1. Asymmetric Orbifold toward Heterotic Anomalous U(1) GUT
Toshifumi Yamashita　
(Nagoya University)
09 Nov. mini workshop
working with
　　 Nagoya U : Maekawa, Moriyama, Teraguchi
Takei, Ito, Kuwakino
　　 NCTU (Taiwan) :　Keijiro Takahashi

2. Introduction anomalous U(1) GUT GUTs from the Heterotic string.
N. Maekawa & T.Y. & ・・・ (’01-’04)
an interesting SUSY-GUT e.g. E6*SU(2)H
DTS, PD, Yukawa hierarchy, SUSY flavor etc.
anomaly : assumed to be cancelled by GS.
to be checked
We aim to embed it into string.
Asymmetric
orbifold
GUTs from the Heterotic string.
examined in '80 exhaustively
simple cpt.
no adjoint Higgs
Kac-Moody level

3. Plan Introduction anomalous U(1) GUT overview
Narain cpt. & Modular inv. & orbifold
Models
Summary

4. Grand Unified Theories
Unification of Gauge Group
: non-Abelian
Charge Quantization
Unification of Matter
SU(5)
SO(10) E6 + + = + + =
wrong GUT relation

5. Grand Unified Theories
sector in E6 model
E-twisting
M.Bando and T.Kugo
mass
massless :
all come from and
mild hierarchy in sector !!
large mixing

6. Grand Unified Theories
Unification of Gauge Group
: non-Abelian
Charge Quantization
How about Higgs ?
Proton decay
Doublet-Triplet (DT) Splitting Problem

7. SUSY-GUT fascinating extension of SM hierarchal Yukawa?
unifications of forces and of matter
charge quantization
stabilization of the weak scale
gauge coupling unification (GCU)
Higgs?
SUSY, (FCNC)?
Proton decay?

8. Anomalous U(1) gauge symmetry
Low energy theory of string theory
Anomaly Green-Schwartz mechanism
Froggatt-Nielsen (FN) mechanism
: FN field
: cutoff scale
ex)

9. Anomalous U(1) gauge symmetry
Low energy theory of string theory
Anomaly Green-Schwartz mechanism
Froggatt-Nielsen (FN) mechanism
SUSY-zero (holomorphic-zero) mechanism
superpotential
: holomorphic
does not appear in
Only positively charged operators can appear.
This strongly constrain !!

10. Anomalous U(1) gauge symmetry
Low energy theory of string theory
Anomaly Green-Schwartz mechanism
Froggatt-Nielsen (FN) mechanism
SUSY-zero (holomorphic-zero) mechanism
Ex)
Adjoints of SO(10) :
Only the linear terms in A’ are relevant for EOM.

11. Anomalous U(1) gauge symmetry
Low energy theory of string theory
Anomaly Green-Schwartz mechanism
Froggatt-Nielsen (FN) mechanism
SUSY-zero (holomorphic-zero) mechanism
Ex)
Adjoints of SO(10) :
The # of non-zero xi labels the vacuum
N=3 : Dimopoulos-Wilczek VEV
DTS

12. Anomalous U(1) GUT SUSY-GUT w/ anomalous U(1) symmetry Ansatz
N.Maekawa & T.Y. & …
SUSY-GUT w/ anomalous U(1) symmetry
Ansatz
“generic” interaction
introduce all possible interactions (including NROs)
assume all the coefficients are O(1)
vacuum
(cutoff scale)
: GUT singlet operator, : U(1)A charge
: breaking scale of U(1)A
charge assignment ＝ def. of model

13. Anomalous U(1) GUT Generic interaction GCU FN mechanism
N.Maekawa & T.Y. & …
Generic interaction
FN mechanism
SUSY-zero mechanism
Natural Gauge Coupling Unification
hierarchal Fermion Yukawa
DW type of adjoint VEV
DT Splitting
proton decay
GCU

14. Anomalous U(1) GUT fascinating extension of SM
N.Maekawa & T.Y. & …
hierarchal Yukawa?
fascinating extension of SM
unifications of forces and of matter
charge quantization
stabilization of the weak scale
gauge coupling unification (GCU)
Higgs?
SUSY, (FCNC)?
horizontal symm.
Proton decay?
Almost all the problem can be solved!!

15. Anomalous U(1) GUT examples of E6 model two charged adjoint
N.Maekawa & T.Y. & …
examples of E6 model
PTP 107, 1201 (2002)
N.Maekawa and T.Y.
E6 * Z2
PLB561, 273(2003)
N.Maekawa
two charged adjoint
6 – 3 or 5 – 2 generations
E6 * SU(2)

16. Plan Introduction anomalous U(1) GUT overview
Narain cpt. & Modular inv. & orbifold
Models
Summary

17. related to doubled geometry?
overview
anomalous U(1) GUT
-- E6 model
charged adjoint Higgs
higher Kac-Moody level
Het, M, F, …
related to doubled geometry?
diagonal embedding
asymmetric orbifold
-- Narain cpt.
asymmetric treatment of
left- and right-moving string
need careful consistency check

18. overview Kac-Moody algebra Diagonal embedding : structure constant
ex) mode exp. of world sheet current
Diagonal embedding w/

19. overview # of generation model cpt. of or SO(32) Hetero w/ Wilson line
e.g. J.Erler (1996)
(in known models)
Z.Kakushadze & H.Tye (1996)
model
cpt. of or SO(32) Hetero w/ Wilson line
Z.Kakushadze & H.Tye (1996)
claimed
w/ adjoint Higgs, 3 generations
& non-Abelian Hidden gauge
Only one model is possible.
Narain compactification
(flat instead of CY, for simplicity)

20. Plan Introduction anomalous U(1) GUT overview
Narain cpt. & Modular inv. & orbifold
Models
Summary

21. Narain cpt. & Modular inv. & orbifold
string w/ compactification

22. Narain cpt. & Modular inv. & orbifold
Heterotic string
from
modular inv.
left
right 4D 6D
16D
"compactified"
on or Spin(32)/Z2 Lie lattice
here?
no geometrical interpretation
Narain compactification
discard geometric interpretation also in cpt. 6D
gauge symm w/ rank 22 is possible : ex)

23. Narain cpt. & Modular inv. & orbifold
Modular invariance
red blue
shift
The modular transformations,
closed string 1-loop amp.
do not change the torus.
The amplitude should be
invariant under these tr..

24. Narain cpt. & Modular inv. & orbifold
partition function
: NS
: R

25. Narain cpt. & Modular inv. & orbifold
partition function
even T S
self-dual
(or )

26. Narain cpt. & Modular inv. & orbifold
geometric compactification
ex.) A2 lattice
winding
momentum
This momentum lattice is
actually even & self-dual.
proof

27. Narain cpt. & Modular inv. & orbifold
Narain lattice
Cf.) geometric
: simple roots of a
(simply laced)
Lie group.
winding
momentum
R= /2 ( ; 0)
R= 2-1/2
for pR=0
: Cartan matrix
This lattice leads to
gauge symmetry.
This does not,
except for SU(2).

28. Narain cpt. & Modular inv. & orbifold
ESDL generating technique
left
right

29. Narain cpt. & Modular inv. & orbifold
ESDL generating technique
more
left
right
left
right

30. Narain cpt. & Modular inv. & orbifold
ZN for simplicity
diagonal embedding
4D N =1 SUSY
Manifold / discrete symm.
impose identity under (discrete) orbifold action
twist (rotation)
shift
(reflection)
ex.) A2 lattice
: Z2,3,6
1/3 twist
: does not change the lattice

31. Narain cpt. & Modular inv. & orbifold
ZN for simplicity
diagonal embedding
4D N =1 SUSY
Manifold / discrete symm.
impose identity under (discrete) orbifold action
twist (rotation)
shift
(reflection)
ex.) A2 lattice
: Z2,3,6
additional string states
: does not change the lattice

32. Narain cpt. & Modular inv. & orbifold
partition function
additional string state w/ different B.C. S
Modular tr. S
S2=1 T

33. Narain cpt. & Modular inv. & orbifold
partition function
ex) Z3
T :
S : S
Modular tr. S
S2=1 T

34. Narain cpt. & Modular inv. & orbifold
partition function
ex) Z3
T :
S :
Rules on are established for symm. orbifold.
Modular inv.
P.F. is inv. if each sector satisfies the tr. property.
For asymm. orbifold, not yet.
Z.Kakushadze
& H.Tye (1996)

35. Narain cpt. & Modular inv. & orbifold
partition function
ex) Z3
T :
S :
Modular inv.
We make and then define by S, T.

36. Narain cpt. & Modular inv. & orbifold
partition function
ex) Z3
T :
S :
Modular inv.
The remaining S, T give non-trivial conditions.

37. Plan Introduction anomalous U(1) GUT overview
Narain cpt. & Modular inv. & orbifold
Models
Summary

38. Models 4D SUSY 4D left right 4D N =1 SUSY zero modes SO(8)LC repr.
phase
even

39. combined with phaseless RM to form gauge field.
Models
diagonal embedding
: Z3 4D
left
E E E6
combined with phaseless RM to form gauge field.
right
level 3 E6

40. Models lattice for right-mover E6 A2 A2 A2 D4 A2 4D left right
4D N =1 SUSY
: Z3 E6
A2 A2 A2
D4 A2

41. Models possible models A2 A2 E6 E6 E6 E6 A2 A2 E6 E6 E6 A2 A2 A2 D4
Z.Kakushadze
& H.Tye (1996)
left
A2 A2
E E E6
right E6
1 neutral adj.
5 – 2 generations Z6
A2 A2
E E E6
A2 A2 A2
Z6 : vector-like
Z3 : 9 generations D4
E E E6
D A2

42. Summary We are working on Heterotic GUTs
to derive Anomalous U(1) GUT from string.
to get 4D N =1 SUSY E6 model
w/ adjoint Higgs & 3 generations
E63 lattice via Narain compactification
diagonal embedding in asymmetric orbifold
Future works
We want two charged Adjoint Higgs
towards Anomalous U(1) GUT

43. Back-up Slides

44. Narain cpt. & Modular inv. & orbifold
ESDL generating technique
(248)
(78,1)+(1,8)+[(27,3)+c.c.]
(78;1)+(1;78)+[(27;27)+c.c.]
left
(78;1,1,1)+(1;8,1,1)+[(1;3,3,3)+c.c.]
+[(27;3,3,1)+c.c.]
right
(78,1,1;1)+(1,1,1;8)+[(1,27,27;3)+c.c.]
+[(27,27,27;1)+(27,1,27;3)+(27,27,1;3)
+(27,27,27;1)+(27,1,27;3)+(27,27,1;3)]
left
right

45. Narain cpt. & Modular inv. & orbifold
ESDL generating technique
(248)
(78,1)+(1,8)+[(27,3)+c.c.]
(78;1)+(1;78)+[(27;27)+c.c.]
left
(78;1,1,1)+(1;8,1,1)+[(1;3,3,3)+c.c.]
+[(27;3,3,1)+c.c.]
right
(78,1,1;1)+(1,1,1;8)+[(1,27,27;3)+c.c.]
+[(27,27,27;1)+(27,1,27;3)+(27,27,1;3)
+(27,27,27;1)+(27,1,27;3)+(27,27,1;3)]
left
right