PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 49–60 DOI: https://doi.org/10.2298/PIM161217003D

POSITIVE ENERGY UNITARY IRREDUCIBLE REPRESENTATIONS OF THE SUPERALGEBRA osp 1 8, ℝ

Vladimir Dobrev^{a} and Igor Salom^{b}

Abstract. We continue the study of positive energy (lowest weight) unitary irreducible representations of the superalgebras osp 1 2n, ℝ . We present the full list of these UIRs. We give a proof of the case osp 1 8, ℝ .

1. Introduction

Recently, superconformal field theories in various dimensions are attracting more interest, in particular, due to their duality to AdS supergravities. This makes the classification of the UIRs of these superalgebras very important. Until recently only those for D 6 were studied since in these cases the relevant superconformal algebras satisfy [24] the Haag–Lopuszanski–Sohnius theorem [19]. Thus, such classification was known only for the D 4 superconformal algebras su 2, 2 N [17] (for N 1), [10–13] (for arbitrary N). More recently, the classification for D 3 (for even N), D 5, and D 6 (for N 1, 2) was given in [23] (some results are conjectural), and then the D 6 case (for arbitrary N) was finalized in [8].

On the other hand the applications in string theory require the knowledge of the UIRs of the conformal superalgebras for D 6. Most prominent role play the superalgebras osp 1 2n . Initially, the superalgebra osp 1 32 was put forward for D 10 [18, 26]. Later it was realized that osp 1 2n would fit any dimension, though they are minimal only for D 3, 9, 10, 11 (for n 2, 16, 16, 32, resp.) [2,3,16]. In all cases we need to find at first the UIRs of osp 1 2n, ℝ the study of which was started in [15] and [9]. Later, in [14], we finalized the UIR classification of [15] as Dobrev–Zhang–Salom (DZS) Theorem. In [14] we proved the DZS Theorem for osp 1 6 .

2010 Mathematics Subject Classification: Primary 17B10; Secondary 17B35,81R05.

Key words and phrases: unitary representation, Verma module, singular vector, subsingular vector.

aSupported by COST Actions MP1210 and MP1405, and by Bulgarian NSF Grant DFNI T02/6.

bSupported by COST Action MP1405 and Serbian Ministry of Science and Technological Development, grant OI 171031.

Communicated by Zoran Rakić.

In the present paper, we prove the DZS Theorem for osp 1 8 . For the lack of space, we refer to [14, 15] for extensive literature on the subject.

2. Preliminaries on representations

Our basic references for Lie superalgebras are [20–22], although in this exposition we follow [15].

The even subalgebra of osp 1 2n, ℝ is the algebra sp 2n, ℝ with maximal compact subalgebra u n su n u 1 .

We label the relevant representations of by the signature

(2.1)  χ  d; a_{1} , …, a_{n 1}

where d is the conformal weight, and a_{1}, …, a_{n 1} are non-negative integers which are Dynkin labels of the finite-dimensional UIRs of the subalgebra su n (the simple part of ).

In [15] were classified (with some omissions to be spelled out below) the positive energy (lowest weight) UIRs of following the methods used for the D 4, 6 conformal superalgebras, cf. [8, 10–13], resp. The main tool was an adaptation of the Shapovalov form [25] on the Verma modules V^{χ} over the complexification ℂ osp 1 2n of .

We recall some facts about ^{ℂ} osp 1 2n (denoted B 0, n in [20, 21]) as used in [15]. The root systems are given in terms of δ_{1} …, δ_{n}, δ_{i}, δ_{j} δ_{ij} , i, j 1, …, n. The even and odd roots systems are [20, 21]

Δ¯0 δ_{i} δ_{j} , 1 i j n; 2δ_{i}, 1 i n , Δ¯1 δ_{i}, 1 i n

(we remind that the signs are not correlated). We shall use the following distinguished simple root system [20,21] Π δ_{1} δ_{2}, …, δ_{n 1} δ_{n}, δ_{n} , or, introducing standard notation for the simple roots,

Π α_{1}, …, α_{n} , α_{j} δ_{j} δ_{j 1} , j 1, …, n 1, α_{n} δ_{n}.

The root α_{n} δ_{n} is odd, the other simple roots are even. The Dynkin diagram is

_{1}  _{n 1}  _{n} .

The black dot is used to signify that the simple odd root is not nilpotent. In fact, the superalgebras B 0, n osp 1 2n have no nilpotent generators unlike all other types of basic classical Lie superalgebras [20, 21].

The corresponding to Π positive root system is

(2.2)  Δ ; ¯0 δ_{i} δ_{j} , 1 i j n; 2δ_{i}, 1 i n , Δ ; ¯1 δ_{i}, 1 i n

We record how the elementary functionals are expressed through the simple roots:

δ_{k} α_{k} α_{n}.

From the point of view of representation theory, more relevant is the restricted root system, such that

Δ̄^{}Δ̄ ; ¯0 Δ ; ¯1, Δ̄ ; ¯0 α Δ ; ¯0 \frac{1}{2}α Δ ; ¯1 δ_{i} δ_{j} , 1 i j n

POSITIVE ENERGY UIRS OF osp 1 8, ℝ 51

The superalgebra osp 1 2n, ℝ is a split real form of osp 1 2n and has the same root system.

The above simple root system is also the simple root system of the complex simple Lie algebra B_{n} (dropping the distinction between even and odd roots) with Dynkin diagram

_{1}  _{n 1}  _{n} .

Naturally, for the B_{n} positive root system we drop the roots 2δ_{i}

Δ_{B_{n}}^{}δ_{i} δ_{j} , 1 i j n; δ_{i}, 1 i n Δ̄^{}

This shall be used essentially below.

Besides (2.1), we shall use the Dynkin-related labelling:

Λ, α_{k}^{}a_{k}, 1 k n,

where α_{k}^{}2α_{k} α_{k}, α_{k} , and the minus signs are related to the fact that we work with lowest weight Verma modules (instead of the highest weight modules used in [22]) and to Verma module reducibility w.r.t. the roots α_{k} (this is explained in detail in [13, 15]).

Obviously, a_{n} must be related to the conformal weight d which is a matter of normalization so as to correspond to some known cases. Thus, our choice is

a_{n} 2d a_{1} a_{n 1} .

The actual Dynkin labelling is given by m_{k} ρ Λ, α_{k}^{}where ρ ^{}is given by the difference of the half-sums ρ¯0, ρ¯1 of the even, odd, resp., positive roots (cf. (2.2)

ρ ρ¯0 ρ¯1 n \frac{1}{2} δ_{1} n \frac{3}{2} δ_{2}  \frac{3}{2}δ_{n 1} \frac{1}{2}δ_{n},
ρ¯0 nδ_{1} n 1 δ_{2}  2δ_{n 1} δ_{n},
ρ¯1 \frac{1}{2} δ_{1}  δ_{n} .

Naturally, the value of ρ on the simple roots is 1: ρ, α_{i}^{}1, i 1, …, n. Unlike a_{k} ℤ_{}for k n, the value of a_{n} is arbitrary. In the cases when a_{n} is

also a non-negative integer, and then m_{k} ℕ (for all k) the corresponding irreps are the finite-dimensional irreps of (and of B_{n}).

Having in hand the values of Λ on the basis, we can recover them for any element of ^{}. We shall need only Λ, β^{}for all positive roots β as given in [15]

Λ, δ_{i} δ_{j} ^{}Λ, δ_{i} δ_{j}  a_{i}  a_{j 1}
Λ, δ_{i} δ_{j} ^{}Λ, δ_{i} δ_{j}  2d a_{1}  a_{i 1} a_{j}  a_{n 1}
(2.3)  Λ, δ_{i}^{}Λ, 2δ_{i} 2d a_{1}  a_{i 1} a_{i}  a_{n 1}
Λ, 2δ_{i} ^{}Λ, δ_{i} d \frac{1}{2} a_{1}  a_{i 1} a_{i}  a_{n 1}

To introduce Verma modules we use the standard triangular decomposition

ℂ ^{}^{}

where ^{}, ^{}, resp., are the subalgebras corresponding to the positive, negative, roots, resp., and denotes the Cartan subalgebra.

We consider lowest weight Verma modules, so that V^{Λ} U ^{}v_{0} where U ^{}is the universal enveloping algebra of ^{}, and v_{0} is a lowest weight vector v_{0} such that

Zv_{0} 0, Z ^{}; Hv_{0} Λ H v_{0}, H .

Further, for simplicity we omit the sign , i.e., we write p v_{0} V^{Λ} with p U ^{}.

Adapting the criterion of [22] (which generalizes the BGG-criterion [1] to the super case) to lowest weight modules, one finds that a Verma module V^{Λ} is reducible w.r.t. the positive root β iff the following holds [15]

(2.4)  ρ Λ, β^{}m_{β},  β Δ^{}, m_{β} ℕ.

If a condition from (2.4) is fulfilled, then V^{Λ} contains a submodule which is a Verma module V^{Λ^{}}with shifted weight given by the pair m, β: Λ^{}Λ mβ. The embedding of V^{Λ^{}}in V^{Λ} is provided by mapping the lowest weight vector v_{0}^{}of V^{Λ^{}} to the singular vector v_{s}^{m,β} in V^{Λ} which is completely determined by the conditions

Xv_{s}^{m,β} 0, X ^{},
Hv_{s}^{m,β} Λ^{}H v_{0}, H , Λ^{}Λ mβ.

Explicitly, v_{s}^{m,β} is given by a polynomial in the positive root generators [4, 13]

v_{s}^{m,β} P^{m,β} v_{0}, P^{m,β} U ^{}.

Thus, the submodule I^{β} of V^{Λ} which is isomorphic to V^{Λ^{}}is given by U ^{}P^{m,β} v_{0}. Note that the Casimirs of ^{ℂ} take the same values on V^{Λ} and V^{Λ^{}}. Certainly, (2.4) may be fulfilled for several positive roots (even for all of them).

Let Δ_{Λ} denote the set of all positive roots for which (2.4) is fulfilled, and let us denote Ĩ^{Λ} _{β Δ_{Λ}} I^{β}. Clearly, Ĩ^{Λ} is a proper submodule of V^{Λ} . Let us also denote F^{Λ} V^{Λ} Ĩ^{Λ} .

Further we shall use also the following notion. The singular vector v_{1} is called descendant of the singular vector v_{2} ℂv_{1} if there exists a homogeneous polynomial P_{12} in U ^{}so that v_{1} P_{12}v_{2}. Clearly, in this case we have: I^{1} I^{2} where I^{k} is the submodule generated by v_{k}.

The Verma module V^{Λ} contains a unique proper maximal submodule I^{Λ} ( Ĩ^{Λ} ) [1, 22]. Among the lowest weight modules with lowest weight Λ there is a unique irreducible one, denoted by L_{Λ} , i.e., L_{Λ} V^{Λ} I^{Λ} . (If V^{Λ} is irreducible, then L_{Λ} V^{Λ} .)

It may happen that the maximal submodule I^{Λ} coincides with the submodule Ĩ^{Λ} generated by all singular vectors. This is, e.g., the case for all Verma modules if rank 2, or when (2.4) is fulfilled for all simple roots (and, as a consequence, for all positive roots). Here we are interested in the cases when Ĩ^{Λ} is a proper submodule of I^{Λ} . We need the following notion.

Definition 2.1. [1, 6, 7] Let V^{Λ} be a reducible Verma module. A vector v_{ssv} V^{Λ} is called a subsingular vector if v_{su} Ĩ^{Λ} and Xv_{su} Ĩ^{Λ} , for all X ^{}

Going from the above more general definitions to we recall that in [15] it was established that from (2.4) follows that the Verma module V^{Λ χ} is reducible if one of the following relations holds (following the order of (2.3

(2.5a)  ℕ m_{ij}^{}j i a_{i}  a_{j 1}
(2.5b)  ℕ m_{ij}^{}2n i j 1 a_{j}  a_{n 1} a_{1}  a_{i 1} 2d
(2.5c)  ℕ m_{i} 2n 2i 1 a_{i}  a_{n 1} a_{1}  a_{i 1} 2d
(2.5d)  ℕ m_{ii} n i \frac{1}{2} 1 a_{i}  a_{n 1} a_{1}  a_{i 1} d .

Further we shall use the fact from [15] that we may eliminate the reducibilities and embeddings related to the roots 2δ_{i}. Indeed, since m_{i} 2m_{ii}, whenever (2.5d) is fulfilled also (2.5c) is fulfilled.

For further use we introduce notation for the root vector X_{j}^{}^{}, j 1, …, n, corresponding to the simple root α_{j}. Naturally, X_{j}^{}^{}corresponds to α_{j}.

Further, we notice that all reducibility conditions in (2.5a) are fulfilled. In particular, for the simple roots from those condition, (2.5a) is fulfilled with β α_{i} δ_{i} δ_{i 1} , i 1, …, n 1 and m_{i}^{}m_{i,i 1}^{}1 a_{i}. The corresponding submodules I^{α_{i}} U ^{}v_{s}^{i}, where Λ_{i} Λ m_{i}^{}α_{i} and v_{s}^{i} X_{i}^{}^{1 a_{i}} v_{0}. These submodules generate an invariant submodule which we denote by I_{c}^{Λ} Ĩ^{Λ} . Since these submodules are nontrivial for all our signatures in the question of unitarity instead of V^{Λ} , we shall consider also the factor-modules F_{c}^{Λ} V^{Λ} I_{c}^{Λ} F^{Λ} . We shall denote the lowest weight vector of F_{c}^{Λ} by Λ_{c} and the singular vectors above become null conditions in F_{c}^{Λ} , i.e., X_{i}^{}^{1 a_{i}} Λ_{c} 0, i 1, …, n 1.

If the Verma module V^{Λ} is not reducible w.r.t. the other roots, i.e., (2.5b,c,d) are not fulfilled, then F_{c}^{Λ} F^{Λ} is irreducible and is isomorphic to the irrep L_{Λ} with this weight.

In fact, for the factor-modules reducibility is controlled by the value of d, or in more detail:

The maximal d coming from the different possibilities in (2.5b) are obtained for m_{ij}^{}1 and they are

d_{ij} n \frac{1}{2} a_{j}  a_{n 1} a_{1}  a_{i 1} i j ,

the corresponding root being δ_{i} δ_{j}.

The maximal d coming from the different possibilities in (2.5c,d), resp., are obtained for m_{i} 1, m_{ii} 1, resp., and they are:

d_{i} n i \frac{1}{2} a_{i}  a_{n 1} a_{1}  a_{i 1} , d_{ii} d_{i} \frac{1}{2},

the corresponding roots being δ_{i}, 2δ_{j}, resp.

There are some orderings between these maximal reduction points [15]:

(2.6)  d_{1} d_{2}  d_{n},
d_{i,i 1} d_{i,i 2}  d_{in} ,
d_{1,j} d_{2,j}  d_{j 1,j} ,
d_{i} d_{jk} d_{ℓ},  i j k ℓ.

Obviously the first reduction point is

d_{1} n 1 \frac{1}{2} a_{1} a_{n 1} .

3. Unitarity

The first results on the unitarity were given in [15], and then improved in [14]. Thus, the statement below should be called Dobrev–Zhang–Salom Theorem.

Theorem DZS 1. All positive energy unitary irreducible representations of the superalgebras osp 1 2n, ℝ characterized by the signature χ in (2.1) are obtained for real d and are given as follows:

d n 1 \frac{1}{2} a_{1}  a_{n 1} d_{1}, a_{1} 0,
d n \frac{3}{2} \frac{1}{2} a_{2}  a_{n 1} d_{12}, a_{1} 0, a_{2} 0,
d n 2 \frac{1}{2} a_{2}  a_{n 1} d_{2} d_{13}, a_{1} 0, a_{2} 0,
d n 2 \frac{1}{2} a_{3}  a_{n 1} d_{2} d_{13}, a_{1} a_{2} 0, a_{3} 0,
d n \frac{5}{2} \frac{1}{2} a_{3}  a_{n 1} d_{23} d_{14}, a_{1} a_{2} 0, a_{3} 0,
d n 3 \frac{1}{2} a_{3}  a_{n 1} d_{3} d_{24} d_{15}, a_{1} a_{2} 0, a_{3} 0,
d n 1 κ \frac{1}{2} a_{2κ 1}  a_{n 1} , a_{1}  a_{2κ} 0, a_{2κ 1} 0,
κ \frac{1}{2}, 1, …, \frac{1}{2} n 1 ,
d n \frac{3}{2} κ \frac{1}{2} a_{2κ 1}  a_{n 1} , a_{1}  a_{2κ} 0, a_{2κ 1} 0,
d n 1 2κ \frac{1}{2} a_{2κ 1}  a_{n 1} , a_{1}  a_{2κ} 0, a_{2κ 1} 0,
d \frac{1}{2} n 1 , a_{1}  a_{n 1} 0
d \frac{1}{2} n 2 , a_{1}  a_{n 1} 0
d \frac{1}{2}, a_{1}  a_{n 1} 0
d 0, a_{1}  a_{n 1} 0

where the last case is the trivial one-dimensional irrep.

The theorem was partially proved [15], while in [14] was given a sketch of a proof, and the case n 3 was proved. We are going to give a proof for osp 1 8 .

4. The case of osp 1 8

For n 4 formula (2.6) simplifies to

d_{1} d_{12} d_{2} d_{23} d_{3} d_{34} d_{4}
d_{13} d_{24}
d_{14}

In the case of osp 1 8 Theorem DZS reads:

Theorem 4.1. All positive energy unitary irreducible representations of the superalgebras osp 1 8, ℝ characterized by the signature χ in (2.1) are obtained for real d and are given as follows

d 3 \frac{1}{2} a_{1} a_{2} a_{3} d_{1}, a_{1} 0,
d \frac{5}{2} \frac{1}{2} a_{2} a_{3} d_{12}, a_{1} 0, a_{2} 0,
d 2 \frac{1}{2} a_{2} a_{3} d_{2} d_{13}, a_{1} 0, a_{2} 0,
d 2 \frac{1}{2}a_{3} d_{2} d_{13}, a_{1} a_{2} 0, a_{3} 0
d \frac{3}{2} \frac{1}{2}a_{3} d_{23} d_{14}, a_{1} a_{2} 0, a_{3} 0
d 1 \frac{1}{2}a_{3} d_{3} d_{24}, a_{1} a_{2} 0, a_{3} 0
d \frac{3}{2} d_{23} d_{14}, a_{1} a_{2} a_{3} 0
d 1 d_{3} d_{24}, a_{1} a_{2} a_{3} 0
d \frac{1}{2} d_{34}, a_{1} a_{2} a_{3} 0,
d 0 d_{4}, a_{1} a_{2} a_{3} 0

where the last case is the trivial one-dimensional irrep.

Proof. For d d_{1} there are no singular vectors and we have unitarity. At d d_{1} there is a singular vector of weight δ_{1} α_{1} α_{2} α_{3} α_{4} [5, 15]:

(4.1)  v_{δ_{1}}^{1} _{k_{1} 0}^{1} _{k_{2} 0}^{1} _{k_{3} 0}^{1} b_{k_{1} ,k_{2} ,k_{3}} X_{1}^{}^{1 k_{1}} X_{2}^{}^{1 k_{2}} X_{3}^{}^{1 k_{3}}
X_{4}^{}X_{3}^{}^{k_{3}} X_{2}^{}^{k_{2}} X_{1}^{}^{k_{1}} v_{0} ^{1,δ_{1}} v_{0},
b_{k_{1} ,k_{2} ,k_{3}}  1 ^{k_{1} k_{2} k_{3}} a_{1} k_{1} \frac{2 a_{1} a_{2}}{1 a_{1} a_{2} k_{2}} \frac{3 a_{1} a_{2} a_{3}}{3 a_{1} a_{2} a_{3} k_{3}}
where H^{s} Ĥ_{1} Ĥ_{2}  Ĥ_{s}, and a basis in terms of simple root vectors only is

used. This singular vector is nontrivial for a_{1} 0 and must be eliminated to obtain a UIR. Below d d_{1} this vector is not singular but has negative norm and thus there is no unitarity for a_{1} 0. On the other hand for a_{1} 0 and any d vector (4.1) is descendant of the compact root singular vector X_{1}^{}v_{0} which is already factored out for a_{1} 0.

Thus, below we discuss only the cases with a_{1} 0, when we have unitarity for d d_{12} \frac{5}{2} \frac{1}{2} a_{2} a_{3} . Then at the next reducibility point d d_{12}, we have a singular vector corresponding to the root δ_{1} δ_{2} α_{1} 2α_{2} 2α_{3} 2α_{4} which is given by

v_{δ_{1} δ_{2}}^{1}  \frac{1}{2 2a_{2} a_{3}} (4.2)
\frac{1}{2} Y_{4}Y_{3}X_{3}^{}X_{2}^{}^{2}X_{1}^{}\frac{1}{4} Y_{4}^{2} X_{3}^{}^{2} X_{2}^{}^{2}X_{1}^{}Y_{4}^{2}X_{3}^{}X_{23}^{}X_{2}^{}X_{1}^{}a_{2}

2 Y_{4}Y_{2}X_{23}^{}X_{1}^{}a_{2} a_{2} 1 Y_{4}^{2}X_{23}^{}X_{23}^{}X_{1}^{}a_{2} a_{2} 1

Y_{4}Y_{3}X_{23}^{}X_{2}^{}X_{1}^{}a_{3} 2 2 Y_{3}Y_{2}X_{2}^{}X_{1}^{}a_{2} a_{3} 1 a_{2} a_{3} 2 Y_{3}^{2} X_{2}^{}^{2}X_{1}^{}a_{2} a_{3} 1 a_{2} a_{3} 2

4 Y_{2}^{2}X_{1}^{}a_{2} a_{2} 1 a_{2} a_{3} 1 a_{2} a_{3} 2

Y_{23}X_{2}^{}X_{1}^{}2a_{2} 1 a_{2} a_{3} 1 a_{2} a_{3} 2 Y_{4}Y_{2}X_{3}^{}X_{2}^{}X_{1}^{}2a_{2} a_{3} 2

\frac{1}{4} Y_{34}X_{3}^{}X_{2}^{}^{2}X_{1}^{}2a_{2} 2a_{3} 3 Y_{24}X_{23}^{}X_{1}^{}a_{2} a_{2} 1 2a_{2} 2a_{3} 3

\frac{1}{2} Y_{34}X_{23}^{}X_{2}^{}X_{1}^{}a_{3} 2a_{2} a_{2} a_{3} 1 2

\frac{1}{2} Y_{24}X_{3}^{}X_{2}^{}X_{1}^{}a_{3} 2a_{2} a_{2} a_{3} 2 2

a_{2} 1 a_{2} a_{3} 2

2 Y_{4}Y_{3}X_{13}^{}X_{2}^{}Y_{34}X_{13}^{}X_{2}^{}2 Y_{4}Y_{3}X_{23}^{}X_{12}^{}Y_{34}X_{23}^{}X_{12}^{}

2 Y_{4}Y_{2}X_{3}^{}X_{12}^{}Y_{24}X_{3}^{}X_{12}^{}2 Y_{4}Y_{1}X_{3}^{}X_{2}^{}Y_{14}X_{3}^{}X_{2}^{}

a_{2} a_{2} 1 a_{2} a_{3} 2 4 Y_{4}Y_{2}X_{13}^{}2 Y_{24}X_{13}^{}4 Y_{4}Y_{1}X_{23}^{}2 Y_{14}X_{23}^{}

a_{2} 1 a_{2} a_{3} 1 a_{2} a_{3} 2

4 Y_{3}Y_{2}X_{12}^{}2 Y_{23}X_{12}^{}4 Y_{3}Y_{1}X_{2}^{}2 Y_{13}X_{2}^{}8 Y_{2}Y_{1} a_{2} 4a_{2}Y_{12} v_{0}

where the root vector X_{jk}^{}corresponds to the compact root δ_{j} δ_{k 1} α_{j} α_{j 1} α_{k}, Y_{k} corresponds to the odd (noncompact) root δ_{k} α_{k} α_{k 1} α_{n}, (thus Y_{4} X_{4}^{}), Y_{jk} corresponds to the even noncompact root δ_{j} δ_{k}. In (4.2) it is more convenient to use a PBW type of basis with the compact roots X_{...}^{}to the right of the noncompact roots Y_{...}. The norm of (4.2) is

64a_{2} a_{2} 1 ^{2} a_{2} 2 a_{2} a_{3} 1 a_{2} a_{3} 2 ^{2} a_{2} a_{3} 3
2d a_{2} a_{3} 4 2d a_{2} a_{3} 5  2a_{2} a_{3} 2 ^{2}.

For d d_{12}, a_{1} 0, a_{2} 0 the singular vector (4.2) is nontrivial and gives rise to a invariant subspace which must be factored out for unitarity. For d \frac{5}{2} \frac{1}{2} a_{2} a_{3} , the vector (4.2) is not singular, but has negative norm and there is no unitarity for a_{2} 0, except at the isolated unitary point d 2 \frac{1}{2} a_{2} a_{3} d_{2} d_{13} where vector (4.2) has zero norm and can not spoil the unitarity. For that value of d there is a singular vector v_{δ_{2}}^{1} of weight δ_{2} α_{2} α_{3} α_{4} [5, 15]:

v_{δ_{2}}^{1} _{k_{1} 0}^{1} _{k_{2} 0}^{1} b_{k_{1} ,k_{2}} X_{2}^{}^{1 k_{1}} X_{3}^{}^{1 k_{2}}
X_{4}^{}X_{3}^{}^{k_{2}} X_{2}^{}^{k_{1}} v_{0} ^{1,δ_{2}} v_{0},
b_{k_{1} ,k_{2}}  1 ^{k_{1} k_{2}} \frac{a_{2} k_{1}}{1 a_{2} a_{3} k_{2}}

which has to be factored out for unitarity for a_{2} 0, while for a_{2} 0 it is descendant of the compact vector X_{2}^{}v_{0}.

Overall no further unitarity is possible for a_{2} 0, thus below we consider only the cases a_{1} a_{2} 0. Then the singular vectors above are descendants of compact root singular vectors X_{1}^{}v_{0} and X_{2}^{}v_{0}, thus, there is no obstacle for unitarity for d 2 \frac{1}{2}a_{3} d_{2} d_{13} (for a_{1} a_{2} 0). The next reducibility point is d d_{13} d_{2}.

POSITIVE ENERGY UIRS OF osp 1 8, ℝ 57

The singular vector for d d_{13} and m 1 has weight δ_{1} δ_{3} α_{1} α_{2} 2α_{3} 2α_{4}: v_{δ_{1} δ_{3}}^{1}

4a_{1} Y_{4}Y_{3}X_{3}^{}X_{12}^{}2a_{1} Y_{4}^{2} X_{3}^{}^{2}X_{12}^{}2 a_{1} a_{2} 1 Y_{4}^{2}X_{3}^{}X_{23}^{}X_{1}^{}

4a_{1} a_{1} a_{2} 1 Y_{4}^{2}X_{3}^{}X_{13}^{}4 a_{3} 1 Y_{4}Y_{3}X_{23}^{}X_{1}^{}8a_{1} a_{3} 1 Y_{4}Y_{3}X_{13}^{} 4 a_{1} a_{2} a_{3} 2 Y_{4}Y_{2}X_{3}^{}X_{1}^{}8a_{1} a_{1} a_{2} a_{3} 2 Y_{4}Y_{1}X_{3}^{} 8a_{3} a_{1} a_{2} a_{3} 2 Y_{3}Y_{2}X_{1}^{}4a_{3} a_{1} a_{2} a_{3} 2 Y_{3}^{2}X_{2}^{}X_{1}^{} 8a_{1}a_{3} a_{1} a_{2} a_{3} 2 Y_{3}^{2}X_{12}^{}2 a_{1} a_{3} 1 a_{2} a_{3} 1 2 Y_{34}X_{23}^{}X_{1}^{} 4a_{1} a_{1} a_{3} 1 a_{2} a_{3} 1 2 Y_{34}X_{13}^{} 2 a_{1} a_{2} 2 a_{1} a_{2} a_{3} 2 Y_{24}X_{3}^{}X_{1}^{} 4a_{1} a_{1} a_{2} 2 a_{1} a_{2} a_{3} 2 Y_{14}X_{3}^{} 4 a_{1} a_{2} 2 a_{3} a_{1} a_{2} a_{3} 2 Y_{23}X_{1}^{} a_{1} a_{2} 2a_{3} 2 Y_{34}X_{3}^{}X_{2}^{}X_{1}^{}2a_{1} a_{1} a_{2} 2a_{3} 2 Y_{34}X_{3}^{}X_{12}^{} 8a_{1} a_{1} a_{2} 2 a_{3} a_{1} a_{2} a_{3} 2 Y_{13} 16a_{1}a_{3} a_{1} a_{2} a_{3} 2 Y_{3}Y_{1}

2 Y_{4}Y_{3}X_{3}^{}X_{2}^{}X_{1}^{}Y_{4}^{2} X_{3}^{}^{2}X_{2}^{}X_{1}^{}v_{0}.

For a_{1} a_{2} 0 it is descendant of the compact root singular vector X_{1}^{}v_{0}. However, there is a subsingular vector

v_{2,13}^{ss}

2a_{3} Y_{23}Y_{1} 2a_{3} Y_{13}Y_{2} 2a_{3} Y_{3} Y_{12} 4a_{3} Y_{3}Y_{2}Y_{1} 2 Y_{4}Y_{3}Y_{2}X_{13}^{}

Y_{34}Y_{2}X_{13}^{}Y_{24}Y_{3}X_{13}^{}Y_{4}Y_{23}X_{13}^{}2 Y_{4}Y_{3}Y_{1}X_{23}^{}Y_{34}Y_{1}X_{23}^{}Y_{14}Y_{3}X_{23}^{}

Y_{4}Y_{13}X_{23}^{}2 Y_{4}Y_{2}Y_{1}X_{3}^{}Y_{24}Y_{1}X_{3}^{}Y_{14}Y_{2}X_{3}^{}Y_{4} Y_{12} X_{3}^{}v_{0}

with the norm 16a_{3} a_{3} 3 2d a_{3} 2 2d a_{3} 3 2d a_{3} 4 . This vector must be factorized in order to obtain UR at d d_{2} d_{13}. But below this value, the vector v_{2,13}^{ss} above has negative norm if a_{3} 0 and there is no unitarity, except at the isolated unitary point d \frac{3}{2} \frac{1}{2}a_{3} d_{23} d_{14}. At that value of d there is a singular vector of weight δ_{2} δ_{3} α_{2} 2α_{3}:

v_{δ_{2} δ_{3}}^{1} 2 a_{3} 1 Y_{4}Y_{3}X_{23}^{}2 a_{3} 1 Y_{4}Y_{2}X_{3}^{}2a_{3} a_{3} 1 Y_{3}^{2}X_{2}^{}
a_{3} 1 Y_{34}X_{23}^{}a_{3} 1 Y_{24}X_{3}^{}\frac{1}{2} 2a_{3} 1 Y_{34}X_{3}^{}X_{2}^{}

2a_{3} a_{3} 1 Y_{23} 4a_{3} a_{3} 1 Y_{3}Y_{2} Y_{4}Y_{3}X_{3}^{}X_{2}^{}\frac{1}{2} Y_{4}^{2} X_{3}^{}^{2}X_{2}^{}v_{0} (4.3) with the norm 16a_{3} a_{3} 1 ^{2} a_{3} 2 2d a_{3} 2 2d a_{3} 3 . For a_{3} 0, singular vector (4.3) should be factored for unitarity, while for a_{3} 0 it is descendant of the compact singular vectors.

In the same range for a_{3} 0 at d d_{3} 1 \frac{1}{2}a_{3} there is a singular vector of weight δ_{3} α_{3} α_{4}:

(4.4)  v_{δ_{3}}^{1} _{k 0}^{1} 1 ^{k} a_{3} k X_{3}^{}^{1 k} X_{4}^{}X_{3}^{}kv_{0}  ^{1,δ_{3}} v_{0}

which must be factored out for unitarity.

On the other hand, for a_{1} a_{2} a_{3} 0 all (sub)singular vectors above are descendants of the compact singular vectors X_{k}^{}v_{0}, k 1, 2, 3, and there is no obstacle for unitarity for d \frac{3}{2} d_{23} d_{14}. For a_{3} 0 and d \frac{3}{2} there is also a singular vector of weight δ_{1} δ_{4}:

v_{δ_{1} δ_{4}}^{1} 4 Y_{4}Y_{2}X_{1}^{}2 Y_{4}^{2}X_{23}^{}X_{1}^{}2 Y_{4}Y_{3}X_{2}^{}X_{1}^{}
Y_{4}^{2}X_{3}^{}X_{2}^{}X_{1}^{}3 Y_{34}X_{2}^{}X_{1}^{}6 Y_{24}X_{1}^{}v_{0}

but it is also descendant of compact singular vectors. Finally, for d \frac{3}{2} there is a subsingular vector of weight δ_{1} δ_{2} δ_{3} δ_{4}:

(4.5)  v_{δ_{1} δ_{2} δ_{3} δ_{4}}^{ss}  _{i,j,k,ℓ 1}^{4} ϵ^{ijkℓ} Y_{i}Y_{j} Y_{k}Y_{ℓ}v_{0}

where ϵ^{ijkℓ} is the totally antisymmetric symbol so that ϵ^{1234} 1. The norm of the vector (4.5) is 2304 1 d d 3 2d 1 2d . Thus, for \frac{3}{2} d 1 there is no unitarity since then the vector (4.5) has negative norm. In all cases there will be no unitarity for d 1, except possibly when a_{1} a_{2} a_{3} 0 to which we restrict below. At d d_{3} d_{24} 1 there are the singular vector (4.4) and the singular vector of weight δ_{2} δ_{4} α_{2} α_{3} 2α_{4}:

v_{δ_{2} δ_{4}}^{1} a_{3} 2 Y_{34}X_{2}^{}2 Y_{4}Y_{3}X_{2}^{}Y_{4}^{2}X_{3}^{}X_{2}^{}v_{0}

both of which are descendants of compact singular vectors. At d d_{3} d_{24} 1, there is also a subsingular vector

v_{δ_{2} δ_{3} δ_{4}}^{ss} Y_{2}Y_{3}Y_{4} Y_{4}Y_{3}Y_{2} \frac{1}{3} _{i,j,k 2}^{4} ϵ^{ijk} Y_{i}Y_{j} Y_{k} v_{0}

of the norm 144d d 1 2d 1 . It is not an obstacle for unitarity for d 1, but for d 1. Thus, there is no unitarity for d 1 except at the isolated unitary point d \frac{1}{2} d_{34}. At that point all (sub)singular vectors above are descendants of compact singular vectors. Yet there is the singular vector

v_{δ_{3} δ_{4}}^{1} \frac{1}{2}Y_{4}^{2}X_{3}^{}2Y_{3}Y_{4} Y_{34} v_{0}

with the norm 8d 2d 1 . It is not an obstacle for unitarity for d \frac{1}{2}, but for d \frac{1}{2}. Thus, there is no unitarity for d \frac{1}{2} except at the isolated point d d_{4} 0 a_{1} a_{2} a_{3} where we have the trivial one-dimensional UIR since all possible states are descendants of factored out singular vectors. □

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Institute of Nuclear Research and Nuclear Energy (Received 19 07 2016) Bulgarian Academy of Sciences Sofia, Bulgaria

dobrev@inrne.bas.bg

Institute of Physics University of Belgrade Belgrade, Serbia

isalom@phy.bg.ac.rs