BPS Z(2) monopoles and N=2 SU(n) superconformal field theories on the Higgs branch

0.24 MB·English

by  Marco A. C. Kneipp

Z
BPS monopoles and
2
= 2 SU(n) superconformal field theories
N
2 on the Higgs branch
1
0
2

t
c
O
6 Marco A.C. Kneipp1 and Paulo J. Liebgott2
2

Universidade Federal de Santa Catarina (UFSC),
]
h Departamento de F´ısica, CFM,
t
- Campus Universit´ario, Trindade,
p
e 88040-900, Florian´opols, Brazil.
h
[

1
v Abstract
3
4 We obtain BPS Z monopole solutions in Yang-Mills-Higgs theories with the
2
2
gauge group SU(n) broken to Spin(n)/Z by a scalar field in the n n representa-
7 2
⊗
. tion. We show that the magnetic weights of the so-called fundamental Z monopoles
0 2
1 correspond to the weights of the defining representation of the dual algebra so(n)∨,
2 and the masses of the non-fundamental BPS Z monopoles are equal to the sum of
2
1
the masses of the constituent fundamental monopoles. We also show that the vacua
:
v responsible for the existence of these Z monopole are present in the Higgs branch
i 2
X of a class of = 2 SU(n) superconformal field theories. We analyze some dualities
N
r these monopoles may satisfy.
a
PACS: 14.80.Hv, 11.15.-q, 02.20.Sv
[email protected]
[email protected]
1 Introduction
Electromagnetic duality in Yang-Mills-Higgs theories was initially proposed by Goddard,
Nuyts, and Olive (GNO) [1] in gauge theories with gauge group G spontaneously broken
to G , by a scalar field φ in an arbitrary representation, in such a way that π (G/G )
0 2 0
is nontrivial, which allows the existence of monopoles solutions. Soon after, Montonen
and Olive (MO) duality was proposed [2] considering a theory with gauge group SU(2)
spontaneously broken to U(1) by a scalar field φ in the adjoint representation. Since then,
monopole’s solutions and the electromagnetic duality has been studied mainly when the
scalar field responsible for the symmetry breaking is in the adjoint representation. In
this case, the unbroken gauge group G necessarily has a U(1) factor which guarantees
0
that π (G/G ) = Z and the theory can have monopole solutions which we shall call Z
2 0
monopoles. On the other hand, much less is known when G is semisimple and therefore φ
0
necessarily can not be in the adjoint representation. In these cases, a nontrivial π (G/G )
2 0
Z Z
will be a cyclic group or a product of cyclic groups and the monopoles are called
n n
monopoles. Therefore, Z monopoles are relevant for GNO duality when G semisimple,
n 0
Z
which had renewed interest in the geometric Langlands program [3]. These monopoles
n
were analyzed for example in [1][4][5] and more recently in [6].
One of the main motivations for the study of monopoles and electromagnetic dualities
is their possible application to the problem of confinement in QCD. Following the ideas of
’t Hooft and Mandelstam, the formation of chromoelectric flux tubes in QCD must be due
Z
to a monopole condensate. However it is not yet clear if these monopoles are monopoles,
Z
monopoles, or Diracmonopoles. In the last years, the ideas of ’t Hooft and Mandelstam
n
Z
were applied to supersymmetric non-Abelian theories with monopoles. In particular in
Z
[7][8] it was analyzed the confinement of monopoles by the formation of magnetic flux
Z
tubesor strings insoft broken = 4 super Yang-Millstheories withanarbitrarysimple
n
N Z
gauge group. It was shown that the tensions of these strings satisfy the Casimir scaling
n
law in the BPS limit, which is believed to be the behavior that the chromoelectric flux
Z
tubes in QCD must satisfy[9]. This result indicates that these strings can be dual to
n
QCD chromoelectric strings.
Z
In order to understand the properties of the monopoles, in [6] we obtained explicitly
n
the asymptotic form of the Z monopoles in SU(n) Yang-Mills-Higgs theories with the
2
gauge group broken to Spin(n)/Z by a scalar in the n n representation of SU(n) or its
2
⊗
symmetric part. In order to obtain these asymptotic forms we generalized the construction
in [4] using the fact that the magnetic weights of the monopoles in this theory must belong
to the cosets Λ (Spin(n)∨) or λ∨ + Λ (Spin(n)∨) corresponding to the two topological
R 1 R
Z Z
sectors associated to the group . It is important to note that a monopole is not its
2 2
Z Z
own antiparticle and the fact that monopoles are associated to topological sectors
2 2
does not imply that they carry non-additive magnetic charges as we will explain in sections
2 and 3. We constructed the monopole solutions considering two symmetry breakings of
the algebra su(n) to so(n): one in which so(n) is invariant under outer automorphism and
another in which it is invariant under Cartan automorphism. In both cases we associated a
su(2) subalgebra, subject to some constraints, to each weight of the defining representation
1
of the dual algebra so(n)∨ and constructed explicitly the Z monopoles called fundamental
2
monopoles. Using linear combinations of the generators of these su(2) subalgebras we
were able to construct other su(2) subalgebras and the corresponding Z monopoles called
2
non-fundamental.
Z
In this paper we write the vacuum solution and the asymptotic forms for the
2
monopolesinterms of singlets andtripletswith respect tothe corresponding su(2)subalge-
bras. We calculate the masses for the BPS monopoles and obtained that the fundamental
BPS Z monopoles have same masses equal to 4πv/e, where v is the norm of the Higgs
2
Z
vacuum. On the other hand, the masses of the non-fundamental monopoles are the
2
sum of the masses of the constituent fundamental monopoles which is consistent with the
interpretation that the non-fundamental monopoles should be multimonopoles composed
Z
of non-interacting fundamental monopoles, similarly to what happens for the monopoles
[10].
Exactelectromagneticdualityisexpectedtohappeninsuperconformaltheories(SCFTs),
with vanishing β function, like = 4 Super Yang-Mills theories[11], = 2 SU(2) Super
N N
Yang-Mills theories with N = 4 flavors [12], etc. More recently, with the works [13][14],
F
Z
there was some renew interest with the study of dualities in SCFTs. The monopoles
2
can not exist in = 4 Super Yang-Mills theories, where all scalars are in the adjoint
N
Z
representation. Therefore, in order to analyze some possible dualities that monopoles
2
may satisfy in SCFTs, we consider = 2 SU(n) super Yang-Mills theories with a hy-
N
permultiplet in the n n representation, which has vanishing β function and which we
⊗
will denote by = 2′ SCFTs. We showed that its potential accept the vacua solutions
N
discussed in the previous sections. These vacua correspond to certain points of the Higgs
Z
branch where the monopoles can exist. That is different from the Coulomb branch
2
where the gauge symmetry is generically broken broken to the maximal torus U(1)r (or
to K U(1) in some specific points) and there are Z monopoles/dyons everywhere on the
×
Z
Coulomb branch. It is interesting to note that the BPS equations for the monopoles
2
Z
does not result on vanishing of any supercharges. Therefore, even the BPS monopoles
2
satisfying the first order BPS equations are in long = 2 massive supermultiplets, like
N
the massive gauge fields in this theory. We also showed that this = 2′ SCFT can have
Z Z N
an Abelian Coulomb phase with monopoles and monopoles. From the results we
2
Z
obtained, we discussed some possible dualities the monopoles may satisfy.
2
This paper is organized as follows: we start in Sec. 2 giving a short review of our
Z
generalized construction of spherically symmetric monopole’s asymptotic forms. Then,
n
in Sec. 3 we obtained explicitly the asymptotic form of the Z monopoles in SU(n) Yang-
2
Mills-Higgs theories with the gauge group broken to Spin(n)/Z by a scalar in the n n
2
⊗
representation. We consider two vacua configurations which break su(n) to so(n) where for
the first configuration so(n) is invariant under Cartan automorphism and for the second
configuration it is invariant under outer automorphism. In Sec. 4 we calculate the BPS
Z
masses for the fundamental and non-fundamental monopoles. In Sec. 5, we show that
2
the vacua responsible for the breaking SU(n) to Spin(n)/Z belong to the Higgs branch
2
of a = 2 SU(n) SCFT and therefore this theory can have these Z monopoles. Finally,
2
N
Z
in Sec. 6 we discuss some possible dualities these monopoles can satisfy.
2
2
Z
2 General properties of monopoles
n
In this section we shall recall some of the principal results of Z monopoles and fix some
n
conventions. For more details see [6]. Let us start considering a Yang-Mills theory with
gauge group G which we shall consider to be simple and simply connected. Let us also
consider that the theory has a scalar field φ in a representation R(G) and φ is a vacuum
0
configuration which spontaneously breaks G to G such that π (G/G ) is nontrivial, and
0 2 0
therefore allows the existence of monopoles. Let us denote3 by g the algebra formed by
the generators of G and g the generators of G . Note that in general, the elements of the
0 0
Cartan subalgebra (CSA) of g do not necessarily belong to the CSA of g. Therefore, we
0
shall denote by H and E respectively the CSA’s generators and the step operators of g,
i α
and h and f the corresponding generators of g . We shall adopt the convention that in
i α 0
the Cartan-Weyl basis, the commutation relations read
[H ,E ] = (α)iE , (1)
i α α
2α H
[E ,E ] = · .
α −α
α2
We shall denote by α , i = 1, 2, ..., r = rankg, the simple roots of g and λ , i =
i i
1, 2, ..., r, the fundamental weights of g. Moreover, we shall denote
2α 2λ
α∨ = i, λ∨ = i, (2)
i α2 i α2
i i
the simple coroots and fundamental coweights respectively. They are, simple roots and
fundamental weights of the dual algebra g∨ and satisfy the relations α λ∨ = α∨ λ = δ .
i· j i · j ij
The asymptotic condition φ = 0 for finite energy configurations implies that asymp-
i
D
totically we can write
φ(θ,ϕ) = g(θ,ϕ)φ , (3)
0
where θ andϕ are theangular spherical coordinatesand g(θ,ϕ) G. Then, theasymptotic
∈
form of the magnetic field of the monopoles can be written as [1]
x
B (θ,ϕ) = i g(θ,ϕ)ω hg(θ,ϕ)−1 (4)
i
2er3 ·
where ω is a real vector called magnetic weight and h belongs to the CSA of g .
i 0
Note that when the gauge group G is broken by a scalar field in the adjoint represen-
tation, the unbroken gauge group G always have a U(1) factor generated by the scalar
0
field vacuum solution Φ = φ T and we can define an Abelian magnetic charge for the
0 0a a
monopole associated to this U(1) factor
1 1
g = dS Tr(B Φ) = dS Tr B g(θ,ϕ)Φ g(θ,ϕ)−1 .
Φ ˛ i i Φ ˛ i i 0
0 0
| | | |
S∞2 S∞2 (cid:0) (cid:1)
3We shall adopt the convention of using capital letters to denote Lie groups and lower letters for Lie
algebras.
3
On the other hand, when φ is not in the adjoint representation, we can not define the
above charge, but we can define magnetic charges associated to the CSA generators h of
a
the unbroken group G as4
0
2π
g = dS Tr B g(θ,ϕ)h g(θ,ϕ)−1 = ω . (5)
a ˛ i i a e a
S∞2 (cid:0) (cid:1)
Therefore, these magnetic charges are proportional to the components of the magnetic
weight associated to a monopole.
Considering that G is semisimple, it can be written as
0
G = G /K(G )
0 0 0
where G0 is the universal covering group off G0 and K(G0) is the kernel of the homomor-
phism G G . One can show that K(G ) is a discrete subgroup of the center of G ,
0 0 0 0
→
which wfe will call Z(G ). Therefore, when G is semisimple, the topological charge sectors
0 0
of the tfheory are associated to f
f
π (G/G ) = π (G ) = K(G ) Z(G ). (6)
2 0 1 0 0 0
⊂
Hence, π (G/G ) is a cyclic group Z , or product of cyclic groups, and the monopoles are
2 0 n f
Z
called monopoles.
n
Now, the center of a group G is a discrete group isomorphic to the classes
0
Z(G ) = exp[2πiΛ (G∨) h], exp 2πi λ∨ +Λ (G∨) h , ... , (7)
0 fr 0 · τ(0) r 0 ·
..., exp 2πi λ∨ +Λ (G∨) h ,
f (cid:8) τn(0) r (cid:2) 0 (cid:0)· (cid:1) (cid:3)
whereΛ (G∨)istherootlattice(cid:2)ofG(cid:0)∨, thedualgroup(cid:1)ofG(cid:3)(cid:9), andthefundamentalcoweights
r 0 0 0
λ∨ are associated to the notes of the extended Dynkin diagram of G related to the node
τq(0) 0
0 by a symmetry transformation, as explained in detail in [16]. The relation (7) is due to
the fact that the quotient Λ (G∨)/Λ (G∨) can be represented by the cosets
w 0 r 0
Λ (G∨), λ∨ +Λ (G∨), λ∨ +Λ (G∨), ... ,λ∨ +Λ (G∨). (8)
r 0 τ(0) re 0 τ2(0) r 0 τn(0) r 0
Since K(G ) Z(G ), the topological charge sectors (6) are associated to the elements of
0 0
⊂
(7) which are in the kernel of the homomorphism G G .
0 0
→
The group elemfent g(θ,ϕ) must satisfy the relation [1]
f
g(π,0)−1g(π,2π) = exp[2πiω h] K(G ) Z(G ) (9)
0 0
· ∈ ⊂
where exp denotes the exponential mapgping in G0. Hence, exp[2πfiω ·h] must be in one
of the classes of (7) associated to K(G ). Therefore, the magnetic weights ω must be only
0
f
4Remgember that when we have a monopole solution, the unbrokenggroup is not fixed but varies with
the space direction within G by conjugation g(θ,ϕ)G0g(θ,ϕ)−1 [24]
4
in the cosets associated to the kernel K(G ) and the Z monopoles will be in the same
0 n
topological sector if their associated magnetic weights ω are in the same coset [6]. The
coset Λ (G∨) corresponds to the trivial element of the group Z and monopoles with
r 0 n
1 Z
magnetic weights in this coset belongs to the trivial topological sector. Note that two
n
monopoles in the same topological sector, i.e. they are associated to magnetic weights ωA
and ωB in the same coset, does not imply that they are the same monopole since they have
different asymptotic magnetic fields (4), unless ωA = ωB.
Let us now consider a generator
β h
Tβ = ·
3 2
such that β is a vector that belongs to one of the cosets associated to K(G ), that is, β
0
β β
can be a magnetic weight. Let us also consider that exist other two generators T ,T / g
which together with Tβ form an su(2) subalgebra 1 2 ∈ 0
3
Tβ,Tβ = iǫ Tβ,
i j ijk k
h i
which we shall denote su(2) . Since exp[2πiβ h] K(G ), then exp[2πiqβ h] K(G )
β 0 0
where q Z. Therefore, qβ is also in one of·the∈cosets associated to K(G·). ∈Since we
0
∈
are interested in the study of fundamental monopoles, we shall consider solutions with
g g
spherically symmetric asymptotic forms. As in [6], from these generators, we shall obtain
explicit monopole asymptotic forms with spherical symmetry using a generalization of the
construction in [4], writing the group element g(θ,ϕ) as
g(θ,ϕ) = exp( iϕqTβ)exp( iθTβ)exp(iϕqTβ), (10)
− 3 − 2 3
which satisfy
g(π,0)−1g(π,2π) = exp[2πiqβ h] K(G ). (11)
0
· ∈
Therefore, the monopole associated to this group element has magnetic weight ω = qβ.
Hence, for each integer q and su(2) subalggebra with Tβ satisfying condition (9), we can
β 3
Z
associate a monopole. A very important difference from the construction in [4], is that
n
in our construction the monopole topological sectors are associated to the cosets (8) and
not to the integer q and therefore, monopoles associated to magnetic weights with same
integer q are not necessarily in the same topological sector. As a consequence, from our
generalized construction we obtain much more solutions. One can think the monopoles
associated to an su(2) and with q > 1 as superpositions of q monopoles with q = 1
β
| | | | | |
associated to the same su(2) . Similarly to [4], we consider that a monopole associated
β
to a su(2) subalgebra with a negative integer q is the antimonopole of the monopole
β
−
with positive integer q and associated to the same su(2) . It is interesting to note that in
β
Z
particular, a monopoles andits antiparticle are in the same topological sector, but if one
2
has magnetic weight qβ the other has qβ and therefore they have different asymptotic
−
Z
magnetic fields (4). Hence, a monopoles is not its own antiparticle.
2
5
Using the identity, for i = j,
6
exp(iaT )T exp( iaT ) = (cosa)T +(sina)ǫ T , (12)
j i j i ijk k
−
where T , i = 1,2,3 form an arbitrary su(2) subalgebra, we can rewrite the asymptotic
i
form (4) for the magnetic field with ω = qβ and g(θ,ϕ) given by (10) as
qx
B(q)(θ,ϕ) = i Tβcosθ +sinθ Tβcosqϕ+Tβsinqϕ . (13)
i er3 3 1 2
h (cid:16) (cid:17)i
One can obtain this asymptotic form from the gauge field configuration [4]
i
W (θ,ϕ) = g(θ,ϕ)Wstringg(θ,ϕ)−1 (∂ g(θ,ϕ))g(θ,ϕ)−1, (14)
i i − e i
with
Wstring = Wstring = 0,
r θ
β
qT (1 cosθ)
Wstring = 3 − ,
ϕ − er sinθ
which gives
W (θ,ϕ) = 0, (15a)
r
1
β β
W (θ,ϕ) = T cosqϕ T sinqϕ , (15b)
θ er 2 − 1
q (cid:16) (cid:17)
W (θ,ϕ) = Tβsinθ cosθ Tβcosqϕ+Tβsinqϕ . (15c)
ϕ − er 3 − 1 2
h (cid:16) (cid:17)i
Z
3 monopoles in SU(n) Yang-Mills-Higgs theories
2
Let us consider an Yang-Mills-Higgs theory with gauge group SU(n) and a scalar field φ
in the direct product representation n n of SU(n). In order to exist Z monopoles, in [6]
2
⊗
we found vacuum solutions φ which break
0
Spin(n)
SU(n) (16)
Z
→
2
for n 3, where Spin(n) is the covering group of SO(n) and associated to the algebra
≥
so(n). We considered two different vacua: for one vacuum, the unbroken so(n) is the
subalgebra of su(n) invariant under Cartan automorphism and for the second vacuum,
so(n) is the subalgebra invariant under outer automorphism and in this case n must be
odd. In both cases, the kernel K(G ) = Z is associated to the cosets
0 2
Λ (Spin(n)∨), λ∨ +Λ (Spin(n)∨), (17)
r 1 r
where λ is a fundamental weight of the so(n) subalgebra, using the convention of [6]. The
1
firstcosetisassociatedtothetrivialtopologicalsector. Asexplainedindetailinsection6of
6
[6], if we consider two Z monopoles with magnetic weights ω(A) and ω(B) belonging to the
2
coset λ∨+Λ (Spin(n)∨) and therefore belonging to the non-trivial topological sector, then
1 r
themonopolecomposedbythesetwomonopoleswillhavemagneticweightω(A)+ω(B) which
belongs to Λ (Spin(n)∨) (since 2λ∨ Λ (Spin(n)∨)) and hence to the trivial topological
r Z 1 ∈ r
sector. It means that the monopole carries an additive magnetic charge, since it is
2
Z
proportional to its magnetic weight, and the topological charge of a monopole is related
2
to the exponential of its magnetic weight by Eq. (7).
Before to consider these two symmetry breakings, let us obtain some Lie algebra results
for the n n representation, which will be useful in the next sections. Let us denote by
⊗
e , l = 1,2, ...,n, the weight states of the n-dimensional representation of su(n). In this
l
| i
representation, the generators of su(n) can be written in terms of the n n matrices E
ij
×
with components (E ) = δ δ or
ij kl ik jl
E e = e . (18)
ij j i
| i | i
Inthiscase, thebasiselementsoftheCSAofsu(n)correspondtothetracelesscombinations
E E , fori = 1,2, ...,n 1. ThegeneratorE , i = j, isthestepoperatorassociated
ii i+1,i+1 ij
− − 6
to the root e e , where e is an orthonormal vector in the n-dimensional vector space.
i j i
−
In the representation n n, the weight states are e e , i,j = 1,2, ...,n and the
i j
⊗ | i⊗| i
generators can be written as
E = E 1+1 E .
ij ij ij
⊗ ⊗
In this representation, for a root β = e e of su(n), we can associate a su(2) subalgebra
i j β
−
β H 1
Tβ = · = (E E ),
3 2 2 ii − jj
E +E 1
Tβ = β −β = (E +E ), (19)
1 2 2 ij ji
E E 1
Tβ = β − −β = (E E ).
2 2i 2i ij − ji
Adopting the notation i,j e e , we can define the weight states
i j
| i ≡ | i⊗| i
1
0 = ( j,j i,i ),
| iβ,1 √2 | i−| i
i
0 = ( j,j + i,i ), (20)
| iβ,2 √2 | i | i
1
0 = ( i,j + j,i ),
| iβ,3 √2 | i | i
where 0 is eigenvector of Tβ with vanishing eigenvalue and one can check
| iβ,i i
β
T 0 = i ǫ 0 . (21)
i | iβ,j ijk| iβ,k
k
X
7
Remembering that for an arbitrary Lie algebra, a weight state T of the adjoint represen-
i
| i
tation is associated to a generator T through the relation
i
T T = i f T = [T ,T ] ,
i j ijk k i j
| i | i | i
k
X
where f are the structure constants of the algebra. Therefore, from Eq. (21) we can
ijk
conclude that the weight states (20) form an adjoint or triplet representation of the su(2)
β
subalgebra (19) and we can associate 0 to Tβ.
| iβ,j j
3.1 so(2m + 1) invariant under outer automorphism
Let us consider first the case where so(2m+1) is the invariant subalgebra of su(2m+1)
under outer automorphism. In this case, the CSA of so(2m + 1) is inside the CSA of
su(2m+1)as explained in detail in [6]. The vacuum configurationwhich breaks su(2m+1)
to this so(2m+1) subalgebra is [6]
2m+1
v
φ = ( 1)l+1 l,2m+2 l , (22)
0
√2 − | − i
l=1
X
where v is a real constant.
Z
Let us now analyze the possible monopole solutions of the theory. Since for the
2
moment we are interested in the so-called fundamental monopoles, we shall consider that
q = 1. The monopoles associated to the non-trivial topological sector must have magnetic
weights β in the coset λ∨ +Λ (Spin(n)∨). This condition is written in terms of coweights
1 r
and coroots of the subalgebra so(2m+1). We showed that this condition can be written
in terms of roots of su(2m+1) as
m−1
β c (α +α ) +(2c +1)(α +α ) (23)
i i 2m+1−i m m m+1
∈
" #
i=1
X
where c are arbitrary integers and α are simple roots of su(2m+1). On the other hand,
i i
the monopoles associated to the trivial topological sector must have magnetic weights β in
the coset Λ (Spin(n)∨). This condition can be written in terms of roots of su(2m+1) as
r
m−1
β c (α +α ) +2c (α +α ), (24)
i i 2m+1−i m m m+1
∈
" #
i=1
X
withc beingintegers. Therefore, β canonlybeintheparticularsubspaceofΛ (SU(2m+1))
i r
which is the union of the subspaces given by conditions (23) and (24). In order to construct
su(2) subalgebras, we consider that β is a root of su(2m+1)in this subspace. In this case,
β
we can consider a su(2) subalgebra of the form of (19) which satisfies all the properties
β
8
discussed before. The only roots of su(2m+1) which satisfy condition (23) of being in the
non-trivial sector, are [6]
(α +α +...+α ) , p = 1,2,...,m. (25)
p p+1 2m+1−p
±
On the other hand, there is no root of su(2m+1) which satisfy condition (24). We con-
structed other su(2) subalgebras associated to other elements in the cosets (17). However,
β
in all the cases we found, the generators were always linear combination of the generators
Z
of (19). Therefore, we called fundamental monopoles, the monopoles associated to the
2
su(2) subalgebras (19) with β being one of the 2m roots (25), similarly to the nomen-
βp
Z
clature used in [10] for the -monopoles. All these fundamental monopoles are in the
non-trivial topological sector. These 2m roots can be written as the weights of the 2m-
dimensional defining representation of so(2m+1)∨ = sp(2m). Therefore, we can say that
Z
the fundamental monopoles are associated to this representation.
2
Using the fact that the simple roots of su(2m+1) can be written as α = e e , we
p p p+1
−
can write these 2m roots, or magnetic weights, as
β = e e ,
p p 2m+2−p
−
for p = 1,2,...,m,m+ 2,m+ 3,...,2m+ 1. We can write the generators of the su(2)
β
subalgebra (19) associated to β in the n n representation as
p
⊗
1
Tβp = (E E ),
3 2 p,p − 2m+2−p,2m+2−p
1
Tβp = (E +E ), (26)
1 2 p,2m+2−p 2m+2−p,p
1
Tβp = (E E ),
2 2i p,2m+2−p − 2m+2−p,p
and the corresponding weight vectors
1
0 = ( 1)p+1 ( 2m+2 p,2m+2 p p,p ),
p,1
| i − √2 | − − i−| i
i
0 = ( 1)p+1 ( 2m+2 p,2m+2 p + p,p ), (27)
p,2
| i − √2 | − − i | i
1
0 = ( 1)p+1 ( p,2m+2 p + 2m+2 p,p ),
p,3
| i − √2 | − i | − i
which are in the adjoint representation of the su(2) subalgebra (26) and satisfy (21).
βp
We can write the vacuum configuration (22) as
φ = 0 +v 0 , (28)
0 p,0 p,3
| i | i
where
v
0 = ( 1)l+1 l,2m+2 l ,
p,0
| i √2 − | − i
l6=p,2m+2−p
X
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