De Sitter in Supergravity and String Theory Diederik Roest (RUG) THEP national seminar November 20, 2009

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

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! Extra dimensions Many vacua (~ )? Dualities Branes & fluxes Super- symmetry String theory has many implications: 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 Scalar field with fluxes and branes simple comp.

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? Do these follow from string theory compactifications? Beyond flux compactifications: Gauge fluxes Geometric fluxes Non- geometric fluxes

2. Gauged supergravity

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?

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?

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.

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 stable dS in various theories [3] [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] Requirements for dS in gauged supergravity? Relations between different models? Relation to string theory compactifications?

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 n V =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): 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 = G 1 x G 2 x … Most often considered but not unique! f ® MNP ; ® = ( + ; ¡ ) ; M = ( 1 ;:::; 12 ) : [1: Schön, Weidner ‘06]

N=4 gauged supergravity Crucial for moduli stabilisation: If gauge group is direct product of factors G = G 1 x G 2 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. V ( Á ; ~ ' ) = eÁV 0 ( ~ ' ) [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 = G 1 x G 2 given by [1] SO(4), SO(3,1) or SO(2,2). Gauge factors specified by [2]- coupling constant g 1,2 - embedding parameter h 1,2 (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 n V vector multiplets and n H hyper multiplets Global symmetry* SL(2) x SO(2, n V -1) x SO(4, n H ) Scalars in cosets of global symmetry Vectors in fundamental rep. of SO(2, n V ), 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]

Truncations N=8: SO(5,3) SO(4,4) N=4: 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) N=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 h 1,2 = 0 embedding parameter h 1,2 = 1 unstable stable 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 4D gauged supergravity 4D gauged supergravity Which of these two sets contain De Sitter vacua? Compactification with gauge and (non-)geometric fluxes

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: H mnp ; Q m np : F mnp ; P m np :

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 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 f mnp + = ² mnpqrs F qrs ; f + m np = Q m np : f mnp ¡ = ² mnpqrs H qrs ; f ¡ m np = P m np :

Fate of dS in compactifications? This year it was shown that one can build up gaugings of the form G = G 1 x G 2 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 4D gauged supergravity 4D gauged supergravity Which of these two sets contain De Sitter vacua? New ones [2] Known ones [1] G + £ G ¡ G + n G ¡ [1: Dibitetto, Linares, D.R. - in progress] [2: De Carlos, Guarino, Moreno ’09] Compactification with gauge and (non-)geometric fluxes

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 non- geometric fluxes?

Truncations N=8: SO(5,3) SO(4,4) N=4: 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) N=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 h 1,2 = 0 embedding parameter h 1,2 = 1 unstable stable 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 4D gauged supergravity 4D gauged supergravity Compactification with gauge and (non-)geometric fluxes Which of these two sets contain De Sitter vacua? New ones [2] Known ones [1] G + £ G ¡ G + n G ¡ [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 non- geometric 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!