The supersymmetric affine Yangian
Revised: May
The supersymmetric a ne Yangian
Matthias R. Gaberdiel 0 1 4
Wei Li 0 1 2
Cheng Peng 0 1 3
Hong Zhang 0 1 2
Zurich 0 1
Switzerland 0 1
0 182 Hope Street, Providence , RI 02912 , U.S.A
1 100190 Beijing , P.R. China
2 Institute of Theoretical Physics, Chinese Academy of Sciences
3 Department of Physics, Brown University
4 Institut fur Theoretische Physik, ETH Zurich
The a ne Yangian of gl1 is known to be isomorphic to W1+1, the W algebra that characterizes the bosonic higher spin  CFT duality. In this paper we propose some of the de ning relations of the Yangian that are relevant for the N = 2 superconformal version of W1+1. Our construction is based on the observation that the N = 2 superconformal W1+1 algebra contains two commuting bosonic W1+1 algebras, and that the additional generators transform in biminimal representations with respect to these two algebras. The corresponding a ne Yangian can therefore be built up from two a ne Yangians of gl1 by adding in generators that transform appropriately.
Conformal and W Symmetry; Higher Spin Symmetry; Quantum Groups

1 Introduction
2
3
4
5
2.1
2.2
3.1
3.2
3.3
5.1
5.2
5.3
5.4
The minimal and conjugate minimal representation
4.2 Identifying the representations
The Yangian at generic parameters
The generators in minimal representations 5.1.1
Other minimal generators
The generators in conjugate minimal representations
The OPEs with e and f
The N = 2 algebra
6
Conclusions
A Additional relations of the free eld theory
B The de ning relations
B.1 The OPE like description
B.2 The mode relations
B.3 The initial conditions
C Supercharge constraints
free) conformal eld theory [1{3], where string theory is expected to contain a higher spin
theory [4]. At this point in moduli space the large symmetry algebra underlying string
{ 1 {
theory (see [5{7] for indirect evidence) is expected to become visible. This is also the place
where the integrability of the theory should be most easily discerned.
In the context of AdS3, the emergence of a higher spin symmetry at the tensionless
point was recently seen quite explicitly in [8], see also [9, 10] for attempts to observe this
directly from a worldsheet perspective. In that case, the dual 2d conformal eld theory
of string theory on AdS3
S
3
T4, the symmetric orbifold of T4, was shown to contain a
W1 symmetry algebra. This is the hallmark of the duality between higher spin theories
on AdS3 and 2d CFT's [11{13], see [14] for a review.
On the other hand, there has also been progress in understanding the integrable
structure of string theory on AdS3 [15{18], and it would be very interesting to relate the higher
spin and integrable symmetries. Integrable theories are usually distinguished by having
a Yangian symmetry, and one may therefore try to identify the relevant Yangian in the
explicit higher spin description. This was recently done [19, 20] for the bosonic toy model
of [13], where the generators of W1
[ ], the symmetry algebra of the higher spin theory,
were explicitly identi ed with those of the a ne Yangian of gl1. (The underlying
isomorphism was rst noted in [21, 22], generalizing the construction of [23], and independently
by [24] and [25{27], see also [28] for further generalizations. The a ne Yangian of gl1 is
also isomorphic to the spherical degenerate double a ne Hecke algebra SHc of [29], and
was also constructed independently in [30].)
In this paper we show how to construct the Yangian algebra corresponding to the
N = 2 superconformal generalisation of W1
. Our approach is partially inspired by the
fact that the underlying higher spin algebra shs[ ] contains two commuting bosonic higher
spin subalgebras hs[ ]
hs[1
]. Subsequently, it was suggested in [31, section 11.1] that
this relation may also be true for the full W1
W1(N =2)[ ] actually has this structure, i.e. that it contains two decoupled W1
(N =2)[ ] algebra. We begin by showing that
[ ] algebras.
(This analysis relies on the precise form of the de ning structure constant of the W1
algebra that was identi ed in [32].) We then show that the additional generators that have
to be added to the two bosonic W1 algebras in order to generate the full W1
algebra transform in what one may call biminimal representations with respect to the two
W1 algebras. (This is to say, they transform as a minimal representation with respect to
one, and as an antiminimal representation with respect to the other; here \antiminimal"
means that it is the conjugate representation to the minimal representation.) The basic
idea of our construction is then to add generators to the two a ne Yangians of gl1 that
have these transformation properties.
The main technical di culty of this approach comes from the fact that the
description of conjugate minimal representations in terms of the a ne Yangian was not known.
The a ne Yangian viewpoint gives rise to an elegant description of representations in
terms of plane partitions [19], see also [33], but this language only applies to the
\box"representations, but not to those made of \antiboxes". However, the biminimal
representations that are relevant for the above extension always involve also antibox
representations. We propose a general formula for the description of antibox representations in
terms of plane partitions, see section 3.2. With this insight we can then propose some
of the commutation relations of the two sets of a ne Yangian generators of gl1 with the
{ 2 {
additional modes, and thus undertake the rst steps towards de ning the supersymmetric
generalisation of the a ne Yangian.
The paper is organized as follows. In section 2 we show that the W1(N =2)[ ] algebra
contains (and can be built up from) two commuting bosonic W1 algebras. We identify
the additional generators that need to be added, and in particular, their representation
properties with respect to the two bosonic W1 algebras. In section 3 we review the
relevant minimal representation of the a ne Yangian, and explain how to describe the
conjugate representation. In section 4 we analyze the N
where the W1(N =2)[ ] algebra has a free eld realization, in terms of which also the two
bosonic W1 algebras can be identi ed. In particular, we can make an explicit ansatz
for the additional generators that need to be added and compute their commutation and
= 2 construction for
= 0,
anticommutation relations for
= 0. In section 5, we then deform these relations away
from the free eld point (
= 0), using as a guiding principle our insight into the correct
description of the minimal and conjugate minimal representations. We furthermore test our
ansatz by comparing to the free eld limit, and by showing that the additional generators
lead to states in the correct representations. Our conclusions and avenues for future work
are outlined in section 6. There are two appendices: in appendix A, we have spelled out
some of the free eld relations that we did not want to put in the main part of the text,
and in appendix B we have summarized the de ning relations of the supersymmtric a ne
Yangian we have found.
Note added: as we were in the nal stages of this work we were made aware of [39]
which contains some overlap with section 2 of our paper.
2
Building up the N
= 2 W1 algebra
In this section we explain that the W1(N =2)[ ] algebra contains two bosonic W1
as mutually commuting subalgebras, one at
=
and one at
= 1
[ ] algebras
. Note that it
is known, see e.g. eq. (202) in [14], that the N = 2 higher spin algebra can be written in
this manner
shs[ ](bos) = hs[ ]
hs[1
However, it is not obvious whether this will also be true for the full quantum W1(N =2)[ ]
algebra.1 This viewpoint will be important below because it will allow us to construct the
full W1(N =2)[ ] algebra starting with these bosonic subalgebras.
2.1
Decoupling the bosonic subalgebras
algebra as WN(N;k=2), i.e. to express both
In order to understand how this comes about, it is convenient to parametrise the W1(N =2)[ ]
and c in terms of N and k as
c
c(NN;k=2) =
3N k
1For the case of the N = 2 W3 algebra this was already noted in [34, 35]; however, for general this is
not known.
(2.1)
(2.2)
{ 3 {
see e.g. [32] for our conventions. As is also explained there, the WN;k
spin h = 2 elds: the stress energy tensor T , and the primary spin h = 2 eld W
W is also primary with respect to the u(1) current J , but T is not; however, we can de ne
the decoupled spin 2 eld via
T~ = T
: J J : :
3
2c
The modes of these two elds then satisfy the commutation relations
[T~m; T~n] = (m
n)T~m+n +
m(m2
1) m; n
(c
1)
12
[T~m; Wn] = (m
n)Wm+n
[Wm; Wn] = (m
n)
2n2
(c
1) T~m+n +
2
where we are using the same conventions as in [32], and the last identity is directly read
o from eq. (2.14) of that paper. To nd the two commuting Virasoro algebras, we make
the ansatz
~
Tm = a+L+m + a Lm ;
Wm = b+L+m + b Lm ;
and demand that L
commute with one another and each lead to a Virasoro algebra with
central charge c
where c+
c . In particular, it follows that
(c
1) = c+ + c ;
0 = b+c+ + b c :
These two conditions x the coe cients a
and b uniquely, and one nds that the two
solutions are
a
= 1 ;
b
=
c22;2
4
(c22;2)2 + 32
n2
(c
1)
:
The corresponding central charges are then
c+ =
(c
b
1)b
b+ ;
(c
b
1)b+
b+ :
cN;k = (N
h
from [32], one nds that c equals cN;k and ck;N , respectively, where
is the central charge of the bosonic WN;k algebra (without the additional u(1) current).
Note that the full (decoupled) stress energy tensor T~, de ned in eq. (2.3), is indeed the
sum of L+ and L ,
~
Tm = L+m + Lm ;
as also follows from (2.7). In particular, this implies that the total central charge of the
N = 2 algebra must equal  the \+1" comes from the u(1) factor we have divided out 
cN;k + ck;N + 1 =
(3k
1)N
k
1
(N =2)) algebra contains at least two
Virasoro primary
elds. It seems plausible that among these spin s elds, we can always
nd two elds W (s) such that2
[Lm; Wn(s) ] = (s
1)m
n) W m(s+)n ;
[Lm; Wn(s) ] = 0 :
(2.12)
Since all W (s)+ elds commute with L , the same must be true for their commutator, and
hence the VOA generated by L+ and the W (s)+
elds must close. (Obviously, a similar
statement also holds for L
and the W (s)
elds.) Furthermore, the commutator of W (s)+
with W (t)
must vanish since, with respect to L+, say, W (t)
behaves like the identity
eld and hence does not give rise to a nontrivial commutator. Thus, if for each spin s
there are two elds such that (2.12) holds, it follows that the W1
two commuting bosonic W1
and the other generated by L
[ ] algebras, one generated by the elds L+ and the W (s)+,
and the W (s) . Given that their central charges equal cN;k
(N =2)[ ] algebra contains
and ck;N , it is very plausible that the relevant W1 algebras are just the bosonic WN;k and
Wk;N algebras, respectively, i.e. that
WN(N;k=2)
WN;k
Wk;N :
In order to con rm this we would have to construct the relevant
elds and determine
their C334 structure constants, but we have not attempted to do so here. Note that this
structure also nicely re ects the Z2
Z2
realized by N $ k.
2.2
Character analysis
Next we want to understand the additional generators that need to be added to the two
bosonic W1
venient to ad[d] aalgseinbgrlaes firneeorbdoesrotno geelndertaoteWW1(N1=2)[ ]. Then the corresponding vaccum
(N =2)[ ]. For the following it will be
concharacter equals
Z2 symmetry of the WN(N;k=2) algebra that is
0(q) =
Y1 (1 + qn+ 12 )2n
n=1
(1
qn)2n ;
00(q) = Y
1
1
Y
s=1 n=s (1
(1 + qn+ 12 )2
qn)(1
qn+1) ;
fb =
1
Y
n=1 (1
1
qn) :
pp =
1
Y
n=1 (1
1
qn)n
;
2We have checked explicitly that this is the case for spin s = 3.
{ 5 {
since the vacuum character of the W1(N =2) algebra is
and a single free boson contributes
We want to organise this character in terms of W1+1[ ]
W1+1[1
character of each W1+1[ ] algebra is described by a plane partition, see e.g. [19, 20]
]. The vacuum
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
where R runs over all representations that appear in
nite tensor powers of the two
biminimal representations, and RT is the conjugate representation to RT , with T denoting
the transpose of R. Since R involves in general box and antibox representations, and since
the wedge character of such a mixed representation is simply the product of the wedge
character of the box representation and that of the antibox representation, the above
identity follows from
S
1
n=1
Y (1 + yqn+ 12 )n =
X yjSj (Swedge) [ ](q)
ST
(wedge) [1 ](q) ;
where S runs over all Young diagrams (labelling say boxrepresentations), ST is the
transpose Young diagram (labelling now antibox representations), and jSj denotes the number
of boxes in S. The rst few cases are explicitly (see [36] for the general method for how to
derive them)
1
n=1
and hence the two algebras account precisely for the denominator of 0(q) in eq. (2.14).
The numerator of (2.14) corresponds to the fermionic excitations, and is accounted for in
terms of biminimal representations of the two algebras3 together with their tensor powers.
Since for each representation R of W1+1[ ] the character is of the form
this amounts to the condition that
(wedge)(q) pp(q) ;
R
(wedge)(q) =
(wedge)(q) =
(wedge)(q) =
(wedge)(q) =
(wedge)(q) =
(wedge)(q) =
q
h
1
(1
(1
(1
(1
(1
q
q2h
q)(1
q2h+1
q)(1
q)(1
q3h+1
q)2(1
q2)
q2)
q3h
q2)(1
q3)
q3)
q3h+3
q)(1
q2)(1
q3) :
{ 6 {
3The biminimal representations that are relevant here are \minimal" with respect to one factor, and
\conjugateminimal" with respect to the other.
There are therefore two such representations, namely
\minimal"{\conjugateminimal" and \conjugateminimal"{\minimal".
1
2
h + h^ =
1
2
h^ =
1
2
1
2
(1 + ) +
It is then straightforward to check (2.20) explicitly,4 provided that we take the box
repre
] to be the ones with conformal dimensions
respectively, so that the total conformal dimension equals
thus reproducing the conformal dimension of the supercharge.
HJEP05(218)
3
The minimal and conjugate minimal representation
As we have seen in the previous section, the additional generators that need to be added
to the two bosonic W1+1 algebras transform in biminimal representations with respect to
these two algebras. For the following it will be important to describe these representations
from the viewpoint of the a ne Yangian.
Recall from [19, 20] that the de ning relations of the a ne Yangian can be written as
(2.27)
(2.28)
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
and
e(z) f (w)
f (w) e(z) =
1
3
(z)
z
w
(w)
;
e(z) e(w)
f (z) f (w)
(z) e(w)
(z) f (w)
'3(z
'3 1(z
'3(z
'3 1(z
w) e(w) e(z)
w) f (w) f (z)
w) e(w) (z)
w) f (w) (z) ;
where ` ' means equality up to terms that are regular at z = 0 or w = 0. Here the elds
are expanded in terms of modes as
e(z) =
1
X
j=0
zje+j1 ;
f (z) =
1
X
j=0
zfj+j1 ;
(z) = 1 + 3
1
X
j=0
zj+j1 ;
and the function '3(z) is de ned by
'3(z) =
(z + h1)(z + h2)(z + h3)
(z
h1)(z
h2)(z
h3)
=
z3 + 2z
z3 + 2z + 3 :
3
The hi parameters satisfy h1 + h2 + h3 = 0, and we have de ned
2 = h1h2 + h2h3 + h1h3 ;
3 = h1h2h3 :
The structure of these OPEs can be summarised by the diagram of gure 1.
4We have done this up to q10. It should not be too hard to prove this analytically, but we have not
attempted to do so.
{ 7 {
ϕ3(Δ)
e
ϕ3(Δ)
ψ
f
In terms of the conformal eld theory language, the hi parameters and
As is explained in [19, 20], the di erent states of the vacuum representation are
described by plane partitions where the eigenvalues of i on the con guration
are given by
(z) =
1 +
'3(z
h( )) ;
corresponding states are still given by (3.11), except that now the in nite product (over the
in nitely many boxes de ning the asymptotic con guration) must be suitably regularized.
3.1
The minimal representation
The simplest nontrivial representations are the minimal representations whose asymptotic
box con guration consists of a single row of boxes extending along either x1, x2 or x3. For
our analysis above, the minimal representation corresponding to an asymptotic single box
in the x2 direction will play a central role.5 Its ground state has the charges
(u) =
1 +
=
1 +
Expanding out in inverse powers of u, this is of the form
(u) = 1 +
5It is the one whose conformal dimension equals (2.27) in the classical limit. Note that selecting out x2
breaks the S3 symmetry of the a ne Yangian to Z2.
{ 8 {
Comparing with (3.6) we read o
1 =
1
h2
;
As will become clear from the free eld analysis of the next section, the above minimal
representations will not su ce. In fact, the biminimal representations that appear in (2.19)
have the property that they are minimal with respect to one W1 algebra, but conjugate
minimal (or antiminimal) with respect to the other. Thus in order to describe these
generators we also need to understand how to describe the conjugate minimal representation
from the a ne Yangian perspective. In the following we shall make a general proposal for
how this works.
charges given by
Given a box representation described by '(u) (where again '(u) does not include
the vacuum factor 0(u)), we claim that the corresponding conjugate representation has
The sign cance of this shift is that it turns the vacuum factor 0(u) = (1 + 0u 3 ) into
Thus the full eigenvalue function of the antibox representation is
0(u) ' 1( u
0 3) =
0(v) '(v)
1
;
where v =
u
0 3 ,
and therefore indeed just the inverse of
(u) =
0(u)'(u). (Note that the shift (and
sign) transformation from u to v is just the spectral ow (and scaling) automorphism of
the a ne Yangian, see e.g. sections 2.2 and 2.3 of [19].)
We can check this proposal explicitly by checking whether the rst few W0s charges
change correctly  recall that for the conjugate representation the charges of the odd spin
W0s generators must have the opposite sign, while the even spin generators are the same.
Suppose then that '(u) describes a given representation with eigenvalues j, i.e. we have
the expansion
sentation is
(1 + 0u 3 ) '(u) = 1 +
u
Then, according to the above proposal, the power series expansion of the conjugate
repreThis predicts that the conjugate representation has the charges
0 = 0 ;
1 =
1 ;
2 = 2 ;
3 =
{ 9 {
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
HJEP05(218)
Together with the form of the spin s = 3 charge from [20]
it follows that the value of W03 on the conjugate representation equals
as expected. This is a fairly nontrivial consistency check of this proposal.
In particular, for the conjugate minimal representation from above we nd
while for s
3 we nd the simple closed form expression
s = h
2
s 2 (1
As an aside, the above analysis now allows us to check the nite N and k corrections
to the conformal dimension (2.28). According to [19, 20], the conformal dimensions with
respect to the coupled theory (where the u(1) generator has not been removed) equals, see
eq. (3.15)
h =
1
2 2 =
1
2
1
0h1h3 =
1 +
1
2
N
N + k
where we have used the dictionary (3.10). Note that this is true both for the minimal
representation, as well as the antiminimal representation, see eq. (3.24). We should also
mention in passing that the decoupled conformal dimension is then, see eq. (5.67) of [19]
hdec =
2
1
2
2
1 =
and hence agrees indeed with the conformal dimension of h(f; 0) in the coset, see, e.g.,
eq. (2.13) of [13].
For the problem at hand, however, we should work with the coupled conformal
dimension (since we are dealing with W1+1 rather than just W1). Furthermore, for the N = 2
construction, we consider two plane partitions that correspond to
and 1
, i.e., for the
second W1 algebra we should exchange the roles of N and k. Note that, according to the
dictionary of eq. (3.10), this only a ects
0, but not the values of hi. Thus we shall work
with the same values of hi for both a ne Yangians, but distinguish the two a ne Yangians
by setting
0 = N ;
^0 = k :
;
;
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
Note that, using (3.10), this relation can be written as or equivalently as
If we then add (3.26) to the corresponding expression with N $ k, we nd altogether
1
2
1
2
htot =
1
as desired. This is another highly nontrivial consistency check for this construction to
work also at the quantum level.
It may also be worth noting that the Z2
Z2 symmetry of the W1(N =2) algebra [32]
has a nice geometric interpretation in this setting. First of all, the S3 symmetry of each of
the two a ne Yangians is broken down to a common Z2 symmetry that exchanges the x1
and x3 direction. In addition, there is the symmetry exchanging the roles of the two a ne
Yangians, which corresponds to the N $ k transformation.
3.3.1
Representations
One can similarly understand how the two minimal N = 2 representations appear from
the above bosonic viewpoint. The relevant representations have conformal dimensions
N
2(N + k + 1)
k
2(N + k + 1)
h(f; 0; N ) =
;
and
h 0; f; (N + 1) =
;
(3.32)
see, e.g. eq. (3.9) of [32]. From the above perspective, these representations correspond
to the representation that has an in nite row of boxes along the x1 direction for either
of the two plane partitions. Indeed, it follows from [19, 20] that the relevant conformal
dimensions are (cf. eq. (3.26))
and
h =
1
2(N + k + 1)
:
Note that these conformal dimensions are higher than those in (3.32), with the di erence
in both cases being equal to
This is the contribution of the overall u(1) generator that was added in (2.16). In general,
the representations of the N = 2 a ne Yangian will therefore be described by in nite box
con gurations extending in the x1 and x3 direction for both plane partitions. (The
representations where the boxes extend along the x3 direction are again the nonperturbative
representations, in analogy to what happens in the bosonic case, see [37].)
(3.29)
(3.30)
(3.31)
For the following it will sometimes be important to compare our ansatz with an explicit
free eld realization. Recall that at
= 0 (or
= 1), the W1
(N =2) algebra has a free eld
construction in terms of free complex fermions and bosons. More explicitly, the neutral
bilinears in the fermions, i.e. the elds of the form
generate W1+1[0], while the neutral bilinears in the bosons, i.e. the elds of the form
j
j
(4.1)
(4.2)
(4.3)
(4.4)
HJEP05(218)
give rise to W1[1]. (Here j and
j are the complex fermions, and
j and
j are the
complex bosons.) On the other hand, the fermionic generators are linear combinations of
the form
and
j
j
From the viewpoint of the two bosonic W1 algebras, i.e. W1+1[0] and W1[1], these
generators transform in the `biminimal' representation. Indeed, the fermion eld j corresponds
son
(with respect to W1+1[0]) to the representation (f; 0) in the coset language, while the
boeld @ j describes (with respect to W1[1]) the representation (f; 0). Thus the two
fermionic elds above correspond to the states in
(f; 0)
(f; 0)
and
(f; 0)
(f; 0) ;
respectively. (Here we have used that the complex conjugate fermion transforms in (f; 0),
and correspondingly for the complex conjugate boson.)
In terms of the description in terms of plane partitions, this means that the rst
fermionic generators act as an addition of an in nite row of boxes with respect to the
rst plane partition  the one corresponding to W1+1[0]  while it acts as an addition
of an in nite row of antiboxes with respect to the second plane partition  the one
corresponding to W1[1].
4.1
The a
ne Yangian generators at
= 0
We can use this free eld realisation to construct the relevant a ne Yangian generators for
this special case, and work out their commutation and anticommutation relations. In the
next section we shall explain how to modify these relations as we move away from
= 0.
We recall from [20] (see also [19]) that for the bosonic W1+1[0] algebra, the
corresponding a ne Yangian generators can be de ned as
r =
er =
fr =
X
X
m+1 im : ;
and
m2Z i
m2Z i
xs =
X
X( m
where the free fermion modes are denoted by
im and
i . As was shown there, these
m
generators satisfy the a ne Yangian algebra of [19] for 2 =
1 and 3 = 0 with 3 0 = 0.
In terms of the hi parameters, this corresponds to the case
see eq. (3.10) above. We also need a description for the a ne Yangian generators associated
to W1[1], and they are given as
^r =
e^r =
f^r =
X
X
m2Z i
X
m2Z i
X
X
m2Z i
X( m)r 1 : i
1 m mi :
m + 1 r 1
: 1i m mi : :
(m + 1)( m)r 2 +
m + 1 r 1
: i
This leads to the a ne Yangian with hi being given by (4.8); the only di erence to the
sense for e^r, f^r with r
case of W1[0] above is that now
1 and ^r with r
2. However, we can at least formally extend
these de nitions to include also e^0 and ^1 by setting i
0
with m = 0 from all of these expressions. (Similarly, we could de ne f^0 by setting 0i
The generator e^0 is then the
1 mode of a nonlocal eld with spin 1. One checks by an
explicit calculation that it satis es the correct commutation relation with the ^r modes,
0, i.e., by dropping the term
in particular (see eq. (4.13) of [20])
[ ^1; e^0] = 0 ;
[ ^2; e^0] = 2e^0 ;
[ ^3; e^0] = 6e^1
2e^0 :
For the fermionic generators we now make the ansatz
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
where s = 12 ; 32 ; : : :, and we de ne the generating functions by
i.e. the term with s = 12 corresponds to the supercharge, etc. Obviously, the algebra also
contains the corresponding +3=2 modes, which we may de ne via conjugation as
and
and
where the corresponding generating functions are
x(z) =
1
X xs z s 1=2 ;
s=1=2
x(z) =
1
X xs z s 1=2 :
s=1=2
xs
W (s3+=21)+ ;
and
xs
and
xys = ys :
Their treatment is similar to that of the xr and xr generators, and is therefore relegated
to appendix A.
It is now straightforward to work out the commutation and anticommutation relations
of these generators. For example, one nds
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
(4.28)
(4.29)
as well as
0 = [ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + [ r+1; xs] [ r; xs+1]
0 = [er+1; xs] [er; xs+1] + [er; xs]
0 = [fr+1; xs] [fr; xs+1] ;
0 = [ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] [ r+1; xs] + [ r; xs+1]
0 = [er+1; xs] [er; xs+1] [er; xs]
0 = [fr+1; xs] [fr; xs+1] :
On the other hand, the commutation relations with the hatted modes are
0 = [ ^r+2; xs] 2[ ^r+1; xs+1] + [ ^r; xs+2] 3[ ^r+1; xs] + 3[ ^r; xs+1] + 2[ ^r; xs] (4.27)
0 = [e^r+1; xs] [e^r; xs+1] 2[e^r; xs]
0 = [f^r+1; xs] [f^r; xs+1] [f^r; xs]
and
In addition, there are the initial conditions
Note that the hatted generators only start with ^2, i.e. the modes ^0 and ^1 are initially
not de ned in (4.9). We have added the mode ^1 by hand  the result also agrees with
what one obtains upon extending the de nition of ^r in (4.9) to r = 1, see the comments
above  and de ned it so that it satis es (4.27) and (4.30) for r
1. However, (4.27)
and (4.30) are then not compatible with [ ^0; xs] = 0 and [ ^0; xs] = 0. The reason for this
will become clear below: the deformed relations, see appendix B.2, contain an additional
contribution that survives (for r = 0) since
1. With this correction term the
^
above results are then also compatible with the recursion relations of appendix B.2 for
r = 0. For the unhatted modes, these problems do not arise, and in fact the commutator
with 2 is determined using (4.21) and (4.24) with r = 0 as
[ 2; xs] = xs
2xs+1 ;
[ 2; xs] = xs + 2xs+1 :
(4.34)
Finally, for the anticommutator of the xs and xr we nd
0 = fxi+2; xjg
2fxi+1; xj+1g + fxi; xj+2g + fxi+1; xjg
fxi; xj+1g
2fxi; xjg : (4.35)
It is also convenient to de ne
with the initial condition that
fxr; ysg = Pr+s ;
P1 =
1
2
where N is the number of complex free bosons and fermions. The Pr modes satisfy a
number of relations that are also spelled out in appendix A.
4.2 Identifying the representations
The discussion around eq. (4.4) suggests that the fermionic generators transform in a
minimal representation with respect to one W1 algebra, but in the conjugate minimal
with respect to the other. We can now verify this also more explicitly.
Let us rst analyse the generators described by x. The eigenvalues of r on the state
i
1=2j0i  this is the relevant state for the description of x1=2j0i  equals
1 =
1 ;
2 = 1 ;
3 =
1 ;
(4.38)
(4.30)
(4.31)
(4.32)
(4.33)
(4.36)
(4.37)
where the rst few
r generators are explicitly, see eq. (4.5)
1 =
1, h2 = 1 and h3 = 0 (with 3 0 = 0) this agrees then with the charges
of the minimal representation, see (3.15).
On the other hand, the charges of the state i
of xs) are
1(x) = 1 ;
2(x) = 1 ;
3(x) = 1 :
These are not the charges of the minimal representation, but rather that of the conjugate
minimal representation. Indeed, evaluating (3.24) for h1 =
1, h2 = 1, h3 = 0 with
3 0 = 0 we nd 1 = 1, 2 = 1 and
3 = 1, which reproduces indeed (4.42).
Incidentally, the situation is precisely reverse with respect to the hatted modes. In
that case, we need to evaluate the charges
1=2j0i (that is relevant for the description
^2 = 2 X
X : i
First consider the state i 1j0i (that is relevant for the xs modes), for which we nd
This is then of the form (3.15) with h1 =
1, h2 = 1, h3 = 0 and 3 ^0 =
1; thus the xr
generators transform in the minimal representation with respect to the hatted modes. On
the other hand, on the state
see eq. (4.13), the charges equal
i 1j0i that is relevant for the description of the xs modes,
^2(x) = 2 ;
^3(x) = 4 :
Now, this does not correspond to the minimal representation, i.e. it does not match (3.15)
with h1 =
1, h2 = 1, h3 = 0 and 3 ^0 =
1, but rather corresponds to the conjugate
minimal representation, i.e., it agrees with (3.24) for h1 =
1, h2 = 1, h3 = 0 with
1. The situation for the y and y generators is similar; we have summarized the
representation properties of these generators in the table 1.
5
The Yangian at generic parameters
Next we want to make a proposal for how the algebra should be deformed away from the
special point
= 0, see eq. (4.8). Our guiding principle is that, with respect to the two
bosonic a ne algebras, denoted by Y and Y^ respectively in the following, the fermionic
generators sit in \biminimal" representations.
generator
unhatted modes Y
x
x
y
y
minimal
conj. minimal
conj. minimal
minimal
hatted modes Y^
conj minimal
minimal
minimal
conj. minimal
The generators in minimal representations
Let us begin with studying the generators that transform in minimal (rather than
conjugate minimal) representations. As we have seen above, the generator xs transforms in
the minimal representation of Y. By analogy with the construction of the bosonic a ne
Yangian, the operation of adding xs should therefore change the eigenvalue of the
modes
by  this is (u) in eq. (3.13) without the \vacuum" factor 0(u) = (1 + 0u 3 )
HJEP05(218)
This then suggests that (4.21) should become
'2(u) =
whose modes  this can be deduced as in [20], see the discussion around eq. (2.12) there
 then satisfy
2[ r+1; xs+1] + [ r; xs+2] + h2 [ r+1; xs] [ r; xs+1] + h1h3 rxs = 0 : (5.3)
Note that this reduces to (4.21) for h2 = 1 and h1h3 = 0.
Before we proceed further, we can test this proposal by working out the charges of
the state that is created by x1=2 from the vacuum. Recall that on the vacuum state the
bosonic and fermionic modes satisfy
eij0i = i;0e0j0i ;
xij0i = i;1=2 x1=2j0i ;
xij0i = i;1=2 x1=2j0i :
(5.4)
We want to con rm that the state x1=2j0i has the charges of the minimal representation of
Y. We postulate that the initial condition (4.33) is now modi ed to
Then it follows that
0 x 12 j0i = N x 12 j0i
1 x 12 j0i =
1
h2 x 12 j0i :
(5.1)
(5.2)
(5.5)
(5.6)
(5.7)
In order to determine the higher charges we deduce from (5.3) that
(h1 + h3), the last equation can now be rewritten as
These charges then agree precisely with eq. (3.15), thus con rming that our ansatz (5.2)
leads to states with the correct charges.
We also make the ansatz where here and in the following e(z) x(w) f (z) x(w)
G( ) and H( ) are functions that will be constrained further below, see section 5.3.
Other minimal generators
The construction works similarly for the generators yr, which behave like the conjugate
operators to xr, i.e. they are like the fr modes relative to er in the bosonic a ne Yangian.
Because of that we expect them to satisfy the inverse OPE, cf. eq. (3.4) and (3.5)
Given the simple relation between (5.2) and (5.14), we postulate that also the yanalogues
of (5.11) and (5.12) only involve simple inverses. The structure of the OPEs of x and y
with respect to the unhatted elds can then be summarized by the diagram of gure 2.
The analysis is completely analogous for the two elds x and y with respect to the hatted elds, and the structure of the corresponding OPEs can thus be similarly realized, see gure 3.
5.2
The generators in conjugate minimal representations
A more interesting case are the OPEs of the unhatted elds with x and y, or equivalently,
that of the hatted elds with x and y. For concreteness, let us describe the former case in
G(Δ) ϕ2(Δ)
x
ϕ3(Δ)
ψ
ϕ3−1(Δ)
f
H(Δ)
H(Δ)
H−1(Δ)
G(Δ)
ϕ2(Δ)
eˆ
ϕ3(Δ)
ϕ3(Δ)
ˆ
ψ
ϕ3−1(Δ)
ˆ
f
detail. Since the charges of the conjugate minimal representation are described by (3.16),
the natural ansatz for the OPE is thus
By the usual arguments, this then leads to the commutation relations
[ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + ( h2 + 2 0 3) [ r+1; xs] [ r; xs+1]
which reduces indeed to (4.24) in the free eld limit. Again, before proceeding further, we
should check that this gives the correct charges on the corresponding states. In analogy
to (5.5) we now postulate
(5.15)
(5.16)
(5.17)
HJEP05(218)
ϕ3(Δ)
ϕ3−1(Δ)
G−1(−Δ − σx3ψ0) ϕ2−1(−Δ − σ3ψ0) H−1(−Δ − σ3ψ0)
ψ
ˆ
ψ
ϕ3(Δ)
ϕ3−1(Δ)
f
ϕ2−1(Δ)
f
y¯
y
x¯
G(Δ)
gures 2 and 3 for comparison.
Then we nd, using (5.16)
= (1
h1h3 0)x1=2j0i
1 x1=2j0i =
x1=2j0i
1
h2
2 x1=2j0i = [ 2; x1=2] j0i = (h2
2 0 3)[ 1; x1=2] j0i + h1h3 0x1=2 j0i
3 x1=2j0i = (h2
thus giving the correct charges of the antiminimal representation, see eq. (3.24). The
structure of the corresponding OPEs can therefore be summarized as in gure 4. The
situation for the hatted elds with respect to x and y is completely analogous and summarized
in gure 5.
5.3
The OPEs with e and f
Since the eld appears in the OPE of the e and the f eld, see eq. (3.1), we can also deduce constraints on the OPE of the e and f eld with x from that with . To this end, { 20 {
G(Δ)
x
ϕ3(Δ)
ϕ3−1(Δ)
ψ
ˆ
ψ
f
y
G−1(−Δ − σx¯3ψˆ0) ϕ2−1(−Δ − σ3ψˆ0) H−1(−Δ − σ3ψˆ0)
H(−Δ − σ3ψˆ0) ϕ2(−Δ − σ3ψˆ0) G(−Δ − σ3ψˆ0)
ϕ3(Δ)
ϕ3−1(Δ)
gures 2 and 3 for comparison.
we recall the ansatz from eqs. (5.11) and (5.12)
Note that, just like the identities of eqs. (3.2) { (3.5), these relations cannot be exactly
correct, but are only true up to terms that are regular at either z = 0 or w = 0, see the
discussion around eq. (5.15) in [20]. Applying this identity twice we nd that
e(z1) f (z2) x(w)
f (z2) e(z1) x(w)
G(z1
G(z1
w) H(z2
w) H(z2
w) x(w) e(z1)f (z2)
w) x(w) f (z2) e(z1) :
Subtracting the two equations from one another and using (3.1) we thus deduce that
(z1)x(w)
(z2)x(w)
z1
z2
G(z1
w) H(z2
w)
x(w) (z1)
z1
z2
x(w) (z2) :
Next we apply (5.2) to the lefthandside, from which we deduce that this equals
Thus it follows that
'2(z1
w)x(w) (z1)
z1
'2(z2
z2
w)x(w) (z2) :
'2(z1
z1
w)
z2
'2(z2
z1
w)
z2
G(z1
w) H(z2
w)
z1
z2
:
(5.21)
(5.22)
(5.23)
(5.24)
(5.25)
(5.26)
(5.27)
HJEP05(218)
Because these identities are only true up to regular terms, this implies that the functions
G( ) and H( ) have to satisfy the identity
The most natural ansatz that is compatible with the free eld limit (see below) is
G( ) H( ) = '2( ) :
G( ) =
+ h2
h1
;
H( ) =
h3
;
which reduces indeed to the correct free eld answers, see eqs. (4.22) and (4.23). However,
this ansatz cannot be right since there are two boxdescendants of the state generated by
x1=2j0i  this follows from the bosonic structure of the minimal representation  and
hence the function G( ) must have two poles [38]. In fact, one can use the representation
theory to constrain the function G( ) (and hence H( )) further, but this goes beyond the
scope of the present paper and will be described elsewhere [38].
Our starting point in section 2 was the W1(N =2)[ ] algebra, and we can now try to identify
its generators with those of the supersymmetric a ne Yangian. The vacuum character, see
eq. (2.14), contains two spin s = 1 elds: the u(1) generator of the N = 2 superconformal
algebra, as well as the extra bosonic generator that we added by hand to the W1
vacuum character (2.15). This free boson should be completely decoupled, and its zero
mode be identi ed with the central generator
U0
1 + ^1 :
Obviously, U0 commutes with all er, fr, e^r and f^r generators, and because of the relations
we have imposed, it also commutes with the xs and xs generators.
We also know that the total Mobius generators correspond to
L 1 = e1 + e^1 ;
L1 =
f1
f^1 ;
L0 =
2 + ^2 ;
1
2
and it is thus natural to assume that the
1 modes of the decoupled boson are
The u(1) generator of the N = 2 algebra, on the other hand, should be identi ed with
U 1 = e0 + e^0 ;
U1 =
f0 + f^0 :
J0 = ( 3 ^0) 1
( 3 0) ^1 ;
In terms of commutators this leads to
HJEP05(218)
and
J 1 = ( 3 ^0) e0
( 3 0) e^0 ;
J1 =
( 3 ^0) f0 + ( 3 0) f^0 :
Then [Jm; Un] = 0 for m; n 2 f0; 1g, and we nd
c(N =2)
3
m m; n ;
with
c(N =2) = 3 ( 3)2 0 ^0 ( 0 + ^0) :
In terms of the dictionary of [20], this central charge then equals
c(N =2) =
3N k
;
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
in agreement with the N = 2 central charge, see eq. (2.11). The other bosonic generators
can be similarly identi ed: for each integer s, there are two decoupled bosonic elds, see
the discussion below (2.12), and they can be identi ed with the a ne Yangian generators
of the two bosonic a ne Yangians, using the dictionary of [20].
This leaves us with the fermionic generators. The lowest fermionic generators are the
supercharge generators Gr , which, at the free point, can be identi ed with
It would be natural to postulate this identi cation also for generic hi. However, there is
a problem with this proposal. The N = 2 generators should commute with the decoupled
free boson described by Un. But even at the free point one nds
The origin of this problem is that the e^0 generator corresponds to the
see also the discussion below eq. (4.11). The fact that e^0 is nonlocal at the free point
is an artefact of the free limit  for generic , both e0 and e^0 describe the
1 mode of
local elds, as follows from the discussion in [20]. One may therefore suspect that the fact
that [U0; x1=2] 6= 0 is purely a free eld artefact, but this is not the case: as we show in
appendix C, with the above identi cations, this problem persists for generic .
We believe that the resolution of this problem is that we need to correct the identi
cation (5.39) by (nonlocal) correction terms. The fact that such nonlocal correction terms
appear is maybe not surprising in view of the fact that also in the bosonic setting nonlocal
correction terms were required for the identi cation of the spin 3 and 4 elds, see [20]. The
relevant analysis is, however, rather cumbersome, and we leave it to future work.
6
In this paper we have found some of the de ning relations of the Yangian algebra that
is expected to be isomorphic to W1
[ ], the N = 2 superconformal version of W1
We have extensively used the fact that W1(N =2)[ ] contains two commuting bosonic W1
algebras, each of which in turn is isomorphic to an a ne Yangian of gl1. The additional
generators transform in biminimal representations with respect to these two W1
gebras. We have shown how this translates into explicit commutation relations for the
additional Yangian generators  our main technical advance is the description of the
conjugate representations, see section 4.2. This has allowed us to make a proposal for at least
some of the de ning relations of the N = 2 superconformal a ne Yangian. We have also
checked  in fact this was an important guiding principle  that these relations reduce
to the expected identities in the free eld case (
= 0).
There are many open questions which we hope to address in the future. First of all,
it would be nice to construct the representation theory of this Yangian algebra, see [38]
for rst steps in this direction; this will involve two plane partitions on which the various
generators should have some natural action. (The two a ne Yangians of gl1 act separately
on each, while the additional biminimal generators generate in nite rows of boxes (and
antiboxes), connecting the two plane partitions.) Among other things, this would allow us
to prove the consistency of our construction and to nd the remaining relations. It would
also be interesting to establish the dictionary to the W1
generalizing the construction of [20] to the current context, and to explore the various
duality symmetries this picture suggests. Note that the construction selects out one of the
three directions of each plane partition, thereby breaking the S3 symmetry [37] of each a ne
Yangian of gl1 to a Z2 symmetry that exchanges the remaining two directions. Together
with the exchange symmetry of the two a ne Yangians, the N = 2 a ne Yangian therefore
has a Z2
Z2 symmetry. We hope to come back to these questions in the near future.
(N =2)[ ] algebra in more detail,
Acknowledgments
We thank Tomas Prochazka and Miroslav Rapcak for sending us [39] prior to publication,
and Tomas Prochazka for comments on the draft. We also thank Yang Lei for help with
drawing the gures. The work of MRG is (partly) supported by the NCCR SwissMAP,
funded by the Swiss National Science Foundation. WL is grateful for support from the
\Thousand talents grant" and from the \Max Planck Partnergruppe". The work of CP is
supported by the US Department of Energy under contract DESC0010010 Task A. HZ is
partially supported by the General Financial Grant from the China Postdoctoral Science
Foundation, with Grant No. 2017M611009. MRG thanks Beijing University for hospitality
during the very
nal stages of this work. We gratefully acknowledge the hospitality of the
Galileo Galilei Institute for Theoretical Physics (GGI) for hospitality, and INFN for partial
nancial support during the program \New Developments in AdS3/CFT2 Holography".
A
Additional relations of the free
eld theory
In addition to the free eld relations that were given in the main body of the text, see
eqs. (4.21){(4.33), the commutation relations of ys and ys are
0 = [ r+2; ys]
2[ r+1; ys+1] + [ r; ys+2] + [ r+1; ys] [ r; ys+1]
0 = [er+1; ys] [er; ys+1]
(A.1)
(A.2)
0 = [ r+2; ys] 2[ r+1; ys+1] + [ r; ys+2] [ r+1; ys] + [ r; ys+1]
are given explicitly by
as well as (for r
2)
Pr =
Pr =
X
X
The Pr modes that were de ned in eq. (4.36), and their corresponding barred version
0 = [Pi+2; xj] 2[Pi+1; xj+1] + [Pi; xj+2] + [Pi+1; xj] [Pi; xj+1] 2[Pi; xj]
0 = [Pi+2; yj] 2[Pi+1; yj+1] + [Pi; yj+2] + [Pi+1; yj] [Pi; yj+1] 2[Pi; yj]
0 = [Pi+2; ej] 2[Pi+1; ej+1] + [Pi; ej+2] ([Pi+1; ej] [Pi; ej+1])
0 = [Pi+2; fj] 2[Pi+1; fj+1] + [Pi; fj+2] ([Pi+1; fj] [Pi; fj+1])
0 = [Pi+2; e^j] 2[Pi+1; e^j+1] + [Pi; e^j+2] + 3([Pi+1; e^j] [Pi; e^j+1]) + 2[Pi; e^j]
0 = [Pi+2; f^j] 2[Pi+1; f^j+1] + [Pi; f^j+2] + 3([Pi+1; f^j] [Pi; f^j+1]) + 2[Pi; f^j]
0 = [Pi+2; xj] 2[Pi+1; xj+1] + [Pi; xj+2] [Pi+1; xj] + [Pi; xj+1] 2[Pi; xj]
0 = [Pi+2; yj] 2[Pi+1; yj+1] + [Pi; yj+2] [Pi+1; yj] + [Pi; yj+1] 2[Pi; yj]
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
HJEP05(218)
0 = [Pi+2; f^j] 2[Pi+1; f^j+1] + [Pi; f^j+2] ([Pi+1; f^j] [Pi; f^j+1])
0 = [Pi+2; ej] 2[Pi+1; ej+1] + [Pi; ej+2] + ([Pi+1; ej] [Pi; ej+1])
0 = [Pi+2; fj] 2[Pi+1; fj+1] + [Pi; fj+2] + ([Pi+1; fj] [Pi; fj+1]) :
B
The de ning relations
Yangian.
B.1
The OPE like description
In this appendix we collect some of the de ning relations of the supersymmetric a ne
HJEP05(218)
(A.22)
(A.23)
(A.24)
(A.25)
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
where '2(u) is de ned in eq. (5.1). For the y elds we nd
In terms of modes, these identities are
[ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + h2 [ r+1; xs] [ r; xs+1] + h1h3 rxs = 0
[ ^r+2; xs] 2[ ^r+1; xs+1] + [ ^r; xs+2] (h2
2 ^0 3) [ ^r+1; xs] [ ^r; xs+1]
+ (h1 + ^0 3)(h3 + ^0 3)[ ^r; xs] h1h3 ^rxs = 0
[ r+2; xs] 2[ r+1; xs+1] + [ r; xs+2] + ( h2 + 2 0 3) [ r+1; xs] [ r; xs+1]
[ r+2; ys] 2[ r+1; ys+1] + [ r; ys+2] + h2 [ r+1; ys] [ r; ys+1]
+ ^0 3( ^0 3
h2)[ ^r; ys] + h1h3 ^rys = 0
[ ^r+2; ys] 2[ ^r+1; ys+1] + [ ^r; ys+2] (h2
2 ^0 3) [ ^r+1; ys] [ ^r; ys+1]
[ r+2; ys] 2[ r+1; ys+1] + [ r; ys+2] (h2
2 0 3) [ r+1; ys] [ r; ys+1]
For completeness, we also give the mode relations of the bosonic generators [19, 20], which
we write only for the unhatted generators (but which hold similarly also for the hatted
generators).
3fej; ekg = [ej+3; ek] 3[ej+2; ek+1] + 3[ej+1; ek+2] [ej; ek+3]
3ffj; fkg = [fj+3; fk] 3[fj+2; fk+1] + 3[fj+1; fk+2] [fj; fk+3]
3f j; ekg = [ j+3; ek] 3[ j+2; ek+1] + 3[ j+1; ek+2] [ j; ek+3]
3f j; fkg = [ j+3; fk] 3[ j+2; fk+1] + 3[ j+1; fk+2] [ j; fk+3]
In addition, they satisfy the Serre relations
B.3
The initial conditions
The modi ed initial conditions, generalizing (4.33) are
Sym(j1;j2;j3)[ej1; [ej2; ej3+1]] = 0 ;
Sym(j1;j2;j3)[fj1; [fj2; fj3+1]] = 0 :
Furthermore,
and
Finally, the initial relations of the bosonic generators are
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
(B.15)
(B.16)
(B.17)
(B.18)
(B.19)
Supercharge constraints
We start by assuming that the decoupled u(1) generator commutes with the x1=2 generator,
which we would like to identify with the supercharge, see eq. (5.39),
We will deduce two identities from this assumption, using the commutation relations we
have postulated. First we consider
For the lefthand side we then use
from which it follows, using again (C.1), that
tator with 3 + ^3,
= 2[(e0
e^0); x1=2] = 4[e0; x1=2] :
The other identity can be derived similarly, except that now we consider the
commu[[( 3 + ^3); x1=2]; (e0 + e^0)] = [[( 3 + ^3); (e0 + e^0)]; x1=2] + [[(e0 + e^0)]; x1=2]; ( 3 + ^3)]
= [[( 3 + ^3); e0 + e^0]; x1=2] :
Now the relevant charge relations are
where d is some constant, and
This then leads to the identity
[ 3 + ^3; x1=2] = (10 + 2h1h3 0)x3=2 + d x1=2
(10 + 2h1h3 0)[x3=2; e0 + e^0] = 6[e1 + e^1; x1=2] + 2 3 0[e0; x1=2] + 2 3 ^0[e^0; x1=2]
= 6[ 3 ^0 e0
3 0 e^0; x1=2] + 2 3 0[e0; x1=2]
+2 3 ^0[e^0; x1=2]
where we have also used that
[e1 + e^1; x1=2] = [L 1; x1=2] = [J 1; x1=2] = [ 3 ^0 e0
3 0 e^0; x1=2] ;
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
N
N + k
1 ;
7! 1
(C.10)
one may
HJEP05(218)
as follows from the fact that x1=2
G+3=2 is the mode of a primary eld of spin 3=2 with
charge +1, i.e., from the relations of the N = 2 superconformal algebra
Gm+r ;
r
Gm+r :
(C.9)
n
2
The two identities (C.4) and (C.7) are only compatible provided that
(10 + 2h1h3 0) = (4
4h1h3 0) ;
i.e. h1h3 0 =
= 0, the analogue of (5.40) vanishes, [e0 + e^0; x1=2] = 0.
is then
at
which corresponds to the case
= 1. Given the usual symmetry
wonder why
= 1 appears rather than
= 0. In fact, if one repeats the same analysis for
xr instead of xr, the same analysis goes through, except that the compatibility condition
= 0. This is also mirrored by the fact that, in the explicit free eld calculation
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