N=2 -> 0 super no-scale models and moduli quantum stability

0.39 MB·

by  Costas Kounnas

LPTENS–16/09, CPHT–RR080.122016 December 2016
N = 2 → 0 super no-scale models
and moduli quantum stability
Costas Kounnas1 and Herv´e Partouche2
1 Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure ,
7 †
1 24 rue Lhomond, F–75231 Paris cedex 05, France
0
[email protected]
2
n 2 Centre de Physique Th´eorique, Ecole Polytechnique, CNRS, Universit´e Paris-Saclay
a F–91128 Palaiseau cedex, France
J
[email protected]
2

]
h Abstract
t
-
p
e Z
We consider a class of heterotic = 2 0 super no-scale -orbifold models. An
h 2
N →
[ appropriate stringy Scherk-Schwarz supersymmetry breaking induces tree level masses to all

1 massless bosons of the twisted hypermultiplets and therefore stabilizes all twisted moduli.
v At high supersymmetry breaking scale, the tachyons that occur in the = 4 0 parent
5 N →
theories are projected out, and no Hagedorn-like instability takes place in the = 2 0
4
N →
5 models (for small enough marginal deformations). At low supersymmetry breaking scale, the
0
stability of the untwisted moduli is studied at the quantum level by taking into account both
0
. untwisted and twisted contributions to the 1-loop effective potential. The latter depends on
1
0 the specific branch of the gauge theory along which the background can be deformed. We
7
derive its expression in terms of all classical marginal deformations in the pure Coulomb
1
: phase, and in some mixed Coulomb/Higgs phases. In this class of models, the super no-
v
i scale condition requires having at the massless level equal numbers of untwisted bosonic and
X
twisted fermionic degrees of freedom. Finally, we show that = 1 0 super no-scale
r Z N →
a models are obtained by implementing a second orbifold twist on = 2 0 super
2
Z N →
no-scale -orbifold models.
2
Unit´e mixte du CNRS et de l’Ecole Normale Sup´erieure associ´ee `a l’Universit´e Pierre et Marie
†
Curie (Paris 6), UMR 8549.
1 Introduction
In string theory, even when starting classically in a flat four-dimensional background, the
vacuum energy induced at the quantum level is hard to reconcile with the present cosmo-
logical constant. When supersymmetry is hardly broken, the 1-loop effective potential is
generically of order M4, where M is the string scale, which is far too large. On the con-
s s
trary, if supersymmetry is exact, the quantum potential vanishes identically at least at the
perturbative level, or leads non-perturbatively to an anti de Sitter vacuum with restored
supersymmetry. A priori more promising, the no-scale models [1] consist somehow of an
intermediate situation. At the classical level, these backgrounds realize in flat space a spon-
taneous breaking of supersymmetry at a scale m , which is a flat direction of the tree level
3/2
potential. However, if the order of magnitude of the quantum effective potential is dictated
by m , it happens to be generically too large. Moreover, the quantum potential induces
3/2
tadpoles for the classical moduli fields, including the dilaton and the “no-scale modulus”
parameterized by m , which are responsible for a destabilization of the flat background.
3/2
Some exceptions however exist, at least at the 1-loop level, when the spontaneous break-
ing of supersymmetry arises via “coordinate dependent compactification” [2,3], the stringy
version of the Scherk-Schwarz mechanism [4]. This total breaking of supersymmetry can
be implemented on initially = 4, 2 or 1 heterotic or type II orbifold models, as well as
N
on orientifold theories [5], or on marginally deformed fermionic constructions [6,7]. In this
framework, some theories referred as super no-scale models [8–11] induce an exponentially
suppressed 1-loop vacuum energy, whose order of magnitude can easily be of order (or lower
than) the presently observed cosmological constant. Type II [12] and orientifold [13] theories
with exactly vanishing vacuum energy at 1-loopeven exist. In all known examples, the mod-
els arise at extrema of the quantum effective potential, with respect to all directions that are
lifted. However, the question of the stability of the non-flat directions must be addressed. In
other words, does the model sit at a minimum, maximum or saddle point of its potential ?
This problem has been addressed for the super no-scale models realizing the = 4 0
N →
spontaneous breaking of supersymmetry [10] and is reconsidered for less symmetric theories
in the present work, such as models implementing an = 2 0 breaking.
N →
In Ref. [10], one considers the = 4 0 no-scale models for given internal metric,
N →
antisymmetric tensor and Wilson lines background expectation values. Supposing the point
1
in moduli space is such that no mass scale below m exist, we denote cM the lowest mass
3/2 s
scale above m . In this case, the 1-loop effective potential takes the form [14]
3/2
V1N-l=oo4p→0 = ξ(nF −nB)m43/2 +O c2Ms2m23/2e−cMs/m3/2 , (1.1)
(cid:0) (cid:1)
where the gravitino mass m scales inversely to the volume involved in the stringy Scherk-
3/2
Schwarz mechanism and n ,n are the numbers of massless fermionic and bosonic degrees
F B
of freedom. The m4 dominant contribution arises from the light towers of Kaluza-Klein
3/2
states, whose masses are of order m , while ξ > 0 depends on moduli other than the
3/2
dilaton and m . For the quantum vacuum energy and tadpoles to be exponentially small,
3/2
one can focus on the models satisfying the super no-scale condition n = n [8–10]. In
F B
this case, the 1-loop vacuum energy is of the order of the observed cosmological constant,
provided the gravitino mass m is about 2 orders of magnitude smaller than the scale cM .
3/2 s
However, switching onsmallmarginaldeformations, collectively denotedY, aroundthepoint
inclassical moduli spacewestartedwith, oneinduces new massscaleslower thanm . Some
3/2
ofthen +n initiallymassless statesacquiresmall masses. Whenthemass scalesYM reach
F B s
the order of m , the exponential contributions in Eq. (1.1) are (m4 ), thus correcting
3/2 O 3/2
n ,n which now take new integer values. In other words, n ,n are effectively functions
F B F B
of Y which interpolate between different integer values corresponding to distinct massless
spectra.
To study the local stability of an = 4 0 super no-scale model [10] around a point in
N →
moduli space characterized by integer n and n , one has to expand these two functions at
F B
quadraticorder inY. Duetotheunderlying = 4structure, themoduli deformationsY are
N
Wilson lines along T6. The result is that those which are associated to non-asymptotically
free gauge group factors become tachyonic at 1-loop. They condense, break spontaneously
the associated gauge symmetry which enters a Coulomb branch, and induce a destabilization
of the vacuum. On the contrary, the Wilson line associated to asymptotically free gauge
group factors become massive and are dynamically attracted to Y = 0. The Wilson lines
associated to conformal groups remain massless.
Z
In Sect. 2, we consider = 2 0 super no-scale theories realized as -orbifolds of
2
N →
= 4 0 no-scale models. At the level of exact = 2 supersymmetry, the twisted
N → N
hypermultiplets introduce new moduli fields living on a quaternionic manifold. We show
that the implementation of the stringy Scherk-Schwarz mechanism can always be chosen so
2
that all twisted moduli acquire a tree level mass of order m and are no more marginal in
3/2
the non-supersymmetric theory. In these models, the super no-scale condition amounts to
having classically equal numbers of massless untwisted bosonic and twisted fermionic degrees
of freedom. Moreover, the tachyons that appear at tree level in the parent = 4 0
N →
theory [3] when m is of the order of M are automatically projected out in the = 2 0
3/2 s
N →
models. In other words, when m is large and local perturbations of other moduli are
3/2
allowed, no Hagedorn-like instability occurs.
In Sect. 3, we study the local stability of the untwisted moduli in this class of super
no-scale models. The analysis generalizes that of Ref. [10] by taking into account, in the
1-loop effective potential =2 0, the contributions arising from the twisted fermions. The
V1N-loop→
expression of =2 0, which we determine atsecond order inmoduli fields, isdistinct ineach
V1N-loop→
branch of the gauge theory along which the classical background can be deformed. To be
more specific, we derive the quantum potential as a function of all moduli fields in the pure
Coulomb branch, as well as in some mixed Coulomb/Higgs branches. Moreover, we show
that in this class of models, because all moduli fields are untwisted, the number of marginal
deformations in any branch of the gauge theory is universal once the model is compactified
down to two dimensions.
Z
In Sect. 4, we show that = 2 0 super no-scale -orbifold models can automatically
2
N →
Z
lead descendent = 1 0 super no-scale theories, by implementing a second orbifold
2
N →
twist. However, we argue that the analysis of the background stability must be generalized
to include new twisted moduli deformations.
A summary of our hypothesis and results can be found in the conclusion, Sect. 5.
2 A class of N = 2 → 0 super no-scale models
In this section, we construct = 2 0 super no-scale backgrounds, keeping in mind the
N →
goal of Sect. 3, which is to study their stability at the quantum level. In the framework of
Z
heterotic -orbifold compactifications, at the exact = 2 level, the models admit special
2
N
Ka¨hler moduli belonging to vector multiplets arising in the untwisted sector. Quaternionic
deformations belonging to hypermultiplets also exist and occur generically in both untwisted
and twisted sectors. In the following, we highlight a particular class of models characterised
3
by an implementation of the stringy Scherk-Schwarz supersymmetry breaking that lifts clas-
sically all moduli of the twisted sector. These models aregeneric in the sense that bothtypes
of moduli, special Ka¨hler and quaternionic, are allowed. However, they are also particular,
since the complex structure of the internal space cannot be deformed away from the orbifold
point, as follows from the non-existence of twisted deformations [15].
We consider heterotic no-scale models on T2 T4/Z , where the = 2 0 sponta-
2
× N →
neous breaking of supersymmetry is implemented by a coordinate dependent compactifica-
tion on T2. We denote the spacetime, T2 and T4 coordinates as X0,1,2,3, X4,5 and X6,7,8,9,
respectively. For notational convenience, we restrict to the case where the stringy Scherk-
Schwarz mechanism is implemented along the compact direction X4 only, which is supposed
to be large, for m to be lower than M . Moreover, even if it is not necessary, we will quote
3/2 s
our results in the case where the second direction of T2 is also large. Our aim is twofold :
First, we want models that develop a super no-scale structure. In terms of the 1-loop
•
partition function Z, the effective potential can be expressed as an integral over the funda-
mental domain of SL(2,Z),
F
M4 d2τ
=2 0 = s Z, (2.1)
V1N-loop→ −(2π)4 2τ2
Z 2
F
where τ = τ +iτ is the Techmu¨ller parameter. As long as a model sits at a point in moduli
1 2
space where no mass scale is lower than m , the untwisted and twisted sectors both yield
3/2
contributions as shown in Eq. (1.1), so that
V1N-l=oo2p→0 = ξ(nuF +ntF −nuB −ntB)m43/2 +O c2Ms2m23/2e−cMs/m3/2 . (2.2)
(cid:0) (cid:1)
In the above expression, nu,nu are the numbers of massless untwisted fermions and bosons,
F B
while nt,nt are their counterparts in the twisted sector. For a model to be super no-scale,
F B
we require
nu +nt = nu +nt . (2.3)
F F B B
Second, we want the precise implementation of the coordinate dependent compactifi-
•
cation to imply the twisted moduli present at the = 2 level to be lifted at tree level.
N
Note that in the present work, the Z twist is non-freely acting and gives a priori rise to
2
massless states in the twisted sector. This situation is to be contrasted with the simpler one,
already studied in Ref. [10], where the Z twist on T4 also shifts the direction X5 of T2. In
2
4
this case, there are no fixed points and the twisted states are automatically super massive
(the strings are stretched along X5), even at the = 2 level.
N
A representative model
The starting point to construct the simplest model that realizes the above goal is the E E
8 8
×
heterotic string compactified on T2 T4. The stringy Scherk-Schwarz mechanism can be
×
introduced by implementing a Z orbifold shift along X4, while a Z orbifold twist acts
2 2
on X6,7,8,9. The 1-loop partition fonction is
1
Z = Z H
2 G
H,G
X (cid:2) (cid:3)
1 1 1 Γ h
= S a;h S h;H Z(F) a;H O(n.c.) 2,2 g Z H Z¯ H O¯(E8), (2.4)
2 2 2 b;g ′ g;G 4,0 b;G 2,2 η2η¯2 4,4 G 0,8 G 0,8
(cid:2) (cid:3)
H,G h,g a,b
X X X (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
where Z(F) a and O(n.c.) are the conformal blocks arising from the left-moving worldsheet
4,0 b 2,2
fermions an(cid:2)d(cid:3)non-compact spacetime coordinates in light cone gauge, while Z¯0,8 HG O¯0(E,88) is
the contribution of the 16 additional right-moving bosonic degrees of freedom, (cid:2) (cid:3)
Z(F) a;H = ( 1)a+b+ab θ ab 2 θ ab++GH θ ab−GH , O(n.c.) = 1 ,
4,0 b;G − η2 η −η 2,2 τ η2η¯2
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) 2
(cid:2) (cid:3)
Z¯0,8 HG = 12 θ¯η¯γδ6 6 θ¯ γδη¯++HG θ¯ γδη¯−−HG , O¯0(E,88) = 21 θ¯η¯γδ′8′ 8 . (2.5)
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
γ,δ γ ,δ
(cid:2) (cid:3) X X′ ′
In our conventions, the spin structures a,b, the twists H,G and γ,δ,γ ,δ are integer mod-
′ ′
ulo 2, while our definitions of the Dedekind η and Jocobi θ α functions can be found in
β
Ref. [16]. The conformal block associated to the T4/Z2 direct(cid:2)io(cid:3)ns is
1
(1+Γ+ +Γ ) if (H,G) (0,0),
4,4 −4,4 η4η¯4 ≡

 16η2η¯2
Z4,4 HG =  (1+Γ+4,4 −Γ−4,4) θ 1 2θ¯ 1 2 if (H,G) ≡ (0,1), (2.6)
 0 0

(cid:2) (cid:3)
16η2η¯2
(cid:2) (cid:3) (cid:2) (cid:3)
if H 1,
 θ 0 2θ¯ 0 2 ≡
 1 G 1 G
 − −


where Γ+ and Γ are the c(cid:2)ontri(cid:3)but(cid:2)ions(cid:3)of the 4-torus zero modes that are even or odd
4,4 −4,4
under the Z twist. As explained in the Appendix, they satisfy Γ+ = Γ . Finally, the T2
2 4,4 −4,4
5
coordinates contribution involves the shifted lattice
Γ2,2 hg (T1,U1) = (−1)gm4 q21|pL|2q¯21|pR|2, (2.7)
(cid:2) (cid:3) mX4,m5
n4,n5
where h,g are integer modulo 2, q = e2iπτ and
1 1
p = U m m +T n + h +T U n ,
L 1 4 5 1 4 1 1 5
√2ImT ImU − 2
1 1 (cid:20) (cid:21)
(cid:0) (cid:1)
1 1
p = U m m +T¯ n + h +T¯ U n , (2.8)
R 1 4 5 1 4 1 1 5
√2ImT ImU − 2
1 1 (cid:20) (cid:21)
(cid:0) (cid:1)
in terms of integer momenta m ,m and winding numbers n ,n , as well as the internal
4 5 4 5
metric and antisymmetric tensor, through the Ka¨hler and complex structure moduli
i G G G2 +G
T = i G G G2 +B , U = 44 55 − 45 54 . (2.9)
1 44 55 − 45 54 1 G
p 44
q
The = 2 0 spontaneous breaking is implemented by coupling the lattice shift h,g
N →
to the spin structure a,b, where a = 0 (a = 1) corresponds to spacetime bosons (fermions).
This is done by inserting in the partition function the modular invariant sign [3]
S a;h = ( 1)ga+hb+gh. (2.10)
b;g −
(cid:2) (cid:3)
Note that the light spectrum must have vanishing winding numbers along the large direc-
tion X4, which implies h = 0 (see Eq. (2.8)) and S = ( 1)ga. If nothing else is introduced
−
in the partition function, the initially massless bosons (a = 0) don’t see the breaking, while
the massless fermions (a = 1) acquire a mass of order m . In this case, the number of
3/2
massless fermions is always vanishing and the model has no chance to be super no-scale. To
remedy this fact, we insert in Z another modular invariant sign [17]
S h;H = ( 1)gH+hG, (2.11)
′ g;G −
(cid:2) (cid:3)
which for h = 0 yields SS = ( 1)g(a+H), so that :
′
−
In the untwisted sector, H = 0, the situation is as before. The = 2 0 breaking
• N →
induces a tree level mass m to the massless fermions, while the massless bosons are not
3/2
modified.
In the twisted sector however, H = 1, the situation is reversed. The = 2 0
• N →
breaking induces a tree level mass m to the massless bosons, while the massless fermions
3/2
6
are not modified.
Therefore, we have
nu = 0, nt = 0, (2.12)
F B
and the super no-scale condition (2.3) becomes
nt = nu . (2.13)
F B
In other words, in the 1-loop effective potential, we want the contribution of the untwisted
massless sector, which is purely bosonic, to compensate that of the twisted massless sector,
which is purely fermionic. Note that the consequences of the SS insertion in a partition
′
function extend far beyond the particular example we consider. They are valid in any
Z
heterotic -orbifold model, where the stringy Scherk-Schwarz mechanism is implemented
2
along the untwisted directions.
The spectrum
In order to see how things work in detail, we write the partition function (2.4) in terms
of SO(2N) affine characters
θ 0 N +θ 0 N θ 0 N θ 0 N
O = 0 1 , V = 0 − 1 ,
2N 2ηN 2N 2ηN
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
θ 1 N +( i)Nθ 1 N θ 1 N ( i)Nθ 1 N
S = 0 − 1 , C = 0 − − 1 . (2.14)
2N 2ηN 2N 2ηN
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
In the untwisted sector, H = 0, we find
Z 0 = O(n.c.) Z 0 O¯ O¯ +V¯ ( 1)GV¯ +C¯ ( 1)GC¯ +S¯ S¯ O¯(E8)
G 2,2 4,4 G 12 4 12 − 4 12 − 4 12 4 0,8
(cid:16) (cid:17)
(cid:2) (cid:3) (cid:2) (cid:3)
O 0 O ( 1)GV +V O O 0 C ( 1)GC +S S
× 2,2 0 4 − 4 4 4 − 2,2 1 4 − 4 4 4
n (cid:16) (cid:17) (cid:16) (cid:17)
(cid:2) (cid:3) (cid:2) (cid:3)
O 1 C ( 1)GS +S C +O 1 O ( 1)GO +V V , (2.15)
− 2,2 0 4 − 4 4 4 2,2 1 4 − 4 4 4
(cid:16) (cid:17) (cid:16) (cid:17) o
(cid:2) (cid:3) (cid:2) (cid:3)
where we have defined characters associated to the shifted T2 as
Γ h +( 1)gΓ h 1
O2,2 hg = 2,2 0 2η−2η¯2 2,2 1 = η2η¯2 q21|pL|2q¯21|pR|2, (2.16)
(cid:2) (cid:3) (cid:2) (cid:3)
(cid:2) (cid:3) kX4,m5
n4,n5
7
with momentum m redefined as 2k +g in the expressions of p ,p in Eq. (2.8). Similarly,
4 4 L R
one obtains in the twisted sector, H = 1,
Z 1 = O(n.c.) Z 1 O¯ ( 1)GC¯ +V¯ S¯ +C¯ O¯ +S¯ ( 1)GV¯ O¯(E8)
G 2,2 4,4 G 12 − 4 12 4 12 4 12 − 4 0,8
(cid:16) (cid:17)
(cid:2) (cid:3) (cid:2) (cid:3)
O 0 C O +S ( 1)GV +O 0 O S +V ( 1)GC
× − 2,2 0 4 4 4 − 4 2,2 1 4 4 4 − 4
n (cid:16) (cid:17) (cid:16) (cid:17)
(cid:2) (cid:3) (cid:2) (cid:3)
+O 1 O C +V ( 1)GS O 1 C V +S ( 1)GO . (2.17)
2,2 0 4 4 4 − 4 − 2,2 1 4 4 4 − 4
(cid:16) (cid:17) (cid:16) (cid:17) o
(cid:2) (cid:3) (cid:2) (cid:3)
Some remarks are in order :
Due to the large volume of T2, the sector h = 1, which yields non-vanishing winding
•
number n + 1, leads contributions of order e πτ2ImT1/4ImU1 1 for τ . Thus, all
4 2 − ≪ ∈ F
conformal blocks proportional to O 1 and O 1 will not be considered explicitly from
2,2 0 2,2 1
now on. (cid:2) (cid:3) (cid:2) (cid:3)
The massless spectrum arises from the conformal blocks proportional to O 0 , for
• 2,2 0
vanishing momenta and winding numbers along T2. In the untwisted (twisted) sec(cid:2)to(cid:3)r, as
announced before, it is bosonic (fermionic). It is accompanied by towers of light bosonic
(fermionic) Kaluza-Klein states, with momenta 2k and m .
4 5
The remaining light spectrum arises from the conformal blocks proportional to O 0 .
• 2,2 1
It is composed of towers of fermionic (bosonic) Kaluza-Klein states, with momenta 2k4(cid:2)+(cid:3)1
and m , which are superpatners of the above mentioned states, with mass degeneration
5
lifted. The mass gap in these sectors is the gravitino mass m , which satisfies
3/2
U 2M2
m2 = | 1| s . (2.18)
3/2 ImT ImU
1 1
It vanishes in the large T2 volume limit, ImT + , U finite, where supersymmetry is
1 1
→ ∞
recovered.
The untwisted sector
In order to realize a super no-scale model, we first count the massless states in the untwisted
sector, H = 0. In a theory where the breaking of supersymmetry is spontaneous, there
cannot be any physical tachyon when the order of magnitude of m is lower than M . We
3/2 s
thus have
1
Z 0 +Z 0 = (nu 0)(qq¯)0 + , (2.19)
2 0 1 levelmatched B − ···
(cid:12)
(cid:0) (cid:2) (cid:3) (cid:2) (cid:3)(cid:1)
(cid:12)
(cid:12) 8
where the ellipsis account for the contributions of nu fermionic superpartners of mass m
B 3/2
and all more massive states. However, only Z 0 needs to be expanded, since the sector
0
h = 0 in Z 0 vanishes, as can be seen in the se(cid:2)co(cid:3)nd line of Eq. (2.15). Not that this fact
1
is not speci(cid:2)fic(cid:3)to the present model. It arises from the supersymmetry breaking sign S and
the identity
1
S a;0 ( 1)a+b+abθ a 2θ a+0 θ a 0 = θ 1 2θ 1 2 = 0. (2.20)
2 b;g − b b+1 b−−1 − 1−g g
a,b
X (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Since Z 0 is the partition function of the parent = 4 0 no-scale model, it is more
0 N →
naturally(cid:2) (cid:3)expressed in terms of SO(8) and SO(16) affine characters, using
V = O V +V O , O¯ = O¯ O¯ +V¯ V¯ , S¯ = C¯ C¯ +S¯ S¯ . (2.21)
8 4 4 4 4 16 12 4 12 4 16 12 4 12 4
Defining G(T4) the gauge symmetry group arising from the T4 lattice on the right-moving
side of the string, and reminding that O¯ +S¯ = O¯(E8), we find
16 16 0,8
1 1 8 1
Z 0 = +2+2+dimG(T4) +dim(E E )+ (q) . (2.22)
2 0 2 τ q¯ 8 × 8 O
2 (cid:18) (cid:19)
(cid:2) (cid:3)
(n.c.)
The first 2 in the parentheses comes from O and account for the bosonic part of the
2,2
= 4 supergravity multiplet, while the second 2 is the dimension of the U(1)2 right-moving
N
gauge symmetry arising from O 0 . We thus have
2,2 0
(cid:2) (cid:3)
nu = 4 dimG(T4) +500 . (2.23)
B
(cid:0) (cid:1)
In order to find the representations in which the nu untwisted massless states are orga-
B
nized, we expand
1+ N+ +( 1)G(N +4) q¯+ dimG(T4) 4
Z 0 = − − ··· where N = − . (2.24)
4,4 G q4/24q¯4/24 ± 2
(cid:2) (cid:3)
(cid:2) (cid:3)
N are non-vanishing if the Γ lattices moduli sit at enhanced symmetry points, while 4 is
± ±4,4
the rank of G(T4). In these notations, we find
1 1
Z 0 +Z 0 = O(n.c.) O 0 O¯(E8)
2 0 1 2,2 2,2 0 q4/24q¯4/24 0,8
(cid:0) (cid:2) (cid:3) (cid:2) (cid:3)(cid:1) (cid:2) (cid:3)¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
V O O O +S S +O V V V +C C
4 4 12 4 12 4 4 4 12 4 12 4
×
n h i
+ V O (cid:0)N+q¯O¯ O¯ (cid:1)+O V (cid:0)(N +4)q¯O¯ O¯(cid:1) + , (2.25)
4 4 12 4 4 4 − 12 4
···
h i o
9