On the unitarity of gauged non-compact world-sheet supersymmetric WZNW models

0.21 MB·English

by  Jonas Bjornsson

On the unitarity of gauged non-compact
world-sheet supersymmetric WZNW models
Jonas Bj¨ornsson1 and Stephen Hwang2
Department of Physics
Karlstad University
9
0 SE-651 88 Karlstad, Sweden
0
2

n
a
J
Abstract
1
2
In this paper we generalize our investigation of the unitarity of non-compact

] WZNW models connected to Hermitian symmetric spaces to the N=1 world-sheet
h
t supersymmetric extension of these models. We will prove that these models have
-
p a unitary spectrum in a BRST approach for antidominant highest weight represen-
e tations if the level and weights of the gauged subalgebra are integers. We will find
h
[ new critical string theories in 7 and 9 space-time dimensions.

3
v
1 Introduction
8
7
5
In a previous paper [1] we considered strings on group manifolds connected to Hermitian
3
. symmetric spaces of non-compact type. These are constructed by starting with a non-
2
0 compact group G which has maximal compact subgroup H with a one-dimensional center
8
Z(H). Then, the coset space G/H is a Hermitian symmetric space of non-compact type.
0
: All these spaces have been classified. They have G being SU(p,q), SO(p,2), SO (2n)
v ∗
i and Sp(2p,R). Furthermore, there are two connected to exceptional groups, E6 (E6 14)
X |−
and E (E ). As the maximal compact subgroup has a one-dimensional center, one
7 7 25
r |−
a can use these groups to construct non-trivial space-time backgrounds where strings can
propagate. Write the maximal compact subgroup as H = H Z(H) and construct the
′
×
coset G/H .
′
This background has a compact direction which plays the rˆole of time. One can
formulate this as a WZNW model, as was done in [2] and prove it to be unitary using
a BRST approach [1]. If one assumes that the weights and level of the subalgebra are
[email protected]
[email protected]
1
integers3, the spectrum is unitary. This is not the case for the Goddard-Kent-Olive coset
construction [1]. In a realistic string model one would take the infinite cover so that
the time is uncompactified. Furthermore, we do not consider sectors corresponding to
spectral flow, which was originally proposed in [3] and further elaborated in refs. [4]-[6]
for the SL(2,R) string. Such sectors would be crucial in constructing an S-matrix unitary
string theory, as modular transformations mix different spectrally flowed sectors for the
SL(2,R) string.
In this paper we will extend our analysis and consider the = 1 world-sheet super-
N
symmetric extension of the WZNW model [7] [8]. This model has bosonic and fermionic
degrees of freedom. These two sectors mix in a trivial way and can be decoupled by re-
defining the affine generators. Constructing coset models of these theories was first done
in[9]forthebosoniccaseandasimple = 1supersymmetric extensionofWZNWmodel.
N
The generalization to other world-sheet supersymmetric WZNW models was considered
in [10] [11] and is, therefore, called the Kazama-Suzuki coset construction. We will in
this paper use a BRST approach to the coset construction [12], as this formulation gives
unitarity for the bosonic case [1]. The BRST formulation of the Kazama-Suzuki model
was considered in [13] and [14].
The relevant representations which we consider here are so-called antidominant high-
est weight representations. In addition, we will assume that the level and weights of the
corresponding subalgebra are integers. We will find that the = 1 world-sheet super-
N
symmetric extension of the WZNW-model has a unitary spectrum. At the end of the
paper we will give a table of the relevant coset models. Some of these models will by
themselves yield criticality. These occur in seven dimensions: SU(3,1)/SU(3) at level
k = 6 and SO(3,2)/SO(3) at level k = 6; in nine dimensions: SU(4,1)/SU(4) at
− −
level k = 40 and SO(4,2)/SO(4) at level k = 32.
− −
In our previous paper [1], we also considered the necessity of our assumed representa-
tions. The proof given there is, however, not in general complete. It holds when the rank
of g is two (see below for explanation of notation). We will in a forthcoming publication
discuss more general cases.
Let us introduce some notations and basic definitions. We will in general follow the
conventions and notations used in [15]. Denote by g and h the Lie algebras corresponding
to the non-compact group G and its maximal compact subgroup H, which admit a Hermi-
tian symmetric space of the form G/H. Let gC and hC denote the corresponding complex
Lie algebras. We will always take the rank r of g to be greater than one and g to be
g
simple. One knows that h has a one-dimensional center, thus one can split h as h u(1).
′
We choose hC such that it is a regular embedding in gC i.e. using the Cartan-Wey⊕l basis,
the Cartan elements of hC, as well as generators corresponding to positive/negative roots,
3The level of the algebra ˆg is integer, or if the Dynkin index is different from one, also half integer.
2
are all in the corresponding decomposition of gC.
Denote by ∆ all roots, ∆+/ the positive/negative roots, ∆ the simple roots, ∆ the
− s c
compact roots, ∆+ = ∆ ∆+ the compact positive roots, ∆ the non-compact roots
c c ∩ n
and ∆+ the positive non-compact roots. We take the long roots to have length √2. Let
n
α ∆ and define the coroot by α∨ = 2(α,α)−1α. Let α(i) ∆+ denote the simple roots.
W∈hen we need to distinguish between different root system∈s, we denote by ∆g and ∆h′
the roots in gC and hC, respectively. Furthermore, we use capital letters, A,B,..., and
′
small letter, a,b,..., to denote elements in the algebra g and h respectively. Denote by
′
JA a generic generator of gˆC which in a Cartan-Weyl basis is
Ji = Hi, i = 1,...,r ;
g
Jα = Eα, α ∆. (1.1)
∈
We fix the basis of the root space such that the highest root is non-compact. In addition,
one can choose the basis of roots such that if α ∆g then the first component is zero
∈ c
and the other r 1 components are, in general, non-zero. This, furthermore, yields an
g
isomorphism betw−een ∆g and ∆h′. g denotes the dual Coxeter number of g. It is well-
c g∨
known that one can choose a basis such that there is a unique non-compact simple root
andifα ∆+ thenthecoefficient ofthenon-compact simple rootisalways oneinasimple
∈ n
root decomposition of α. Dynkin diagrams and relations between positive non-compact
roots of the Lie-algebras are presented in the appendix of [16] and, with our notation, in
[1]. For the embedding of h in g, one defines the Dynkin index of an embedding
′
(θ(g),θ(g))
Ih′ g = , (1.2)
⊂ (θ(h′),θ(h′))
where, for regular embeddings, θ(h) is the highest root of h in g. The conformal anomaly
′ ′
for the G/H WZNW model is
′
c = k −gg∨ dim(g) + 1 dim(g) κ−gh∨′ dim(h′) 1 dim(h), (1.3)
tot ′
k 2 − κ − 2
(cid:0) (cid:1) (cid:0) (cid:1)
where κ is defined as
κ Ih′ gk. (1.4)
≡ ⊂
Let us use operator product language to write down the relations for the algebra. The
operator product expansions (OPE’s) of the = 1 superaffine algebra are
N
JA(z)JB(w) kκAB(z w)−2 +fAB JC(w)(z w)−1 (1.5)
∼ − C −
JA(z)λB(w) fAB λC(w)(z w)−1 (1.6)
∼ C −
1 for R fermions
λA(z)λB(w) ∼ kκAB(z −w)−1 × 1 z + w for NS− fermions , (1.7)
(cid:26) 2 w z −
(cid:0)p p (cid:1)
3
where means equality up to regular terms, fAB and κAB are the structure constants
∼ C
and killing form, respectively, for the corresponding finite dimensional Lie algebra. The
non-zero values in a Cartan-Weyl basis of g are
fiα = αiδ
β α,β
fαβ = α δ
i i∨ α+β,0
fαβ = e δ
γ α,β α+β,γ
κij = Gij
2
καβ = δ , (1.8)
α+β,0
(α,α)
where Gij (α(i)∨,α(j)∨) and e is non-zero iff α + β is a root. The correspondence
α,β
≡
between OPE’s and brackets are4
1
[A ,B ] = dww∆B+n 1 dzz∆A+m 1 (A(z)B(w)), (1.9)
m n (2πi)2 − − R
I I
where ∆ is the conformal dimension of the operator and is the radial ordering of the
R
operators
A(z)B(w) z > w
(A(z)B(w)) = | | | | , (1.10)
R ( )ǫAǫB B(w)A(z) z < w
(cid:26) − | | | |
where ǫ denote the Grassman parity of the operator. The radial ordering satisfies
(A(z)B(w)) : A(w)B(w) : +A(z)B(w), (1.11)
R ∼
where : : means normal ordering. The element corresponding to the center, uˆ(1), is
H = Λ Hi , (1.12)
m (1)i m
where Λ are fundamental weights which are defined by Λ ,α(j)∨ = δj. H satisfies
(i) (i) i m
(cid:16) (cid:17)
H ,E α = 0, α ∆+
m n± ∈ c
(cid:2)Hm,En±β(cid:3) = ±Em±β+n, β ∈ ∆+n. (1.13)
One can also define new cu(cid:2)rrents (cid:3)
1
JˆA(z) = JA(z)+ fA : λBλC(z) : (1.14)
2k BC
which make the OPE’s between λA and JˆA consist of only regular terms. Furthermore,
these currents satisfy an affine algebra of level k g .
∨
−
4This is for bosonic operators and Ramond fermions.
4
2 Coset construction, the BRST approach
The Kazama-Suzuki coset construction is defined by considering states Φ in a repre-
| i
sentation of the supersymmetric affine algebra and then gauging a subalgebra by the
conditions
Jhˆ′ Φ = λˆh′ Φ = 0, (2.1)
+ | i +| i
where Jhˆ′ and λˆh′ are generic annihilation operators in the subalgebra hˆ. In the BRST
+ + ′
approach one introduces instead a BRST operator to define the coset space [13] [14]. One
needs to introduce, in addition to the ghost sector, an auxiliary sector with currents J˜a(z)
for the subalgebra hˆ . The BRST current is5
′
j1 = c Jˆa 1 fa λBλC +Jˆ˜a 1 fa λ˜bλ˜c +γ λa +λ˜a
BRST a − 2k BC − 2κ bc a
(cid:18) (cid:19)
(cid:16) (cid:17)
1
fab c c bc +fab c γ βc, (2.2)
− 2 c a b c a b
where c and γ are fermionic and bosonic ghosts, respectively, with corresponding mo-
a a
menta ba and βa. They satisfy6
ca(z)bb(w) ∼ δab(z −w)−1
γa(z)βb(w) ∼ δab(z −w)−1. (2.3)
The OPE of the BRST current defined in eq. (2.2) with itself yields only regular terms
if κ + κ˜ = 0. Thus, we take the level of the extra sector κ˜ to be κ. To get the
−
physical subspace we need to add contributions to this BRST charge arising from the
superconformal symmetry. One finds
j = : c Jˆa 1 fa λBλC +Jˆ˜a 1 fa λ˜bλ˜c : + : γ λa +λ˜a :
BRST a − 2k BC − 2κ bc a
(cid:18) (cid:19)
(cid:16) (cid:17)
1
+ : cT : + : γG : fab : c c bc : +fab : c γ βc :
−2 c a b c a b
1 1
: ∂ccb : : γ2b+ : ∂cγβ : + : cγ∂β : : c∂γβ :
− − 2 −2
3 1
+ : c∂bac : + : c∂βaγ : + : cβa∂γ : + : ∂cbac :
a a a a
2 2
+ : ∂cβaγ : : γbaγ : : γ∂βac : : ∂γβac :, (2.4)
a a a a
− − −
5Here we have assumed that h′ is simple. If it is not one has for each simple part the BRST current
as in eq. (2.2). Henceforth, we will present equations which hold for the case when h′ is simple as the
generalization to the semisimple case is straightforward.
6In a mode expansion, γ is chosen to have hermiticity properties γ† = γ , γ† = γ ,
a,m i,m i,m α,m −α,m
β† = β and β† = β .
i,m − i,m α,m − −α,m
5
where
1
T(z) = κ : JˆAJˆB : + : ∂λAλB :
AB
2k
+ 1 κ (cid:16): Jˆ˜aJˆ˜b : + : ∂λ˜aλ˜b : +(cid:17)T (z) (2.5)
ab ′
2κ˜
1 (cid:16) 1 (cid:17)
G(z) = κ : JˆAλB(z) : f : λAλBλC(z) :
AB ABC
k −6k
(cid:18) (cid:19)
+ 1 κ : Jˆ˜aλ˜b(z) : 1 f : λ˜aλ˜bλ˜c(z) : +G(z), (2.6)
ab abc ′
κ˜ −6κ˜
(cid:18) (cid:19)
where T (z) and G(z) originates from a unitary superconformal algebra making the con-
′ ′
formalanomaly equalto15 asthe corresponding BRSTcharge isonlynilpolent forc = 15.
Furthermore, we have in the BRST charge introduced superconformal ghosts and
corresponding momenta. These have the non-zero OPE’s
c(z)b(w) (z w)−1
∼ −
γ(z)β(w) (z w)−1. (2.7)
∼ −
From the BRST current one can define a few useful charges, one is the zero mode for the
Virasoro algebra,
Ltot = [Q,b ]
0 0
= L + m : b c : +m : ba c :
0 m m m m,a
− −
m Z
X∈ (cid:0) (cid:1)
+ r : β γ : +r : βa γ : +a , (2.8)
r r r r,a g
− −
r Z+ν
∈X (cid:0) (cid:1)
and the other corresponds to the Cartan subalgebra of the Lie algebra
Htot,i = Q,bi
0 0
= (cid:2)Hˆi +H(cid:3)ˆ˜i (α,α)αi : λ αλα : (α,α)αi : λ˜ αλ˜α :
0 0 − 4k −m m − 4κ˜ −m m
− −
mX,α∈∆ mX,α∈∆c
+ αi : bα c : + αi : βα γ :, (2.9)
n n,α r r,α
− −
mX,α∈∆c r∈Z+Xν,α∈∆c
where a = 1/2 or a = 0 is a normal ordering constant and ν = 0 or ν = 1/2 for the
g g
−
Neveu-Schwarz and Ramond sectors, respectively.
6
3 The statespace
We consider here only discrete highest weight representations of the non-compact real
forms. To be more precise, we consider the representations which have a highest weight
that is antidominant. This subclass of highest weights will be defined below. The rep-
resentations are constructed by using redefined currents where the world-sheet fermions
and affine currents decouple. Furthermore, we work in the complex algebra and choose a
specific hermitian conjugation of the operators to get representations of the relevant real
form. A hermitian conjugation rule which yields the correct representations is
Eˆα † = Eˆ α
n − n−
−
(cid:16) (cid:17)
λαn † = −λ−nα (3.1)
−
(cid:0) (cid:1)
where α ∆ . The rest of the operators satisfy
n
∈
(A n)† = An. (3.2)
−
The highest weight state in both the Neveu-Schwarz and Ramond sectors is of the form
0;µ,µ˜ 0;µ 0;µ˜ 0 . (3.3)
| i ≡ | i⊗| i⊗| ighost
For the Neveu-Schwarz sector, we have
A 0;µ,µ˜ = 0, (3.4)
m
| i
for m > 0, where A denotes a generic operator. For m = 0 one has
Aα 0;µ = 0
0 | i
Hˆi 0;µ = µi 0;µ ,
0| i | i
bi 0 = b 0 = 0 (3.5)
0| ighost 0| ighost
where α ∆ , A = E,c,b and i = 1,...,r. We have here defined
+
∈ { }
(α,α)
cα = καβc = c (3.6)
m 2 m,β m,−α
and similary for the bosonic ghosts.
For the highest weight state in the Ramond sector the difference from the Neveu-
Schwarz case comes from the zero modes for the world-sheet fermions and bosonic ghosts.
Therepresentationsareconstructedinthesamewayasforthebosonicgeneratorswiththe
7
exception of the fermions corresponding to the Cartan subalgebra. For these we introduce
creation and annihilation operators by linear combinations
1
λ0a˜,± =  √12 (cid:16)±λ′02a˜−1 +iλ′20a˜(cid:17) rg = 2Z (3.7)
 √2 ±λ′20a˜ +iλ′20a˜+1 rg = 2Z+1
(cid:16) (cid:17)

where a˜ = 1,...,[r /2]7. Here, λi are linear combinations of 1 λj such that
g ′0 √ k 0
−
λi,λj = δi,j. (3.8)
′0 ′0 −
h i
From these operators one can construct a set of states by defining a highest weight state
that satisfies
λa˜,+ 0;µ = λα 0;µ = 0. (3.9)
0 | i 0 | i
This construction also generalizes to the bosonic ghosts,
γα 0 = βα 0 = γa˜,+ 0 = βa˜,+ 0 = 0, (3.10)
0 | ighost 0 | ighost 0 | ighost 0 | ighost
where γ0a˜,+ and β0a˜,+ are constructed as in eq. (3.7), α ∈ ∆+c and a˜ = 1,...,[rh′/2]. For
r odd, there is one extra component of λi which is left out of the above construction.
g ′
This gives an extra two-fold degeneration, which is of no importance to us and will be
disregarded in the following.
The highest weight state satisfies
Htot,i 0;µ,µ˜ = µi +µ˜i +hi 0;µ,µ˜
0 | i g | i
g h′
(cid:0) (cid:1)
Ltot 0;µ,µ˜ = C2 + C2 +a 0;µ,µ˜ , (3.11)
0 | i 2k 2κ˜ g | i
!
where a is 1/2 or 0 and hi is 2ρi or ρi +ρi for the Neveu-Schwarz and the Ramond
g − g ˆh g h′
sectors, respectively8. The states are now constructed by applying the redefined creation
operators on the highest weight state. As the fermions and ghosts describe a free theory
the corresponding statespaces are simple. For the redefined currents the representations
are unchanged, except that the level is shifted to k g and κ˜ g for the gˆ- and hˆ˜-sector,
− g∨ − h∨
7[] denotes the integer part.
8The contribution to hi has different sources in the different sectors, for the Neveu-Schwarz sector it
g
arises from the fermionic ghosts and from the Ramond sector it arises from the world-sheet fermions.
8
respectively. Thus, unitary representations with a highest weight µˆ˜, called dominant
integral, satisfy for the hˆ˜ -sector
′
µ˜i 0 i = 2,...,r
g
≥
κ˜ ≥ gh∨′ +(θh′,µ). (3.12)
and antidominant representations for the gˆ-sector are representations, with a highest
weight µˆ that satisfies
µi < ρi i = 1,...,r
g
−
k < (θ ,µ). (3.13)
g
The Verma modules corresponding to antidominant weights are irreducible, as can be
deduced from the Shapovalov-Kac-Kazhdan determinant [17] [18].
For well-known reasons, one does not consider the cohomology of the corresponding
BRST charge of the current defined in eq. (2.4), but the relative cohomology defined by
the conditions
Q Φ = 0 NS and R sectors
| i
bi Φ = 0 NS and R sectors
0| i
b Φ = 0 NS and R sectors
0
| i
β Φ = 0, R sector and r 2Z+1 (3.14)
0 g
| i ∈
where i = 2,...,r .
g
4 Unitarity
The mass shell condition is of the form
(µ,µ+2ρg) (µ˜,µ˜+2ρh′)
+ +N +l +a = 0, (4.1)
′ g
2k 2κ˜
(cid:18) (cid:19)
where l 0 originates from some unitary SCFT. Following [1], define
′
≥
rg
χ(gˆ,hˆ′ SVir)(τ,φ,θ) Tr exp 2πiτ Ltot exp i θ Htot,i+iφH ( 1)∆Ngh (.4.2)
⊕ ≡ 0 i 0 0 −
" " # #
i=2
(cid:2) (cid:0) (cid:1)(cid:3) X
∆N is the ghostnumber of the state relative to the highest weight state for the ghosts.
gh
The traceistaken over allstatesin gˆ hˆ˜′ SCFT ghost. A denotes thestatespace
Hµˆ×Hµˆ˜ ×Hl′ ×H′ H...
of A. Therefore, the character can be decomposed into separate parts
χ(gˆ,hˆ′ SVir)(τ,φ,θ) = e2πiτagχ1(τ,φ,θ)χhˆ˜′(τ,θ), (4.3)
⊕
9
where we have defined
χ1(τ,φ,θ) χgˆ(τ,φ,θ)χSCFT (τ)χghost(τ,θ). (4.4)
≡
As is well known [19], the character defined in eq. (4.2) only gets contributions from non-
trivialBRSTinvariant states. However, since weareinterested intherelativecohomology,
where the eigenvalues of Ltot and Htot,i are zero, one needs to project onto such states.
0 0
We need, therefore, to consider the function
rg
dτ (gˆ,hˆ′ SVir)(τ,φ) dτ (dθ ) χ(gˆ,hˆ′ SVir)(τ,φ,θ) . (4.5)
⊕ i ⊕
B ≡
Z Z Z Yi=2 n o
The τ- and θ -integrations are formal integrations which project onto the τ- and θ -
i i
gˆ,hˆ˜′ SVir
independent part, making the character compatible with eq. (3.14). “ ⊕ ”(τ,φ)
B
is called the generalized branching function. This definition was first introduced in [20]
and extended in [21]. As in [1], define another function, the signature function
Σ(ˆg,hˆ′ SVir) (τ,φ,θ)
⊕
rg
≡ Tr′ exp 2πiτ Lt0ot exp i θiH0tot,i +iφH0 (−1)∆Ngh , (4.6)
" " # #
i=2
(cid:2) (cid:0) (cid:1)(cid:3) X
The prime on the trace indicates that the trace is taken with signs i.e. a state with
positive (negative) norm contributes with a positive (negative) sign in the trace. The
corresponding coset signature function is defined as
rg
dτ (ˆg,ˆh′ SVir)(τ,φ) dτ (dθ ) Σ(ˆg,hˆ′ SVir)(τ,φ,θ) . (4.7)
⊕ i ⊕
S ≡
Z Z Z Yi=2 n o
The relevance of these functions to the present problem was stated is (cf. Lemma 2 in
[1]).
Lemma 1 Q is unitary if, and only if,
Hµˆµ˜ˆ
dτ (ˆg,ˆh′ SVir)(τ,φ) (ˆg,hˆ′ SVir)(τ,φ) = 0. (4.8)
⊕ ⊕
B −S
Z h i
Here Q denotes the irreducible subspace of gˆ hˆ˜′ SCFT ghost consisting of
Hµˆµ˜ˆ Hµˆ ×Hµˆ˜ ×Hl′ ×H′
non-trivial states in the relative cohomology.
We can now state our main result.
10