Effective supergravity descriptions of superstring cosmology

Antoine Van Proeyen

K.U. Leuven

Barcelona, IRGAC, July 2006

From strings to supergravity Landscape of vacua of string theories is a landscape

of supergravities The basic string theories have a supergravity as field

theory approximation. Also after the choice of a compact manifold one is

left with an effective lower dimensional supergravity with number of supersymmetries determined by the Killing spinors of the compact manifold

Fluxes and non-perturbative effects lead to gauged supergravities

Not every supergravity is interpretable in terms of strings and branes (yet).

Plan1. Cosmology, supergravity and cosmic strings,

or the problem of uplifting terms2. Supergravities: catalogue, geometries, gauged supergravities3. N=1 and N=2 supergravities: multiplets, potential4. Superconformal methods: or: Symplification of

supergravity by using a parent rigid supersymmetric theory5. Examples

• cosmic string in N=1 and N=2• other embeddings of manifolds• producing (meta)stable de Sitter vacua

6. Final remarks

Old conflict of supergravity with cosmology:

the cosmological constant The smallness of the cosmological constant

already lead to the statement ‘The world record of discrepancy of theory and ‘The world record of discrepancy of theory and experiment’. experiment’. Indeed, any model contains the Planck mass, i.e. 10120 eV versus cosmol.constant 10-4 eV

But a main problem is now: the sign of the cosmological constant.

Supersymmetry is not preserved for any solution with positive cosmological constant. Why ?

Supersymmetry with (anti) de Sitter

Preserved Susy: must be in a superalgebra

Nahm superalgebras:bosonic part is (A)dS £ R

AdS: compact R-symmetry group:

dS: non-compact R ! pseudo-SUSYneeds negative kinetic energies

Supergravity Supergravity is the field theory corresponding

to superstring theory. For calculations it is useful to find an effective

supergravity description Needs ‘uplifting’ terms’, e.g. in KKLT

mechanism Other example: effective theory of cosmic

string

Cosmic string solution

r

D = g D = 0

Abrikosov, Nielsen, Olesen

BPS solution: ½ susy

The cosmic string model

A supergravity model for the final state after the D3-brane – anti-D3-brane annihilation : a D1 string

‘FI term’ represents brane-antibrane energy. In general: a positive term in the potential

P. Binétruy, G. Dvali, R. Kallosh and AVP, ‘FI terms in supergravity and cosmology’, hep-th/0402046AVP, Supergravity with Fayet-Iliopoulos terms and R-symmetry, hep-th/0410053.

Cosmic string solution

r

D = g D = 0

Abrikosov, Nielsen, Olesen

BPS solution: ½ susy

represents energy of D3 brane systemtachyon ↔ field

G. Dvali, R. Kallosh and AVP, hep-th/0312005

Effective supergravity description:

2. Supergravities

d 32 24 20 16 12 8 4

11 M

10 IIA IIB I

9 N=2 N=1

8 N=2 N=1

7 N=4 N=2

6 (2,2) (2,1) (1,1) (2,0) (1,0)

5 N=8 N=6 N=4 N=2

4 N=8 N=6 N=5 N=4 N=3 N=2 N=1

SUGRA SUGRA/SUSY SUGRA SUGRA/SUSY

See discussion inAVP, hep-th/0301005

Dimensions and # of supersymmetries

What is determined by specifying dmension d and N (i.e.Q) ?

What remains to be determined ? 32 ≥ Q > 8: Once the field content is determined:

kinetic terms determined. Gauge group and its action on scalars to be determined. Potential depends on this gauging

Q = 8: kinetic terms to be determined Gauge group and its action on scalars to be determined. Potential depends on this gauging

Q = 4: (d=4, N=1): potential depends moreover on a superpotential function W.

With > 8 susys: symmetric spaces

The map of geometries

4 susys: Kähler: U(1) part in isotropy group

8 susys: very special,

special Kähler and quaternionic-Kähler

SU(2)=USp(2) part in holonomy group

U(1) part in holonomy group

Gauge group

Number of generators = number of vectors. This includes as well vectors in supergravity

multiplet and those in vector multiplets (cannot be distinguished in general)

The gauge group is arbitrary, but to have positive kinetic terms gives restrictions on possible non-compact gauge groups.

3. N=1 and N=2 supergravities

Multiplets with spin · 2

N=1 N=2graviton m. (2, ) graviton m. (2, , ,1)

vector mult. (1, ) vector mult. (1, , ,0,0)(special) Kähler

geometry

chiral mult. ( , 0,0)Kähler

geometry

hypermult. ( , ,0,0,0,0)quaternionic-

Kähler

Bosonic termsMultiplets with spin · 2

N=1 N=2graviton m. (2, ) graviton m. (2, , ,1)

vector mult. (1, ) vector mult. (1, , ,0,0)

chiral mult. ( , 0,0) hypermult. ( , ,0,0,0,0)

Potentials

In N=1 determined by the holomorphic superpotential W() and by the gauging (action of gauge group on scalar manifold)- Isometries of Kähler manifold of the scalar manifold

can be gauged by vectors of the vector multiplets In N¸ 2 only determined by gauging

(action of gauge group on hypermultiplets and vector multiplet scalars)- Vector multiplet scalars are in adjoint of gauge group. - Hypermultiplets: isometries can be gauged by the

vectors of the vector multiplets

N=1 Potential

General fact in supergravity

V = fermions ( fermion) (metric) ( fermion)

F-term D-term

gravitino gaugino chiral fermions

depends on superpotential W, and on K (MP

-2 corrections)

Fayet-Iliopoulos termFayet-Iliopoulos term

depends on gauge transformations and on K + arbitrary constant (for U(1) factors)

4. Superconformal methods

Superconformal idea, illustration Poincaré gravity

Superconformal formulation for N=1, d=4 R-symmetry in the conformal approach The potential

or: Simplification of supergravity by using a parent rigid supersymmetric theory

The idea of superconformal methods Difference susy- sugra: the concept of multiplets

is clear in susy, they are mixed in supergravity Superfields are an easy conceptual tool

Gravity can be obtained by starting with conformal symmetry and gauge fixing.

Before gauge fixing: everything looks like in rigid supersymmetry + covariantizations

Poincaré gravity by gauge fixing• scalar field (compensator)

conformal gravity:

See: negative signature of scalars ! Thus: if more physical scalars: start with (– ++...+)

First action is conformal invariant, Scalar field had scale transformation (x)=D(x)(x)

choice determines MP

dilatational gauge fixing

Superconformal formulation for N=1, d=4

superconformal group includes dilatations and U(1) R-symmetry

Super-Poincaré gravity = Weyl multiplet: includes (auxiliary) U(1) gauge field + compensating chiral multiplet

Corresponding scalar is called ‘conformon’: Y Fixing value gives rise to MP :

U(1) is gauge fixed by fixing the imaginary part of Y, e.g. Y=Y*

Superconformal methods for N=1 d=4(n+1) – dimensional Kähler manifold with conformal symmetry (a closed homothetic Killing vector ki)

(implies a U(1) generated by kj Jj i )

Gauge fix dilatations and U(1)

n-dimensional Hodge-Kähler manifold

Potential (in example d=4, N=1)

F-term potential is unified by including the extra chiral multiplet:

D-term potential: is unified as FI is the gauge transformation of the compensating scalar:

5. Examples

• cosmic string in N=1 and N=2• other embeddings of manifolds• producing stable de Sitter vacua

N=1 supergravity for cosmic string N=1 supergravity consists of :

- pure supergravity: spin 2 + spin 3/2 (“gravitino”)

- gauge multiplets: spin 1 + spin ½ (“gaugino”) vectors gauge an arbitrary gauge group

- chiral multiplets: complex spin 0 + spin ½ in representation of gauge group

For cosmic string setup:- supergravity

- 1 vector multiplet : gauges U(1) (symmetry Higgsed by tachyon)

- 1 chiral multiplet with complex scalar: open string tachyon: phase transformation under U(1)

Cosmic string solution1 chiral multiplet (scalar ) charged under U(1) of a vector multiplet (W ), and a FI term

r

D = g D = 0

C = r C = r( 1- MP-2)

leads to deficit angle

Abrikosov, Nielsen, Olesen

BPS solution: ½ susy

Amount of supersymmetry

FI term would give complete susy breaking Cosmic string: ½ susy preservation Supersymmetry completely restored far from

string: FI-term compensated by value of scalar field.

Embedding in N=2

Augmenting the supersymmetry

Consistent reduction of N=2 to N=1

Trivial in supersymmetry: any N=2 can be written in terms of N=1 multiplets → one can easily put some multiplets to zero.

Not in supergravity as there is no (3/2,1) multiplet This is non-linearily coupled to all other multiplets

Multiplets with spin · 2

N=1 N=2graviton m. (2, ) graviton m. (2, , ,1)

vector mult. (1, ) vector mult. (1, , ,0,0)

chiral mult. ( , 0,0) hypermult. ( , ,0,0,0,0)

Consistent truncation and geodesic motion

Sigma models: scalars of N=2 theory- e.o.m.: geodesic motion

- consistent truncation: geodesic submanifold: any geodesic in the submanifold is a geodesic of the ambient manifold

Consistent truncation

N=2 consistent truncations have been considered in details in papers of L. Andrianopoli, R. D’Auria and S. Ferrara, 2001

The truncation of N=2 The quaternionic-Kähler manifold MQK must

admit a completely geodesic Kähler-Hodge submanifold MKH : MKH ½ MQK

for the 1-dimensional quaternionic-Kähler:

Multiplets with spin · 2

N=1 N=2graviton m. (2, ) graviton m. (2, , ,1)

vector mult. (1, ) vector mult. (1, , ,0,0)

chiral mult. ( , 0,0) hypermult. ( , ,0,0,0,0)

P-terms to D-terms

Triplet P-terms of N=2 are complex F-terms and real D-terms of N=1

Projection to N=1 should be ‘aligned’ such that only D-terms remain to get a BPS string solution

Ana Achúcarro, Alessio Celi, Mboyo Esole, Joris Van den Bergh and AVP, D-term cosmic strings from N=2 supergravity, hep-th/0511001

Embeddings more general

What we have seen: only a part of the scalar manifold is non-trivial in the solution.

In this way a simple model can still be useful in a more complicated theory: the subsector that is relevant for a solution is the same as for a more complicated theory

Such situations have been considered recently for more general special geometries (N=2 theories) in a recent paper: ‘Tits-Satake projections of homogeneous special geometries’, P. Fré, F. Gargiulo, J. Rosseel, K. Rulik, M. Trigiante and AVP

(meta)stable de Sitter vacua

N=1: many possibilities, see especially KKLT-like constructions in A. Achúcarro, B. de Carlos, J.A. Casas and L. Doplicher, De Sitter vacua from uplifting D-terms in effective supergravities from realistic strings, hep-th/0601190

N=2 much more restrictive. Remember: potential only related to gauging.

Stable de Sitter vacua from N=2 supergravity

A few models have been found in d=5, N=2 (Q=8) that allow de Sitter vacua where the potential has a minimum at the critical point B. Cosemans, G. Smet, 0502202

P. Frè, M. Trigiante, AVP, 0205119

A few years ago similar models where found for d=4, N=2

These are exceptional: apart from these constructions all de Sitter extrema in theories with 8 or more supersymmetries are at most saddle points of the potential

Main ingredients

- Symplectic transformations (dualities) from the standard formulation in d=4 - Tensor multiplets (2-forms dual to vector multiplets) in d=5, allowing gaugings that are not possible for vector multiplets

Non-compact gauging Fayet-Iliopoulos terms (again the uplifting!)

6. Final remarks

There is a landscape of possibilities in supergravity, but supergravity gives also many restrictions.

For consistency, a good string theory vacuum should also have a valid supergravity approximation.

For cosmology, the supergravity theory gives computable quantities.

We have illustrated here some general rules, but also some examples that show both the possibilities and the restrictions of supergravity theories.