Distribution of Vacua in Calabi-Yau Compactification Yuji - PowerPoint PPT Presentation

Distribution of Vacua in Calabi-Yau Compactification Yuji Tachikawa (Particle Theory Group, Univ. of Tokyo, Hongo) based on JHEP 01 (2006) 100 [hep-th/0510061] by Tohru Eguchi and YT March, 2006 @ UTAP , Hongo 0/68

CONTENTS ⇒ 1. On the Landscape & the Swampland ⋄ 2. Flux Compactification ⋄ 3. Statistics of Vacua ⋄ 4. Monodromy and Vacuum Density ⋄ 5. Summary & Comments 1/68

1. String Theory Landscape & Swampland ⋄ Quantization of gravity • because it’s challenging • because it will be needed soon ⇐ spectral index of primordial fluctuation ⋄ Candidates (generally covariant + quantum mechanical): • String (or M) theory • Loop Quantum Gravity ... Pure metric theory. 2/68

String / M theory ⋄ Not originally meant to quantize gravity World lines ⇒ World sheets ⋄ Consistency ⇒ 10 D + graviton ⋄ Many higher-dim’l solitons, branes , which support gauge fields ⋄ 3/68

Compactification 10D ⇒ 4D Minkowski + very small 6D space ⋄ ⋄ Many consistency conditions. ⋄ Semi-realistic models: • Supersymmetrized Standard Models + • Hidden sector for dynamically breaking SUSY • Axion, etc.. which is a triumph for string theory. Presence of Moduli . ⋄ 4/68

Status ⋄ No experimental tests. ⋄ Rich as a theoretical model natural setting for various QFT phenomena • (ADHM, Seiberg-Witten, Montonen-Olive duality etc.) natural setting for various higher-dim’l SUGRA • microscopic account of entropy of BPS black holes • predicted many nontrivial mathematical results • ⋄ Unified most of the research on QFT & SUGRA practitioners 5/68

Moduli Fields Neutral , light field with only Planck-suppressed interaction ⋄ ⋄ How light ? ⇒ massless or SUSY br. or Hubble Corresponding to the ‘ moduli ’ of the compactification manifold ⋄ moduli (pl.) modulus (singl.) : ⋄ parameter(s) in the pure math jargon. VEV of moduli field determines ⋄ the shape & size of the internal manifold. Shape & size determines the Yukawa/gauge couplings. ⋄ 6/68

Moduli Problem Massless scalar ⇒ 5th force ⋄ ⋄ Susy breaking will make them massive ∼ M sb , • Overproduced in preheating • decay after BBN etc. Need to make it much heavier ! ⋄ 7/68

Moduli Fixing in String theory ⋄ Vexing problems for a long time ⇐ Consistency forbids introduction of potentials by hand Flux compactification + D-brane Instanton Correction saved the ⋄ day. ⋄ Roughly speaking, Flux inside internal mfd. ⇒ Tend to spread • D-brane wrapping inside internal mfd. ⇒ Tend to shrink • ⇒ Shape & Size fixed. 8/68

# of choices of flux are HUGE !! ⋄ • Holes in Calabi-Yau: 100 ∼ 200 • Flux per hole is integral, • with upper bound ∼ 100 ⇒ 100 100 of choices ⋄ Flux given ⇒ Moduli fixed ⇒ Shape & size fixed ⇒ Yukawa & gauge coupling Huge # of densely-distributed realizable couplings. ⋄ Huge landscape of 4d vacua . ⋄ 9/68

Really? ⋄ Opinion varies: • Yet-to-unknown consistency condition ⇒ unique solution ? ⇑ • Let’s analyze models at hand statistically ! ⇓ • Any 4d Lagrangian can be UV-completed with gravity ! 10/68

Swampland [Vafa] Q. Which 4d Lagrangian is OK ? ⋄ we’d like to argue without the long detour into 10d string, Calabi-Yau, fluxes and all that messy stuffs. ⋄ Anomaly cancellation. ⇒ Certain gauge groups & matter contents are not allowed. ⋄ Upperbound on the rank of gauge groups ⋄ Gravity should be weaker than gauge coupling [Arkani-Hamed-Motl-Nicolis-Vafa, hep-th/0601001] ⋄ Positivity of certain dimension > 4 operators ⇐ Causality. [Adams-Arkani-Hamed-Dubovsky-Nicolis-Rattazzi , hep-th/0602178] 11/68

CONTENTS � 1. On the Landscape & the Swampland ⇒ 2. Flux Compactification ⋄ 3. Statistics of Vacua ⋄ 4. Monodromy and Vacuum Density ⋄ 5. Summary & Comments 12/68

2. Flux Compactification d = 4 , N = 1 Supergravity { Q α , Q β } = γ µ ⋄ αβ P µ • ( g µν , ψ µ ) ( A a µ , λ a • α ) ( ψ i α , φ i ) • ⋄ P µ gauged ⇒ Q α gauged φ i are complex scalars, G i ¯ ⋄  and V restricted in d 4 x √ g � �  + V ( φ, ¯ � φ ) ∂ µ φ i ∂ µ ¯ φ ¯  ( φ, ¯ G i ¯ φ ) 13/68

K ( φ, ¯ φ ) : K¨ ahler potential , W ( φ ) : superpotential ⇒ ⋄  ( φ, ¯ φ ) = ∂ i ¯  K ( φ, ¯ G i ¯ ∂ ¯ φ ) , φ ) = e K � φ ) − 3 | W ( φ ) | 2 � V ( φ, ¯ G i ¯  D i W ( φ ) ¯ W ( ¯  ¯ D ¯ D i W ( φ ) = ( ∂ i + ( ∂ i K )) W ⋄ K¨ ahler transformation: K ( φ, ¯ φ ) → K ( φ, ¯ φ ) + f ( φ ) + ¯ f ( ¯ φ ) W ( φ ) → e − f ( φ ) W ( φ ) D i W ( φ ) → e − f ( φ ) D i W ( φ )  and V ( φ, ¯ leaves G i ¯ φ ) invariant. 14/68

10d IIB supergravity e − φ , C , g µν , H NSNS [ µνρ ] = ∂ [ µ B NSNS H RR [ µνρ ] = ∂ [ µ B RR , νρ ] , νρ ] F [ µνρστ ] = ∂ [ µ C νρστ ] with constraint F [ µνρστ ] = ǫ µνρσταβγδǫ F [ αβγδǫ ] , +fermions � C (4) ∧ H NSNS ∧ H RR An important coupling: (3) (3) 15/68

Branes � dx µ A µ ⋄ point-like objects couple to A µ via worldline ⋄ objects extended in p -direction couple to ( p + 1) -form fields via � dx µ 0 · · · dx µ p C [ µ 0 ··· µ p ] worldvolume • ⇐ C D(-1) brane = D-instanton B NSNS • ⇐ F1 brane = string B RR • ⇐ D1 brane = D-string • ⇐ C (4) D3 brane � (3) ⇒ H NSNS ∧ H RR has D3-brane charge C (4) ∧ H NSNS ∧ H RR ⋄ (3) 16/68

Calabi-Yau compactification ⋄ 10=4+6 ⋄ 6-dimensional CY = the holonomy SU (3) ⊂ SO (6) x 1 , x 2 , x 3 , x 4 , x 5 , x 6 → z 1 , z 2 , z 3 , ¯ z ¯ z ¯ z ¯ 1 , ¯ 2 , ¯ 3 , ⇒ CY : complex mfd ahler form ω , everywhere nonzero (3 , 0) form Ω with K¨ ⋄ 6d spinor 4 = 3 ⊕ 1 under SU (3) ⇒ 1/4 of SUSY remain ⇒ Type IIB/CY : N = 2 ⇒ breaks SUSY to N = 1 No gauge fields ⇒ put D-branes ⋄ 17/68

Moduli in CY compactification CYs come in various topological types : ⋄ h 1 , 1 two -cycles, h 1 , 1 four -cycles • 2 h 1 , 2 + 2 three -cycles: • A 0 , A 1 , . . . , A h 12 and B 0 , B 1 , . . . , B h 12 so that A i · B j = δ ij and A i · A j = B i · B j = 0 CYs can be continuously deformed , parametrized by ⋄ � • ω ∧ ω : sizes of four-cycles for i = 1 , . . . , h 11 ρ i = C i � • z i = Ω : periods of three-cycles for i = 1 , . . . , h 12 A i 18/68

g mn ( ρ i , z i ) ⋄ The metric of CY varies as ρ i and z i : ⇒ 10d metric : ds 2 = η µν dx µ dx ν + g mn ( ρ i ( x µ ) , z i ( x µ )) dx m dx n � dx 10 � g (10) R (10) ⇒ ⋄ Plug this into S = � dx 4 � g (4) R (4) + S = � � dx 4 � g (4) g µν  + dx 4 � g (4) g µν (4) G ′  ∂ µ ρ i ∂ ν ¯ ρ ¯  ∂ µ z i ∂ ν ¯ z ¯  + (4) G i ¯ i ¯ � ρ i combines with C (4) to become a complex scalar ⋄ C i � � ρ i C (4) ω ∧ ω + complexified = i C i C i 19/68

h 11 + h 12 massless complex scalars in total ⋄ ρ i : called size moduli or K¨ ahler moduli • z i : called shape moduli or complex structure moduli • Axio-dilaton τ = ie − φ + C (0) is also a modulus. ⋄ 20/68

Superpotentials for Moduli Just compactifying on CY leads to W = 0 ⇒ V = 0 . ⋄ ⋄ Masses to all moduli ⇒ We need W depending all variables τ , ρ i , z i . Fluxes give W for τ and z i ’s • Instanton corrections give W for ρ ’s • [Kachru-Kallosh-Linde-Trivedi hep-th/0301240] ⋄ Let’s see each in detail. 21/68

Flux superpotential Type IIB has 2-form potentials B NSNS and B RR ⋄ with 3-form field strengths H NSNS and H RR Quantized fluxes through three-cycles ⋄ ⋄ They give rise to � Ω ∧ ( H RR + τH NSNS ) W = CY h 12 �� � � � � � ( H RR + τH NSNS ) − = Ω Ω ( H RR + τH NSNS ) A i B i B i A i i =0 h 12 ) − ∂F � � � z i ( N RR + τN NSNS ( M RR + τM NSNS = ) i i i i ∂z i i =0 22/68

Comments h 12 ) − ∂F � � z i ( N RR + τN NSNS ( M RR + τM NSNS � W = ) i i i i ∂z i i =0 This depends on string coupling and shape , not on the size . ⋄ ⋄ N i and M i are the number of fluxes, hence integers Linear in Fluxes . ⋄ 23/68

This form for W : obtainable by a standard KK reduction ; ⋄ ⋄ or, from the domain-wall tension [Gukov]: • Wrap ( p, q ) 5-brane on A i : ⇒ a BPS domain wall in 4d point of view. � � ⇒ The tension should be � W | ∞ − W | −∞ � from 4d SUGRA. � � � � � � � • The tension is � ( p + τq ) Ω � , from the ( p, q ) -brane action. � � � � A i • p units of H RR and q units of H NSNS through B i . ⇒ W ! 24/68

Constraint on N i and M i � C (4) ∧ H NSNS ∧ H RR in type IIB sugra. ⋄ A term � C (4) . ⋄ Of course there is a coupling D3 � C (4) to Orientifold planes. ⋄ Another coupling − O3 ⇒ EOM for C (4) leads � H RR ∧ H NSNS # O 3 = # D 3 + h 12 � � N RR i M NSNS − M RR i N NSNS � = # D 3 + i i i =0 ⋄ # O3 is fixed by the geometry of CY. 25/68