General Analysis of LARGE General Analysis of LARGE Volume - PowerPoint PPT Presentation

General Analysis of LARGE General Analysis of LARGE Volume Scenarios with String Volume Scenarios with String Loop Moduli Stabilisation Loop Moduli Stabilisation Michele Cicoli Michele Cicoli Department of Applied Mathematics and Theoretical Physics Department of Applied Mathematics and Theoretical Physics University of Cambridge University of Cambridge SP08, Upenn, 29 May 2008 SP08, Upenn, 29 May 2008 Based on: 1) M. Cicoli, J. Conlon, F. Quevedo arXiv:0708.1873 [hep-th] 2) M. Cicoli, J. Conlon, F. Quevedo arXiv:0805.1029 [hep-th] 3) M. Cicoli, F. Quevedo (in preparation)

Outline Outline • Brief Review of Moduli Stabilisation in IIB Brief Review of Moduli Stabilisation in IIB • SUGRA SUGRA • General Analysis of LARGE Volume General Analysis of LARGE Volume • Scenarios Scenarios • General Structure of String Loop Corrections General Structure of String Loop Corrections • in IIB SUGRA in IIB SUGRA • Inclusion of String Loop Corrections Inclusion of String Loop Corrections • • Calabi Calabi- -Yau examples Yau examples • • Conclusions Conclusions •

Type IIB CY Flux Compactifications Type IIB CY Flux Compactifications • • Low energy limit to N=1 4D Type IIB SUGRA via dimensional Low energy limit to N=1 4D Type IIB SUGRA via dimensional reduction reduction Need to know f, W, K! Need to know f, W, K! • Ubiquitous presence of moduli: massless uncharged scalar particles es • Ubiquitous presence of moduli: massless uncharged scalar particl with effective gravitational coupling that would give rise to long ng with effective gravitational coupling that would give rise to lo range unobserved fifth forces range unobserved fifth forces Need to stabilise them! Need to stabilise them! • Closed string moduli: U (complex structure), S (axio- -dilaton), T dilaton), T • Closed string moduli: U (complex structure), S (axio (Kä (K ähler) hler) • • Open string moduli: Wilson lines, D3 and D7 moduli Open string moduli: Wilson lines, D3 and D7 moduli • • Turn on background 3- Turn on background 3 -form fluxes (GKP) form fluxes (GKP) D U W = D D s W =0 =0 fixes U and S moduli fixes U and S moduli supersymmetrically supersymmetrically D U W = s W • No- -scale structure flat potential for T moduli at tree scale structure flat potential for T moduli at tree level level • No T moduli still unfixed! T moduli still unfixed! • • Need to study perturbative versus non- Need to study perturbative versus non -perturbative corrections! perturbative corrections!

Perturbative vs vs Non perturbative Non perturbative Perturbative negligible negligible • • In general: In general: where for where for and with and with Neglect loop corrections and get Neglect loop corrections and get

LARGE Volume LARGE Volume • • Fix T moduli in a natural Fix T moduli in a natural way way and V V ~exp( τ i i τ • Set W W 0 ~ 1 1- -10 10 and ~exp(a a i ) • Set 0 ~ i ) α ’ α ’ and non and non- -perturbative corrections compete naturally to give perturbative corrections compete naturally to give an exponentially large volume AdS minimum that breaks an exponentially large volume AdS minimum that breaks SUSY SUSY α ’ perturbative α • Need to up- -lift to dS (anti lift to dS (anti- -D3, D D3, D- -terms, F terms, F- -terms, non terms, non- -perturbative ’ corr corr.) .) • Need to up • Simplest realisation of LVS: � � P P 4 with h 11 =2 • Simplest realisation of LVS: 4 [1,1,1,6,9] with h 11 =2 [1,1,1,6,9] • Generalisation: Swiss- -cheese CY cheese CY (F (F 11 , � � P P 4 4 � P P 4 4 for h 11 =3 ) • Generalisation: Swiss 11 , [1,3,3,3,5], � [1,1,3,10,15] for h 11 =3 ) [1,3,3,3,5], [1,1,3,10,15]

Mass scales scales Mass for V V ~ (for ~ 10 10 15 ) ( 15 l l s 6 ) 6 s NB Get NB Get a a robust effective field theory robust effective field theory and generate and generate hierarchies hierarchies (axionic, weak weak and neutrino scale)!!! and neutrino scale)!!! (axionic,

Some open questions questions in LVS in LVS Some open • No general analysis general analysis: : what happens for an arbitrary what happens for an arbitrary CY? CY? • No ⇒ LARGE Volume ⇒ LARGE Volume Claim Claim • No inclusion inclusion of of loop corrections for an arbitrary loop corrections for an arbitrary CY CY • No (except except Swiss Swiss- -cheese cheese CY CY by by BHP) BHP) ( ⇒ general structure of string loop corrections in IIB SUGRA ⇒ general structure of string loop corrections in IIB SUGRA • Tension between moduli moduli stabilisation stabilisation and and chirality chirality • Tension between ⇒ string loop corrections can give a solution: ⇒ string loop corrections can give a solution: � � P P 4 4 [1,3,3,3,5] [1,3,3,3,5] • Inflation flatness spoiled by loop corrections • Inflation flatness spoiled by loop corrections • No detectable tensor modes detectable tensor modes • No ⇒ string loop corrections can give a solution: ⇒ string loop corrections can give a solution: K3 Fibration K3 Fibration ??? ???

LARGE Volume Claim LARGE Volume Claim → skip it long → Proof: 40 : 40 pages pages long skip it! ! Proof

Necessary and Sufficient Necessary and Sufficient Conditions for LARGE Volume Conditions for LARGE Volume ξ > 0 > 1 ξ • h 12 > h 11 > 0 • h 12 > h 11 > 1 • At least one blow- -up mode (point up mode (point- -like singularity) with a non like singularity) with a non- - • At least one blow perturbative superpotential (guaranteed since the cycle is rigid!) perturbative superpotential (guaranteed since the cycle is rigid !) perturbative effects, volume by α α ’ • • Blow- Blow -up modes fixed by non up modes fixed by non- -perturbative effects, volume by ’ corrections + W W np corrections + np • For N N small blow- -up modes, there are still L=(h up modes, there are still L=(h 11 -N N small -1) moduli which 1) moduli which • For small blow 11 - small - are sent large (e.g. fibration moduli) are sent large (e.g. fibration moduli) their non- -perturbative corrections are switched off perturbative corrections are switched off their non • Get L flat directions! • Get L flat directions! • These directions are usually lifted by string loop corrections • These directions are usually lifted by string loop corrections L moduli lighter than the volume! L moduli lighter than the volume!

String Loop Corrections to K String Loop Corrections to K • • Explicit calculation known only for unfluxed toroidal orientifolds Explicit calculation known only for unfluxed toroidal orientifolds as as (BHK) (BHK) where where is due due to to the the exchange exchange of KK of KK strings between strings between D7s and D3s and D7s and D3s and is is due is due to to the the exchange exchange of Winding of Winding strings between intersecting strings between intersecting D7s D7s NB Complicated dependence NB Complicated dependence on the U moduli BUT on the U moduli BUT simple dependence simple dependence on on the T moduli! the T moduli!

Generalisation to CY Generalisation to CY • Generalisation to Calabi Calabi- -Yau Yau three three- -folds folds (BHP) (BHP) • Generalisation to where either where either or or × Z 2 × In fact for fact for T T 6 6 /(Z /(Z 2 Z 2 ): In 2 ): Conjecture for Conjecture for transverse to D7 transverse to D7 an arbitrary CY! CY! an arbitrary intersection of D7s intersection of D7s

Low Energy Approach Low Energy Approach • • The reduced The reduced DBI action DBI action contains contains a a term like term like τ around its Expand τ • around its VEV VEV • Expand • Read off the gauge off the gauge coupling coupling • Read • Loop corrected kinetic terms • Loop corrected kinetic terms • Analogy with charged scalar scalar fields fields • Analogy with charged

Example: : � � P P 4 4 Example [1,1,1,6,9] [1,1,1,6,9] • • Match the conjecture for Match the conjecture for • Loop corrections from corrections from the the conjecture for conjecture for � � P P 4 • Loop 4 [1,1,1,6,9] [1,1,1,6,9] (BHP) (BHP) α ’ the α NB Loops Loops are are leading with leading with respect to respect to the ’ corrections corrections! ! NB Low Energy Interpretation: Low Energy Interpretation : OK! OK! • Match the conjecture also for conjecture also for � � P P 4 • Match the 4 [1,1,2,2,6] [1,1,2,2,6]

Extended No No- -scale scale Structure Structure Extended 1 and Proof: : Expand Expand K K - and use homogeneity use homogeneity! ! Proof -1 α ’ The loop corrections to V are subleading with respect to the α ’ ones BUT ones BUT The loop corrections to V are subleading with respect to the are crucial to stabilise the SM cycle or to lift the L flat directions!!! are crucial to stabilise the SM cycle or to lift the L flat dire ctions!!!

General formula formula for for the 1 the 1 loop loop General corrections to V V corrections to Homogeneity allows us to simplify it to: : NB We know NB We know the the Homogeneity allows us to simplify it to sign of of these these sign corrections!!! !!! corrections δ K and δ NB Everything Everything in in terms terms of of K K ii K W W !!! !!! NB ii and