Transcript How stable are extra dimensions?

De Sitter in Supergravity
and String Theory
Diederik Roest (RUG)
THEP national seminar
November 20, 2009
Outline
1.
2.
3.
4.
5.
Introduction
Gauged supergravity
De Sitter in supergravity
De Sitter in string theory
Conclusions
(family tree)
(compactifications)
1. Introduction
Strings




Quantum gravity
No point particles, but small strings
Unique theory
Bonus: gauge forces
Unification of four forces of Nature?
…and then some!
String theory has
many implications:
Supersymmetry
Extra
dimensions
Many vacua
(~10500)?
Branes
& fluxes
Dualities
How can one extract
4D physics from this?
Compactifications
Stable compactifications

Simple compactifications yield
massless scalar fields, so-called moduli,
in 4D.

Would give rise to a new type of
force, in addition to gravity and
gauge forces. Has not been observed!

Need to give mass terms to these
scalar fields (moduli stabilisation).

Extra ingredients of string theory,
such as branes and fluxes, are crucial!
energy
simple
comp.
with fluxes
and branes
Scalar field
Flux compactifications




Lots of progress in understanding moduli stabilisation in string
theory (2002-…)
Using gauge fluxes one can stabilise the Calabi-Yau moduli
Classic results:
– IIB complex structure moduli stabilised by gauge fluxes [1]
– IIB Kahler moduli stabilised by non-perturbative effects [2]
– All IIA moduli stabilised by gauge fluxes [3]
But:
– Vacua are supersymmetric AdS (i.e. have a negative
cosmological constant)
[1: Giddings, Kachru, Polchinski ’02]
[2: Kachru, Kallosh, Linde, Trivedi ’03]
[3: DeWolfe, Giryavets, Kachru, Taylor ’05]
String cosmology


Two periods of accelerated expansion: very early universe and
present time.
Does string theory have anything to say
about this? In other words, where is
De Sitter in the string theory landscape?
dS in supergravity and string theory?





Supergravity as an effective description of string theory
compactifications.
Effect of fluxes etc: gauged supergravity.
Do gauged supergravities have dS vacua?
Gauge fluxes
Do these follow from string theory
compactifications?
Beyond flux compactifications:
Non-geometric
fluxes
Geometric fluxes
2. Gauged supergravity
Supertheories
Supersymmetry
Supergravity
Global supersymmetry:
 Relates spin-0,1 bosons and spin-1/2 fermions
 In 4D one can have up to N = 4 supersymmetries
 Only in ten dimensions and lower
 Favorable UV behaviour
 Perhaps we are going to see N = 1 at LHC?
Gauged
supergravity
Supertheories
Supersymmetry
Supergravity
Gauged
supergravity
Local supersymmetry:
 Relates spin-0,1,2 bosons and spin-1/2,3/2 fermions
 Necessarily includes spin-2 graviton
supergravity
 In 4D one can have up to N = 8 supersymmetries
 Only in eleven dimensions and lower
 Relevant for theories of quantum gravity?
Supertheories
Supersymmetry



Supergravity
Gauged
supergravity
Supergravity has many scalar fields that could be used for e.g.
cosmology. A priori massless scalar fields.
Only possibility of introducing masses is via specific scalar
potential energies
Fully specified by gaugings: part of the global symmetries are
made local. Depends on global symmetry and number of vectors
gauged supergravity.
Scalar potential
Generically gives rise to negative potential energy. Corresponding
vacuum is Anti-De Sitter space (AdS). Scalar potentials of
gauged supergravity play important role in AdS/CFT
correspondence.
By careful finetuning one can also build scalar potentials that are
interesting for cosmology, e.g. with a positive potential energy.
Corresponding vacuum is De Sitter space (dS).
dS in gauged supergravity?
Positive and negative results in different flavours N = 2…8 of
supergravity:
 N = 4, 8: unstable dS with η = O(1) [1]
 N = 2: stable dS [2]
 no-go theorems for stableRequirements
dS in various for
theories
dS in [3]
gauged
supergravity? Relations between
different models? Relation to
string theory compactifications?
[1: Kallosh, Linde, Prokushkin, Shmakova ’02]
[2: Fré, Trigiante, Van Proeyen ’02]
[3: De Wit, Van Proeyen, (…) '84, '85, Gomez-Reino, (Louis), Scrucca ’06, ’07, ’08]
3. De Sitter in supergravity
N=4 supergravity
Effective theory of
type I / heterotic string theory on T^6 or
type II / M-theory on K3 x T^2 or with orientifolds.
Key ingredients:
 Supergravity plus nV=6 vector multiplets
 Global symmetry SL(2) x SO(6, 6)
 Scalars in cosets of global symmetry
 Vectors in fundamental rep. of SO(6, 6),
and into e-m dual under SL(2).
N=4 gauged supergravity
Possible gaugings classified by parameters [1] which are a doublet
under
SL(2):
®
f
MNP
;
® = (+ ; ¡ ) ;
M = (1; : : : ; 12) :
Electric and magnetic gaugings.
Subject to a set of quadratic constraints that impose Jacobi
identities and orthogonality of charges. Possible solution:
direct product of simple factors with certain angles
G = G1 x G2 x …
Most often considered but not unique!
[1: Schön, Weidner ‘06]
N=4 gauged supergravity
Crucial for moduli stabilisation:
If gauge group is direct product of factors
G = G1 x G2 x …
they must have different SL(2) angles [1]
("duality or De Roo-Wagemans angles")
If angles are equal, the scalar potential has runaway directions:
V (Á; '~) = eÁ V0 ( '~)
Impossible to stabilise moduli in dS.
[1: De Roo, Wagemans ’85]
De Sitter in N=4

Known De Sitter vacua in N = 4: split up in two six-dimensional
gauge factors G = G1 x G2 given by [1]
SO(4), SO(3,1) or SO(2,2).
Gauge factors specified by [2]
- coupling constant g1,2
- embedding parameter h1,2

(Plus some exceptional cases with 3+9 split.)
All unstable: tachyonic directions with -2 < η < 0.
No stable De Sitter vacua are expected for N ≥ 4 - proof ? [3]


[1: De Roo, Westra, Panda ’06]
[2: D.R., Rosseel – in progress]
[3: Gomez-Reino, (Louis), Scrucca ’06, ’07, ’08]
N=2 supergravity
Effective theory of
type I / heterotic string theory on K3 x T^2 or
type II / M-theory on Calabi-Yau manifold.
Key ingredients:
 Supergravity plus nV vector multiplets and nH hyper multiplets
 Global symmetry* SL(2) x SO(2, nV-1) x SO(4, nH)
 Scalars in cosets of global symmetry
 Vectors in fundamental rep. of SO(2, nV), and into e-m dual
under SL(2)
N=2 gauged supergravity
Gaugings in vector sector are similar to the N=4 case.
Differences with N=4 due to hyper sector:
 Choice to gauge isometries of hypers as well
 Possible to gauge SO(2) or SO(3) even if hypers are absent
(“Fayet-Iliopoulos parameters”)
Lower amount of supersymmetry allows for more multiplets and
hence for more possible gaugings.
Stable dS in N=2
In contrast to N=4 case, there are a few “mysterious” examples of
stable dS in N=2 [1].
Example:
 Take SL(2) x SO(2,4) x SO(4,2), i.e. six vectors.
Gauge SO(1,2) x SO(3) with different SL(2) angles.
Leads to stable dS if gauge group acts on hypers as well.
[1: Fré, Trigiante, Van Proeyen ‘02]
unstable
N=8:
SO(5,3)
SO(4,4)
embedding
parameter h1,2 = 0




Truncations
unstable
N=4:
stable
N=2:
SO(4) x SO(4)
SO(4) x SO(3,1)
SO(3,1) x SO(3,1)
SO(3,1) x SO(2,2)
SO(2,2) x SO(2,2)
SO(2,1) x SO(2)H
SO(2,1) x SO(3)H
SO(2,1)H x SO(3)H
embedding
parameter h1,2 = 1
Family tree of dS relations: (almost) all known models related
[1]!
Explains stable N=2 from unstable N=4
Possible to derive FI terms from N=4 gaugings
Also gives rise to new stable N=2 cases! [1: D.R., Rosseel – in progress]
4. De Sitter in string theory
Compactifications
“Vanilla” compactifications lead to ungauged supergravities:
e.g.
on torus
(N=8)
with orientifold
(N=4)
on Calabi-Yau
(N=2)
on CY with orientifold
(N=1)
Problem of massless moduli in 4D, no scalar potential!
Need to include additional “bells and whistles” on internal manifold
M.
Flux compactifications
Additional “ingredients” consistent with N=4 compactifications:
 Gauge fluxes
(electro-magnetic field lines in M)
 Geometric fluxes
(non-trivial Ricci-curvature on M)
 Non-geometric fluxes
(generalisation due to T-duality)
Higher-dimensional origin?
10D string
theory
Compactification with gauge and
(non-)geometric fluxes
4D gauged
supergravity
4D gauged
supergravity
Which of these two sets contain De Sitter vacua?
IIB with O3-planes
Convenient duality frame: can always be reached by T-duality
transformations. Only allowed fluxes:
 NS-NS gauge and non-geometric fluxes:
Hmnp ;

Qm n p :
R-R gauge and non-geometric fluxes:
Fm n p ;
Pm n p :
Relation to N=4 gauged
Half of structure constants are sourced by these fluxes:
(where SO(6,6) index splits up in (m,m) indices)

Electric gaugings sourced by R-R gauge and NS-NS nongeometric fluxes:
f

mnp
+
= ² m n pqr s F qr s ;
f + m n p = Qm n p :
Magnetic gaugings sourced by NS-NS gauge and R-R nongeometric fluxes:
f
mnp
¡
= ² m n pqr s H qr s ;
Structure constants
related to fluxes
f¡
m
np
= Pm n p :
Structure constants
unrelated to fluxes
Fate of dS in compactifications?
This year it was shown that one can build up gaugings of the form
G = G1 x G2 in this way [1].
But these fluxes are not enough to build up any of the products of
simple gauge groups with dS vacua [2].


Only gauge fluxes:
(nilpotent)2
(CSO(1,0,3)2)
Gauge and non-geometric fluxes:
(non-semi-simple)2
(CSO(1,2,1)2 = ISO(1,2)2)
where CSO(p,q,r) is a (contraction)r of SO(p,q+r).
[1: D.R. ’09, Dall’Agata, Villadoro, Zwirner ‘09]
[2: Dibitetto, Linares, D.R. - in progress]
Higher-dimensional origin?
10D string
theory
Compactification with gauge and
(non-)geometric fluxes
New ones
G
n G[2]¡
+
4D gauged
supergravity
4D gauged
supergravity
Known
G £ones
G [1]
+
¡
Which of these two sets contain De Sitter vacua?
[1: Dibitetto, Linares, D.R. - in progress]
[2: De Carlos, Guarino, Moreno ’09]
5. Conclusions
Conclusions





Modern cosmology requires accelerated expansion
De Sitter in extended supergravity and link to string theory
Careful tuning of scalar potential in gauged supergravity
Relations between supergravity models with dS vacua?
Higher-dimensional origin in terms of gauge, geometric or nongeometric fluxes?
unstable
N=8:
SO(5,3)
SO(4,4)
embedding
parameter h1,2 = 0




Truncations
unstable
N=4:
stable
N=2:
SO(4) x SO(4)
SO(4) x SO(3,1)
SO(3,1) x SO(3,1)
SO(3,1) x SO(2,2)
SO(2,2) x SO(2,2)
SO(2,1) x SO(2)H
SO(2,1) x SO(3)H
SO(2,1)H x SO(3)H
embedding
parameter h1,2 = 1
Family tree of dS relations: almost all known models related!
Explains stable N=2 from unstable N=4
Also gives rise to new, possibly stable N=2 cases
Possible to derive FI terms from N=4 gaugings
[1: D.R., Rosseel - in progress]
Higher-dimensional origin?
10D string
theory
Compactification with gauge and
(non-)geometric fluxes
New ones
G
n G[2]¡
+
4D gauged
supergravity
4D gauged
supergravity
Known
G £ones
G [1]
+
¡
Which of these two sets contain De Sitter vacua?
[1: Dibitetto, Linares, D.R. - in progress]
[2: De Carlos, Guarino, Moreno ’09]
Conclusions








Family tree of supergravity models with dS vacua
“Non-trivial” and stable N=2 models follow from
“trivial” and non-stable N=4 models
New unstable N=4 and stable N=2 models
Higher-N origin to Fayet-Iliopoulos terms
Higher-dimensional origin in terms of gauge, geometric or nongeometric fluxes?
Leads to none of known N=4 models!
Semi-direct instead of direct product gaugings?
Other compactifications?
Thanks for your attention!
Off to the drinks!