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Effective supergravity descriptions of superstring cosmology

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

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Effective supergravity descriptions of superstring cosmology

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

  2. 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).

  3. Plan • Cosmology, supergravity and cosmic strings, or the problem of uplifting terms • Supergravities: catalogue, geometries, gauged supergravities • N=1 and N=2 supergravities:multiplets, potential • Superconformal methods: or: Symplification of supergravity by using a parent rigid supersymmetric theory • Examples • cosmic string in N=1 and N=2 • other embeddings of manifolds • producing (meta)stable de Sitter vacua • Final remarks

  4. 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 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 ?

  5. 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

  6. 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

  7. |f | x r D = gx D = 0 Cosmic string solution Abrikosov, Nielsen, Olesen BPS solution: ½ susy

  8. 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/0402046 AVP, Supergravity with Fayet-Iliopoulos terms and R-symmetry, hep-th/0410053.

  9. |f | x r x represents energy of D3 brane system tachyon ↔ field f D = gx D = 0 G. Dvali, R. Kallosh and AVP, hep-th/0312005 Effective supergravity description: Cosmic string solution Abrikosov, Nielsen, Olesen BPS solution: ½ susy

  10. 2. Supergravities Dimensions and # of supersymmetries See discussion inAVP, hep-th/0301005

  11. 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 determinedGauge 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.

  12. 8 susys: very special, special Kähler and quaternionic-Kähler U(1) part in holonomy group SU(2)=USp(2) part in holonomy group The map of geometries • With > 8 susys: symmetric spaces • 4 susys: Kähler: U(1)part in isotropy group

  13. 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.

  14. 3. N=1 and N=2 supergravities

  15. Bosonic terms

  16. Potentials • In N=1 determined by the holomorphic superpotentialW() 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

  17. dchiral fermions dgravitino dgaugino F-term D-term depends on gauge transformations and on K + arbitrary constant xa (for U(1) factors) Fayet-Iliopoulos term N=1 Potential V = åfermions (dfermion) (metric) (dfermion) • General fact in supergravity depends on superpotential W, and on K (MP-2 corrections)

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

  19. 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

  20. scalar field (compensator) conformal gravity: f Poincaré gravity by gauge fixing dilatational gauge fixing • First action is conformal invariant, • Scalar field had scale transformation df (x)=LD(x)f(x) choicedetermines MP See: negative signature of scalars ! Thus: if more physical scalars: start with (– ++...+)

  21. 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*

  22. Gauge fix dilatations and U(1) n-dimensional Hodge-Kähler manifold Superconformal methods for N=1 d=4 (n+1) – dimensional Kähler manifold with conformal symmetry (a closed homothetic Killingvector ki) (implies a U(1) generated by kj Jj i)

  23. 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:

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

  25. 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)

  26. |f | x BPS solution: ½ susy r D = gx D = 0 C= r C= r( 1- x MP-2) leads to deficit angle Cosmic string solution 1 chiral multiplet (scalar f ) charged under U(1) of a vector multiplet (Wm ), and a FI term x Abrikosov, Nielsen, Olesen

  27. 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.

  28. Embedding in N=2 Augmenting the supersymmetry

  29. 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

  30. Consistent truncation and geodesic motion • Consistent truncation • 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=2 consistent truncations have been considered in details in papers of L. Andrianopoli, R. D’Auria and S. Ferrara, 2001

  31. 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:

  32. 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

  33. 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

  34. (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.

  35. A few years ago similar models where found for d=4,N=2 P. Frè, M. Trigiante, AVP, 0205119 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 • 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

  36. 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!)

  37. 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.

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