Wilson Line Induced 3-Form Fluxes
in Heterotic CY Compactifications
Marco Zagermann
(Leibniz Universität Hannover)
Berlin, August 28, 2014
Samstag, 30. August 2014
Based on:
Apruzzi, Gautason, Parameswaran, MZ (2014)
(to appear)
Samstag, 30. August 2014
Outline
1. Introduction
2. Complete intersection Calabi-Yaus (CICYs)
3. Computing HWL on CICYs
4. Explicit examples
5. Conclusions
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1. Introduction
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A particularly fruitful and popular region
of the “string theory landscape”:
Heterotic Calabi-Yau compactifications
• Grand Unification naturally built in
• Technically very “clean”:
- No D-branes and O-planes
- No RR sector
- Lowest α‘ corrections simple and well known
- Powerful tools from algebraic geometry
Samstag, 30. August 2014
Almost perfect realizations of (MS)SM sector
Gauge group breaking:
E.g. Braun,He, Ovrut, Pantev (2005)
Bouchard, Donagi (2005)
Anderson, Gray, He, Lukas (2009)
Anderson, Gray, Lukas, Palti (2012)
E8 × E8
Vector bundle (nonstandard Emb.)
i¯j
Fij = F¯i¯j = 0, g Fi¯j = 0
GHidden × GGUT
Wilson lines
� � �
F = 0, P exp i γ A �= 0
G�Hidden × GSM
Samstag, 30. August 2014
More difficult:
Moduli stabilization
E.g. de Alwis, Cicoli,
Westphal (2013)
• No RR-fluxes, at most H-flux
Wflux =
�
Ω3 ∧ H
Can fix at most complex structure moduli
• Dilaton stabilization via gaugino condensation
Wg.c. ∼ e−aS
• For integer quantized H (i.e.
gen.
�Wflux � = O(1)
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1
4π 2 α�
�
Σ3
H = n ):
DS W = 0 stabilizes S at
Rohm, Seiberg,
strong coupling! Dine,Witten
(1985)
Some suggested alternatives:
( 1 ) Use non-CY backgrounds with dJ2 �= 0
Wtree =
�
Ω3 ∧ (H +
i
dJ2 )
2
Cardoso, Curio, Dall‘Agata, Lüst (2003)
Problem: Model building not well understood
( 2 ) World sheet instantons
Curio, Krause, Lüst (2005)
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( 3 ) Use gauge field background:
α�
(ω
3YM
4
H = dB −
− ω3L )
�
�
ω3YM ≡ tr A ∧ F − 13 A ∧ A ∧ A
(analogous for ω3L )
Two candidates:
E8 × E8
Hol. vector bundle
(non-standard Emb.)
(a)
GHidden × GGUT
Wilson lines
�
GHidden
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× GSM
(b)
( a ) Use holomorphic vector bundle with NSE
Anderson, Gray, Lukas, Ovrut (2010, 2011,2013)
�Wflux � = 0
Generically not all c.s. moduli may be stabilized
(see also de Alwis, Cicoli, Westphal (2013))
Samstag, 30. August 2014
( b ) Use Wilson lines: H = H
WL
=
α� WL
− 4 ω3YM
For some Σ3 one may have non-vanishing CS-invariants:
1
4π 2 α�
�
Σ3
H
WL
=
1
− 16π2
�
WL
ω
Σ3 3YM
=O
� �
1
p
(p = order of fund. group corresponding to the WL)
H
WL
is fractionally quantized (and a top. invariant)
�Wflux � = O(0.1 . . . 0.01) possible (for mod. large p)
May ameliorate strong coupling problem of
Gukov, Kachru, Liu,
non-pert. dilaton stabilization
Mc Allister (2003)
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Possible issues and side effects of fractional H-flux:
• Generically high scale SUSY breaking:
M3/2 ∼ MGUT
de Alwis, Cicoli, Westphal (2013)
• Consistency with assumed CY-background unclear
Dine, Rohm, Seiberg, Witten (1985)
Lopes Cardoso, Curio, Dall‘Agata, Lüst (2003)
Frey, Lippert (2005)
Chatzistavrakidis, Lechtenfeld, Popov (2012)
• Possible global 2D worldsheet anomalies
Witten (1985)
Gukov, Kachru, Liu, McAllister (2003)
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Presence of HWL can have important consequences!
Important point:
For a given CY and a given set of WL‘s, H
be chosen, but is completely determined!
WL
cannot
Need to develop tools for computing HWL for a
given CY and WL background
→ Topic of this talk
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We focus on a simple class of CY spaces that are also
particularly interesting for SM phenomenology:
Complete intersection Calabi-Yau spaces
(CICY‘s)
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2. Complete intersection CY‘s (CICY‘s)
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CICYs:
Common zero loci of a set of homogeneous
polynomials in products of projective spaces
CP
CY
P1 (z) = 0
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P2 (z) = 0
n
CICYs:
Common zero loci of a set of homogeneous
polynomials in products of projective spaces
Simplest example: Fermat quintic
�
X1,101 = z ∈ CP4 |
(h1,1 , h1,2 ) = (1, 101)
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�
5
z
i=1 i = 0
�5
General form described by a configuration matrix
n1
CP
CP n2
..
.
m11
m21
..
.
m12
m22
···
···
..
.
CP
nk
mk1
mk2
···
m1l
m2l
mkl
i.e., l polynomials Pi (i = 1, . . . l) in the ambient space
CPn1 × . . . × CPnk
with mji powers of the coordinates of CPnj
Ricci-flatness (i.e. CY-ness)
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�l
i=1
mji = nj + 1
CP n1
CP n2
..
.
m11
m21
..
.
m12
m22
CP nk
mk1
mk2
···
···
..
.
···
m1l
m2l
mkl
�l
i=1
mji = nj + 1
Examples:
( 1 ) Quintic:
X1,101
�l
�
i=1
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�
�
�5
4
= z ∈ CP | P1 (z) = i=1 z5i = 0
�
CP4 |5
m1i = 5 = 4 + 1 = n1 + 1
CP n1
CP n2
..
.
m11
m21
..
.
m12
m22
CP nk
mk1
mk2
···
···
..
.
m1l
m2l
mkl
···
�l
i=1
mji = nj + 1
Examples:
( 2 ) Split bicubic:
X
19,19
�
�
= (t, ζ, η) ∈ CP × CP × CP | P1 (t, ζ) = P2 (t, η) = 0
P1 (t, ζ) =
1
t1 (ζ13
+
ζ23
+
2
ζ33 )
2
+ t2 ζ 1 ζ 2 ζ 3
P2 (t, η) = t2 (η13 + η23 + η33 ) + t1 η1 η2 η3
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CP 1
CP 2
CP 2
1
3
0
1
0
3
Wilson lines
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Wilson lines
= a group homomorphism:
WL : π1 (CY) → G,
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� � �
γ→
� P exp i γ A
Wilson lines
CICY‘s are simply connected! π1 (CY) = {1}
→ No Wilson lines possible
Samstag, 30. August 2014
Wilson lines
CICY‘s are simply connected! π1 (CY) = {1}
→ No Wilson lines possible
Mod out by freely acting discrete isometry group Γ
CY/Γ → π1 (CY/Γ) = Γ
Wilson lines possible on CY/Γ
Samstag, 30. August 2014
Wilson lines
CICY‘s are simply connected! π1 (CY) = {1}
→ No Wilson lines possible
Mod out by freely acting discrete isometry group Γ
CY/Γ → π1 (CY/Γ) = Γ
Wilson lines possible on CY/Γ
Nice byproduct: CY/Γ has much smaller Hodge numbers
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Typical form of discrete symmetries Γ
Γ=R×S∼
= Zn+1 × Zn+1
Action of R and S on CPn generated by
R
gR : za �→ ω a za ,
S
gS : za �→ za+1
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ω=e
2πi
n+1
(Rotations)
(Cyclic permutations)
Examples:
( 1 ) Quintic:
Γ = R × S = Z5 × Z5
X1,101 /Γ = X1,5 with π1 (X1,5 ) = Γ = Z5 × Z5
( 2 ) Split bicubic:
X
Samstag, 30. August 2014
19,19
/Γ = X
Γ = R × S = Z3 × Z3
3,3
3,3
π
(X
) = Γ = Z3 × Z3
with 1
3. Computing HWL on CICYs
Samstag, 30. August 2014
For HWL �= 0 there has to be at least
one 3-cycle Σ3 with
�
�
α
WL
H
=
−
Σ3
4
�
WL
ω
=
�
0
3YM
Σ3
Our approach:
Compute CS-invariants on a well-understood class of
explicitly known 3D submanifolds of the CICYs that
exhaust the 3rd homology group
→ Special Lagrangian submanifolds (sLags) based
on antiholomorphic isometric involutions
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Special Lagrangian submanifolds (sLags)
= real 3D submanifolds, Σ, of a CY manifold with
J2 |Σ = 0,
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iθ
2
Im(e Ω3 )|Σ = 0
Special Lagrangian submanifolds (sLags)
= real 3D submanifolds, Σ, of a CY manifold with
J2 |Σ = 0,
iθ
2
Im(e Ω3 )|Σ = 0
• are volume minimizing in their homology class
• Fixed point sets of isometric antiholomorphic
involutions, σ, on CY are sLags:
σ 2 = id,
σ(J) = −J,
σ(Ω) = eiθ Ω3
Σσ := Fix(σ) = a sLag
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For a CICY: ∃ two simple classes of σ‘s with nontrivial
fixed point sets:
“A-type”:
• σA : ����
z �→ z
∈CPn
• σAU = U−1 ◦ σA ◦ U
U : CPn → CPn & symmetry of polynomials Pi
“C-type”:
n
n
CY
⊂
.
.
.
×
CP
×
CP
×
.
.
.
For
� �� � � �� �
�z
�w
• σC : (z, w) �→ (w, z)
•
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(U1 ,U2 )
σC
=
−1
−1
(U1 , U2 )
◦ σC ◦ (U1 , U2 )
Examples:
( 1 ) Quintic:
X1,101 = {z ∈ CP4 |
Only A-type sLags
�5
5
z
i=1 i = 0}
( 2 ) Split bicubic:
X
19,19
�
�
= (t, ζ, η) ∈ CP × CP × CP | P1 (t, ζ) = P2 (t, η) = 0
1
2
A-type and C-type sLags
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2
Next step:
• Classify on which A/C-type sLags the WLs on
the quotient space CY/Γ may project nontrivially
• Only for those sLags, the CS invariants could
possibly be non-zero and need to be computed
• A crucial criterion:
The action of Γ on a given sLag Σ
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Case 1:
R , S : Σ �→ Σ�
CY
Σ
CY/R, CY/S
R, S
Modding out
R,S
Σ‘
Σ = Σ‘
WL due to R,S projects trivially to Σ on CY/Γ
No CS-invariant on Σ
Can ignore such sLags
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Case 2:
R or S : Σ �→ Σ
CY
R or S
Σ
CY/R, CY/S
Modding out
R or S
Σ/R, Σ/S
WL due to R,S may project nontrivially to Σ/R, Σ/S
Nontrivial CS-invariant on Σ/R, Σ/S not ruled out
CS-invariant on Σ/R , Σ/S needs to be computed
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This eliminates already many sLags as
potential carriers of fractional H-flux!
Compute
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�
Σ/Γ
H
WL
only for the remaining sLags
This eliminates already many sLags as
potential carriers of fractional H-flux!
Compute
�
Σ/Γ
H
WL
only for the remaining sLags
But how?
Samstag, 30. August 2014
If Σ/Γ is a Seifert fibration or decomposes into
Seifert fibrations upon “torus decomposition”, one can
use formulas from Nishi (1998), Kirk, Klassen (1993)
to compute the CS invariants
Samstag, 30. August 2014
Seifert fibrations
= Smooth 3D manifolds that can be viewed as a
circle fibration over a 2D orbifold base B
circle fibers
possible orbifold singularities
2D base B
= Important building blocks in the classification of
E.g. Hatcher (2007), Brin (2007)
3D manifolds!
Examples:
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T3 , S2 × S1 , S3 , S3 /Zn , . . .
Torus decomposition into Seifert fibrations
3D manifold that is not a
Seifert fibration, but can be
decomposed into Seifert
fibrations by cutting along
2-tori
Seifert fibrations
with 2-torus boundaries
Samstag, 30. August 2014
In all our examples, the relevant sLags turn out to
be Seifert fibrations or can be torus decomposed
into Seifert fibrations
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4. Explicit examples
Samstag, 30. August 2014
( 1 ) Quintic:
X1,5 = X1,101 / Z5 × Z5
���� ����
R
S
π1 (X1,5 ) = R × S = Z5 × Z5
One finds:
• Only one A-type sLag needs to be checked
• It has topology S3 /Z10
• Only WLs related to S can project nontrivially on it
• For WL that preserves
we compute �
Gvis = SU(3)c × SU(2)L × SU(2)R × U(1)2
3 2
WL
H
=
k
Σ
5
Samstag, 30. August 2014
�Wflux � =
3 2
c5k
( 2 ) Split bicubic:
X3,3 = X19,19 / Z3 × Z3
���� ����
R
π1 (X
3,3
S
) = R × S = Z3 × Z3
We find:
• A-type sLags do not have right topology to
support a CS invariant
• C-type sLags do not have the right topology
to even support a Z3 WL
HWL = 0
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�Wflux � = 0
5. Conclusions
Samstag, 30. August 2014
• Wilson lines are important ingredients for realistic
heterotic CY model building
• Wilson lines�can give rise to fractionally quantized
3-form flux Σ HWL (fractional CS-fluxes)
• Fractional CS-fluxes may have important phenomenological consequences (moduli stabilization, high susy
breaking scale) and may raise self-consistency issues
• Started to develop tools for computing HWL for a
given WL and CY setup
Samstag, 30. August 2014
• Some steps could be done in a general way,
some still required case by case study.
There is probably room for improvement!
• Computed HWL for two concrete examples and
found strong constraints on the possible HWL
Samstag, 30. August 2014
“You cannot simply put a D-brane
anywhere you want!”
R. Blumenhagen
Samstag, 30. August 2014
“You cannot simply put a D-brane
anywhere you want!”
R. Blumenhagen
“You cannot simpy assume fractional
CS-terms anywhere you want!”
Samstag, 30. August 2014
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