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arXiv:hep-th/9412196v1 23 Dec 1994

TUM–HEP–213/94

hep-th/9412196

September 1994

Low–Energy Properties of (0,2) Compactifications∗

Peter Mayr and Stephan Stieberger

Institut f¨ ur Theoretische Physik

Physik Department

Technische Universit¨ at M¨ unchen

D–85747 Garching, FRG

∗Invited talk given by S. Stieberger at the 28th International Symposium on the Theory of

Elementary Particles in Wendisch–Rietz, August 30–September 3, 1994. Supported by the

Deutsche Forschungsgemeinschaft and the EU under contract no. SC1–CT92–0789.

Abstract. We investigate the low–energy properties of a Z Z12 orbifold with con-

tinuous Wilson line moduli. They give rise to a (0,2) superstring compactification.

Their K¨ ahler potentials and Yukawa couplings are calculated. We study the discrete

symmetries of the model and their implications on the threshold corrections to the

gauge couplings as well as for string unification.

String theory is the only known theory which consistently unifies all interactions. To make

contact with the observable world one constructs the field–theoretical low–energy limit of a

given ten–dimensional string theory. One possibility to get a four–dimensional effective N=1

supergravity theory is to compactify six of the ten dimensions on an internal Calabi–Yau man-

ifold (CYM) [1] or its singular limits, the toroidal orbifolds [2]. In general CYMs are Ricci–flat

K¨ ahler manifolds. If the spin connection is identified with the gauge connection the gauge group

is always E6× E8. There are alternative embeddings of the spin connection involving stable,

irreducible, holomorphic SU(4) or SU(5) bundles which result in the gauge groups SO(10)×E8

or SU(5)×E8, respectively [3]. CYMs with E6gauge group1have matter representation in the

27, 27 of E6. In addition there can be many singlets. Those of them which are in the adjoint

representation 8 of the SU(3) are related to the bundle of endomorphisms End T, where T is the

holomorphic tangent bundle over the CYM K. For the standard embedding the number of the

27 and 27 generations is given by topological invariants, the number of independent harmonic

(2,1) forms and (1,1) forms on K, respectively. On the other hand the number of E6singlets

from the 8 is the dimension2of H1(EndT). Of course, to make contact with the Standard Model

one would like to obtain the gauge group SU(3) × SU(2) × U(1). One way to achieve this is to

give Planck–scale vacuum expectation values to certain components of 27 or 27 matter fields.

This is an explicit symmetry breaking lifting the flat directions in the superpotential [3, 4]. On

the CYM this can be understood as a deformation of the bundle that describes the embedding

of the spin connection into the gauge group to a new, stable bundle. In general this leads to

1We will drop the second E8 factor since it couples only gravitionally to the E6 and therefore plays the rˆ ole of

the hidden sector gauge group giving rise to gaugino condensation.

2H1(EndT) is the space of infinitesimal deformations of the complex structure of T. Its dimension depends on

the complex structure of K and therefore on the (2,1)–moduli.

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a (0,2) superconformal symmetry on the world–sheet. If the CYM is not simply connected the

group can also be broken by non-trivial holonomies of gauge fields in the internal directions

such that the vacuum state gauge field Aa

icannot necessarily be gauged away, even though the

field strength Fa

ijvanishes everywhere in order that supersymmetry (SUSY) remains unbroken

[5, 3]. Therefore the Wilson loop

U(γ) = P e

−i?

γ

TaAa

idxi

, (1)

with γ being a closed path on K and Tathe E6group generators, can represent an element

of E6different from unity. Let us denote the subgroup generated by the elements U(γ) for a

fixed vacuum state gauge field Aa

iby H. At energies below the compactification scale, E6will

be spontaneously broken to the subgroup G of E6 which commutes with H. This symmetry

breaking is due to the effective Higgs vacuum expectation values (vevs) of the order of the

compactification scale

HIJ= Pexp

−i

?

γ

(Ta)IJAa

idxi

(2)

in the adjoint 78 representation of E6. If the CYM has an Abelian fundamental group, inde-

pendent Wilson loops have to commute and therefore U(γ) takes the form

Ui= e

i

rkG0

?

I=1

aI

iHI

,(3)

where the HIare generators of the Cartan subalgebra of the original unbroken gauge group

G0= E6,E8and the real parameters aI

iare the Wilson line moduli corresponding to a breaking

direction in the root space. Toroidal orbifolds have six independent non–contractible loops

which give rise to six Wilson lines aI

i[6, 7]. Since a group generator of the unbroken gauge

group specified by its root α has to commute with H, the roots of the unbroken gauge group

have to fulfill the condition:

rkG0

?

I=1

αIaI

i= 0 mod 2π , i = 1,...,6. (4)

Note that the above sum is proportional to the mass of the vector boson corresponding to

α after an adjoint symmetry breaking. The various 27 and 27 representations of E6split into

representations w.r.t. the unbroken gauge group. In models with Wilson lines the usual relations

arising from the organization of states in E6multiplets are less stringent since only the singlets

of the combined action of the holonomy and gauge transformations survive as massless states

[5]. This gives rise to an elegant solution of the doublet–triplet splitting problem usually present

in GUTs and even in (2,2)–string models. Moreover also the usual Yukawa coupling unification

of GUTs is absent. On the other hand gauge coupling unification is still present since the gauge

bosons come from the single adjoint 78 representation of E6. Heterotic string compactifications

on a CYM represent a (2,2) superconformal field theory (SCFT) on the world–sheet with central

charges (c,¯ c) = (6,9) together with free fields (in light cone gauge the remaining free left–handed

gauge fermions, one complex left–moving and right–moving boson and one complex right–moving

fermion). Since the Wilson lines only couple to the free fields describing the unbroken gauge

degrees of freedom, the right–handed N=2 SCA is completely unaffected by them. On the other

hand the left–handed N=2 algebra can be broken by a non–standard embedding, but not by the

Wilson lines [8]. Anyway this (0,2) SCFT is enough to ensure N=1 space–time supersymmetry

under certain additional conditions [9].

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The model we want to study is a Z Z12orbifold with the torus lattice Λ6= SU(3) × F4. In

the complex basis the twist has the eigenvalues θ = exp[1/12(4,1,−5)]. The twist embedding

in the gauge lattice Λ16= E8× E′

resulting gauge group is U(1)2×SU(3)3×SU(4)′2×U(1)′2. The continuous Wilson lines in the

first E8are chosen to be:

8is chosen to be Θ = Z Z(2)

3

× Z Z(6)

12× Z Z′(2)

3

× Z Z′(2)

2

[10]. The

aI

1= (λ,µ;0,0;0,0;0,0)

w.r.t to the two weights d1= (1/√2,1/√6),d2= (−1/√2,1/√6) of SU(3). The corresponding

gauge group is

;aI

2= (−µ,λ − µ;0,0;0,0;0,0) withλ,µ ∈ IR , (5)

U(1)2× SU(3)3× SU(4)′2× U(1)′2

U(1)8× SU(4)′2× U(1)′2

U(1)6× SU(4)′2× U(1)′2

At special values of λ,µ the gauge group is enhanced. Away from these points the gauge fields

become massive. This fact is known as the stringy Higgs effect. Their mass is governed by the

Wilson line moduli and the compactification radius R

,

,

,

λ,µ ∈ Z Z ∧ λ + µ ∈ 3Z Z ,

λ,µ ∈ Z Z ∧ λ + µ / ∈ 3Z Z ,

λ,µ / ∈ Z Z .

(6)

M2

X∼λ2− λµ + µ2

R2

. (7)

A N=1 supergravity up to second order derivatives in space–time is completely characterized

by three functions: the K¨ ahler potential, the superpotential W and the so–called f–function.

The kinetic terms for the massless fields are encoded in the K¨ ahler potential. The superpotential

contains the Yukawa couplings as well the gauge kinetic terms specified by the f–function. The

latter determines the tree–level gauge coupling g−2

three functions for some (0,2) orbifold compactifications. In particular we are interested in their

moduli dependence. Specifically they will depend on the six–dimensional moduli T and U as well

as on the complex Wilson line modulus A (containing µ and λ). Therefore we will concentrate

on the kinetic terms Ki¯j=∂2K(Φa,¯Φa)

∂Φi¯∂Φj

for (Φa∈ T,U,A) and on the corresponding dependence

of the Yukawa couplings. The superpotential reads

a

= Refa. Our aim is to determine these

W = habcAaAbAc+ Yijk(T,U,A)σiσjσk+ h′

Here habcare the Yukawa couplings between three untwisted string states Aawhich are constant.

The coupling Yijk(T,U,A) between three strings σifrom the twisted sector is moduli–dependent

due to world–sheet instantons [11]. The last coupling between two strings from the twisted sector

and one from the untwisted sector is in general moduli–dependent as well [11, 12]. One should

however keep in mind that this simple cubic renormalizable form is corrected after integrating out

the heavy string states. These corrections are non–polynomial in the charged fields. The cubic,

renormalizable superfield couplings dictated by E6group theory are dijkΣiΣjΣk, dabcΩaΩbΩc,φ3

and φΣiΩiwith Σirepresenting a 27 and Ωaa 27, respectively. The last coupling is important

for neutrino masses as well as for symmetry breaking if e.g. a Σifield acquires a vev of the order

of MPlanck[3]. After the breaking of E6those fields which are not invariant under the twist and

the Wilson line action must be set to zero. The scalar φ in the φΣiΩicoupling cannot be any

moduli field: this coupling would give rise to mass terms for the 27 generations and thus violate

the relation between topology and the number of generations3. The superpotential does not get

any corrections from sigma model perturbation theory as well as from string loop corrections.

ijaσiσjAa. (8)

3There is another way to see this: φ being a (1,1)–modulus would violate the Peccei–Quinn symmetry which

is supposed to hold at least pertubatively. On the other hand there can be couplings as Uφφ with U being a

(2,1)–modulus [3]. An example for a non-renormalizable coupling is f(T,U)ΣiΩi, which generates a µ–term [13].

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The only potential correction to it comes from world–sheet instantons and non–perturbative

string effects. From instanton corrections there can arise UV–divergent (2727)K(K ≥ 2) terms

which destabilize the vacuum due to a non–vanishing beta–function. The conditions for which

such couplings are absent have been determined in [14].

To get the moduli– and Wilson line dependence of the K¨ ahler potential and the Yukawa

couplings one has to start with a supersymmetric non–linear sigma model rather than performing

the calculation in field theory. We consider the non–linear sigma–model action [15, 16]:

S =

?

C2dzd¯ z

6

?

i,j=1

¯∂Xi(gij+bij+1

4aI

iaIj)∂Xj+

16

?

I=1

6

?

i=1

¯∂XiaiI∂XI+

16

?

I,J=1

¯∂XI(GIJ+BIJ)∂XJ(9)

with the chirality constraint¯∂XI= 0,PI

possibly break the gauge group via the Wilson line mechanism. As explained before they carry

an index of the Cartan subalgebra. The remaining matter fields can be neglected. These fields

can be easily included in our results along the lines of [17]. The complete action with all matter

fields contains in addition also a world–sheet sigma–model anomaly. After discarding the Wilson

lines independent part (9) can be rewritten in complex coordinates Zi= X(2i−1)+UX(2i),¯Z¯i=

X(2i−1)+¯UX(2i), i = 1,2 with U = −1

−i

2√3

3i√3 and

˜T=

√3

3(2λ − µ) .

The action and the moduli–space are identical to those of a four–dimensional Z Z3orbifold [17]

at the special points T12 = 0,T21 = A,T1 ≡˜T,T2 ≡ TE⊥

from a

SU(2)×SU(2)×U(1)coset to a

consistent with the symmetries of that moduli space [18] can be written4:

R= 0. It contains only those gauge fields which could

2+i

2

√3 being the fixed (2,1)–modulus:

S =

?

C2dzd¯ z (˜T¯∂¯Z¯1∂Z1+ A¯∂¯Z¯1∂Z2+ TE⊥

8

¯∂¯Z2∂Z2+ hc.) (10)

with TE⊥

8=2

2b + i√3(R2+1

8|A|2) ,

A=µ + i

(11)

8, where the moduli–space collapses

SU(2,2)

SU(2,1)

U(1). Therefore the most general K¨ ahler potential being

K = −ln[(−i˜T + i¯˜T)(−iTE⊥

8+ i¯TE⊥

8) − |A|2] .(12)

Similar one derives the K¨ ahler potential of the other orbifolds [20, 21].

In the following we want to evaluate the Yukawa coupling between twisted matter fields:

YFi,Fj,Fk≡ lim

|x|→∞|x|4h?σ+

Fi(x, ¯ x)σ+

Fj(1,1)σ−−

Fk(0,0)? . (13)

Here σ+

(f,F) satisfying θf = f +2πw ,ΘF = F +2πW and Θ˜F =˜F +2πW +2πwiaiwith w ∈ Λ6,W ∈

Λ16. It is defined via its operator product expansion with the coordinate differentials ∂Z(z, ¯ z)

and creates a twisted string at the world–sheet insertion z = x, where the local monodromy

becomes [11]:

F(x, ¯ x) denotes a twist field with conformal weight h corresponding to the fixed point F =

Xi(e2πiz,e−2πi¯ z)=(θi

jXj)(z, ¯ z) ,

(ΘI

XI(e2πiz,e−2πi¯ z)=

JXJ)(z, ¯ z) .

(14)

One observes that the local monodromy conditions do not feel the Wilson line whereas the

global monodromy conditions become Wilson line dependent [21]. The number of fixed points

4This was recently shown by a different method in [19].

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˜F is not changed in the presence of continuous Wilson lines [7]. The fixed points are subject

to the fixed–point selection rule Fi+ Fj− (1 + Θ)Fk∈ Λ. Using results of [12] one arrives at

[20, 21]:

YFi,Fj,Fk(˜T,A) ∼ (detg)

1

4

?

? v∈(1−θ2)(fk−fj+w),w∈Λ6

? u∈(1−Θ2)(Fk−Fj+Λ16+wiai)⊥

e

iπ(T|v|2+A¯ vu+TE⊥

8

|u|2). (15)

Again, we discarded the Wilson line independent parts which can be obtained from [12]. u,v are

the complexified components v = v1+ iv2and u = u1+ iu2of ? v,? u, respectively. One observes

that the Wilson lines produce additional hierarchies between the Yukawa couplings.

Let us now turn to the topic of threshold corrections to the gauge couplings. It can be shown

that the expression of [22] can be simplified in the N=2 sector to [20, 21]:

△a(˜T,¯˜T,A,¯ A) = b(1,Θ3)

a

(A = 0)

?

˜Γ

d2τ

τ2

?

k1,k2∈Z Z

Z4d(τ, ¯ τ,˜T,¯˜T,A,¯ A,k1,k2)Ca(τ,k1,k2) .(16)

with

Z4d=

?

n1,n2∈Z Z

m1,m2∈Z Z

e

−πτ2

Im(˜ T−i√3/8|A|2)ImU|˜TUn2+˜Tn1−U(m1+Ak1−1

2Ak2)+m2−1

2Ak1|2e2πiτ(m1n1+m2n2)

(17)

and a holomorphic moduli–independent function Ca(τ,k1,k2). b(1,Θ3)

coefficient of the (1,Θ3) sector for A = 0. The integrand is invariant under Γ0(2) as it is

required by modular invariance. The evaluation of (16) is rather cumbersome and is the subject

of [20, 21]. Instead let us discuss the discrete symmetries of △a(˜T,¯˜T,A,¯ A). One finds the

following symmetries [18]:

a

(A = 0) is the β–function

˜T −→−1

˜T + 1

˜T

, A −→A

˜T

˜T −→

−→

−→

A

A

A + 2U ,˜T −→˜T + U − λ¯U − µU

A + 2 ,˜T −→˜T + U − λ − µ¯U

(18)

In addition one has A → A − 2µ − 2λU together with the corresponding transformation on˜T.

There are also some fixed directions in the (λ,µ) or A plane, along them a shift in A is not

accompanied by a shift in the field b. E.g.: along µ = 0 or λ = 0 the shifts λ → λ + 6 or

µ → µ + 6 respectively, lead to the same theory. Eqs. (18) are also the symmetries of the N=2

spectrum. We want to stress that the symmetries of the K¨ ahler potential are in general not the

symmetries of the threshold corrections since the truncation performed in these cases violates

modular invariance. Therefore one has to be careful in identifying automorphic functions of

SU(2,1) and threshold corrections. For more details see [21].

Finally we want to discuss the implications of (16) for gauge coupling unification in string

theory. At string tree–level all gauge couplings are related to the gravitional coupling by the

well–known equations

g2

aka= 4πα′−1GN= g2

string, ∀ a , (19)

valid at Mstring. Here α′is the inverse string tension and kais the Kac–Moody level of the group

factor labeled by a. The scale Mstringcan be determined to be Mstring= 0.52gstring× 1018GeV

5