De Sitter in Supergravity and String Theory

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De Sitter in Supergravityand String Theory

• Diederik Roest (RUG)THEP national
  seminarNovember 20, 2009

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Outline

1. Introduction
2. Gauged supergravity
3. De Sitter in supergravity (family tree)
4. De Sitter in string theory (compactifications)
5. Conclusions

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1. Introduction
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Strings

• Quantum gravity
• No point particles, but small strings
• Unique theory
• Bonus gauge forces
• Unification of four forces of Nature?

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and then some!
String theory has many implications
How can one extract 4D physics from this?
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Compactifications
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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!

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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 022 Kachru,
Kallosh, Linde, Trivedi 033 DeWolfe,
Giryavets, Kachru, Taylor 05
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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?

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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?
• Do these follow from string theory
  compactifications?
• Beyond flux compactifications

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2. Gauged supergravity
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Supertheories
Super- symmetry
Gauged supergravity
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?

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Supertheories
Super- symmetry
Gauged supergravity
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?

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Supertheories
Super- symmetry
Gauged supergravity
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.

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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).

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dS in gauged supergravity?

• Positive and negative results in different
  flavours N 28 of supergravity
• N 4, 8 unstable dS with ? O(1) 1
• N 2 stable dS 2
• no-go theorems for stable dS in various theories
  3

Requirements for dS in gauged supergravity?
Relations between different models? Relation to
string theory compactifications?
1 Kallosh, Linde, Prokushkin, Shmakova 022
Fré, Trigiante, Van Proeyen 023 De Wit, Van
Proeyen, () '84, '85, Gomez-Reino, (Louis),
Scrucca 06, 07, 08
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3. De Sitter in supergravity
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N4 supergravity

• Effective theory of type I / heterotic string
  theory on T6 or type II / M-theory on K3 x T2
  or with orientifolds.
• Key ingredients
• Supergravity plus nV6 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).

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N4 gauged supergravity

• Possible gaugings classified by parameters 1
  which are a doublet under SL(2)
• 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
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N4 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
• Impossible to stabilise moduli in dS.

1 De Roo, Wagemans 85
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De Sitter in N4

• 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 39 split.)
• All unstable tachyonic directions with -2 lt ?
  lt 0.
• No stable De Sitter vacua are expected for N 4
  - proof? 3

1 De Roo, Westra, Panda 062 D.R., Rosseel
in progress3 Gomez-Reino, (Louis), Scrucca
06, 07, 08
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N2 supergravity

• Effective theory of type I / heterotic string
  theory on K3 x T2 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)

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N2 gauged supergravity

• Gaugings in vector sector are similar to the N4
  case.
• Differences with N4 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.

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Stable dS in N2

• In contrast to N4 case, there are a few
  mysterious examples of stable dS in N2 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
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Truncations
unstable
unstable
stable
N8 SO(5,3)SO(4,4)
N4 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)
N2 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 0
embedding parameter h1,2 1

• Family tree of dS relations (almost) all known
  models related 1!
• Explains stable N2 from unstable N4
• Possible to derive FI terms from N4 gaugings
• Also gives rise to new stable N2 cases!

1 D.R., Rosseel in progress
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4. De Sitter in string theory
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Compactifications

• Vanilla compactifications lead to ungauged
  supergravities
• e.g. on torus (N8) with
  orientifold (N4) on Calabi-Yau (N2) on
  CY with orientifold (N1)
• Problem of massless moduli in 4D, no scalar
  potential!
• Need to include additional bells and whistles
  on internal manifold M.

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Flux compactifications

• Additional ingredients consistent with N4
  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)

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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?
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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
• R-R gauge and non-geometric fluxes

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Relation to N4 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
  non-geometric fluxes
• Magnetic gaugings sourced by NS-NS gauge and R-R
  non-geometric fluxes

Structure constants related to fluxes
Structure constants unrelated to fluxes
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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,qr).

1 D.R. 09, DallAgata, Villadoro, Zwirner
092 Dibitetto, Linares, D.R. - in progress
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Higher-dimensional origin?
10D string theory
Compactification with gauge and (non-)geometric
fluxes
4D gauged supergravity
4D gauged supergravity
New ones 2
Known ones 1
Which of these two sets contain De Sitter vacua?
1 Dibitetto, Linares, D.R. - in progress2
De Carlos, Guarino, Moreno 09
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5. Conclusions
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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 non-geometric fluxes?

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Truncations
unstable
unstable
stable
N8 SO(5,3)SO(4,4)
N4 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)
N2 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 0
embedding parameter h1,2 1

• Family tree of dS relations almost all known
  models related!
• Explains stable N2 from unstable N4
• Also gives rise to new, possibly stable N2 cases
• Possible to derive FI terms from N4 gaugings

1 D.R., Rosseel - in progress
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Higher-dimensional origin?
10D string theory
Compactification with gauge and (non-)geometric
fluxes
4D gauged supergravity
4D gauged supergravity
New ones 2
Known ones 1
Which of these two sets contain De Sitter vacua?
1 Dibitetto, Linares, D.R. - in progress2
De Carlos, Guarino, Moreno 09
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Conclusions

• Family tree of supergravity models with dS vacua
• Non-trivial and stable N2 models follow from
• trivial and non-stable N4 models
• New unstable N4 and stable N2 models
• Higher-N origin to Fayet-Iliopoulos terms
• Higher-dimensional origin in terms of gauge,
  geometric or non-geometric fluxes?
• Leads to none of known N4 models!
• Semi-direct instead of direct product gaugings?
• Other compactifications?

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Thanks for your attention!

• Off to the drinks!