Download PDF - String Phenomenology 2014

Double Field Theory and
Non-Geometric Backgrounds:
Dimensional Reduction and
Cosmological Applications
DIETER LÜST (LMU, MPI)
String-Pheno Conference 2014: ICTP Trieste, July 11, 2014
Donnerstag, 10. Juli 14
1
Double Field Theory and
Non-Geometric Backgrounds:
Dimensional Reduction and
Cosmological Applications
DIETER LÜST (LMU, MPI)
O. Hohm, D.L., B. Zwiebach, arXiv:1309.2977;
F. Hassler, D.L., arXiv:1401.5068;
F. Hassler, D.L., S. Massai, arXiv:1405.2325
String-Pheno Conference 2014: ICTP Trieste, July 11, 2014
Donnerstag, 10. Juli 14
1
Outline:
I) Introduction
II) Double Field Theory and Dimensional
Reduction on Non-geometric Backgrounds
III) De Sitter and Inflation
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I) Introduction
Non-geometric backgrounds are generic within the
landscape of string „compactifications“.
Several interesting features:
● They are only consistent in string theory.
● Make use of string symmetries, T-duality
T-folds,
● Left-right asymmetric spaces
Asymmetric orbifolds
(Kawai, Lewellen, Tye, 1986; Lerche, D.L. Schellekens, 1986, Antoniadis, Bachas, Kounnas, 1987; Narain, Sarmadi,Vafa, 1987;
Ibanez, Nilles, Quevedo, 1987; ......, Faraggi, Rizos, Sonmez, 2014)
● Effective gauged supergravity
(Dall‘Agata,Villadoro, Zwirner, 2009; Nicolai, Samtleben,2001; Dibitetto, Linares, Roest, 2010; ...)
● Are related to non-commutative/non-associative geometry
(Blumenhagen, Plauschinn; Lüst, 2010; Blumenhagen, Deser, Lüst, Rennecke, Pluaschin, 2011;
Condeescu, Florakis, Lüst; 2012, Andriot, Larfors, Lüst, Patalong, 2012))
● They can be potentially used for the construction
of de Sitter vacua and inflation
(Hertzberg, Kachru, Taylor, Tegmark, 2007; Caviezel, Köerber, Körs, D.L., Wrase, 2008;
Danielsson, Haque, Shiu, Underwood,Van Riet, 2009; Daniellson, Haque, Koerber, Shiu,Van Riet, 2011;
Dall‘Agata, Inverso, 2012;Blabäck, Danielsson, Dibitetto, 2013; Damian, Diaz-Barron, Loaiza-Brito, Sabido, 2013)
Donnerstag, 10. Juli 14
SUGRA
CY
Flux comp.
DFT
Non-Geometric spaces
Non-geometric spaces
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Double Field Theory
II) Non-geometry & double field theory
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II) Non-geometry & double field theory
Closed string background fields: Gij , Bij ,
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II) Non-geometry & double field theory
Closed string background fields: Gij , Bij ,
Doubling of closed string coordinates and momenta:
˜i, X i)
- Coordinates: O(D,D) vector
X M = (X
- Momenta: O(D,D) vector
winding
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pM = (˜
pi , pi )
← T-duality →
momentum
II) Non-geometry & double field theory
Closed string background fields: Gij , Bij ,
Doubling of closed string coordinates and momenta:
˜i, X i)
- Coordinates: O(D,D) vector
X M = (X
pM = (˜
pi , pi )
- Momenta: O(D,D) vector
winding
Generalized metric: HM N =
5
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✓
← T-duality →
ij
G
Bik Gkj
momentum
ik
Gij
G Bkj
Bik Gkl Blj
◆
II) Non-geometry & double field theory
Closed string background fields: Gij , Bij ,
Doubling of closed string coordinates and momenta:
˜i, X i)
- Coordinates: O(D,D) vector
X M = (X
pM = (˜
pi , pi )
- Momenta: O(D,D) vector
winding
Generalized metric: HM N =
✓
← T-duality →
ij
G
Bik Gkj
momentum
ik
Gij
G Bkj
Bik Gkl Blj
T-duality - O(D,D) transformations:
HM N !
They contain:
Donnerstag, 10. Juli 14
P
⇤M
Q
HP Q ⇤N
Bij ! Bij + 2⇡⇤ij ,
5
, ⇤ 2 O(D, D)
R ! L2s /R
◆
Non-geometric backgrounds & non-geometric fluxes:
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Non-geometric backgrounds & non-geometric fluxes:
- Non-geometric Q-fluxes: spaces that are locally still
Riemannian manifolds but not anymore globally.
(Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht (2005); Dabholkar, Hull, 2005)
Transition functions between two coordinate patches are
given in terms of O(D,D) T-duality transformations:
C. Hull (2004)
Di↵(MD )
!
O(D, D)
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Non-geometric backgrounds & non-geometric fluxes:
- Non-geometric Q-fluxes: spaces that are locally still
Riemannian manifolds but not anymore globally.
(Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht (2005); Dabholkar, Hull, 2005)
Transition functions between two coordinate patches are
given in terms of O(D,D) T-duality transformations:
C. Hull (2004)
Di↵(MD )
!
O(D, D)
- Non-geometric R-fluxes: spaces that are even
locally not anymore manifolds.
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Example: Three-dimensional flux backgrounds:
Fibrations: 2-dim. torus that varies over a circle:
2
TX 1 ,X 2
,! M
3
,!
1
SX 3
The fibration is specified by its monodromy properties.
Metric, B-field of T : HM N (X )
2
3
S1
O(2,2) monodromy:
1
HM N (X 3 + 2⇡) = ⇤O(2,2) HP Q (X 3 ) ⇤O(2,2)
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Example: Three-dimensional flux backgrounds:
Fibrations: 2-dim. torus that varies over a circle:
2
TX 1 ,X 2
,! M
3
1
SX 3
,!
The fibration is specified by its monodromy properties.
Metric, B-field of T : HM N (X )
2
3
S1
O(2,2) monodromy:
Complex structure
1
HM N (X 3 + 2⇡) = ⇤O(2,2) HP Q (X 3 ) ⇤O(2,2)
⌧ of T 2 :
Kähler parameter ⇢ of T 2 :
Donnerstag, 10. Juli 14
a⌧ (X 3 ) + b
⌧ (X + 2⇡) =
c⌧ (X 3 ) + d
3
0
3
0
a
⇢(X
)
+
b
⇢(X 3 + 2⇡) = 0
c ⇢(X 3 ) + d0
7
Torus
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Torus with non-constant B-field (H-flux),
B-field is patched together by a B-field
(gauge) transformation: B ! B + 2⇡H
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Non geometric torus, metric is patched
together by a T-duality transformation: Gij ! Gij
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Non geometric torus, metric is patched
together by a T-duality transformation: Gij ! Gij
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3-dimensional fibration:
⌧ (X + 2⇡) =
3
1
⌧ (X 3 )
2
TX
1X2
1
SX
3
S1
Twisted torus with f-flux
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3-dimensional fibration:
1
⇢(X 3 )
⇢(X + 2⇡) =
3
2
TX
1X2
S1
1
SX
3
Non-geometric space with Q-flux
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(i) (Non-)geometric backgrounds with parabolic
monodromy and single 3-form fluxes:
Chain of four T-dual spaces:
Flat torus
with constant
H-flux
Tx1
Twisted
torus with
f-flux
Tx2
Nongeometric
space with
Q-flux
Tx3
Nongeometric
space with
R-flux
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(i) (Non-)geometric backgrounds with parabolic
monodromy and single 3-form fluxes:
Chain of four T-dual spaces:
Flat torus
with constant
H-flux
Tx1
Twisted
torus with
f-flux
Tx2
Nongeometric
space with
Q-flux
Tx3
Nongeometric
space with
R-flux
Linear B-field
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(i) (Non-)geometric backgrounds with parabolic
monodromy and single 3-form fluxes:
Chain of four T-dual spaces:
Flat torus
with constant
H-flux
Tx1
Twisted
torus with
f-flux
Tx2
Nongeometric
space with
Q-flux
Tx3
Nongeometric
space with
R-flux
Linear B-field
Linear metric
(Nilmanifold)
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Donnerstag, 10. Juli 14
(i) (Non-)geometric backgrounds with parabolic
monodromy and single 3-form fluxes:
Chain of four T-dual spaces:
Flat torus
with constant
H-flux
Tx1
Linear B-field
Twisted
torus with
f-flux
Tx2
Nongeometric
space with
Q-flux
Tx3
Nongeometric
space with
R-flux
non-linear metric and B-field
Linear metric
(Nilmanifold)
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(i) (Non-)geometric backgrounds with parabolic
monodromy and single 3-form fluxes:
Chain of four T-dual spaces:
Flat torus
with constant
H-flux
Tx1
Linear B-field
Twisted
torus with
f-flux
Tx2
Nongeometric
space with
Q-flux
Tx3
Nongeometric
space with
R-flux
non-linear metric and B-field
Linear metric
(Nilmanifold)
no local metric and B-field
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(ii) (Non-)geometric backgrounds with elliptic
monodromy and multiple, (non)-geometric fluxes.
They can be described in terms of twisted tori and
(a)symmetric freely acting orbifolds.
A. Dabholkar, C. Hull (2002, 2005)
C. Hull; R. Read-Edwards (2005, 2006, 2007, 2009)
D. Lüst, JHEP 1012 (2011) 063, arXiv:1010.1361,
C. Condeescu, I. Florakis, D. Lüst, JHEP 1204 (2012), 121, arXiv:1202.6366
C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arXiv:1307.0999
In general not T-dual to a geometric space!
Only consistent in string theory, respectively in DFT.
The fibre torus depends on the third coordinate in a more
complicate way.
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Double field theory action:
W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)
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Double field theory action:
W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)
• O(D,D) invariant effective string action containing
momentum and winding coordinates at the same time:
Z
2D
2 0
SDFT = d X e
R
X M = (˜
xm , xm )
R = 4HM N @M 0 @N 0 @M @N HM N
1 MN
+ H
@M HKL @N HKL
8
4HM N @M 0 @N
+ 4@M HM N @N
1 MN
KL
H
@
H
@L HM K
N
2
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Donnerstag, 10. Juli 14
0
0
Double field theory action:
W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)
• O(D,D) invariant effective string action containing
momentum and winding coordinates at the same time:
Z
2D
2 0
SDFT = d X e
R
X M = (˜
xm , xm )
R = 4HM N @M 0 @N 0 @M @N HM N
1 MN
+ H
@M HKL @N HKL
8
4HM N @M 0 @N
0
+ 4@M HM N @N
0
1 MN
KL
H
@
H
@L HM K
N
2
• Covariant fluxes of DFT:(Geissbuhler, Marques, Nunez, Penas; Aldazabal, Marques, Nunez)
FABC = D[A EB
M
EC]M , D = E
A
A
M
@
M
.
Comprise all fluxes (Q,f,Q,R) into one covariant expression:
Fabc = Habc ,
F a bc = F a bc ,
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Fc ab = Qc ab ,
F abc = Rabc
DFT action in flux formulation:
SDFT
=
Z

⇣1
0
0
0
2d
AA0
dX e
FA FA0 S
+ FABC FA0 B 0 C 0
S AA ⌘ BB ⌘ CC
4
1
FABC F ABC
6
1 AA0 BB 0 CC 0 ⌘
S
S
S
12
FA F A
(Looks similar to scalar potential in gauged SUGRA.)
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DFT action in flux formulation:
SDFT
=
Z

⇣1
0
0
0
2d
AA0
dX e
FA FA0 S
+ FABC FA0 B 0 C 0
S AA ⌘ BB ⌘ CC
4
1
FABC F ABC
6
1 AA0 BB 0 CC 0 ⌘
S
S
S
12
FA F A
(Looks similar to scalar potential in gauged SUGRA.)
• Strong constraint (string level matching condition):
(CFT origin of the strong constraint: A. Betz, R. Blumenhagen, D. Lüst, F. Rennecke, arXiv:1402.1686)
@M @ · = 0 ,
M
@M f @ g = DA f D g = 0
M
A
Functions depend only on one kind of coordinates.
The strong constraint defines a D-dim. hypersurface (brane) in
2D-dim. double geometry.
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Donnerstag, 10. Juli 14
Implementations of the strong constraint:
(i) DFT on spaces satisfying the strong constraint (SC)
Rewriting of SUGRA, geometric spaces
(ii) DFT on spaces with „mild violation“ of the SC
Non-geometric spaces (Q-flux)
(iii) DFT on spaces with „strong violation“ of the SC
Very non-geometric spaces (R-flux)
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Dimensional reduction of double field theory:
Generalized Scherk-Schwarz compactifications
matching
amplitudes
SUGRA
without fluxes
simplification (truncation)
string theory
D
wi
th
flu
xe
s
(o
nly
su
d SUGRA
embedding
tensor
19
Donnerstag, 10. Juli 14
bs
e
t)
gauged
SUGRA
Dimensional reduction of double field theory:
Generalized Scherk-Schwarz compactifications
matching
amplitudes
D
Double Field Theory
@˜i · = 0
SUGRA
without fluxes
simplification (truncation)
string theory
SFT
SC
wi
th
flu
xe
s
(o
n
ly
su
d SUGRA
embedding
tensor
20
Donnerstag, 10. Juli 14
bs
et
)
gauged
SUGRA
Dimensional reduction of double field theory:
Generalized Scherk-Schwarz compactifications
matching
amplitudes
D
Double Field Theory
@˜i · = 0
SC
SUGRA
without fluxes
simplification (truncation)
string theory
SFT
wi
th
Violation of SC
Non-geometric space
flu
xe
s
(o
nly
su
t)
d SUGRA
embedding
tensor
21
Donnerstag, 10. Juli 14
bs
e
gauged
SUGRA
Dimensional reduction of double field theory:
Generalized Scherk-Schwarz compactifications
matching
amplitudes
D
Double Field Theory
@˜i · = 0
SUGRA
without fluxes
simplification (truncation)
string theory
SFT
wi
th
SC
flu
xe
s(
Violation of SC
Non-geometric space
on
ly
su
bs
et
)
d SUGRA
embedding
tensor
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Donnerstag, 10. Juli 14
gauged
SUGRA
e.o.m.
Vacuum
Dimensional reduction of double field theory:
Generalized Scherk-Schwarz compactifications
matching
amplitudes
D
Violation of SC
Non-geometric space
SC
th
flu
xe
s(
on
ly
su
d SUGRA
embedding
tensor
bs
et
)
gauged
SUGRA
23
Donnerstag, 10. Juli 14
Asymmetric
orbifold CFT
uplift
wi
Double Field Theory
@˜i · = 0
SUGRA
without fluxes
simplification (truncation)
string theory
SFT
e.o.m.
Vacuum
Dimensional reduction of double field theory:
G. Aldazabal, W. Baron, D. Marques, C. Nunez, arXiv:1109.0290;
D. Berman, E. Musaev, D. Thompson, arXiv:1208.0020;
D. Berman, K. Lee, arXiv:1305.2747;
O. Hohm, D. Lüst, B. Zwiebach, arXiv:1309.2977;
F. Hassler, D. Lüst, arXiv:1401.5068.
• Consistent DFT solutions: RM N = 0
• 2(D-d) linear independent Killing vectors:
LKIJ HM N = 0
• DFT and Scherk-Schwarz ansatz gives rise to effective
theory in D-d dimensions:
Se↵ =
Z
⇣
1
dx
ge
R + 4@µ @
Hµ⌫⇢ H µ⌫⇢
12
⌘
1
1
N
HM N F M µ⌫ Fµ⌫
+ Dµ HM N Dµ HM N V
4
8
(D
d) p
2
µ
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Donnerstag, 10. Juli 14
• Effective scalar potential:
1 KL
1
IJ
F I F J K L H + FIKM FJLN HIJ HKL HM N
4
12
V =
•
RM N = 0
V =0
and
K
Minkowski vacua:
MN
V
=
=0
HM N
This leads to additional conditions on the fluxes FIKM .
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The corresponding backgrounds are in general nongeometric and go beyond dimensional reduction of SUGRA.
26
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The corresponding backgrounds are in general nongeometric and go beyond dimensional reduction of SUGRA.
(i) Mild violation of SC:
• Killing vectors violate the SC.
• Patching of coordinate charts correspond to generalized
coordinate transformations that violate the SC.
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Donnerstag, 10. Juli 14
The corresponding backgrounds are in general nongeometric and go beyond dimensional reduction of SUGRA.
(i) Mild violation of SC:
• Killing vectors violate the SC.
• Patching of coordinate charts correspond to generalized
coordinate transformations that violate the SC.
(ii) Strong violation of SC:
• Background fields violate the SC.
However the fluxes have to obey the closure constraint consistent gauge algebra in the effective theory.
M. Grana, D. Marques, arXiv:1201.2924
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Donnerstag, 10. Juli 14
SUGRA
CY
Flux comp.
T
F
D
DFT without the SC non-geometric spaces
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C
S
+
Simplest non-trivial solutions: d=3 dim. backgrounds:
2
TX
1X2
S1
1
SX
3
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Simplest non-trivial solutions: d=3 dim. backgrounds:
2
TX
1X2
S1
1
SX
3
Parabolic background spaces:
Single fluxes:
1
12
123
H123 or f23
or Q3 or R
MN
R
= 0.
These backgrounds do not satisfy
- CFT: beta-functions are non-vanishing at
quadratic order in fluxes.
- Effective scalar potential: no Minkowski minima (
Donnerstag, 10. Juli 14
AdS)
Elliptic background spaces:
Multiple fluxes:
These backgrounds do satisfy RM N = 0 .
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Elliptic background spaces:
Multiple fluxes:
These backgrounds do satisfy RM N = 0 .
• Single elliptic geometric space (Solvmanifold):
2
f13
=
1
f23
=f
R orbifold.
Symmetric ZL
4 ⇥ Z4
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Elliptic background spaces:
Multiple fluxes:
These backgrounds do satisfy RM N = 0 .
• Single elliptic geometric space (Solvmanifold):
2
f13
=
1
f23
=f
R orbifold.
Symmetric ZL
4 ⇥ Z4
• Single elliptic T-dual, non-geometric space:
12
H123 = Q3 = H
R orbifold.
Asymmetric ZL
4 ⇥ Z4
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Elliptic background spaces:
Multiple fluxes:
These backgrounds do satisfy RM N = 0 .
• Single elliptic geometric space (Solvmanifold):
2
f13
=
1
f23
=f
R orbifold.
Symmetric ZL
4 ⇥ Z4
• Single elliptic T-dual, non-geometric space:
12
H123 = Q3 = H
R orbifold.
Asymmetric ZL
4 ⇥ Z4
• Double elliptic, genuinely non-geometric space:
H123 =
=H,
=
L
Asymmetric Z4 orbifold.
12
Q3
2
f13
29
Donnerstag, 10. Juli 14
1
f23
=f
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
⌧ (2⇡) =
1
,
⌧ (0)
30
Donnerstag, 10. Juli 14
1
f 2 + Z,
4
1
H 2 + Z.
4
⇢(2⇡) =
1
⇢(0)
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
⌧ (2⇡) =
1
,
⌧ (0)
30
Donnerstag, 10. Juli 14
Background
1
f2
+ Z , strong
satisfies
4
1constraint
H2
⇢(2⇡) =
4
+ Z.
1
⇢(0)
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
⌧ (2⇡) =
1
,
⌧ (0)
Background
1
f2
+ Z , strong
satisfies
4
1constraint
H2
⇢(2⇡) =
4
+ Z.
1
⇢(0)
Patching is generated by generalized diffeomorphism:
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Donnerstag, 10. Juli 14
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
⌧ (2⇡) =
1
,
⌧ (0)
Background
1
f2
+ Z , strong
satisfies
4
1constraint
H2
⇢(2⇡) =
4
+ Z.
1
⇢(0)
Patching is generated by generalized diffeomorphism:
30
Donnerstag, 10. Juli 14
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
1
,
⌧ (0)
⌧ (2⇡) =
Background
1
f2
+ Z , strong
satisfies
4
1constraint
H2
⇢(2⇡) =
4
+ Z.
1
⇢(0)
Patching is generated by generalized diffeomorphism:
x3 ! x3 + 2⇡
)
x
˜01 =
x
˜2 ,
x
˜02 =
x1 ,
x01 =
x2 ,
x02 =
x
˜1 .
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Donnerstag, 10. Juli 14
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
1
,
⌧ (0)
⌧ (2⇡) =
Background
1
f2
+ Z , strong
satisfies
4
1constraint
H2
⇢(2⇡) =
4
+ Z.
1
⇢(0)
Patching is generated by generalized diffeomorphism:
Patching does
x3 ! x3 + 2⇡
)
x
˜01 =
x
˜2 ,
x
˜02 =
x1 ,
x01 =
x2 ,
x02 =
x
˜1 .
30
Donnerstag, 10. Juli 14
not satisfy strong
constraint
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
1
,
⌧ (0)
⌧ (2⇡) =
Background
1
f2
+ Z , strong
satisfies
4
1constraint
H2
⇢(2⇡) =
4
+ Z.
1
⇢(0)
Patching is generated by generalized diffeomorphism:
Patching does
x3 ! x3 + 2⇡
)
x
˜01 =
x
˜2 ,
x
˜02 =
x1 ,
x01 =
x2 ,
x02 =
x
˜1 .
not satisfy strong
constraint
Corresponding Killing vectors of background:
0
1
B0
B
B0
ˆ
J
K Iˆ = B
B0
B
@0
0
Donnerstag, 10. Juli 14
0
1
0
0
0
0
0
1
3
2 (Hx
1
0
0
0
+ fx
˜ )
3
0
+ fx
˜2 )
0
1
0
0
1
2
2 (Hx
30
0
+ Hx
˜3 )
0
0
1
0
1
3
2 (f x
1
2
2 (f x
1
0
+ Hx
˜2 )C
C
C
0
C
C
0
C
A
0
1
E.g. double elliptic background:
⌧ (x3 )
=
⇢(x3 )
=
=)
⌧0 cos(f x3 ) + sin(f x3 )
,
cos(f x3 ) ⌧0 sin(f x3 )
⇢0 cos(Hx3 ) + sin(Hx3 )
,
cos(Hx3 ) ⇢0 sin(Hx3 )
1
,
⌧ (0)
⌧ (2⇡) =
Background
1
f2
+ Z , strong
satisfies
4
1constraint
H2
⇢(2⇡) =
4
+ Z.
1
⇢(0)
Patching is generated by generalized diffeomorphism:
Patching does
x3 ! x3 + 2⇡
)
x
˜01 =
x
˜2 ,
x
˜02 =
x1 ,
x01 =
x2 ,
x02 =
x
˜1 .
not satisfy strong
constraint
Corresponding Killing vectors of background:
0
1
B0
B
B0
ˆ
J
K Iˆ = B
B0
B
@0
0
Donnerstag, 10. Juli 14
0
1
0
0
0
0
0
1
3
2 (Hx
1
0
0
0
+ fx
˜ )
3
0
0do
Killing
vectors
1
1
3
3
2
˜ ) 2 (f x + H x
˜2 )C
2 (f x + H x
C
C
0
not satisfy0 strong constraint.
C
C
0
0
C
However
their
algebra
A
1
0
0 closes!
1
0
+ fx
˜2 )
0
1
0
0
1
2
2 (Hx
1
30
● There situations, where the strong constraint even for the
background can be violated. - This seems to be the case for
certain very asymmetric orbifolds. C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arXiv:1307.0999
⌧0 cos(f4 x3 + f2 x
˜3 ) + sin(f4 x3 + f2 x
˜3 )
⌧ (x3 , x
˜3 ) =
,
cos(f4 x3 + f2 x
˜3 ) ⌧0 sin(f4 x3 + f2 x
˜3 )
⇢0 cos(g4 x3 + g2 x
˜3 ) + sin(g4 x3 + g2 x
˜3 )
⇢(x3 , x
˜3 ) =
,
cos(g4 x3 + g2 x
˜3 ) ⇢0 sin(g4 x3 + g2 x
˜3 )
Fluxes:
Parameter
f4
f2
g4
g2
1
f4 , g4 2 + Z
8
1
f2 , g2 2 + Z
4
Fluxes
f, f˜
˜
Q, Q
H, Q
f˜, R
R
Asymmetric ZL
⇥
Z
4
2 orbifold with H, f ,Q,R-fluxes.
This (partially?) solves a so far existing puzzle between
effective SUGRA and uplift/string
compactification.
31
Donnerstag, 10. Juli 14
III) De Sitter and Inflation
32
Donnerstag, 10. Juli 14
F. Hassler, D. Lüst, S. Massai, arXiv:1405.2325
III) De Sitter and Inflation
F. Hassler, D. Lüst, S. Massai, arXiv:1405.2325
Effective scalar potential of double elliptic backgrounds:

1 f12 + 2f1 f2 (⌧R2 ⌧I2 ) + f22 |⌧ |4
H 2 + 2HQ(⇢2R ⇢2I ) + Q2 |⇢|4
+
V (⌧, ⇢) = 2
R
2⌧I2
2⇢2I
32
Donnerstag, 10. Juli 14
0
III) De Sitter and Inflation
F. Hassler, D. Lüst, S. Massai, arXiv:1405.2325
Effective scalar potential of double elliptic backgrounds:

1 f12 + 2f1 f2 (⌧R2 ⌧I2 ) + f22 |⌧ |4
H 2 + 2HQ(⇢2R ⇢2I ) + Q2 |⇢|4
+
V (⌧, ⇢) = 2
R
2⌧I2
2⇢2I
The potential is positive semi-definite.
No up-lift is needed!
32
Donnerstag, 10. Juli 14
0
III) De Sitter and Inflation
F. Hassler, D. Lüst, S. Massai, arXiv:1405.2325
Effective scalar potential of double elliptic backgrounds:

1 f12 + 2f1 f2 (⌧R2 ⌧I2 ) + f22 |⌧ |4
H 2 + 2HQ(⇢2R ⇢2I ) + Q2 |⇢|4
+
V (⌧, ⇢) = 2
R
2⌧I2
2⇢2I
0
The potential is positive semi-definite.
No up-lift is needed!
Vacuum structure:
• Minkowski vacua:
⇢?R
= 0,
e.g.
⇢?I
=
s
HQ > 0
H
,
Q
Vmin = 0
H = Q = 1/4
Asymmetric
32
Donnerstag, 10. Juli 14
L
Z4
orbifold.
HQ < 0
H
? 2
? 2
(⇢R ) + (⇢I ) =
,
Q
• de Sitter vacua:
Vmin =
However here the radius R is not stabilized.
33
Donnerstag, 10. Juli 14
4HQ > 0
HQ < 0
H
? 2
? 2
(⇢R ) + (⇢I ) =
,
Q
• de Sitter vacua:
Vmin =
4HQ > 0
However here the radius R is not stabilized.
Another option: SO(2,2) gauging
H2
V (⇢, ⌧ ) = 2 1 + 2(⇢2R
2⇢I
⇢2I )
+ |⇢|
4
H2
H2
2
2
+
(1 + |⇢| )(1 + |⌧ | ) + 2 1 + 2(⌧R2
⇢I ⌧ I
2⌧I
⇢ = ⌧ = i with Vmin = 4H
?
?
⌧I2 ) + |⌧ |4
2
All moduli ⌧ and ⇢ have positive mass square.
33
Donnerstag, 10. Juli 14
Inflation from non-geometric backgrounds:
There are some attractive features for inflation:
34
Donnerstag, 10. Juli 14
Inflation from non-geometric backgrounds:
There are some attractive features for inflation:
• The potentials are positive with quadratic and quartic
couplings that depend on the (non)-geometric fluxes.
No up-lift is needed!
One needs to tune fluxes to obtain slow roll inflation.
( Orbifolds with high order of twist!)
34
Donnerstag, 10. Juli 14
Inflation from non-geometric backgrounds:
There are some attractive features for inflation:
• The potentials are positive with quadratic and quartic
couplings that depend on the (non)-geometric fluxes.
No up-lift is needed!
One needs to tune fluxes to obtain slow roll inflation.
( Orbifolds with high order of twist!)
• The non-trivial monodromies allow for enlarged field
(McAllsiter, Silverstein, Westphal, 2008)
range of the inflaton field.
Realization of monodromy inflation in order to
obtain a visible tensor to scalar ratio
(gravitational waves).
34
Donnerstag, 10. Juli 14
Enlarged field range for parabolic monodromy
⇢ ! ⇢ + 1 or ⌧ ! ⌧ + 1 :
·, ﬂ
...
...
i
≠ 32
≠ 12
1
2
3
2
Infinite field range for ⌧R or ⇢R .
35
Donnerstag, 10. Juli 14
Enlarged field range for elliptic Z4 monodromy
1
1
⇢!
or ⌧ !
:
⇢
⌧
·, ﬂ
i
≠ 12
1
2
Infinite field range for combinations of
⌧R and ⌧I or combinations of ⇢R and ⇢I .
36
Donnerstag, 10. Juli 14
Enlarged field range for elliptic Z6 monodromy
⇢!
1
+1
⇢
or ⌧ !
1
+1 :
⌧
·, ﬂ
i
≠ 12
1
2
3
2
Infinite field range for combinations of
⌧R and ⌧I or combinations of ⇢R and ⇢I .
37
Donnerstag, 10. Juli 14
Simple elliptic model for non-geometric inflation:
Expect fluxes
1
H, Q ⇠
N
Kinetic energy:
Inflaton field:
Lkin

1
= 2 (@⇢R )2 + (@⇢I )2
4⇢I
⇢R
=
2⇢I
Inflaton potential:
V ( , ⇢I ) = V0 (⇢I ) + m (⇢I )
2
V0 (⇢I ) =
H2
2HQ⇢2I + Q2 ⇢4I
,
2⇢2I
+ (⇢I )
m2 (⇢I ) = 4HQ + 4Q2 ⇢2I ,
38
Donnerstag, 10. Juli 14
2
4
(⇢I ) = 8Q2 ⇢2I
Minimization with respect to ⇢I :
)
V0 = 0
Inflaton mass and self-coupling:
m = 4HQ
2
✓
◆
1
?
2
+ ⇢I M s ,
?
⇢I
? 3
2
(⇢
2
2
I)
gs MP 2 =
m
1 + (⇢?I )2
Small
1 2 ?
= 2 M s ⇢I )
gs
?
⇢
small value for I .
39
Donnerstag, 10. Juli 14
2
(MP
=
?
8HQ⇢I
Slow roll inflation with 60 e-foldings and
ns ⇠ 0.967 ,
ns = 1
MP2
✏=
2
m ' 6 ⇥ 10
6
✓
@ V
V
MP ,
r ⇠ 0.133 (BICEP2)
6✏ + 2⌘ ,
◆2
2
0
⌘=
,
1/4
V0
H ' Q ' 10
0
r = 16✏
' 10
5
,
MP2
2
@ V
V
!
' 15MP
MP )
?
⇢I
 10
2
Need very small fluxes (large monodromy N ' 105 )
and sub-stringy value for the volume of the fibre.
40
Donnerstag, 10. Juli 14
IV) Summary
41
Donnerstag, 10. Juli 14
IV) Summary
● DFT allows for consistent reduction on non-geometric
backgrounds that go beyond SUGRA and also beyond
generalized geometry.
41
Donnerstag, 10. Juli 14
IV) Summary
● DFT allows for consistent reduction on non-geometric
backgrounds that go beyond SUGRA and also beyond
generalized geometry.
● Non-geometric backgrounds posses some attractive (generic)
features for string cosmology:
- Uplift to Minkowski or de Sitter
- Elliptic monodromy of finite order
Finite enlargement of field range for inflaton
Suppressed masses and couplings for inflaton
41
Donnerstag, 10. Juli 14
IV) Summary
● DFT allows for consistent reduction on non-geometric
backgrounds that go beyond SUGRA and also beyond
generalized geometry.
● Non-geometric backgrounds posses some attractive (generic)
features for string cosmology:
- Uplift to Minkowski or de Sitter
- Elliptic monodromy of finite order
Finite enlargement of field range for inflaton
Suppressed masses and couplings for inflaton
● Particle physics model building and full moduli stabilization
on non-geometric backgrounds still needs to be further
developed.
41
Donnerstag, 10. Juli 14
IV) Summary
● DFT allows for consistent reduction on non-geometric
backgrounds that go beyond SUGRA and also beyond
generalized geometry.
● Non-geometric backgrounds posses some attractive (generic)
features for string cosmology:
- Uplift to Minkowski or de Sitter
- Elliptic monodromy of finite order
Finite enlargement of field range for inflaton
Suppressed masses and couplings for inflaton
● Particle physics model building and full moduli stabilization
on non-geometric backgrounds still needs to be further
developed.
Thank you very much!
Donnerstag, 10. Juli 14
41

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