Effective supergravity descriptions of superstring cosmology Antoine Van Proeyen K.U. Leuven Barcelona, IRGAC, July 2006

From strings to supergravity n Landscape of vacua of string theories is a landscape of supergravities n The basic string theories have a supergravity as field theory approximation. n 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 n Fluxes and non-perturbative effects lead to gauged supergravities n Not every supergravity is interpretable in terms of strings and branes (yet).

Plan 1. Cosmology, supergravity and cosmic strings, or the problem of uplifting terms 2. Supergravities: catalogue, geometries, gauged supergravities 3. N=1 and N=2 supergravities: multiplets, potential 4. Superconformal methods: or: Symplification of supergravity by using a parent rigid supersymmetric theory 5. 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 world record of discrepancy of theory and experiment’. n The smallness of the cosmological constant already lead to the statement ‘The world record of discrepancy of theory and experiment’. Indeed, any model contains the Planck mass, i.e eV versus cosmol.constant eV n But a main problem is now: the sign of the cosmological constant. n Supersymmetry is not preserved for any solution with positive cosmological constant. Why ?

Supersymmetry with (anti) de Sitter n Preserved Susy: must be in a superalgebra Nahm superalgebras: bosonic part is (A)dS £ R n AdS: compact R- symmetry group: dS: non-compact R ! pseudo-SUSY needs negative kinetic energies

Supergravity n Supergravity is the field theory corresponding to superstring theory. n For calculations it is useful to find an effective supergravity description n Needs ‘uplifting’ terms’, e.g. in KKLT mechanism n 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 n A supergravity model for the final state after the D3-brane – anti-D3-brane annihilation : a D1 string n ‘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/ AVP, Supergravity with Fayet- Iliopoulos terms and R-symmetry, hep- th/

Cosmic string solution r    D = g  D = 0 Abrikosov, Nielsen, Olesen BPS solution: ½ susy   represents energy of D3 brane system tachyon ↔ field  G. Dvali, R. Kallosh and AVP, hep-th/ Effective supergravity description:

2. Supergravities d M 10IIAIIBI 9N=2N=1 8N=2N=1 7N=4N=2 6(2,2)(2,1)(1,1)(2,0)(1,0) 5N=8N=6N=4N=2 4N=8N=6N=5N=4N=3N=2N=1 SUGRA SUGRA/SUSY SUGRA SUGRA/SUSY See discussion in AVP, hep-th/ Dimensions and # of supersymmetries

What is determined by specifying dmension d and N (i.e.Q) ? What remains to be determined ? n 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 n Q = 8: kinetic terms to be determined Gauge group and its action on scalars to be determined. Potential depends on this gauging n Q = 4: (d=4, N=1): potential depends moreover on a superpotential function W.

n With > 8 susys: symmetric spaces The map of geometries n 4 susys: Kähler: U(1) part in isotropy group n 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 n Number of generators = number of vectors. n This includes as well vectors in supergravity multiplet and those in vector multiplets (cannot be distinguished in general) n The gauge group is arbitrary, but to have positive kinetic terms gives restrictions on possible non-compact gauge groups.

Gauged supergravity for applications n Essential progress: stabilisation of moduli by supergravity potentials n N=1: superpotential produced by fluxes -e.g. Gukov-Vafa-Witten result for CY in IIB -Analogous result in heterotic string -Or understand from

Structure of the action n (D=4) with fields of spin 2, 1, 0, 3/2, 1/2 n Kinetic terms determined by some matrices. They describe the geometry n Gauged supergravity: What are the covariant derivatives ? The field strenghts can be non-abelian. Potential V ? If these (from a higher-dimensional theory) get scalar values, these terms produce new contributions to the potential V.

3. N=1 and N=2 supergravities Multiplets with spin · 2 N=1 N=2 graviton 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 terms Multiplets with spin · 2 N=1 N=2 graviton 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 n General fact in supergravity V =  fermions (  fermion) (metric) (  fermion) F-term D-term  gravitino  gaugino  chiral fermions depends on superpotential W, and on K (M P -2 corrections) Fayet-Iliopoulos term   depends on gauge transformations and on K + arbitrary constant   (for U(1) factors)

N=2 Potential P I analogous to D  in N=1: moment maps associated to isometries in manifolds with a symplectic structure. FI terms FI terms are undetermined constants in D or P V =  fermions (  fermion) (metric) (  fermion)  gravitino  hyperino  gaugino Killing vectors on quaternionic-Kähler manifold X I (z) symplectic representation of scalars of vector multiplets

4. Superconformal methods n Superconformal idea, illustration Poincaré gravity n Superconformal formulation for N=1, d=4 n R-symmetry in the conformal approach n The potential or: Simplification of supergravity by using a parent rigid supersymmetric theory

The idea of superconformal methods n Difference susy- sugra: the concept of multiplets is clear in susy, they are mixed in supergravity n Superfields are an easy conceptual tool n Gravity can be obtained by starting with conformal symmetry and gauge fixing. n 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 (– ) n First action is conformal invariant, Scalar field had scale transformation  (x)=  D (x)  (x)  choice determines M P dilatational gauge fixing

Superconformal formulation for N=1, d=4 n superconformal group includes dilatations and U(1) R-symmetry n Super-Poincaré gravity = Weyl multiplet: includes (auxiliary) U(1) gauge field + compensating chiral multiplet n Corresponding scalar is called ‘conformon’: Y n Fixing value gives rise to M P : n 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 k i ) (implies a U(1) generated by k j J j i ) Gauge fix dilatations and U(1) n-dimensional Hodge-Kähler manifold

R-symmetry in conformal approach n R-group is part of the superconformal group. n Should then be gauge-fixed n Isometries act also on compensating scalars n The moment map is the transformation of the compensating scalars under the isometry In conformal formulation: R £ Isom Gauge fixing of R : R £ Isom ! Isom. But the remaining isometries have contribution from R-symmetry

Potential (in example d=4, N=1) n F-term potential is unified by including the extra chiral multiplet: n 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 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 n 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 solution  1 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-  M P -2 ) leads to deficit angle Abrikosov, Nielsen, Olesen BPS solution: ½ susy

Amount of supersymmetry n FI term would give complete susy breaking n Cosmic string: ½ susy preservation n 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 n Trivial in supersymmetry: any N=2 can be written in terms of N=1 multiplets → one can easily put some multiplets to zero. n Not in supergravity as there is no (3/2,1) multiplet n This is non-linearily coupled to all other multiplets Multiplets with spin · 2 N=1 N=2 graviton 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 n 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 n 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 M QK must admit a completely geodesic Kähler-Hodge submanifold M KH : M KH ½ M QK n for the 1-dimensional quaternionic-Kähler: Multiplets with spin · 2 N=1 N=2 graviton 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 n Triplet P-terms of N=2 are complex F-terms and real D-terms of N=1 n 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/

Embeddings more general n What we have seen: only a part of the scalar manifold is non-trivial in the solution. n 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 n 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 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/ n N=2 much more restrictive. Remember: potential only related to gauging.

Stable de Sitter vacua from N=2 supergravity n 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, P. Frè, M. Trigiante, AVP, n A few years ago similar models where found for d=4, N=2 n 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 n - 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 n Non-compact gauging n Fayet-Iliopoulos terms (again the uplifting!)

6. Final remarks n There is a landscape of possibilities in supergravity, but supergravity gives also many restrictions. n For consistency, a good string theory vacuum should also have a valid supergravity approximation. n For cosmology, the supergravity theory gives computable quantities. n We have illustrated here some general rules, but also some examples that show both the possibilities and the restrictions of supergravity theories.