Ui it diR “T V t” dUniversita di Roma “Tor Vergata” …statistics.roma2.infn.it/.../members/Pradisi/MG12PRADISI.pdfTools: (non-geometric) twists and shifts Asymmetric orbifolds

Non-Geometrical Compactificationswith Few Moduliwith Few Moduli

Gianfranco Pradisi

U i it di R “T V t ” dUniversita di Roma “Tor Vergata” andINFN Sezione di Roma “Tor Vergata”

Paris, 16 july 2009

P. Anastasopoulos, M. Bianchi, J.F. Morales, G.P., JHEP 0906:032, 2009 [arXiv:0901.0113 [hep-th]]

G. Pradisi - Non-Geometrical ...

OUTLINEOUTLINE• Motivations• Chiral TwistsChiral Twists• Shifts• Free Fermionic ConstructionsFree Fermionic Constructions• Type IIB N=2 Models • Type IIB N=1 Models• Type IIB N=1 Models• Type I Models

Gepner Map (in progress)• Gepner Map (in progress)• Conclusions


Moti ationsMotivations• Moduli Stabilization

Internal fluxesNon perturbative effects

Hundreds of papers in the lastfew years

Non perturbative effects

• CY with small Hodge Numbers

TriadophiliaA special corner in the landscape? P. Candelas, X. de la Ossa, Y.H. He, B. Szendröi (2007)

P. Candelas, R. Davies (2008)P. Candelas, R. Davies (2008)


y h + hAround 30.000 Hodge number pairs M. Kreuzer, H. Skarke (2004)

y = h11+ h

21

G. Pradisi - Non-Geometrical ... χ = 2 (h11 − h21)

From:P. Candelas, X. de la Ossa, Y.H. He, B. Szendröi (2007)

P. Candelas, R. Davies (2008)


From:P C d l X d l OP. Candelas, X. de la Ossa, Y.H. He, B. Szendröi (2007)

P. Candelas, R. Davies (2008)


IDEA: gauging and twisting a symmetry that exists

M. Mueller, E. Witten (1986)K Narain M Sarmadi C Vafa (1987)

IDEA: gauging and twisting a symmetry that exists only at special values of the moduli space

K. Narain, M. Sarmadi, C. Vafa (1987)M. Dine, E. Silverstein (1997)A. Dabholkar, J. Harvey (1999)M. Bianchi, J. Morales, G.P. (2000)A. Faraggi (2005)Y D li t B J li C K (2008)

Tools: (non-geometric) twists and shifts

Asymmetric orbifolds of Tori: left and right modes live on different

Y. Dolivet, B. Julia, C. Kounnas (2008)Chiral Twists

Asymmetric orbifolds of Tori: left and right modes live on different manifolds

Z2 chiral reflections:

Ii: X i

L→ −X i

L, X i

R→ X i

R, ψi → −ψi , ψi → ψi .

i L L R R

I : X i → X i X i → X i ψi → ψi ψi → ψi

The torus lattice should admit chiral automorphisms(p , p )

Ii: X i

L→ X i

L, X i

R→ −X i

R, ψi → ψi , ψi → −ψi .


p(pL, pR)

The standard way for 4-d Type IIB: S. Elitzur, E. Gross, E. Rabinovici, N. Seiberg (1987)K. Narain, M. Sarmadi, C. Vafa (1987) L. Dixon, V. Kaplunovski, C. Vafa (1987)W. Lerche, D. Luest, A.N. Schellekens, N.P. Warner (1988)

A Lie algebra of rank six

The lattice:

G

The lattice:

Γ6,6(G) ≡ ( pL, pR) : p

L, pR∈ Λ

W(G), p

L− p

R∈ Λ

R(G)

Chiral automorphisms acting for instance on the left-moving modes are elements of the Weyl group of Gelements of the Weyl group of

Typically there is a non-vanishing B-field on these lattices that itself reduce the number of “fixed points”. Our Example: the D6 lattice:

G

reduce the number of fixed points . Our Example: the D6 lattice:

⎛0 −1 0 0 0 0

⎞Bab =

⎜⎜⎜⎜⎜⎜⎜1 0 −1 0 0 0

0 1 0 −1 0 0

0 0 1 0 −1 −1

⎟⎟⎟⎟⎟⎟⎟⎜⎜⎝ 0 0 0 1 0 0

0 0 0 1 0 0

⎟⎟⎠


It admits a description in terms of 12L + 12R free Majorana fermions

Using the characters related to the conjugacy classes of D6g j g y 6

Z6,6(D6) = |O

12|2 + |V

12|2 + |S

12|2 + |C

12|2

We will consider twists that are chiral with generators

6 12 12 12 12

Z42We will consider twists that are chiral with generators

I3456, I1256, I3456, I1256

2

The untwisted moduli (except axion-dilaton) are projected out

|iiL⊗ |ji

R = ψi− 12

|0iL⊗ ψj− 1

2

|jiR

i, j = 1, . . . , 6

Susy is reduced to 1/4 of the original one.


(non-geometric) Shifts

In toroidal compactifications it is possible to “gauge” a shift, Namely a fraction of the lattice vectorsNamely a fraction of the lattice vectors.

Geometric Momentum shift p + δ ⇔ (pL, pR) + (δ,δ)

One dimensional circle: a freely acting orbifold that simply rescale the radius

G. Pradisi - Non-Geometrical ...the radius

Wh bi d ith i t l t S h k S h h iWhen combined with an internal symmetry: Scherk-Schwarz mechanismR. Rohm (1984)P. Fayet (1985)S. Ferrara, C. Kounnas, M. Porrati, F. Zwirner (1989)

• Spontaneous (partial) breaking of supersymmetryS. Ferrara, C. Kounnas, M. Porrati, F. Zwirner (1989)C. Kounnas, B. Rostand (1990)I. Antoniadis, C. Kounnas (1991)E. Kiritsis, C. Kounnas (1997)

• Supersymmetry is recovered in the decompactification limit.

Geometric winding shift: + δ ( ) + (δ δ)Geometric winding shift: w + δ ⇔ (pL, pR) + (δ,−δ)

T-dual to momentum shift for closed strings.

I. Antoniadis, G. D'Appollonio, E. Dudas, A. Sagnotti (1999)


In the presence of branes:In the presence of branes:

• Brane supersymmetry (enhanced symmetry, no chirality) or M-theory breaking.

• Multiplets of branes (reduced rank of the gauge group like a B field)

I. Antoniadis, G. D'Appollonio, E. Dudas, A. Sagnotti (1999)Z. Kakushadze, H. Tye (1999)


C V f E Witt (1996)Non-geometric T-duality preserving shift:(+ δ

C.Vafa, E. Witten (1996)C. Angelantonj, M. Cardella, N. Irges (2006)P. Camara, E. Dudas, T. Maillard, G.P. (2008)(

p+ δ

w + ρ(pL, pR) + (δ + ρ, δ − ρ)⇐⇒

A “rigid” shift does not allow to restore the symmetry. This is our case.

Effect: lifting the twisted states

Massless states survive only if the shift is longitudinal to the chiral “inversion”.


Free fermionic constructionsFree fermionic constructions

A simple setting to combine chiral twists and non-geometric shifts is ti t f f i i t ti

Fermionization of internal left bosonic coordinates

resorting to free fermionic constructions I. Antoniadis, C. Bachas, C. Kounnas (1987)H. Kawai, D. Lewellen, H. Tye (1987)

Fermionization of internal left bosonic coordinates

∂Xi = yi wi i = 1, . . . , 6

and similar for the right ones.

World-sheet supercurrent

G = ψμ ∂Xμ + ψi yi ωi μ = 7, 8

Specify the spin structure for each fermion


A system of twists and shifts is equivalent to a choice of fermionic sets (spin structures). In particular it is sufficient to specify a basis.

I. Antoniadis, C. Bachas, C. Kounnas (1987)

Conditions for the periodicity of the supercurrent

∀i # ψi −# yi −# wi = 0 mod 2 ;∀i # ψ # y # w 0 od ;

∀i # ψi −# yi −# wi = 0 mod 2 .

Modular invariance on the basis bα

n(bα) = 0 mod 8 ;

n(b ∩ b ) = 0 mod 4 ;n(bα∩ b

β) = 0 mod 4 ;

n(bα∩ b

β∩ b

γ) = 0 mod 2 ;

(b b b b ) 0 d 2

n = nL − nR

n(bα∩ b

β∩ b

γ∩ b

σ) = 0 mod 2 ;


Type IIB on the Maximal Torus of SO(12)Type IIB on the Maximal Torus of SO(12)Free fermionic description

F = ψ1...8 y1...6w1...6| ψ1...8 y1...6 w1...6 ,

S ψ S = ψ1...8 ,

S = ψ1...8 .ψ

0 0

Chi l hift d fl ti d t th f i i b t

Z2L σA × Z02L σB × Z2R σC × Z

02L σDThe Projection

Chiral shifts and reflections correspond to the fermionic subsets

I = ψi yi σ = yiwiIi = ψ y σ

i = y w

Ii = ψi yi σ

i = yi wi


Scan of N = 1L + 1R Models

Four additional sets must be added to the “toroidal” basis F, S, S

b (b b ) I i i ¯k k (ψ )3456 ( )i i |(˜ ˜)k k b1 = (b1L, b1R) = I3456 σ

i1i2... σk1k2... = (ψ y)3456 (y w)i1i2...|(y w)k1k2...

b2 = (b2L, b2R) = I1256 σ

j1j2... σl1l2... = (ψ y)1256 (y w)j1j2...|(y w)l1l2...

b1 = (b1R, b1L) = I3456 σ

k1k2... σi1i2... = (y w)k1k2...|(ψ y)3456(y w)i1i2...

b2 = (b2R, b2L) = I1256 σ

l1l2... σj1j2... = (y w)l1l2...|(ψ y)1256(y w)j1j2...2

(2R,2L)

1256(y ) |(ψ y) (y )

The scan is on the possible choices of compatible with supersymmetry and Modular Invariance. Not exhaustive on discrete torsion

ii, ji, ki, li

The massless spectrum

T = G + n V + n HT0 = G2

+ nvV2+ n

hH2

These are non-geometrical models. We can define “effective” Hodge numbers and Euler characteristic asand Euler characteristic as

h11 = nh − 1 h

21 = nv χ = 2(h11 − h21)


Results of the scan


An example: the minimal (1,1) model

Untwisted sector: the chiral shifts let only the supergravity-m and the universal hyper m survive:

p ( , )

universal hyper-m survive:

G2+H

2

Twisted sectors: the shift is ineffective if longitudinal to the inversion. It happens only in the sector

b3b3 = I1234I1234σ24σ24

giving

H2+V

2

The Type IIB model exhibiting the “minimal” massless spectrum content:


Comparison with other models with “few” moduli

• Non L-R Symmetric model hyper-free with N=2 SUGRA and only one vector multiplet. Y. Dolivet, B. Julia, C. Kounnas (2008)

• N=2 (1,1) Type I Model E. Kiritsis, M. Lennek, A.N. Schellekens (2008)

• N=3 models with only 1 vector (2 complex scalars + dilaton) S. Ferrara, C. Kounnas (1989)

• N=2 (2,2) and (1,3) CY models (freely-acting quotients) P. Candelas, R. Davies (2008)

• Geometric (3,3) models C. Vafa, E. Witten (1995)R. Donagi, B. Ovrut, T. Pantev, D. Waldram (2001)V Bouchard R Donagi (2007)V. Bouchard, R. Donagi (2007)P. Camara, E. Dudas, T. Maillard,G.P.(2008)R. Donagi, K. Wendland (2008)E. Kiritsis, M. Lennek, A.N. Schellekens (2008)…


The unoriented projection (Type I)

There is a natural but subtle Omega projection.

TrH ⊗H Ω¡gL ⊗ gR

¢= TrH gTrHL⊗HR

Ω¡g ⊗ g

¢= TrHL

gΩ

is now the “chiral geometric mean” of gΩ gL gR

The unoriented amplitudes correspond to the “new” sets:

b1Ω = I3456σ15 = ψ

3456 y1346w15

b = I σ = ψ1256 y46w1245b2Ω = I1256 σ1245 = ψ y w

b3Ω = I1234 σ24 = ψ

1234 y13w24


On the minimal type IIB model

Th i t t Kl i b ttl j ti li i t ll th

On the minimal type IIB model

The consistent Klein-bottle projection eliminates all the Massless Tadpoles

No (non-geometric) D-branes are needed

Result: the minimal N=1 massless content

1(T +K)l=G

1+ 2C

12(T + )

massless 1+

1


Models with open strings

The general case difficult (thousands of characters)

p g


Just an example:

at the point. SO(12)T 6 /Z2L × Z02L × Z2R × Z

02L

Two Discrete Torsion choicesTwo Discrete Torsion choices

12(T +K)

0,A = G4+ 6V

4An extension invariant

12(T +K)

0,B = G1+ 6V

1+ 25C

1A permutation invariant


The only tadpole condition is satisfied providedThe only tadpole condition is satisfied provided

n + n = 4 → (N = 4) U(n)× U(4− n) SYMn1+ n

2= 4 → (N = 4) U(n)× U(4 n) SYM

B-model: the extended symmetry on the branes is broken in the bulk

G.P., A. Sagnotti, Ya.S.Stanev, 1995


N = 1L + 0R Models

Starting point: breaking all the supersymetries associated to the right movers

L R

Starting point: breaking all the supersymetries associated to the right movers

(−)FRσR or equivalently the sets F, SAn asymmetric

NN = 4L+0R with 18 vector multiplets and SU(2)6 gauge group.The famous

L. Castellani, R. D’Auria, F. Gliozzi, S. Sciuto (1986)L Bluhm R Dolan P Goddard (1987)L. Bluhm, R. Dolan, P. Goddard (1987)H. Kawai, D. Lewellen, H. Tye (1987)L. Dixon, V. Kaplunovsky, C. Vafa (1987)

we add two more elements

b1 = I3456 σ

i1i2... σk1k2... = (ψ y)3456 (y w)i1i2...|(y w)k1k2... ,

( ) ( ) |( ) b2 = I1256 σ

j1j2... σl1l2... = (ψ y)1256 (y w)j1j2...|(y w)l1l2...


Results of the Scan

Generically T = G + n V + n V0 + n C + n C0Generically T0 = G1

+ nvV1+ n

v0V1+ n

cC1+ n

c0C1

NS-NS NS-NSR-R R-R

Generalized D-branes can couple to twisted R-R states.M. Bianchi (2008)


(nv, n0

v;n

c, n0

c) = (14, 0; 5, 0)The example

v v c c

b1 = I3456 σ12 σ45

b2 = I1256 σ36 σ5

SO(12)L× SO(20)

R →£SO(4)2 × SO(2)2

¤L×£SO(2)2 × SO(16)

¤R

SO(16)R → SO(2)× SO(14)

Little groupT0 = G

1+ 14V

1+ 5C

1

with a gauge group SU(2)4 × U(1)2


Gepner Map (in progress) M Bianchi G P M C TimirgaziuGepner Map (in progress)

F = ψ7,8ψ1...6 y1...6w1...6| ψ7,8 ψ1...6 y1...6 w1...6

M. Bianchi, G.P., M.C. Timirgaziu

FIIB = ψ

7,8ψ1...6 y1...6w1...6|ψ7,8 ψ1...6 y1...6w1...6

F ψ7 8ψ1 6 y1 6w1 6| y1 6 w1 6 η1 16φ1 16FHet = ψ

7,8ψ1...6 y1...6w1...6| y1...6w1...6, η1...16φ1...16

Identifying η1...6 → ψ1...6Identifying

It is possible to apply the same twistings as in Type IIB.

η → ψ

Gauge group: η7...16| z φ1...16| z E6

SO(10) E8

• The models are not chiral

• (1 1) corresponds to (27 27)G. Pradisi - Non-Geometrical ...

(1,1) corresponds to (27,27)

ConclusionsConclusions• We have exploited the use of (non-geometric) twists and shifts in the search of calculable Type IIB and Type I string models with few moduli

• We have found a series of

with “effective” Hodge numbers

N = 1L + 1R

(h h )with effective Hodge numbers (h11, h

21)

(n, n) n = 1, 2, 3, 4, 5, 9

(2n, 2n+ 6), (2n+ 6, 2n) n = 0, 1, 2

(2n+ 3, 2n+ 15), (2n+ 15, 2n+ 3) n = 0, 1

with (1,1) model that has no counterpart in CY compactifications.

• We have found also a series of

with relatively few moduli

N = 1L + 0R


with relatively few moduli

“Dreams”Dreams• Type IIB N=1 models with only the universal dilaton chiral multiplet.

• N=2 models with vanishing Hodge numbers

• Phenomenologically N=1 viable models with all the moduli stabilized.

Effects on open string moduli are subtler. Analyzed models arealways non-chiral!

Is there tension between CHIRALITY and MODULI STABILIZATION?

I. Antoniadis, A. Kumar, T. Maillard (2005)M Bi hi E T i (2005)M. Bianchi, E. Trevigne (2005)G. Aldazabal, P. Camara, A. Font, L. Ibanez (2005)F. Marchesano (2006)R. Blumenhagen, S. Moster, E. Plaushinn (2008)


THANK YOU!THANK YOU!


Models with open strings


p g


Just an example:

at the point. SO(12)T 6 /Z2L × Z02L × Z2R × Z

02L

The Modular Invariant can be organized in terms of 64 characters.

Tunt = |χ1|

2 + |χ5|2 + |χ

9|2 + |χ

13|2

Modular Invariant completions depend on the Discrete TorsionModular Invariant completions depend on the Discrete Torsion


Two choices

TA = |χ

1+ χ

17+ χ

35+ χ

49|2 + |χ

5+ χ

21+ χ

39+ χ

53|2

+|χ9+ χ

30+ χ

45+ χ

64|2 + |χ

13+ χ

26+ χ

41+ χ

60|2

TB = χ

1χ1+ χ

18χ2+ χ

33χ3+ χ

52χ4+ χ

5χ5+ χ

22χ6+ χ

37χ7+ χ

56χ8+ χ

9χ9

+χ29

χ10+ χ

47χ11+ χ

61χ12+ χ

13χ13+ χ

25χ14+ χ

43χ15+ χ

57χ16+ χ

17χ17

+χ2χ18+ χ

51χ19+ χ

34χ20+ χ

21χ21+ χ

6χ22+ χ

55χ23+ χ

38χ24+ χ

14χ25

+χ26

χ26+ χ

44χ27+ χ

58χ28+ χ

10χ29+ χ

30χ30+ χ

48χ31+ χ

62χ32+ χ

3χ33

+χ20

χ34+ χ

35χ35+ χ

50χ36+ χ

7χ37+ χ

24χ38+ χ

39χ39+ χ

54χ40+ χ

41χ41

+χ59

χ42+ χ

15χ43+ χ

27χ44+ χ

45χ45+ χ

63χ46+ χ

11χ47+ χ

31χ48+ χ

49χ49

+χ36

χ50+ χ

19χ51+ χ

4χ52+ χ

53χ53+ χ

40χ54+ χ

23χ55+ χ

8χ56+ χ

16χ57

+ + + + + + ++χ28

χ58+ χ

42χ59+ χ

60χ60+ χ

12χ61+ χ

32χ62+ χ

46χ63+ χ

64χ64

A: an extension invariant

B: a permutation invariantG. Pradisi - Non-Geometrical ...

B: a permutation invariant

Massless characters

χ1,χ

2,χ

3,χ

4,χ

17,χ

18,χ

23,χ

24,χ

33,χ

35,χ

38,χ

40,χ

49,χ

52,χ

54,χ

55

Unoriented projection

χ1 = V − S − C + . . . χ

i = 2O − S − C + . . .

Unoriented projection

K = χ1+ χ

17+ χ

35+ χ

49+ χ

5+ χ

21+ χ

39+ χ

53χ1

χ17

χ35

χ49

χ5

χ21

χ39

χ53

+χ9+ χ

30+ χ

45+ χ

64+ χ

13+ χ

26+ χ

41+ χ

60

Unoriented spectra

(TA)massless = |V + 6O − 4S − 4C|2

(T ) |V S C|2 + 15|2O S C|2(TB)massless = |V − S − C|2 + 15|2O − S − C|2

K = (V − S − C) + 3(2O − S − C)


12(T +K)

0,A = G4+ 6V

42 ,

12(T +K)

0,B = G1+ 6V

1+ 25C

1

Open strings2

( )( )A = (χ1+ χ

17+ χ

35+ χ

49)(2n

1n1+ 2n

2n2)

+(χ5+ χ

21+ χ

39+ χ

53)(n2

1+ n2

2+ n2

1+ n2

2)

5 21 39 53 1 2 1 2

+(χ9+ χ

30+ χ

45+ χ

64)(2n

1n2+ 2n

1n2)

+(χ + χ + χ + χ )(2n n + 2n n )+(χ13+ χ

26+ χ

41+ χ

60)(2n

1n2+ 2n

2n1)

M = (χ5+ χ

21+ χ

39+ χ

53)(n

1+ n

2+ n

1+ n

2)

Tadpole is canceled provided

n + n 4 → (N 4) U(n)× U(4 n) SYM

B-model: the extended symmetry on the branes is broken in the bulk

n1+ n

2 = 4 → (N = 4) U(n)× U(4− n) SYM


y yG.P., A. Sagnotti, Ya.S.Stanev, 1995

Chi l Ch t f 6 /0 0Chiral Characters of T 6 /Z2L × Z02L × Z2R × Z

02L

Untwisted Sectorχ1 = (O2O2O2O6 + V2V2V2V6) τ00 + (O2V2V2O6 + V2O2O2V6) τ01 + (V2O2V2O6 + O2V2O2V6) τ02 + (V2V2O2O6 + O2O2V2V6) τ03

χ2 = (O2V2V2O6 + V2O2O2V6) τ00 + (O2O2O2O6 + V2V2V2V6) τ01 + (V2V2O2O6 + O2O2V2V6) τ02 + (V2O2V2O6 +O2V2O2V6) τ03

(V O V O +O V O V ) + (V V O O + O O V V ) + (O O O O + V V V V ) + (O V V O + V O O V )χ3 = (V2O2V2O6 +O2V2O2V6) τ00 + (V2V2O2O6 + O2O2V2V6) τ01 + (O2O2O2O6 + V2V2V2V6) τ02 + (O2V2V2O6 + V2O2O2V6) τ03

χ4 = (V2V2O2O6 +O2O2V2V6) τ00 + (V2O2V2O6 + O2V2O2V6) τ01 + (O2V2V2O6 + V2O2O2V6) τ02 + (O2O2O2O6 + V2V2V2V6) τ03

χ5 = (V2V2V2O6 + O2O2O2V6) τ00 + (V2O2O2O6 + O2V2V2V6) τ01 + (O2V2O2O6 + V2O2V2V6) τ02 + (O2O2V2O6 + V2V2O2V6) τ03

χ6 = (V2O2O2O6 +O2V2V2V6) τ00 + (V2V2V2O6 + O2O2O2V6) τ01 + (O2O2V2O6 + V2V2O2V6) τ02 + (O2V2O2O6 + V2O2V2V6) τ03

χ7 = (O2V2O2O6 + V2O2V2V6) τ00 + (O2O2V2O6 + V2V2O2V6) τ01 + (V2V2V2O6 + O2O2O2V6) τ02 + (V2O2O2O6 + O2V2V2V6) τ03

χ8 = (O2O2V2O6 + V2V2O2V6) τ00 + (O2V2O2O6 + V2O2V2V6) τ01 + (V2O2O2O6 + O2V2V2V6) τ02 + (V2V2V2O6 +O2O2O2V6) τ03

χ9 = (C2C2C2C6 + S2S2S2S6) τ00 + (C2S2S2C6 + S2C2C2S6) τ01 + (S2C2S2C6 + C2S2C2S6) τ02 + (S2S2C2C6 + C2C2S2S6) τ03

χ10 = (C2S2S2C6 + S2C2C2S6) τ00 + (C2C2C2C6 + S2S2S2S6) τ01 + (S2S2C2C6 + C2C2S2S6) τ02 + (S2C2S2C6 + C2S2C2S6) τ03χ10 ( 2 2 2 6 2 2 2 6) 00 ( 2 2 2 6 2 2 2 6) 01 ( 2 2 2 6 2 2 2 6) 02 ( 2 2 2 6 2 2 2 6) 03

χ11 = (S2C2S2C6 + C2S2C2S6) τ00 + (S2S2C2C6 + C2C2S2S6) τ01 + (C2C2C2C6 + S2S2S2S6) τ02 + (C2S2S2C6 + S2C2C2S6) τ03

χ12 = (S2S2C2C6 + C2C2S2S6) τ00 + (S2C2S2C6 + C2S2C2S6) τ01 + (C2S2S2C6 + S2C2C2S6) τ02 + (C2C2C2C6 + S2S2S2S6) τ03

χ13 = (S2S2S2C6 + C2C2C2S6) τ00 + (S2C2C2C6 + C2S2S2S6) τ01 + (C2S2C2C6 + S2C2S2S6) τ02 + (C2C2S2C6 + S2S2C2S6) τ03

(S C C C + C S S S ) + (S S S C + C C C S ) + (C C S C + S S C S ) + (C S C C + S C S S )χ14 = (S2C2C2C6 + C2S2S2S6) τ00 + (S2S2S2C6 + C2C2C2S6) τ01 + (C2C2S2C6 + S2S2C2S6) τ02 + (C2S2C2C6 + S2C2S2S6) τ03

χ15 = (C2S2C2C6 + S2C2S2S6) τ00 + (C2C2S2C6 + S2S2C2S6) τ01 + (S2S2S2C6 + C2C2C2S6) τ02 + (S2C2C2C6 + C2S2S2S6) τ03

χ16 = (C2C2S2C6 + S2S2C2S6) τ00 + (C2S2C2C6 + S2C2S2S6) τ01 + (S2C2C2C6 + C2S2S2S6) τ02 + (S2S2S2C6 + C2C2C2S6) τ03


1 t isted Sector1-twisted Sector

χ17 = (V2S2S2O6 +O2C2C2V6) τ10 + (V2C2C2O6 + O2S2S2V6) τ11 + (O2S2C2O6 + V2C2S2V6) τ12 + (O2C2S2O6 + V2S2C2V6) τ13

(V C C O +O S S V ) + (V S S O +O C C V ) + (O C S O + V S C V ) + (O S C O + V C S V )χ18 = (V2C2C2O6 +O2S2S2V6) τ10 + (V2S2S2O6 +O2C2C2V6) τ11 + (O2C2S2O6 + V2S2C2V6) τ12 + (O2S2C2O6 + V2C2S2V6) τ13

χ19 = (O2S2C2O6 + V2C2S2V6) τ10 + (O2C2S2O6 + V2S2C2V6) τ11 + (V2S2S2O6 +O2C2C2V6) τ12 + (V2C2C2O6 +O2S2S2V6) τ13

χ20 = (O2C2S2O6 + V2S2C2V6) τ10 + (O2S2C2O6 + V2C2S2V6) τ11 + (V2C2C2O6 +O2S2S2V6) τ12 + (V2S2S2O6 +O2C2C2V6) τ13

χ21 = (O2C2C2O6 + V2S2S2V6) τ10 + (O2S2S2O6 + V2C2C2V6) τ11 + (V2C2S2O6 + O2S2C2V6) τ12 + (V2S2C2O6 +O2C2S2V6) τ13

χ22 = (O2S2S2O6 + V2C2C2V6) τ10 + (O2C2C2O6 + V2S2S2V6) τ11 + (V2S2C2O6 + O2C2S2V6) τ12 + (V2C2S2O6 +O2S2C2V6) τ13

χ23 = (V2C2S2O6 +O2S2C2V6) τ10 + (V2S2C2O6 +O2C2S2V6) τ11 + (O2C2C2O6 + V2S2S2V6) τ12 + (O2S2S2O6 + V2C2C2V6) τ13

χ24 = (V2S2C2O6 +O2C2S2V6) τ10 + (V2C2S2O6 +O2S2C2V6) τ11 + (O2S2S2O6 + V2C2C2V6) τ12 + (O2C2C2O6 + V2S2S2V6) τ13

χ25 = (C2O2O2C6 + S2V2V2S6) τ10 + (C2V2V2C6 + S2O2O2S6) τ11 + (S2O2V2C6 + C2V2O2S6) τ12 + (S2V2O2C6 + C2O2V2S6) τ13

χ26 = (C2V2V2C6 + S2O2O2S6) τ10 + (C2O2O2C6 + S2V2V2S6) τ11 + (S2V2O2C6 + C2O2V2S6) τ12 + (S2O2V2C6 + C2V2O2S6) τ13

χ27 = (S2O2V2C6 + C2V2O2S6) τ10 + (S2V2O2C6 + C2O2V2S6) τ11 + (C2O2O2C6 + S2V2V2S6) τ12 + (C2V2V2C6 + S2O2O2S6) τ13

χ28 = (S2V2O2C6 + C2O2V2S6) τ10 + (S2O2V2C6 + C2V2O2S6) τ11 + (C2V2V2C6 + S2O2O2S6) τ12 + (C2O2O2C6 + S2V2V2S6) τ13

χ29 = (S2V2V2C6 + C2O2O2S6) τ10 + (S2O2O2C6 + C2V2V2S6) τ11 + (C2V2O2C6 + S2O2V2S6) τ12 + (C2O2V2C6 + S2V2O2S6) τ13χ29 = (S2V2V2C6 + C2O2O2S6) τ10 + (S2O2O2C6 + C2V2V2S6) τ11 + (C2V2O2C6 + S2O2V2S6) τ12 + (C2O2V2C6 + S2V2O2S6) τ13

χ30 = (S2O2O2C6 + C2V2V2S6) τ10 + (S2V2V2C6 + C2O2O2S6) τ11 + (C2O2V2C6 + S2V2O2S6) τ12 + (C2V2O2C6 + S2O2V2S6) τ13

χ31 = (C2V2O2C6 + S2O2V2S6) τ10 + (C2O2V2C6 + S2V2O2S6) τ11 + (S2V2V2C6 + C2O2O2S6) τ12 + (S2O2O2C6 + C2V2V2S6) τ13

χ32 = (C2O2V2C6 + S2V2O2S6) τ10 + (C2V2O2C6 + S2O2V2S6) τ11 + (S2O2O2C6 + C2V2V2S6) τ12 + (S2V2V2C6 + C2O2O2S6) τ13



χ33 = (S2V2S2O6 + C2O2C2V6) τ20 + (S2O2C2O6 + C2V2S2V6) τ21 + (C2V2C2O6 + S2O2S2V6) τ22 + (C2O2S2O6 + S2V2C2V6) τ23

χ34 = (S2O2C2O6 + C2V2S2V6) τ20 + (S2V2S2O6 + C2O2C2V6) τ21 + (C2O2S2O6 + S2V2C2V6) τ22 + (C2V2C2O6 + S2O2S2V6) τ23

χ35 = (C2V2C2O6 + S2O2S2V6) τ20 + (C2O2S2O6 + S2V2C2V6) τ21 + (S2V2S2O6 + C2O2C2V6) τ22 + (S2O2C2O6 + C2V2S2V6) τ23

χ36 = (C2O2S2O6 + S2V2C2V6) τ20 + (C2V2C2O6 + S2O2S2V6) τ21 + (S2O2C2O6 + C2V2S2V6) τ22 + (S2V2S2O6 + C2O2C2V6) τ23

χ37 = (C2O2C2O6 + S2V2S2V6) τ20 + (C2V2S2O6 + S2O2C2V6) τ21 + (S2O2S2O6 +C2V2C2V6) τ22 + (S2V2C2O6 + C2O2S2V6) τ23

χ38 = (C2V2S2O6 + S2O2C2V6) τ20 + (C2O2C2O6 + S2V2S2V6) τ21 + (S2V2C2O6 +C2O2S2V6) τ22 + (S2O2S2O6 + C2V2C2V6) τ23

χ39 = (S2O2S2O6 + C2V2C2V6) τ20 + (S2V2C2O6 + C2O2S2V6) τ21 + (C2O2C2O6 + S2V2S2V6) τ22 + (C2V2S2O6 + S2O2C2V6) τ23

χ40 = (S2V2C2O6 + C2O2S2V6) τ20 + (S2O2S2O6 + C2V2C2V6) τ21 + (C2V2S2O6 + S2O2C2V6) τ22 + (C2O2C2O6 + S2V2S2V6) τ23

χ41 = (O2C2O2C6 + V2S2V2S6) τ20 + (O2S2V2C6 + V2C2O2S6) τ21 + (V2C2V2C6 +O2S2O2S6) τ22 + (V2S2O2C6 +O2C2V2S6) τ23χ41 = (O2C2O2C6 + V2S2V2S6) τ20 + (O2S2V2C6 + V2C2O2S6) τ21 + (V2C2V2C6 +O2S2O2S6) τ22 + (V2S2O2C6 +O2C2V2S6) τ23

χ42 = (O2S2V2C6 + V2C2O2S6) τ20 + (O2C2O2C6 + V2S2V2S6) τ21 + (V2S2O2C6 +O2C2V2S6) τ22 + (V2C2V2C6 +O2S2O2S6) τ23

χ43 = (V2C2V2C6 +O2S2O2S6) τ20 + (V2S2O2C6 +O2C2V2S6) τ21 + (O2C2O2C6 + V2S2V2S6) τ22 + (O2S2V2C6 + V2C2O2S6) τ23

χ44 = (V2S2O2C6 +O2C2V2S6) τ20 + (V2C2V2C6 +O2S2O2S6) τ21 + (O2S2V2C6 + V2C2O2S6) τ22 + (O2C2O2C6 + V2S2V2S6) τ23

(V S V C +O C O S ) + (V C O C +O S V S ) + (O S O C + V C V S ) + (O C V C + V S O S )χ45 = (V2S2V2C6 +O2C2O2S6) τ20 + (V2C2O2C6 +O2S2V2S6) τ21 + (O2S2O2C6 + V2C2V2S6) τ22 + (O2C2V2C6 + V2S2O2S6) τ23

χ46 = (V2C2O2C6 +O2S2V2S6) τ20 + (V2S2V2C6 +O2C2O2S6) τ21 + (O2C2V2C6 + V2S2O2S6) τ22 + (O2S2O2C6 + V2C2V2S6) τ23

χ47 = (O2S2O2C6 + V2C2V2S6) τ20 + (O2C2V2C6 + V2S2O2S6) τ21 + (V2S2V2C6 +O2C2O2S6) τ22 + (V2C2O2C6 +O2S2V2S6) τ23

χ48 = (O2C2V2C6 + V2S2O2S6) τ20 + (O2S2O2C6 + V2C2V2S6) τ21 + (V2C2O2C6 +O2S2V2S6) τ22 + (V2S2V2C6 +O2C2O2S6) τ23



χ49 = (S2S2V2O6 +C2C2O2V6) τ30 + (S2C2O2O6 +C2S2V2V6) τ31 + (C2S2O2O6 + S2C2V2V6) τ32 + (C2C2V2O6 + S2S2O2V6) τ3349 ( 2 2 2 6 2 2 2 6) 30 ( 2 2 2 6 2 2 2 6) 31 ( 2 2 2 6 2 2 2 6) 32 ( 2 2 2 6 2 2 2 6) 33

χ50 = (S2C2O2O6 +C2S2V2V6) τ30 + (S2S2V2O6 +C2C2O2V6) τ31 + (C2C2V2O6 + S2S2O2V6) τ32 + (C2S2O2O6 + S2C2V2V6) τ33

χ51 = (C2S2O2O6 + S2C2V2V6) τ30 + (C2C2V2O6 + S2S2O2V6) τ31 + (S2S2V2O6 +C2C2O2V6) τ32 + (S2C2O2O6 +C2S2V2V6) τ33

χ52 = (C2C2V2O6 + S2S2O2V6) τ30 + (C2S2O2O6 + S2C2V2V6) τ31 + (S2C2O2O6 +C2S2V2V6) τ32 + (S2S2V2O6 +C2C2O2V6) τ33

χ53 = (C2C2O2O6 + S2S2V2V6) τ30 + (C2S2V2O6 + S2C2O2V6) τ31 + (S2C2V2O6 +C2S2O2V6) τ32 + (S2S2O2O6 +C2C2V2V6) τ33χ53 = (C2C2O2O6 + S2S2V2V6) τ30 + (C2S2V2O6 + S2C2O2V6) τ31 + (S2C2V2O6 +C2S2O2V6) τ32 + (S2S2O2O6 +C2C2V2V6) τ33

χ54 = (C2S2V2O6 + S2C2O2V6) τ30 + (C2C2O2O6 + S2S2V2V6) τ31 + (S2S2O2O6 +C2C2V2V6) τ32 + (S2C2V2O6 +C2S2O2V6) τ33

χ55 = (S2C2V2O6 +C2S2O2V6) τ30 + (S2S2O2O6 +C2C2V2V6) τ31 + (C2C2O2O6 + S2S2V2V6) τ32 + (C2S2V2O6 + S2C2O2V6) τ33

χ56 = (S2S2O2O6 +C2C2V2V6) τ30 + (S2C2V2O6 +C2S2O2V6) τ31 + (C2S2V2O6 + S2C2O2V6) τ32 + (C2C2O2O6 + S2S2V2V6) τ33

(O O C C + V V S S ) + (O V S C + V O C S ) + (V O S C +O V C S ) + (V V C C +O O S S )χ57 = (O2O2C2C6 + V2V2S2S6) τ30 + (O2V2S2C6 + V2O2C2S6) τ31 + (V2O2S2C6 +O2V2C2S6) τ32 + (V2V2C2C6 +O2O2S2S6) τ33

χ58 = (O2V2S2C6 + V2O2C2S6) τ30 + (O2O2C2C6 + V2V2S2S6) τ31 + (V2V2C2C6 +O2O2S2S6) τ32 + (V2O2S2C6 +O2V2C2S6) τ33

χ59 = (V2O2S2C6 +O2V2C2S6) τ30 + (V2V2C2C6 +O2O2S2S6) τ31 + (O2O2C2C6 + V2V2S2S6) τ32 + (O2V2S2C6 + V2O2C2S6) τ33

χ60 = (V2V2C2C6 +O2O2S2S6) τ30 + (V2O2S2C6 +O2V2C2S6) τ31 + (O2V2S2C6 + V2O2C2S6) τ32 + (O2O2C2C6 + V2V2S2S6) τ33

χ61 = (V2V2S2C6 +O2O2C2S6) τ30 + (V2O2C2C6 +O2V2S2S6) τ31 + (O2V2C2C6 + V2O2S2S6) τ32 + (O2O2S2C6 + V2V2C2S6) τ33

χ62 = (V2O2C2C6 +O2V2S2S6) τ30 + (V2V2S2C6 +O2O2C2S6) τ31 + (O2O2S2C6 + V2V2C2S6) τ32 + (O2V2C2C6 + V2O2S2S6) τ33

χ63 = (O2V2C2C6 + V2O2S2S6) τ30 + (O2O2S2C6 + V2V2C2S6) τ31 + (V2V2S2C6 +O2O2C2S6) τ32 + (V2O2C2C6 +O2V2S2S6) τ33

χ64 = (O2O2S2C6 + V2V2C2S6) τ30 + (O2V2C2C6 + V2O2S2S6) τ31 + (V2O2C2C6 +O2V2S2S6) τ32 + (V2V2S2C6 +O2O2C2S6) τ33


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Ui it diR “T V t” dUniversita di Roma “Tor Vergata” …statistics.roma2.infn.it/.../members/Pradisi/MG12PRADISI.pdfTools: (non-geometric) twists and shifts Asymmetric orbifolds