1
(k1 + 2N − 2)
k2 (k1 +k2)
+ (k2 +2N −2) − (k1 +k2 +2N − 2)
.
(2.4)
Note that the dual Coxeter number of S O(2N ) is equal
to (2N − 2) and the dimension of S O(2N ) is given by
21 2N (2N − 1).
Then the Virasoro algebra realized in the coset model (1.1)
[5,16] is summarized by (2.1) together with (2.2) and (2.4).
2.2 Higher spin 4 current (2.1)
The 28 S O(
8
) generators T a are given in Appendix A. Then
the structure constant introduced in the above is given by
f abc
= − 2i Tr[T c T a T b − T c T b T a ].
Then one obtains [T a , T b] = i f abc T c.
The totally symmetric S O(2N ) invariant tensor of rank 4
is defined as [17,18]
T a T b T c + T a T c T b + T c T a T b + T b T a T c
+T b T c T a + T c T b T a = dabcd T d .
That is, one can express the d tensor as1
(2.2)
dabcd
= 21 Tr[T d T a T b T c + T d T a T c T b + T d T c T a T b
+T d T b T a T c + T d T b T c T a + T d T c T b T a ]. (2.7)
Note that one uses Tr(T a T b) = 2δab.
One obtains the product of the structure constants
f abc f abd = 2(2N − 2)δcd ,
and the triple product leads to
f adb f bec f c f a = −(2N − 2) f de f .
Furthermore, one obtains the following non-trivial triple
product between d tensor (2.7) and f tensor (2.5):
dadeb f b f c f cga = − 43 (N − 1)dde f g + 4δd f δeg
+4δdgδe f − 8δdeδ f g
1
− 3 (2N − 5) f d f h f heg
1
− 3 (2N − 5) f dgh f he f .
By multiplying f d f h into (2.10) and rearranging the indices,
one obtains with (2.9)
dabc f f agd f bde f che = 2(2N 2 − 7N + 11) f f gh .
For the index condition f = d in (2.10) together with the
identity (2.8), one obtains
daabc = 2(4N − 1)δbc.
Note that this behavior is different from the one of the unitary
case where the trivial result daabc = 0 arises [19]. One also
has
dabcd dabce = 12[N (2N − 1) + 2]δde.
Let us describe how one can obtain the higher spin current
with the help of the d tensor we introduced. For the second
1 One can consider the rank 3 tensor as dabc = 21 Tr[T a T bT c +
T bT a T c], which is identically zero.
(2.5)
(2.6)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
rank S O(2N ) invariant symmetric tensor δab, one describes
the stress energy tensor in (2.2). According to the
observation of footnote 1, there is no non-trivial third rank S O(2N )
invariant symmetric tensor dabc. Then the next non-trivial
higher spin current can be constructed from the fourth rank
S O(2N ) invariant symmetric tensor dabcd (2.6).
Let us consider the following higher spin 4 current, along
the line of [6,19,20]:
W (
4
)(z) = dabcd [ A1 J a J b J c J d + A2 J a J b J c K d
+ A3 J a J b K c K d
+ A4 J a K b K c K d + A5 K a K b K c K d ](z)
+[ A6 ∂ J a ∂ J a + A7 ∂2 J a J a
+ A8 ∂ K a ∂ K a + A9 ∂2 K a K a
+ A10 ∂ J a ∂ K a + A11 ∂2 J a K a
+ A12 J a ∂2 K a + A13 f abc J a ∂ J b K c
+ A14 f abc J a K b∂ K c + A15 J a J a J b J b
+ A16 K a K a K b K b + A17 J a J a K b K b
+ A18 J a J a J b K b + A19 J a K a K b K b
+ A20 J a J b K a K b](z).
One should obtain the 20 relative (k1, k2, N )-dependent
coefficients. The first five quartic terms in (2.14) can
easily be understood in the sense that they are the only
possible terms from each spin 1 current, J a (z) and K a (z)
using the dabcd tensor. The next seven derivative terms in
(2.14) can be found from the second derivative of stress
energy tensor ∂2T (z). The remaining eight terms can arise
in T T (z).
First of all, the higher spin 4 current should have the
regular terms with the diagonal spin 1 current in the coset model
as follows [6,19,20]:
J a (z) W (
4
)(w) = + · · · , J a (z) ≡ ( J a + K a )(z).
Let us calculate the OPEs between the diagonal spin 1
current and the 20 terms in (2.14) in order to use the
condition (2.15). One can perform the various OPEs by
following the procedures done in the unitary case [21]. Let
us focus on the A1 term in (2.14) which has the
regular OPE with K a (z). Then Eqs. (2.22), (2.23) and (2.24)
of [21] can be used. For example, Eq. (2.24) of [21]
provides the information of the OPE between the J a (z) and the
above A1 term. Using Eqs. (2.11) and (2.8), one can simplify
the fourth-order pole in (2.24) of [21] which was given by
f ab f f f ci dbcde f geh f idg J h (w).
It turns out that we are left with J a (w) with an N
dependent S O(2N ) group theoretical factor. The third-order
pole,
f ab f dbcde( f hdg f f ch J g J e + f heg f f ch J d J g
+ f heg f f dh J c J g)(w) + f ac f dbcde f geh f f dg J b J h (w),
can be simplified with the help of (2.11). We are left with
f abc J b J c(w) in (2.16) with N dependent coefficient factor
which is proportional to ∂ J a (w). Finally the second-order
pole,
(2.14)
(2.15)
(2.16)
(2.17)
−4k1dabcd J b J c J d (w)
+dbcde( f ab f f f cg J g J d J e + f ab f f f dg J c J g J e
+ f ab f f f eg J c J d J g
+ f ac f f f dg J b J g J e + f ac f f f eg J b J d J g
+ f ad f f f eg J b J c J g)(w),
can be simplified further together with (2.10).
Then we obtain the final OPE as follows:
J a (z) dbcde J b J c J d J e(w)
1
= (z − w)4 2(2N − 2)(4N 2 − 14N + 22) J a (w)
1
− (z − w)3 2(4N 2 − 14N + 22) f abc J b J c(w)
1
+ (z − w)2 [−(4k1 + 8(N − 1))dabcd J b J c J d
−(12 + (2N − 2)(2N − 5)) f abc∂ J b J c
+(12 + (2N − 2)(2N − 5)) f abc J b∂ J c](w) + · · · .
There is no first-order pole in Eq. (2.18). One can check the
second-order pole in (2.18) from (2.17).
Let us consider the A2 term in (2.14) where there
exists a K d (z) dependence. Starting from Eqs. (2.23) and
(2.19) of [21] with Eqs. (2.11) and (2.10), one can
simplify the third-order pole, f ac f f dbcde f geh f f dg J h K b(w),
as f abc J b K c(w) with an N dependent factor. Similarly, the
second-order pole,
−3k1dabcd J c J d K b(w) − k2dabcd J b J c J d (w)
+dbcde( f ac f f f dg J g J e K b + f ac f f f eg J d J g K b
+ f ad f f f eg J c J g K b)(w),
can be simplified in terms of several independent terms. It
turns out that in this case also there are no first-order poles.
Therefore, one obtains the following OPE corresponding
to A2 term:
(2.18)
(2.19)
J a (z) dbcde J b J c J d K e(w)
1
= (z − w)3 (4N 2 − 14N + 22) f abc J b K c(w)
1
+ (z − w)2 [−(3k1 + 4(N − 1))dabcd J b J c K d
−k2dabcd J b J c J d + 12 J b J b K a
−12 J a J b K b − 12 f abc∂ J b K c
+(2N − 5) f abc f cde J b J e K d ](w) + · · · .
One can see the second-order pole in (2.20) from (2.19).
Let us consider the A3 term in (2.14). From Eq. (2.22) of
[21], one has the relevant OPEs. For example, the
secondorder pole,
(−2k1dabcd J d K b K c + f ad f f f egdbcde J g K b K c
−2k2dabcd K d J b J c + f ad f f f egdbcde K g J b J c)(w),
can be reexpressed in terms of various independent terms
with the help of the identity (2.10). It turns out that the
relevant OPE coming from (2.21) can be summarized as
(2.20)
(2.21)
1
J a (z) dbcde J b J c K d K e(w) = (z − w)2
× −
2k1 + 43 (N − 1) dabcd J b K c K d
− 2k2 + 43 (N − 1) dabcd J b J c K d
+8 J b K a K b + 4 f abc∂ J b K c
−8 J a K b K b − 4 f abc J b∂ K c
1
− 3 (2N − 5) f abc f cde J d K e K b
1
− 3 (2N − 5) f abc f cde J d K b K e
−8 J b J b K a + 8 J a J b K b
1
− 3 (2N − 5) f abc f cde J e J b K d
1
− 3 (2N − 5) f abc f cde J b J e K d (w) + · · · .
(2.22)
It is useful to realize that this OPE remains the same after the
exchange of J a (w) and K a (w) together with k1 ↔ k2. The
left-hand side is invariant under this transformation because
the d tensor is totally symmetric. The 12 terms in the
secondorder pole can be divided into two groups and each of them
has their own counterpart.
It is straightforward to complete this calculation step by
step. We summarize the remaining 17 OPEs in Appendix
B. Then we have the complete expressions in (2.18), (2.20),
(2.22), and Appendix B.
The higher spin 4 current should transform as a primary
field under the stress energy tensor (2.2). According to the
previous regular condition (2.15), the diagonal spin 1
current J a (z) does not have any singular terms in the OPE with
the higher spin 4 current W (
4
)(w) after we use the results of
Appendix B. Then there are no singular terms in the OPE
between the stress energy tensor in the denominator of the
coset model (1.1) and the higher spin 4 current because the
former is given by J a J a (z). The singular terms can arise
from the OPE between the stress energy tensor in the
numerator of the coset model and the higher spin 4 current.
Therefore, one should have the following condition [6,19,20]:
(2.23)
(2.24)
Tˆ (z) W (
4
)(w)
(z−1w)n , n=3,4,5,6
= 0.
Here the stress energy tensor in the numerator is described
by
1
Tˆ (z) ≡ − 2(k1 + 2N − 2)
1
− 2(k2 + 2N − 2)
Of course, the higher spin 4 current has the standard OPE
(the second- and first-order poles) with stress energy tensor
(2.2) as usual.
Let us calculate the OPE between the stress energy
tensor (2.24) and the A1 term in (2.14). First of all, because
the A1 term does not contain the K a (w) spin 1 current,
one can consider the OPE between the first term of (2.24)
and the A1 term. It is well known that the spin 1 current
J a (w) transforms as a primary field under the first term
of (2.24) (i.e., the stress energy tensor in the first factor
of the numerator). Then one should obtain the OPE J b(z)
dbcde J c J d J e(w) and it turns out that there exists a
nontrivial second-order pole given by −3k1(8N − 2) J c J c(w),
where the identity (2.12) is used. Note that the structure
constant term vanishes due to the presence of dbcde.
Furthermore, one should calculate the OPE between the above stress
energy tensor and the previous expression dbcde J c J d J e(w)
where the order of the singular terms is greater than 2. Then
we are left with −3k1(8N − 2) J c J c(w) by combing the
contribution −2k1(8N − 2) J c J c(w) from the contraction
between the stress energy tensor and J c(w) and the
contribution −k1(8N − 2) J c J c(w) from the OPE between the
stress energy tensor and dbcde J d J e(w). Therefore, the final
total contribution is summarized by −6k1(8N − 2) J c J c(w)
and we present this OPE as follows:
1
Tˆ (z) dbcde J b J c J d J e(w) = − (z − w)4
×12k1(4N − 1) J a J a (w) + O
1
This result in (2.25) shows a behavior different from the
corresponding OPE in the unitary case, because in the latter
there is no contribution from the fourth-order pole because
the above daabc tensor for the SU (N ) group vanishes [19].
Let us move on the A2 term in (2.14). In this case, the spin
1 current K d (w) is present. However, the contribution in the
higher singular terms of the stress energy tensor coming from
the second term of (2.24) vanishes. Then one can calculate
the OPE between the stress energy tensor in the first factor of
the numerator and the A2 term. By using the previous
procedure one can obtain the contribution −2k1(8N − 2) J c K c(w)
from the contraction with J b(w) current and the
contribution −k1(8N − 2) J c K c(w) from the contraction with other
remaining factor dbcde J c J d K e(w). By adding these two, one
obtains the following OPE:
1
Tˆ (z) dbcde J b J c J d K e(w) = − (z − w)4
×6k1(4N − 1) J a K a (w) + O
1
Now let us describe the contribution from the A3 term in
(2.14) where the quadratic K c K d (w) appears. In this case,
one should also calculate the contribution from the stress
energy tensor in the second term in (2.24). As done before,
the contribution from the contraction with the J b(w) spin
1 current is given by −k1(8N − 2)K c K c(w). Similarly the
contribution from the contraction with the remaining factor
is given by −k2(8N − 2) J c J c(w). Then we are left with
+O
One also sees the symmetry under the transformation
J a (z) ↔ K a (z) and k1 ↔ k2.
It is straightforward to perform the other remaining
calculations step by step. We summarize the remaining 17 OPEs
in Appendix C. Then we are left with (2.25)–(2.27), and
Appendix C.
Now one can determine the undetermined coefficient
functions A1, A2, . . . , A20 appearing in the higher spin 4 current
in (2.14). The 23 linear equations are given in Appendix B
explicitly. The eight linear equations are given in Appendix
C. By solving them, one obtains the final expressions in
Appendix D. They depend on k1, k2 and N . The
corresponding coefficients for k1 = 1 are presented in Appendix E.
Appendix F corresponds to the case where k1 = 2N − 2.
2.3 Three-point functions [22] with two scalars where
k1 = 1
The zero modes of the current satisfy the commutation
relations of the underlying finite dimensional Lie algebra
S O(2N ). For the state |(v; 0) , T a corresponds to i K0a and
for the state |(0; v) , T a corresponds to i J0a as follows:
|(v; 0) :
T a ↔ i K0a , |(0; v) :
T a ↔ i J0a .
(2.28)
Note that from the defining equation of the OPEs (2.3), one
obtains
[ Jma , Jnb] = −k1mδabδm+n,0 + f abc J c
m+n, [K ma , Knb]
= −k2mδabδm+n,0 + f abc K mc+n.
(2.29)
In (2.29), the central terms for the zero modes vanish. Recall
that our generators for the S O(2N ) satisfy [T a , T b] =
i f abc T c [6].
The large N ’t Hooft limit is described as [23,24]
2N
N , k2 → ∞, λ ≡ 2N − 2 + k2
The presence of the numerical value −2 in the denominator
of (2.30) is not important in the large N ’t Hooft limit [25].
Compared to the large N = 4 holography in [10,26,27]
where one can obtain the eigenvalue equations from the
several low N values inside the package of [28], one should
analyze both the coefficients and the zero modes of the 20
terms in higher spin 4 current in order to obtain the
corresponding eigenvalue equations.
fixed.
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(2.35)
2.3.1 Eigenvalue equation of the zero mode of the higher
spin 4 current acting on the state |(0; v)
Let us consider the eigenvalue equation of the zero mode of
the A1 term of the higher spin 4 current in (2.14) acting on
the primary state (0; v)
dabcd ( J a J b J c J d )0|(0; v) .
Using the fact that the zero mode is nothing but the product
of each zero mode but the ordering is reversed [19,20], Eq.
(2.31) becomes
dabcd ( J d J c J b J a )0|(0; v) .
Note that the ground state transforms as a vector
representation with respect to J0a , while the zero mode K0a has vanishing
eigenvalue equation [22],
K0a |(0; v) = 0.
Equation (2.32) becomes 1 2N
dabcd (−i )4Tr(T d T c T b T a )|(0; v) .
In order to use the previous identity in (2.6), one can express
the above A1 term as follows:
1 dabcd ( J a J b J c J d + J b J c J a J d + J b J a J c J d
6
+ J c J a J b J d + J a J c J b J d + J c J b J a J d ),
due to the symmetric property of the d tensor. Then the
equivalent expression corresponding to (2.34) with (2.35) can be
written in terms of
(2.36)
(2.37)
1 1 dabcd Tr(T d T c T b T a + T d T a T c T b + T d T c T a T b
2N 6
+T d T b T a T c + T d T b T c T a + T d T a T b T c)
×|(0; v) .
The reason why the extra 21N exists is that one should have
the eigenvalue, not the trace. Using the identity (2.6), one can
reexpress (2.36) as
1 1 dabcd dcbad
2N 3
Here the identity (2.13) is used and we take the large N limit
at the last result in (2.37).
One can analyze the other 19 terms in (2.14). Among them,
the 16 terms which have the K a (z) spin 1 current do not
contribute to the eigenvalue equation, because one can take
the zero mode and change the ordering of the zero modes as
in (2.32). Then one can move the rightmost zero mode K0a to
the right and use the previous condition (2.33). On the other
hand, the remaining A6, A7 and A15 terms can contribute to
the eigenvalue equation.
The zero mode of the A6 term of the higher spin 4 current
acting on the primary state (0; v) is
(∂ J a ∂ J a )0|(0; v) = (∂ J a )0(∂ J a )0|(0; v)
= (− J0a )(− J0a )|(0; v) = J0a J0a |(0; v) ,
(2.38)
where the zero mode of ∂ J a in (2.38) can be obtained from
the usual mode expansion and is given by the zero mode
of − J a . Now using the correspondence (2.28), the above
expression leads to
21N Tr(i T ai T a )|(0; v) = − 21N 2δaa
1 1
2 2N (2N − 1) → −2N ,
= − 2N 2
where the extra factor 21N is considered as in (2.36) and the
large N limit is taken.
Now the final contribution from the zero mode of the A7
term of the higher spin 4 current acting on the primary state
(0; v) is given by
(∂2 J a J a )0|(0; v) = J0a (∂2 J a )0|(0; v) = J0a 2 J0a |(0; v) ,
where the zero mode of ∂2 J a in (2.40) can be obtained from
the usual mode expansion also and is given by the zero mode
of 2 J a . Therefore, one can follow the previous description.
It turns out that
(2.39)
(2.40)
1 1 δabδcd dcbad
2N 3
where the identity (2.12) is used in (2.43). Furthermore, the
A15 term itself behaves as N 0 in Appendix E. Then there is
no contribution at the leading order approximation.
By combining (2.37), (2.39) and (2.41) with the
corresponding coefficients in the large N limit of Appendix E, the
zero mode eigenvalue equation leads to
By following the procedure in the A1 term, one sees that the
above (2.42) can be written as
8N 3 A1 + (−2N ) N 2 12(2λ − 9) A1
5(2λ − 3)
+(−4N )
One can also calculate the same eigenvalue equation at finite
N and k2 corresponding to (2.44), which will appear later.
2.3.2 Eigenvalue equation of the zero mode of the higher
spin 4 current acting on the state |(v; 0)
Let us describe the eigenvalue equation of the zero mode of
the A1 term of the higher spin 4 current in (2.14) acting on
the primary state (v; 0),
dabcd ( J a J b J c J d )0|(v; 0) = dabcd J0d J0c J0b J0a |(v; 0) .
Note that the ground state transforms as a vector
representation with respect to K0a and the singlet condition for the
primary state (v; 0) can be described as [22]
( J0a + K0a )|(v; 0) = 0.
Then Eq. (2.45) is equivalent to
− dabcd J0d J0c J0b K0a |(v; 0) = −dabcd K0a J0d J0c J0b|(v; 0) ,
where Eq. (2.46) is used and the zero mode K0a is moved to
the left. Now the singlet condition is applied to the rightmost
J0b and we are left with
dabcd K0a J0d J0c K0b|(v; 0)
= dabcd K0a K0b J0d J0c|(v; 0) .
One can further take the previous steps and obtains
dabcd K0a K0b K0c K0d |(v; 0) .
Then using the correspondence (2.28), Eq. (2.49) becomes
1
2N
to
dabcd (−i )4Tr(T a T b T c T d )|(0; v) ,
which leads to the previous eigenvalue in (2.37).2
What happens for the A5 term of the higher spin 4 current
in (2.14)? According to the large N behavior of the
coefficient A5, this coefficient behaves as N1 in Appendix E and,
moreover, the analysis of eigenvalue equation leads to N 3
behavior. Therefore, the total power for the large N behavior
is given by N 2 and can be ignored in this approximation.
Let us move on the A6 term. The eigenvalue equation leads
(∂ J a ∂ J a )0|(v; 0)
= J0a J0a |(v; 0)
= − J0a K0a |(v; 0)
= K0a K0a |(v; 0) ,
where the singlet condition (2.46) is used. After using the
correspondence (2.28), this becomes the previous result in
(2.39).
Similarly, the A7 term eigenvalue equation gives
(∂ 2 J a J a )0|(v; 0)
= J0a 2 J0a |(v; 0)
= −2 J0a K0a |(v; 0)
= 2K0a K0a |(v; 0) ,
which leads to (2.41).
(2.48)
(2.49)
(2.50)
(2.54)
(2.55)
2 The eigenvalue equation of the zero mode of the A2 term of the higher
spin 4 current in (2.14) acting on the primary state (v; 0) can be written
as
dabcd ( J a J b J c K d )0|(v; 0) = dabcd K0d J0c J0b J0a |(v; 0) ,
(2.51)
which is equivalent to (2.47) with an extra minus sign due to the
symmetric property of the d symbol. Then we are left with the fact that Eq.
(2.51) is equal to the previous result (2.50) with minus sign.
The eigenvalue equation of the zero mode of the A3 term of the
higher spin 4 current in (2.14) acting on the primary state (v; 0) leads
to
dabcd ( J a J b K c K d )0|(v; 0) = dabcd K0d K0c J0b J0a |(v; 0) ,
(2.52)
which is equal to (2.48) and thus (2.52) becomes Eq. (2.50).
Similarly, the eigenvalue equation of the zero mode of the A4 term
of the higher spin 4 current in (2.14) acting on the primary state (v; 0)
can be described as
dabcd ( J a K b K c K d )0|(v; 0) = dabcd K0d K0c K0b J0a |(v; 0)
(2.53)
= −dabcd K0d K0c K0b K0a |(v; 0) ,
where the singlet condition is used and the above expression (2.53)
leads to (2.50) with an extra minus sign.
(2.59)
(2.60)
(2.62)
(2.63)
(2.64)
by writing the derivative term as the commutator of normal
ordered product. Then the zero mode of Eq. (2.59) is given
by
(K0b J0a J0b J0a − K0b J0b J0a J0a )|(v; 0)
= −(K0b K0a K0b K0a − K0b K0a K0a K0b)|(v; 0) .
Then Eq. (2.60) becomes
(−i )4Tr(T b T a T b T a − T b T a T a T b)|(v; 0) . (2.61)
1
− 2N
1
− 2N
Furthermore, Eq. (2.61) will reduce to
(−i )4i f bacTr(T b T a T c)|(v; 0)
1
= − 2N
(−i )4i f bac 1 Tr(T b T a T c − T b T c T a )|(v; 0) .
2
One can use the identity (2.5) and obtains, together with (2.8),
1
− 2N
(−i )4i f bac 1 2i f bac
2
1
= 2N
1
2(2N − 2) 2N (2N − 1) → 4N 2.
2
Let us focus on the A14 term. One has the relation
f abc J a K b∂ K c(z) = J a K b K a K b(z) − J a K b K b K a (z).
For the A8 and A9 terms of the higher spin 4 current, these
coefficients behave as N from Appendix E in the large N
limit and the corresponding eigenvalues behave as N . Then
the total power of the large N behavior is given by 2 and
these terms can be ignored at the leading order calculation.3
Let us consider the A13 term of the higher spin 4 current
in (2.14). One can easily see that
f abc J a ∂ J b K c(z) = J a J b J a K b(z) − J a J a J b K b(z),
3 Let us describe the next A10 term of the higher spin 4 current in
(2.14). One obtains
(∂ J a ∂ K a )0|(v; 0) = K0a J0a |(v; 0) = −K0a K0a |(v; 0) ,
(2.56)
where Eq. (2.56) is equivalent to Eq. (2.54) with an extra minus sign.
We can also calculate the eigenvalue equation for the A11 term,
(∂2 J a K a )0|(v; 0) = K0a 2 J0a |(v; 0) = −2K0a K0a |(v; 0) .
(2.57)
Equation (2.57) is equivalent to (2.55) with an extra minus sign.
One can continue to calculate the eigenvalue equation corresponding to
the A12 term as follows:
( J a ∂2 K a )0|(v; 0) = 2K0a J0a |(v; 0) = −2K0a K0a |(v; 0) .
(2.58)
Then Eq. (2.58) is the same contribution from A11 term.
The zero mode of (2.64) can be described as
(K0b K0a K0b J0a − K0a K0b K0b J0a )|(v; 0)
= −(K0b K0a K0b K0a − K0a K0b K0b K0a )|(v; 0) .
Then Eq. (2.65) becomes 1
by combining the first two generators. Equation (2.67) is
equivalent to (2.62) and (2.63).
Are there any contributions from the A15– A20 terms?
These coefficients behave as N 0, N12 , N1 , N 0, N1 and N 0,
respectively, from Appendix E. There are no contributions.
Then one obtains the final eigenvalue equation as follows:
W0(
4
)|(v; 0) =
8N 3 A1 − 8N 3
4λ
(λ − 1)
A1
(−i )4Tr(T b T a T b T a − T a T b T b T a )|(v; 0) . (2.66)
the two eigenvalue equations (2.44) and (2.68) lead to
W0(
4
)|(v; 0) = (1 + λ)(2 + λ)(3 + λ)|(v; 0) ,
W0(
4
)|(0; v) = (1 − λ)(2 − λ)(3 − λ)|(0; v) .
(2.65)
(2.67)
A1|(v; 0) .
(2.68)
(2.69)
(2.70)
If one takes the overall normalization factor for the W (
4
)(z)
as A4 rather than A1 as in (2.69), then A4 becomes A4 =
− 125N 3 λ3. In principle, one can calculate the OPE between
W (
4
)(z) and W (
4
)(w) from the explicit 20 terms in (2.14),
although the complete computation of the eighth-order
singular terms is rather involved for general (k2, N ) manually.
Then one expects that the central term, the eighth-order pole
of the above OPE, is given by A24 f (λ, N ) where f (λ, N ) is
a (fractional) function of λ and N (after the large N limit is
taken). That is, our normalization is given by the central term
of the OPE between the higher spin 4 current and itself which
25
behaves as 144N 6 λ6 f (λ, N ) where f (λ, N ) is not known at
the moment.
The above eigenvalues are also observed in [24] by
following the descriptions in [29] where the unitary case is
analyzed.
One of the primaries is given by (v; 0) ⊗ (v; 0) and the
other primary is given by (0; v) ⊗ (0; v) by pairing up
identical representations on the holomorphic and antiholomorphic
sectors in the context of diagonal modular invariant [15]. Let
us denote them as follows:
O+ = (v; 0) ⊗ (v; 0),
O− = (0; v) ⊗ (0; v).
The ratio of the three-point functions, from (2.70), is given
by
O+O+W (
4
) (1 + λ)(2 + λ)(3 + λ)
O−O−W (
4
) = (1 − λ)(2 − λ)(3 − λ)
,
in the notation of (2.71). This is the same form as for the
unitary case [21,30]. In the corresponding unitary bulk
calculation of [15], for λ = 21 , this ratio for generic spin is given
by (−1)s (2s −1) with spin s. One expects that the orthogonal
bulk computation will give rise to the behavior of (2.72).
2.3.3 Eigenvalue equation of the zero mode of the higher
spin 4 current acting on the state |(v; v)
For the primary (v; v) with the condition J0a |(v; v) = 0,
one can calculate the eigenvalue equation [8]. The non-trivial
contributions arise from the A5, A8, and A9 terms. It turns
out that
W0(
4
)|(v; v) = −N 2
48λ2(λ2 + 1)
5(λ − 3)(λ − 1)(2λ − 3)
A1|(v; v) .
(2.73)
(2.71)
(2.72)
In (2.73), Appendix E is used.
2.3.4 Further eigenvalue equations
We also present the eigenvalue equations [30] at finite N and
k2, by using Appendix E, as follows:
6A1
W0(
4
)|(0; v) = (3k2 + 2N − 2)d(1, k2, N )
Of course, the eigenvalue equations (2.74) become (2.44)
and (2.68), respectively, in the large N ’t Hooft limit.
Compared to the unitary case in [30], the above eigenvalues do
not have a simple factorized form. This is because of the fact
that the identities between f and d symbols contain rather
complicated functions of N .
For convenience, we also present the eigenvalue equations
for the spin 2 stress energy tensor (2.2) with k1 = 1,
k2
T0|(0; v) = 2(k2 + 2N − 1) |(0; v) →
(k2 + 4N − 3)
T0|(v; 0) = 2(k2 + 2N − 2) |(v; 0) →
(1 − λ)
2
(1 + λ)
2
Note that the conformal dimension of (0; v) can be obtained
from the formula [4, 8, 23, 31, 32]
1
h(0; v) = 2 (2N −1)
1 1
1+(2N −2) − 1+k2 +(2N − 2)
|(0; v) ,
|(v; 0) .
(2.75)
(2.76)
k2
= 2(k2 + 2N − 1)
,
where the overall factor 21 (2N − 1) is the quadratic Casimir
eigenvalue of the S O (2N ) vector representation. Similarly,
the conformal dimension of (v; 0) can be obtained,
Then the two results (2.76) and (2.77) are coincident with the
ones in (2.75).4
2.4 The OPE between the higher spin 4 current and itself
where k1 = 1, N = 4 and k2 is arbitrary
Let us describe the OPE between the higher spin 4 current
and itself. Because it is rather involved to calculate this OPE
manually, one fixes the value of N and then one can
compute this OPE inside the package of [28]. For fixed N = 4,
which is the lowest value one can consider non-trivially, one
obtains the fourth-order pole of this OPE, by realizing that
the right structure constants should behave according to the
well-known results [33], as follows:
3 42
= 10 ∂ 2 T (w) + (5c + 22)
18(c + 24) W (
4
)(w)
(5c + 22)
Here the central charge reduces to
4k2(k2 + 13)
c(k1 = 1, k2, N = 4) = (k2 + 6)(k2 + 7) ,
which can be obtained from (2.4) by substituting the two
values of k1 = 1 and N = 4. The overall factor can be fixed
as
4 Furthermore, one can write down the eigenvalue equation for the
state |(v; v)
(2N − 1) λ2
T0|(v; v) = 2(k2 + 2N − 2)(k2 + 2N − 1) |(v; v) → 4N |(v; v) .
(2.78)
Note that in the large N ’t Hooft limit the eigenvalue (2.78) reduces to
zero.
The conformal dimension of (v; v) can be obtained as follows:
1
h(v; v) = 2 (2N − 1)
1 1
k2 + (2N − 2) − 1 + k2 + (2N − 2)
(2N − 1)
= 2(k2 + 2N − 2)(k2 + 2N − 1)
.
This looks similar to the unitary case [29]: the overall factor is again the
quadratic Casimir eigenvalue of S O(2N ) in the vector representation.
In the denominator one has (k2 + 2N − 2) and this quantity plus one.
There exists a relation together with (2.76), (2.77) and (2.79),
h(v; v) = h(0; v) + h(v; 0) − 1,
which was also observed in [23]. The identity in (2.80) is checked from
(2.75) and (2.78).
(2.81)
(2.82)
(2.79)
(2.80)
k2
A1(k1 = 1, k2, N = 4) = 2520(k2 + 7)
×
by comparing the coefficient of the first term in the right-hand
side of (2.81).
Let us emphasize that there exists a new primary field in
(2.81) which is given by
(k2 − 1)k2 J a J a J b J b
(2.84)
W (
4
)(z)
1
In other words, there exists a nonzero expression by
combining the fourth-order pole with the first line of (2.81)
with minus sign. Furthermore, one can express the various
nonzero terms as the one in (2.84). One can easily see that
the ten operators except the last operator appear in the
previous higher spin 4 current in (2.14). It is straightforward to
analyze the description appearing in Appendices B and C for
the last operator in (2.84).
Let us further restrict to the simplest case where one can
see the full structure of the corresponding OPE without
losing any terms in the right-hand side. In other words, in the
particular limit where k2 → ∞ corresponding to c = 4, the
structure constants do not vanish. That is, there is no (c − 4)
factor in the right-hand side of the OPE.
Then the higher spin 4 current can be written in terms of
W (
4
)(z)
k1=1,k2→∞,N =4
1√ (dabcd J a J b J c J d
= 2520 3
+18∂ J a ∂ J a − 12 ∂2 J a J a
−3 J a J a J b J b)(z),
(2.85)
by substituting N = 4 and k2 → ∞ limit in Appendix E.
The normalization factor is consistent with the general form
in (2.83). The field contents in (2.85) are given in terms of
the numerator spin 1 current (having the level k1 = 1) of
the coset model. Of course, the stress energy tensor contains
only the first term with k1 = 1 in (2.2) in this limit.
Then one can obtain the corresponding higher spin 4
current from (2.84) by taking the k2 → ∞ limit and it turns out
that
W (
4
)(z)
k1=1,k2→∞,N =4
1
+ 2
C444 ∂ W (
4
) (w)
1
+ (z − w)2
× ∂2
1 5 42
168 2 ∂4T + 36 (5c + 22)
− 130 ∂2T + 356 C444 ∂2W (
4
)
24(72c + 13)
+ (5c + 22)(2c − 1)(7c + 68)
− 53 ∂2T T + 710 ∂4T
(95c2 + 1254c − 10904)
− 6(5c + 22)(2c − 1)(7c + 68)
− 59 ∂2T T + 730 ∂4T
T (T 2
1 (95c2 + 1254c − 10904)
− 2 6(5c + 22)(2c − 1)(7c + 68)
× ∂
Here the central charge coming from (2.82) is given by
c(k1 = 1, k2 → ∞, N = 4) = 4,
from (2.4) by substituting the right numbers. Moreover, the
two structure constants are given by
C444 =
18(c + 24)
(5c + 22)
, C464 =
12(c − 1)(11c + 656)
(2c − 1)(7c + 68)
,
together with (2.88). Note that there are two extra last lines
in (2.87) associated with the new primary higher spin 4
current, compared to the previous result in [33]. Equation (2.89)
already appeared in [33–35].
2.5 Next higher spin currents
In the second-order pole of (2.87), there exists a primary
higher spin 6 current. One can imagine the six products of
spin 1 current with correct contractions of S O(2N ) indices.
Let us consider the higher spin 4 current W (
4
)(z) which
contains dabcd J a J b J c J d (z) and the same higher spin 4
current which contains dd e f g J d J e J f J g(z). Then one has
the second-order pole of this OPE, dabcd dd e f gδdd J a J b J c
J e J f J g(w), by considering the singular term between J d (z)
and J d (w). This gives rise to the term of dabcd dde f g J a
J b J c J e J f J g(w). Then one expects that the higher spin
6 current contains this term and is given by W (
6
)(z) =
dabcd dde f g J a J b J c J e J f J g(z) + · · · . According to the
description of [17,18], the tensorial structure of S O(2N )
symmetric invariant tensor of rank 6 can be determined by the
product of two rank 4 d symbols. Therefore the above
expression can be rewritten in terms of d tensor of rank 6 and one
should have W (
6
)(z) = dabce f g J a J b J c J e J f J g(z) + · · · . It
would be interesting to observe the full expression for the
higher spin 6 current.
3 Higher spin currents with N = 1 supersymmetry in
the stringy coset model with two levels (2N − 2, k2)
In the presence of adjoint fermions coming from the equality
of one of the levels and the dual Coxeter number of S O(2N ),
one can construct the higher spin 27 current which is the
superpartner of the previous higher spin 4 current. In doing this,
the role of the spin 23 current living in the N = 1
superconformal algebra is crucial. The OPE between this N = 1
lowest higher spin multiplet, denoted by (
27 , 4
), is described
using the Jacobi identities.
3.1 Spin 23 , 2 currents and N = 1 superconformal algebra
The spin 23 current can be obtained from the spin 21 current
and spin 1 current as follows [6,16]:
G(z) =
4(N − 1)
(2N − 2 + k2)(4N − 4 + k2)
×
Furthermore, we can express the spin 1 current from the
above spin 21 current satisfying (3.2) as
J a (z) ≡ f abcψ bψ c(z).
It is easy to check that this spin 1 current satisfies the first
equation of (2.3) with k1 = (2N − 2).
Then it is easy to see that there are only two terms in (3.1)
and the relative coefficients can be fixed by using the above
spin 23 current, which should transform as a primary field
under the stress energy tensor (2.2) with k1 = (2N − 2) as
follows:
1 3 1
T (z) G(w) = (z − w)2 2 G(w) + (z − w) ∂ G(w) + · · · .
In other words, the condition (3.4) determines the relative
coefficients of (3.1).
The overall factor in (3.1) can be determined by the
following OPE between the spin 23 current and itself:
1 2c 1
G(z) G(w) = (z − w)3 3 + (z − w) 2T (w) + · · · .
Here the central charge in (3.5) is given by (2.4) with the
condition k1 = (2N − 2).
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
It is useful to write down the following OPEs which will
be used in later calculations:
1 1
Gˆ (z) ψa(w) = (z − w) 2
− 2(Nk2− 1) J a + K a (w) + · · · ,
1
Gˆ (z) J a(w) = − (z − w)2 k2ψa(w)
1
+ (z − w)
1
Gˆ (z) K a(w) = (z − w)2 k2ψa(w)
Compared to the unitary case [36–38], the behavior of relative
coefficient, which is equal to one over three times the level
divided by the dual Coxeter number, occurs in (3.7). See also
[6,16].
3.2 Eigenvalue equation of the zero mode of the higher
spin 4 current
We also present the eigenvalue equations for the spin 2 stress
energy tensor (2.2) with k1 = (2N − 2),
k2(2N − 1)
T0|(0; v) = 8(N − 1)(k2 + 4N − 4) |(0; v)
(1 − λ)
→ 4(1 + λ) |(0; v) ,
(2N − 1)(k2 + 6N − 6)
T0|(v; 0) = 8(N − 1)(k2 + 2N − 2) |(v; 0)
(1 + 2λ)
→ 4 |(v; 0) .
In (3.8), the large N ’t Hooft limit is taken at the final stage.
Note that the conformal dimension of (0; v) can be obtained
from the formula
1
h(0; v) = 2 (2N − 1)
1
(2N − 2) + (2N − 2)
1
− (2N − 2) + k2 + (2N − 2)
k2(2N − 1) ,
= 8(N − 1)(k2 + 4N − 4)
where the overall factor 21 (2N − 1) is the quadratic Casimir
eigenvalue of the S O(2N ) vector representation. Similarly,
(3.13)
(3.14)
the conformal dimension of (v; 0) can be obtained:5
One way to determine the higher spin 27 current is to use the
OPE between the spin 23 current and the higher spin 4 current
in previous section. Note that the corresponding coefficients
at the critical level k1 = (2N − 2) are given in Appendix
F. In other words, from the N = 1 super primary condition
[36,37], one should have
5 Moreover, the eigenvalue equation for the state |(v; v) can be
obtained as follows:
(N −1)(2N −1) λ2
T0|(v; v) = (k2 +2N −2)(k2 +4N −4) |(v; v) → 2(λ+1) |(v; v) .
(3.10)
The conformal dimension of (v; v) in (3.10) can also be obtained as
follows:
1 1 1
h(v; v) = 2 (2N −1) k2 + (2N − 2) − (2N − 2)+k2 +(2N −2)
(N − 1)(2N − 1)
= (k2 + 2N − 2)(k2 + 4N − 4)
.
There exists a relation together with (3.9), (3.13) and (3.11),
h(v; v) = h(0; v) + h(v; 0) − (42(NN −− 11)) .
Here the last term in (3.12) is the ratio of quadratic Casimir eigenvalue
for the vector representation and the dual Coxeter number of S O(2N ).
(3.11)
(3.12)
Gˆ (z) W (
4
)(w)
In order to calculate the second-order pole of (3.15), one
can use the three OPEs in (3.6). The explicit results are given
in Appendix G. Of course, this will give us the final higher
spin 27 current but it is rather non-trivial to simplify in simple
form. Therefore, after we identify the correct field contents
for fixed N = 4, we introduce the undetermined coefficients
and fix them using the previous methods we used in
previous section. That is, the higher spin 27 current should not
have any singular terms with the diagonal spin 1 current and
transform as a primary higher spin current under the stress
energy tensor.
Then one can express the higher spin 27 current as follows
[6,19,20]:
7
W (
2
)(z) = B1 dabcd ψ a J b J c J d (z)
+B2 dabcd f ae f f beg J c J d ψ f K g(z)
+B3 dabcd J a K bψ c K d (z)
+B4 dabcd f ae f f beg K c K d ψ f K g(z)
+B5 J a ψ a J b J b(z) + B6 K a K a ψ b K b(z)
+B7 J a J a ψ b K b(z) + B8 J a J a ψ b J b(z)
+B9 ψ a K a K b K b(z)
+B10 f abc f cde K a K eψ b K d (z)
+B11 J a ψ a K b K b(z) + B12 ψ a J b K a K b(z)
+B13 J a J bψ a K b(z)
+B14 f abc f cde J a J eψ b K d (z)
+B15 J a J b K a ψ b(z)
+X (k2, N )(G T − 18 ∂2G)(z).
(3.16)
The B7 term can be written as (ψ a J b J b K a − 2 f abcψ a ∂
J b K c − (2N − 2)∂2ψ a K a )(z) by moving the field ψ b
to the left. Similarly, the B8 term can be described as
(ψ a J a J b J b+(2N −2)∂2ψ a J a −(2N −2)ψ a ∂2 J a +2(2N −
2)∂ψ a ∂ J a )(z). For the B13 term one obtains (ψ a J b J a K b +
(2N − 2)∂2ψ a K a )(z). For the B14 term one can write down
(3(2N − 2) f abcψ a ∂ J b K c + (2N − 2)2∂2ψ a K a )(z). For the
B15 term, one has (ψ a J b J a K b+ f abcψ a ∂ J b K c)(z) by
moving ψ b to the left. Furthermore, the B2 term and the B4 term
can be simplified using the identity (2.10). For the remaining
other terms, the fermion ψ a can be moved to the leftmost
position without any extra terms because of the properties
of the f and d symbols. The B5, B6, B7, B11, B12, and B13
terms can be seen from G T (z). The B8, B9, and B15 terms
(3.17)
(3.18)
are written in terms of B5, B6, and B13 terms plus derivative
terms, respectively.
Note that the last term in (3.16) is a quasiprimary field in
the sense that the OPE between the stress energy tensor and
this field does not contain the third-order pole. We realize
that this term does not appear for the particular N = 4 case.
We would like to determine the undetermined coefficients
B1–B15 and X (k2, N ) in (3.16). As in (2.15), one should
have the regular condition as follows:
7
J a (z) W (
2
)(w) = + · · · .
In Appendix H, we present the OPEs between the diagonal
spin 1 current and the 15 fields in (3.16). Moreover, the higher
spin 27 current transforms as a primary field under the stress
energy tensor. In other words, one has
7
Tˆ (z) W (
2
)(w)
(2N −2+k2)(4N −4+k2) W (
4
)(w). In doing this, the OPEs in
4(N −1)
(3.6) are crucial. In order to see the presence of higher spin
4, the rearrangement of the normal ordered product should
be taken because the above first-order pole terms contain
unwanted terms. Of course, we do not have to worry about
the extra contractions in the OPEs because we are interested
in the first-order pole as described above.
3.4 The OPEs between the higher spin 27 , 4 currents
It is natural to ask how the OPEs between the higher spin 27
current and the higher spin 4 current arise. They have rather
long expressions for the N = 4 case.
Therefore, one tries to obtain the corresponding OPEs
from the Jacobi identities for the above higher spin currents
and other relevant higher spin currents. We will consider
7 7 7
only the three OPEs, W (
2
)(z) W (
2
)(w), W (
2
)(z) W (
4
)(w)
and W (
4
)(z) W (
4
)(w). What kind of new primary higher
spin currents are present in the right-hand side of OPEs?
7 7
From the OPEs of W (
2
)(z) W (
4
)(w) or W (
4
)(z) W (
2
)(w),
one can think of the presence of a new higher spin 123
current at the first order pole. Furthermore, from the OPE
W (
4
)(z) W (
4
)(w), the new higher spin 6 current can appear
in the second-order pole of this OPE. Note that there is no
new higher spin 7 current in the first-order pole. The reason
is as follows. One can calculate the OPE W (
4
)(w) W (
4
)(z)
in the presence of the new higher spin 7 current at the
firstorder pole, use the symmetry z ↔ w and end up with the
OPE W (
4
)(z) W (
4
)(w). By focusing on the new higher spin
7 current, one realizes that there exists an extra minus sign.
Therefore, the new higher spin 7 current should vanish.
Then one can assign the above two higher spin
currents as one single N = 1 higher spin current, denoted by
(
6 , 123
) where the numbers stand for each spin. From the
OPE in W (
4
)(z) W (
4
)(w), the second-order pole provides a
new higher spin 6 current. Then one can turn to the N = 1
higher spin current denoted by (
121 , 6
). Furthermore, from the
bosonic higher spin 4 current in the previous section, one can
introduce its superpartner whose spin is given by 29 . The
corresponding N = 1 higher spin current is characterized by
(
4 , 9
) using the above notation.
2 7 7
For N = 4, one can calculate the OPE in W (
2
)(z) W (
2
)(w).
By requiring that the seventh-order pole should be equal to
27c , one can determine the coefficient A1 as
k2
A1(k1 = 6, k2, N = 4) = 5040(k2 + 12)
×
(k2 + 2)(k2 + 4)
6(k2 + 6)(k2 + 12)(k2 + 14)(k2 + 16)
.
The fifth-order pole gives 2T (w) and the fourth-order pole
gives ∂ T (w). Similar behaviors arise in (2.87). Let us
describe the third-order pole. One can easily check that the
following quantity together with (3.19):
8(37c + 3)T T + 3(2c − 117)∂ GG
1
(4c + 21)(10c − 7)
3
− 10 (302c − 327)∂2T (w)
is a quasiprimary field. The third-order pole subtracted by
both (3.20) and 130 ∂2T (w) (which is a descendant field) is
a primary field. However, this is not written in terms of the
previous higher spin 4 current. This implies that there exists
a new primary higher spin 4 current. The structure constants
appearing in (3.20) are obtained from the Jacobi identities.
Because we are dealing with the extensions of the N = 1
superconformal algebra, the ∂ GG(w) term appears in
addition to T T (w) and ∂2T (w).
By assuming that the N = 1 OPE between the N = 1
higher spin 27 multiplet contains the N = 1 higher spin
4 , 121 , 6 multiplets, one obtains the complete structure of
these OPEs in a component approach (and N = 1
superspace). They are given in Appendix K in terms of the
central charge and some undetermined structure constants. It
would be interesting to see whether there exist other
additional higher spin currents or not. See also the work in [39]
where the Jacobi identities are used.
(3.19)
(3.20)
3.5 The OPE in the N = 1 superspace
From the three OPEs in a component approaches described
in Appendix K, one summarizes its N = 1 superspace in
simple notation as follows:
where [I] appearing in (3.21) is the N = 1 superconformal
family of the identity operator. According to the field contents
in [40], where k2 is fixed as k2 = 1, the above OPE should
not contain the N = 1 higher spin integer multiplets. See
also [41]. The right-hand side should contain the first, the
second and the fourth terms. It would be interesting to observe
this behavior explicitly. First of all, the single higher spin 4
current should exist by combining the previous two kinds of
higher spin 4 currents under the constraint k2 = 1.
4 Higher spin currents with N = 2 supersymmetry in
the stringy coset model with two levels
(2N − 2, 2N − 2)
The additional adjoint fermions allow us to construct the
spin 1, 23 currents in the N = 2 superconformal algebra.
Furthermore, the additional higher spin 3, 27 currents can be
found explicitly along the line of [42]. The lowest higher spin
3 current of U (
1
) charge 43 is obtained and it can be written in
terms of two adjoint fermions. There exists another N = 2
higher spin multiplet, which consists of the above same spin
contents, (
3, 27 , 27 , 4
) with different U (1) charges. Finally,
the OPE between these two N = 2 higher spin multiplets is
described.
4.1 Spin 1, 23 , 23 , 2 currents and N = 2 superconformal
algebra
Let us introduce the second adjoint fermions which satisfy
the following OPE:
1 1 δab
χ a (z) χ b(w) = − (z − w) 2
It is easy to see that one can express the spin 1 current from
the above spin 21 current with (4.1) as
K a (z) ≡ f abcχ bχ c(z).
This spin 1 current satisfies the second equation of (2.3) with
k2 = (2N − 2).
Then it is straightforward to construct the four generating
currents, denoted by (
1, 23 , 23 , 2
), corresponding to the N = 2
(4.1)
(4.2)
superconformal algebra as follows [43]:
2
J (z) = 3 i ψ a χ a (z),
1
G+(z) = − 6√3(2N − 2)
1
G−(z) = − 6√3(2N − 2)
× ψ a J a − 3 ψ a K a − i χ a K a + 3 i χ a J a (z),
× ψ a J a − 3 ψ a K a + i χ a K a − 3 i χ a J a (z),
1 1
T (z) = − 4(2N − 2) J a J a (z) − 4(2N − 2) K a K a (z)
1 ( J a + K a )( J a + K a )(z).
+ 6(2N − 2)
(4.3)
By realizing that the difference between G+(z) and G−(z)
occurs in the third and fourth terms, under χ a (z) → −χ a (z),
one sees the relation G+(z) ↔ G−(z).
Let us introduce the following spin 1 current by taking the
product of two adjoint fermions:
La ≡ f abcψ bχ c.
1
c = 3 N (2N − 1),
The central charge can be reduced to
which can be obtained from (2.4) by substituting the
corresponding two levels. In order to construct the higher spin
currents, let us introduce the following intermediate spin 2
current:
M1a ≡ dabcd ψ bχ c J d ,
M2a ≡ dabcd ψ bχ c K d ,
M3a ≡ dabcd ψ bχ c Ld ,
together with (3.3), (4.2) and (4.4). Compared to the unitary
case in [44], the contracted indices appear in the two different
adjoint fermions (because of the symmetric d tensor) as well
as the spin 1 currents.
4.2 Higher spin 3, 27 , 27 , 4 currents
From the experience of Sects. 2 and 3, there exist the higher
spin 4 current and the N = 1 higher spin 27 current denoted
by (
27 , 4
); then there are two choices where the above N = 1
higher spin 27 multiplet can arise from the lower two
component currents or higher two component currents. Let us try
to find the higher spin currents by taking the second choice.
By writing the possible candidate terms for the higher spin
3 current, one can think of the product of spin 1 currents (3.3),
(4.2) or (4.4) and the intermediate spin 2 currents in (4.6).
Furthermore, one can think of the product of each component
field in the spin 23 currents living in the N = 2
superconformal algebra. Of course, one should consider the possible
(4.6)
derivative terms. Therefore, one can consider the following
higher spin 3 current [6]:
W (
43
)(z) = a1 J a M1a(z) + a2 K a M1a(z) + a3 La M1a(z)
3
+a10 J a∂ J a(z) + a11 J a∂ K a(z)
+a4 J a M2a(z) + a5 K a M2a(z) + a6 La M2a(z)
+a7 J a M3a(z) + a8 K a M3a(z) + a9 La M3a(z)
+a12 J a∂ La(z) + a13 ∂ J a K a(z) + a14 K a∂ K a(z)
+a15 K a∂ La(z) + a16 ∂ J a La(z)
+a17 ∂ K a La(z) + a18 La∂ La(z)
+a23 (ψa K a)(ψ b K b)(z)
+a19 (ψa J a)(ψ b J b)(z) + a20 (ψa J a)(ψ b K b)(z)
+a21 (ψa J a)(χ b J b)(z) + a22 (ψa J a)(χ b K b)(z)
+a24 (ψa K a)(χ b J b)(z) + a25 (ψa K a)(χ b K b)(z)
+a26 (χ a J a)(χ b J b)(z)
+a27 (χ a J a)(χ b K b)(z) + a28 (χ a K a)(χ b K b)(z).
The U (
1
) charge 43 will be determined later.
As done in previous sections, one can use two
requirements in order to fix the above coefficients. One of them is
the regularity with the diagonal spin 1 current as follows:
J a (z) W 4(
3
)(w) = + · · · .
3
Here the diagonal spin 1 current in (4.8) is the sum of (3.3)
and (4.2). The other is given by the primary condition, which
can be described as follows together with (4.7):
Tˆ (z) W 4(
3
)(w)
3
3i
+ 2 (a7 − a8) (ψ a K a )(χ b J b)
i
+ 2 (a7 − a8) (ψ a K a )(χ b K b)
+(a7 − a8) (χ a J a )(χ b K b) (z).
(4.10)
Note that there exist also a4, a11, a14, a20, a24, and a28
dependent terms (other coefficients depend on these six
coefficients and a7 and a8 after the above two conditions are used)
but they are identically zero, respectively. From the
definitions of (4.6), the first eight terms in (4.10) contain the rank 4
d symbol. One can see the common nonderivative expression
in the third term and sixth term and then one can combine
them with the coefficient (a7 − a8). Similarly, the fifth term
and the seventh term share the common nonderivative term
with the coefficient −(a7 − a8). Furthermore, the
composite fields appearing in (4.10) contain the various derivative
terms (it is obvious that the ninth–twelfth terms do have the
derivative terms and also they can appear from the ordering
for the composite fields) but the precise coefficients will lead
to the vanishing of these derivative terms.
For the extended N = 2 superconformal algebra, there is
one additional condition for the higher spin current which is
the U (
1
) charge (i.e., the coefficient of the first-order pole of
the OPE with the spin 1 current). That is [44],
1
J (z) Wq(
3
)(w) = (z − w) q Wq(
3
)(w) + · · · .
It turns out that the U (
1
) charge is fixed and for q = 43 , there
is a relation a12 = 34i (a7 −a8). This relation is used in (4.10).
For q = − 43 , there is a relation a12 = − 34i (a7 − a8). It is
useful to express the above higher spin 3 current in a
manifestly U (
1
) charge symmetric way. Let us focus on the first
term in (4.10). If one substitutes the definition of M1a in (4.6),
one has f abcψ bψ cdade f ψ d χ e J f (w) where J a is replaced
by the fermions. One substitutes for the J f using the
relation (3.3) and obtains f abcdade f f f gh ψ bψ cψ d χ eψ gψ h (z).
Now move the composite field ψ d χ e to the right. One obtains
f abcdade f f f gh ψ bψ cψ gψ h ψ d χ e(z), which can be written
in terms of 2i f abcdade f f f gh ψ bψ cψ gψ h (ψ d + i χ d )(ψ e −
i χ e)(z) from the symmetric property of dade f . Then the
overall factor is given by 18 (a7 − a8) by considering the numerical
factor − 4i (a7 − a8).6 Let us describe the eighth term which
is the last term which contains the d symbol. So one has
(4.11)
6 Similarly, the second term can be analyzed also. The relevant term can
be written in terms of f abcdadef f f ghχbχcψd χeψ gψh(z), which can
also be expressed as f abcdadef f f ghχbχcψ gψhψd χe(z). Once again
i f abcdadef f f ghiχbiχcψ gψh(ψd +
this can be described as − 2
iχd )(ψe − iχe)(z) as done before. The overall factor of the
second term is given by 2i (a7 − a8). Then the total overall
factor gives 14 (a7 − a8). Intentionally, we rewrite the above as
18 (a7 − a8) f abcdadef f f gh(iχbiχcψ gψh + iψbiψcχ gχ h)(ψd + iχd )
f abcdade f f f gh ψ bχ cψ gχ h ψ d χ e(z), which can be
identified with − 2i f abcdade f f f gh ψ bi χ cψ gi χ h (ψ d + i χ d )(ψ e −
i χ e)(z). By multiplying the overall factor i (a7 − a8),
one obtains 21 (a7 − a8) f abcdade f f f gh ψ bi χ cψ gi χ h (ψ d +
i χ d )(ψ e − i χ e)(z). This can be further rewritten in terms of
18 (a7 − a8) f abcdade f f f gh (ψ bi χ cψ gi χ h + ψ bi χ ci χ gψ h +
i χ bψ cψ gi χ h + ψ bi χ ci χ gψ h )(ψ d + i χ d )(ψ e − i χ e)(z).
Finally, one can summarize the first eight terms in (4.10) that
are given by 18 (a7 − a8)dabcd f ae f f bgh (ψ e + i χ e)(ψ f +
i χ f )(ψ g + i χ g)(ψ h + i χ h )(ψ c + i χ c)(ψ d − i χ d )(z).
Now we are considering the last six terms in (4.10).
The first term is given by −(a7 − a8) f acd ψ a ψ cψ d f be f ψ b
χ eχ f (z). This can be rewritten as − 41 (a7 − a8)( f acd ψ a ψ c
ψ d f be f ψ bχ eχ f + 3 f acd ψ ai χ ci χ d f be f ψ bψ eψ f )(z)
where we use the fact that there exists a minus sign when
the first three factors ψ ai χ ci χ d move to the right. Therefore,
there should be an overall factor 41 . One can analyze the other
four terms.7 Finally, one can summarize the last six terms in
(4.10) are given by − 41 (a7 − a8) f abc f de f (ψ a + i χ a )(ψ b +
i χ b)(ψ c + i χ c)(ψ d + i χ d )(ψ e + i χ e)(ψ f − i χ f )(z).
By putting (a7 − a8) = 1, one obtains the higher spin 3
current with U (
1
) charge 43 as follows:
W 4(
3
)(z) = 18 dabcd f ae f f bgh (ψ e + i χ e)(ψ f + i χ f )
3
×(ψ g + i χ g)(ψ h + i χ h )(ψ c + i χ c)(ψ d − i χ d )(z)
− 41 f abc f de f (ψ a + i χ a )(ψ b + i χ b)(ψ c + i χ c)
×(ψ d + i χ d )(ψ e + i χ e)(ψ f − i χ f )(z).
(4.13)
One can calculate the U (
1
) charges for the adjoint fermions
with (4.3) as follows:
J (z) (ψa + i χ a)(w) = (z −1 w) 31 (ψa + i χ a )(w) + · · · ,
1 1
J (z) (ψa −i χ a )(w) = (z − w) (−1) 3 (ψa −i χ a )(w)+· · · .
(4.14)
Footnote 6 continued
(ψe − iχe)(z) using the property of the d symbol. One can analyze the
other terms up to the seventh term.
7 Let us describe the last term, which is given by −(a7 −
a8) f acd iχaψcψd f bef iχbχeχ f (z). As above, this can be
written as − 41 (a7 − a8)(3 f acd iχaψcψd f bef iχbχeχ f −
f acd iχaχcχd f bef iχbψeψ f )(z).
There are also identities as follows:
f abcψaψbψc f def ψd ψeψ f = 0, f abcψaχbχc f def ψd χeχ f = 0,
f abcχaψbψc f def χd ψeψ f = 0, f abcχaχbχc f def χd χeχ f = 0.
(4.12)
As explained before, Eq. (4.12) can be checked by moving the first three
fermions to the right; there exists a minus sign.
Then it is obvious that the above higher spin 3 current (4.13)
has U (
1
) charge 43 : there exist five factors with U (
1
) charge
13 and one factor with U (
1
) charge − 31 according to (4.14).
For the unitary case [44], one sees the factor f ae f f bgh (ψ e
+i χ e)(ψ f +i χ f )(ψ g +i χ g)(ψ h +i χ h ) and the other factor
is given by dabc f chi (ψ h −i χ h )(ψi −i χ i ) of the U (
1
) charge
− 23 in the nonderivative terms. However, the orthogonal case
contains the different factor dabcd (ψ c + i χ c)(ψ d − i χ d ) of
U (
1
) charge 0 in (4.13).
In order to obtain the other higher spin currents, it is useful
to calculate the following OPEs:
f abc(ψ b + i χ b)(ψ c + i χ c)(w) + · · · ,
G+(z) (ψ a + i χ a )(w) = + · · · ,
G+(z) (ψ a − i χ a )(w) = (z −1 w)
1
× 2√3(2N − 2)
G−(z) (ψ a − i χ a )(w) = + · · · ,
G−(z) (ψ a + i χ a )(w) = (z −1 w)
f abc(ψ b − i χ b)(ψ c − i χ c)(w) + · · · .
(4.15)
(4.16)
We will use this property to calculate the OPEs for the
particular singular terms. One sees the U (
1
) charge conservation
in (4.15).
How does one determine the other higher spin currents
related to the lowest one? Let us recall the following OPE
[44–46]:
G+(z) W 43(
3
)(w) = − (z −1 w) W 37(
27
)(w) + · · · .
Here the higher spin current appears in the first-order pole.
Once we have obtained the first-order pole in the above OPE,
then we obtain the higher spin current. See also the relevant
work in [47]. Because the lowest higher spin 3 current is
written in terms of adjoint fermions, it is better to calculate
the OPE between G+(z) and fermions appearing in (4.13).
According to the observations of (4.15), the spin 23 current
G+(z) has non-trivial OPE with the spin 21 current of U (
1
)
charge − 31 , while the spin 23 current G−(z) has non-trivial
OPE with the spin 21 current of U (
1
) charge 13 . Then it is
obvious that when one calculates the left-hand side of (4.16),
the only non-trivial singular terms appear at the location of
the last factors, (ψ d −i χ d )(w) and (ψ f −i χ f )(w) in (4.13).
This leads to the following higher spin 27 current of U (
1
)
charge 73 :
W 7(
27
)(z) = 2√3(21N − 2) 81 dabcd f ae f f bgh f di j
3
×(ψ e +i χ e)(ψ f +i χ f )(ψ g +i χ g)(ψ h +i χ h )
×(ψ c + i χ c)(ψi + i χ i )(ψ j + i χ j )
×(ψ a +i χ a )(ψ b +i χ b)(ψ c +i χ c)(ψ d +i χ d )
×(ψ e + i χ e)(ψ g + i χ g)(ψ h + i χ h ) (z).
In (4.17), the N dependence appears in the overall factor
rather than the relative coefficients. One easily sees that the
above two expressions preserve the U (
1
) charge by counting
the U (
1
) charge at each factor. In other words, each factor
has a U (
1
) charge of 13 .
From the OPE [44]
G−(z) W 43(
3
)(w) = (z −1 w) W 31(
27
)(w) + · · · ,
one can obtain the other higher spin 27 current of U (
1
) charge
13 . It turns out, from the first-order pole of (4.18), that
(4.17)
(4.18)
W 1(
27
)(z) = 2√3(21N − 2) 81 dabcd f ae f f bgh
3
×[+ f ei j ((ψi − i χ i )(ψ j − i χ j ))(ψ f + i χ f )(ψ g + i χ g)
×(ψ h + i χ h )(ψ c + i χ c)(ψ d − i χ d )
− f f i j (ψ e + i χ e)((ψi − i χ i )(ψ j − i χ j ))(ψ g + i χ g)
×(ψ h + i χ h )(ψ c + i χ c)(ψ d − i χ d )
+ f gi j (ψ e + i χ e)(ψ f + i χ f )((ψi − i χ i )(ψ j − i χ j ))
×(ψ h + i χ h )(ψ c + i χ c)(ψ d − i χ d )
− f hi j (ψ e + i χ e)(ψ f + i χ f )(ψ g + i χ g)
×((ψi − i χ i )(ψ j − i χ j ))(ψ c + i χ c)(ψ d − i χ d )
+ f ci j (ψ e + i χ e)(ψ f + i χ f )(ψ g + i χ g)(ψ h + i χ h )
×((ψi − i χ i )(ψ j − i χ j ))(ψ d − i χ d )](z)
1 1 f abc f de f
− 2√3(2N − 2) 4
×[+ f ai j ((ψi − i χ i )(ψ j − i χ j ))(ψ b + i χ b)
×(ψ c + i χ c)(ψ d + i χ d )(ψ e + i χ e)(ψ f − i χ f )
− f bi j (ψ a + i χ a )((ψi − i χ i )(ψ j − i χ j ))(ψ c + i χ c)
×(ψ d + i χ d )(ψ e + i χ e)(ψ f − i χ f )
+ f ci j (ψ a + i χ a )(ψ b + i χ b)((ψi − i χ i )(ψ j − i χ j ))
×(ψ d + i χ d )(ψ e + i χ e)(ψ f − i χ f )
− f di j (ψ a + i χ a )(ψ b + i χ b)(ψ c + i χ c)
×((ψi − i χ i )(ψ j − i χ j ))(ψ e + i χ e)(ψ f − i χ f )
+ f ei j (ψ a + i χ a )(ψ b + i χ b)(ψ c + i χ c)(ψ d + i χ d )
×((ψi − i χ i )(ψ j − i χ j ))(ψ f − i χ f )](z).
(4.19)
From (4.15), the OPE between G−(z) and (ψ a − i χ a )(w)
does not have any singular terms and the contribution from
this OPE in (4.19) vanishes. Note that the big bracket stands
for the normal ordered product [19,20]. Of course, one can
move those factors to the right in order to simplify further.
Each term has the U (
1
) charge 13 because there are four
factors for the U (
1
) charge 13 and three factors for the U (
1
)
charge − 31 . Totally one has 13 U (
1
) charge.
From the relation [44]
one obtains, by calculating the left-hand side of (4.20) with
(4.3) and (4.17) and reading off the first order pole,
×[+ f ai j ((ψi − i χ i )(ψ j − i χ j ))(ψb + i χ b)(ψc + i χ c)
×(ψd + i χ d )(ψe + i χ e)(ψ g + i χ g)(ψ h + i χ h )
− f bi j (ψa + i χ a)((ψi − i χ i )(ψ j − i χ j ))(ψc + i χ c)
×(ψd + i χ d )(ψe + i χ e)(ψ g + i χ g)(ψ h + i χ h )
+ f ci j (ψa + i χ a)(ψb + i χ b)((ψi − i χ i )(ψ j − i χ j ))
×(ψd + i χ d )(ψe + i χ e)(ψ g + i χ g)(ψ h + i χ h )
− f di j (ψa + i χ a)(ψb + i χ b)(ψc + i χ c)((ψi − i χ i )
×(ψ j − i χ j ))(ψe + i χ e)(ψ g + i χ g)(ψ h + i χ h )
+ f ei j (ψa + i χ a)(ψb + i χ b)(ψc + i χ c)(ψd + i χ d )
×((ψi − i χ i )(ψ j − i χ j ))(ψ g + i χ g)(ψ h + i χ h )
− f gi j (ψa + i χ a)(ψb + i χ b)(ψc + i χ c)(ψd + i χ d )
×(ψe + i χ e)((ψi − i χ i )(ψ j − i χ j ))(ψ h + i χ h )
+ f hi j (ψa + i χ a)(ψb + i χ b)(ψc + i χ c)(ψd + i χ d )
×(ψe + i χ e)(ψ g + i χ g)((ψi − i χ i )(ψ j − i χ j ))](z).
(4.21)
(
7
) (
7
)
W 27 (w) = −W 7 2 (w)
− 3 3
χa→−χa
The properties in (4.15) are used. One can check that the
U (
1
) charge of each term is equal to 43 where there are six
positive ones and two negative ones. In order to obtain the
[p4ri4m]awrhyicchurcraenntb,eonoebtashinoeudldfrcoomns(i4d.e2r1)( Wan43(d4)(4−.1319)∂.W 43(
3
))(z)
Then the higher spin 3, 27 , 27 , and 4 currents are
summarized by (4.13), (4.17), (4.19) and (4.21) with addition of the
derivative of (4.13).
4.3 Other higher spin 3, 27 , 27 , 4 currents
In the description of (4.11), for the opposite U (
1
) charge,
there exists also another solution for the higher spin 3 current.
One obtains the higher spin 3 current of U (
1
) charge − 43 as
follows:
W (
34
) (w) = W 4(
3
)(w)
− 3 3
χa→−χa
.
More explicitly, one can read off the explicit expression
which can be obtained from (4.13) by replacing the second
adjoint fermions with those together with minus sign. It is
obvious that the U (
1
) charge − 43 of this higher spin current
can be seen during this process: five factors of U (
1
) charge
− 53 and one factor with U (
1
) charge 13 .
Let us calculate the other higher spin currents. From the
well-known OPE [44]
G+(z) W −(343) (w) = − (z −1 w) W −(2713)(w) + · · · ,
one obtains the higher spin 27 current together with (4.22)
and (4.23) as follows:
(
7
) (
7
)
W 21 (w) = −W 1 2 (w)
− 3 3
χa→−χa
Note that under the change of χ a → −χ a , the original
U (
1
) charge is changed into the negative one. More
explicitly, one can take this operation in (4.18). Then the left-hand
side of (4.18) leads to the left-hand side of (4.23) with the
help of (4.22) and the right-hand side of (4.18) becomes
(
7
)
W 31 2 (w) χa→−χa . By using the first-order pole from (4.23),
then we are left with (4.24).
Similarly, the OPE [44] with (4.22),
G−(z) W −(343) (w) = (z −1 w) W −(2773)(w) + · · · ,
provides the following result for the higher spin current, by
considering Eq. (4.16), where the operation χ a → −χ a is
taken, and Eq. (4.22),
(4.22)
(4.23)
(4.24)
(4.26)
In other words, the left-hand side of (4.25) is equal to the
lefthand side of (4.16) with the additional operation χ a → −χ a .
We also use Eq. (4.22). Then the right-hand side of (4.25) can
be read off from this relation and we arrive at (4.26).
From the relation [44],
G+(z) W −(2773)(w) = (z −1w)2 73 W −(343) (w)
one sees that the first-order pole of (4.27) leads to
,
χa →−χa
where Eq. (4.20) together with the operation χ a → −χ a is
used. Moreover, Eq. (4.26) is used also. As described before,
the field (4.28) is not a primary under the stress energy tensor.
The primary current is given by (W −(443) + 91 ∂ W −(343) )(w), which
can be obtained from (−W 43(
4
) + 91 ∂ W 43(
3
))(w) by changing of
χ a (w) → −χ a (w).
Therefore, the higher spin 3, 27 , 27 , 4 currents are
summarized by (4.22), (4.24), (4.26), and (4.28). They are obtained
from the higher spin currents appearing in previous
subsection by simple change of the adjoint fermions χ a (z) up to
signs.
4.4 The OPE between the two lowest higher spin currents
in N = 2 superspace
Because the coset with the critical levels has the N = 2
supersymmetry, one can describe the OPE between the two
lowest higher spin multiplets in the N = 2 superspace. Let
us consider the OPE between the two N = 2 lowest higher
spin 3 multiplets where they have two opposite U (
1
) charges.
That is,
W(
43
)(Z1) W(
3
)4 (Z2),
3 − 3
where each four component current, obtained in the previous
subsection, is given by
(4.27)
(4.28)
W(433) ≡
W(
3
)4 ≡
− 3
(
7
) (
7
)
W 4(
3
), W 7 2 , W 1 2 , W 4(
4
)
3 3 3 3
W (
3
) , W ( 271), W ( 277), W (
4
)
− 43 − 3 − 3 − 43
,
.
In principle, in order to obtain the explicit OPE in (4.29),
one should calculate only the four OPEs between the four
component currents living in W(
43
)(Z1) in (4.30) and the
3
(4.29)
(4.30)
lowest component current in W(
3
)4 (Z2) in (4.30), due to the
− 3
N = 2 supersymmetry. See also the relevant work in [48–
50] where the various N = 2 multiplets in different coset
models are studied. From the four OPEs, one can realize that
the right-hand sides of these OPEs should have U (
1
) charges
0, 1, or −1 by adding the U (
1
) charges. Recall that the four
currents characterized by the N = 2 stress energy tensor
T ≡ ( J, G+, G−, T ) of the N = 2 superconformal algebra
have 0, 1, −1, and 0, respectively. It is natural to consider the
right-hand side of (4.29) in terms of the N = 2 stress energy
tensor T(Z2) with its various descendant fields in a minimal
way.
Inside of the package of [51], one can introduce the OPEs,
T(Z1) T(Z2), which is the standard OPE corresponding to
the N = 2 superconformal algebra, T(Z1) W(
43
)(Z2), which
3
is the N = 2 primary condition with U (
1
) charge 43 and
T(Z1) W(
3
)4 (Z2), which is the N = 2 primary condition with
− 3
U (
1
) charge − 43 . All the coefficients appearing in these OPEs
are constants except the central charge c, which is a function
of N in (4.5). Then one can write down the right-hand side
of OPE (4.29) with arbitrary coefficients which depend on
c or N . After using the Jacobi identities, we summarize the
structure constants in Appendix L explicitly. See also the
relevant work in [52].
One expects that there should be present other higher
spin multiplets in the various OPEs. For example, W(
43
)(Z1)
3
W(
43
)(Z2) or W(
3
)4 (Z1) W(
3
)4 (Z2) as in the unitary case [44].
3 − 3 − 3
It would be interesting to obtain these higher spin multiplets
explicitly further.
5 Conclusions and outlook
In the coset model (1.1), we have constructed the higher spin 4
current for general levels. For k1 = 1 with arbitrary N and k2,
the eigenvalue equations of the zero mode of the higher spin
4 current acting on the states are obtained. The corresponding
three-point functions are also determined. The N = 1 higher
spin multiplet characterized by (
27 , 4
) for k1 = 2N − 2 in
terms of adjoint fermions and spin 1 current is obtained. The
two N = 2 higher spin multiplets denoted by (
3, 27 , 27 , 4
)
for k1 = k2 = 2N − 2 in terms of two adjoint fermions are
determined. Some of the OPEs in the N = 1 or N = 2 coset
models are given explicitly.
We consider the possible related open problems as
follows:
• One can also try to obtain the higher spin currents in the
following coset model:
(5.1)
SˆO(2N + 1)k1 ⊕ SˆO(2N + 1)k2
SˆO(2N + 1)k1+k2
.
It seems that the minimum value of N for the non-trivial
existence of the d symbol (and corresponding higher spin 4
current) is given by N = 2. In the present paper, the minimum
value of N is given by N = 4 and the number of independent
fields in the higher spin currents is rather big, implying that
it is rather non-trivial to extract the corresponding OPEs. In
the coset model (5.1), for the N = 2 or N = 3 case, one
expects that one can analyze the OPEs further and observe
more structures in the right-hand sides of the OPEs.
• Further algebraic structures.
In order to observe the algebraic structures living in the
bosonic, N = 1 or N = 2 higher spin multiplets for
generic N (and generic k2), one should calculate the
various OPEs between them manually. In practice, this is rather
involved because, for example, the higher spin 4 current in
the bosonic coset model consists of 20 terms and the number
of OPEs is greater than 200. In [28], one can try to obtain
the various OPEs for the fixed low N values (for example,
N = 4, 5, 6, 7, . . .) and expect the N dependence of structure
constants appearing in the right-hand side of the OPEs.
• N = 2 enhancement of [25].
One considers the critical level condition in [25,53]. It
would be interesting to observe any N = 2 enhancement
or not. One can easily see the breaking of the adjoint
representation in S O(2N + 1) into the adjoint representation
of S O(2N ) plus the vector representation of S O(2N ). The
first step is to construct the N = 2 superconformal algebra
realization.
• The additional numerator factors.
For example, one considers the following coset model
where the extra numerator factor exists in the coset:
SˆO(2N )2N −2 ⊕ SˆO(2N )2N −2 ⊕ SˆO(2N )2N −2 .
SˆO(2N )6N −6
It is an open problem to see whether one constructs the N = 3
superconformal algebra [54] from the three kinds of adjoint
fermions or not. It is non-trivial to obtain the three spin 23
currents satisfying the standard OPEs between them. Then one
can try to obtain the higher spin currents living in the above
coset model (5.2). Furthermore, one can describe another
coset model where the additional numerical factor occurs.
It is an open problem to construct the linear (or nonlinear)
N = 4 superconformal algebra from the four kinds of adjoint
fermions.
(5.2)
• Further identities between f and d tensors of S O(2N ).
One can analyze the various identities involving f and d
tensors by following the description of [17,18]. They will be
useful in order to calculate the OPEs between the higher spin
currents in the context of Sects. 3 and 4.
• Zero mode eigenvalue equations in other representations
There exists an adjoint representation of S O(2N ). It is an
open problem to describe the eigenvalue equations for the
zero mode of the higher spin 4 current acting on the states
associated with the adjoint representation. For the S O(
8
)
generators in the adjoint representation, one has 28 × 28
matrices whose elements are given by the structure constant.
• Marginal operator.
One of the motivations in Sect. 4 is based on the
presence of the perturbing marginal operator [55], which breaks
the higher spin symmetry but preserving the N = 2
supersymmetry. It would be interesting to obtain this operator and
calculate the mass terms with the explicit eigenvalues along
the lines of [56–59]. In the large c limit, the right-hand side
of the OPE has the simple linear terms.
• N = 2 superspace description for the adjoint fermions.
We obtained the two N = 2 higher spin multiplets. It is
an open problem to see whether one can write down the two
adjoint fermions in N = 2 superspace. This will allow us
to write down the N = 2 higher spin multiplets in N = 2
superspace.
• Asymptotic quantum symmetry algebra.
We have obtained the eigenvalue equations and three-point
functions at finite N and k2 in Sect. 2. Along the lines of
[9], it is an open problem to study the asymptotic quantum
symmetry algebra of the higher spin theory on the Ad S3
space. See also [24] where a brief sketch for the large N ’t
Hooft limit is given.
Acknowledgements CA acknowledges warm hospitality from the
School of Liberal Arts (and Institute of Convergence Fundamental
Studies), Seoul National University of Science and Technology. This
research was supported by Basic Science Research Program through
the National Research Foundation of Korea funded by the Ministry of
Education (No. 2016R1D1A1B03931786).
Open Access This article is distributed under the terms of the Creative
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