Exploring Type II Flux Vacua: SUSY, Non-SUSY, and Non-geometric University of Wisconsin, Madison September 2005 W. Taylor (MIT) hep-th/0505160 (w/ O. De Wolfe, A. Giryavets, S. Kachru) hep-th/0508133 (w/ J. Shelton, B. Wecht) 1
Outline 1. Introduction/Motivation 2. IIB vacua 3. IIA vacua 4. Synthesis: non-geometric compactifications 5. Summary + open questions 2
1. Introduction/Motivation Type IIA/IIB string compactification: X 6 − → M 10 M 4 SUSY, no fluxes: X 6 = Calabi-Yau, M 4 = R 4 Generic Calabi-Yau: Moduli Example: ( T 2 ) 3 in type IIB Complex structure: τ K¨ ahler modulus: U = B xy + i × volume Axiodilaton: S = χ + ie − φ Moduli space: manifold of SUSY vacua 3
Moduli stabilization Turn on integrally quantized (topological) fluxes F ( p ) H abc , a 1 ··· a p ⇒ 4D potential � √ g e − 2 φ | H | 2 + | F | 2 + · · · � � V ∼ M 6 is Moduli dependent Problems: A) Runaway moduli ( V ∼ H 2 / volume 2 ) B) Tadpoles � e.g., A 4 ∧ F 3 ∧ H 3 in IIB Chern-Simons term ⇒ ∇ 2 6 A 4 ∼ F 3 ∧ H 3 One solution: Orientifold planes T O p < 0 , D − charge( O p ) < 0 4
Fluxes + O-planes → moduli stabilization Goal: Study “landscape” of string vacua Motivations: • May connect to phenomenology • May connect to cosmology • May shed light on foundational aspects of string theory “Anthropic”/environmental selection issues of limited practical consequence without a better global picture of range of possible vacua, some dynamical principle/definition of string theory Summary of talk: • We know of many flux vacua • There probably exist many many more 5
2. Type IIB flux vacua Consider integral (topological) IIB fluxes: H abc , F abc Two ways to study: (Giddings-Kachru-Polchinski, . . . ) A) 10D SUGRA S → 4D potential V (moduli) B) Superpotential W for 4D SUGRA (Gukov-Vafa-Witten) Begin with A: � √ g � � e − 2 φ ( R + ( ∂φ ) 2 − | H | 2 ) − � | F ( p ) | 2 S = p − A 4 ∧ H 3 ∧ F 3 + δ (6) D3 , O3 ( T D3 , O3 − A 4 ) A 4 tadpole cancellation: � N D 3 + F 3 ∧ H 3 = N O 3 Varying zero-modes (moduli) gives S → V (moduli) where zero-modes of φ, B, g, A p − 1 are moduli 6
B) Analysis using 4D superpotential Potential can be written V = e K ( DWDW − 3 | W | 2 ) where DW = ∂W + ( ∂K ) W and � W = G ∧ Ω (GVW) (depends only on CS moduli, axiodilaton S ) “no-scale” dependence on K¨ ahler moduli: D K WD K W = 3 | W | 2 gives V = e K ( D CS WD CS W ) SUSY solutions: DW = W = 0 7
Summary of IIB vacuum analysis to date • Equations of motion ∂ moduli V = 0 , DW = 0 ⇒ some moduli stabilized • Potential can be written V ∼ | iG (3) − ∗ G (3) | 2 + · · · vol 2 where G (3) = F (3) − SH (3) . iG (3) = ∗ G (3) ⇔ ISD. • Generically stabilizes complex structure moduli, S • SUSY DW = 0 solutions ISD, V = 0 , M 4 = R 4 • K¨ ahler moduli only stabilized nonperturbatively (Denef/Douglas/Florea/Grassi/Kachru) • Tadpole constraint + ISD ⇒ finite # of inequivalent solutions • Statistical analysis of IIB vacua begun (Douglas, Ashok/Douglas, Denef/Douglas, DGKT, . . . ) 8
3. Type IIA flux vacua Can have fluxes H 3 , F 6 , F 4 , F 2 , F 0 (massive IIA) Use Orientifold 6-plane to cancel A 7 tadpole F 0 H 3 + N D 6 = N O 6 Both analysis methods again possible. A) Explicit computation of 4D potential V V ∼ e 2 φ H 2 vol 7 / 3 + e 4 φ F 2 F 2 O 6 vol 2 + e 4 φ 4 vol − e 3 φ 0 vol 3 / 2 + · · · Note: volume dependence allows K¨ ahler stabilization B) Superpotential formalism (Grimm/Louis) � W Q = Ω c ∧ H 3 J c ∧ F 4 − F 0 � � W K = J c ∧ J c ∧ J c + · · · 6 9
Summary of IIA vacuum analysis • K¨ ahler moduli generically stabilized • Some models: all moduli stabilized (DGKT example: T 6 / Z 2 3 ) • Other models: unstabilized axions — needed to cancel anomaly on branes (Camara/Font/Iba˜ nez) • F 4 unconstrained by tadpole ⇒ ∞ # of vacua • No no-scale structure: for SUSY DW = 0 vacua W = 0 ⇒ Minkowski, W � = 0 ⇒ AdS 4 • Exist controlled families of vacua, g → 0 , volume → ∞ • non-SUSY vacua exist in controlled regime SUSY breaking from flux sign change 10
Simple example of IIA vacua: T 6 / Z 2 3 Consider ( T 2 ) 3 with τ = e 2 πi/ 3 Mod out by T : ( z 1 , z 2 , z 3 ) → ( α 2 z 1 , α 2 z 2 , α 2 z 3 ) Q : ( z 1 , z 2 , z 3 ) → ( α 2 z 1 + 1 + α , α 4 z 2 + 1 + α , z 3 + 1 + α ) 3 3 3 Singular limit of CY, χ = 24 , 9 Z 3 singularities h 2 , 1 = 0 , h 1 , 1 = 12 3 K¨ ahler moduli from tori, 9 from singularities Orientifold: fixed plane of σ : z i → − ¯ z i Holomorphic 3-form i 1 √ Ω = 3 1 / 4 dz 1 ∧ dz 2 ∧ dz 3 = 2 ( α 0 + i β 0 ) 11
Moduli of T 6 / Z 2 3 model: A (3) = ξ α 0 , φ (axion , dilaton) 3 ds 2 = � γ i dz i d ¯ z i i =1 3 β i dz i ∧ d ¯ � z i B 2 = i =1 Metric, B -field components γ i β i ⇒ 3 K¨ ahler moduli Remaining 9 K¨ ahler moduli from blow-up modes. Fluxes (quantized): H bg = − p β 0 3 e 1 dz 2 ∧ d ¯ z 2 ∧ dz 3 ∧ d ¯ z 3 + cyclic F bg � � = constant 4 √ Tadpole condition m 0 p = − 2(2 π α ′ ) No tadpole constraint on e i . 12
Can explicitly solve EOM to get B = A (3) = 0 Potential ( v i = constant × γ i , φ ) 3 √ 2 p 2 e 2 φ i ) e 4 φ e 4 φ e 3 φ V = 1 � e 2 i v 2 vol 3 + m 2 vol 2 + ( vol + 2 2 m 0 p 0 vol 3 / 2 i =1 (vol = constant × γ 1 γ 2 γ 3 ) Solving � 1 � 1 / 6 � 3 � � 1 e 1 e 2 e 3 ds 2 = z i , � | e i | dz i d ¯ � � 5 � � 9 κ m 0 � � i =1 � 5 � 1 / 4 e φ = 3 κ 4 | p | . 12 | m 0 e 1 e 2 e 3 | Scaling of solutions for large e i ∼ E : vol ∼ E 3 / 2 e φ ∼ E − 3 / 4 Λ ∼ E − 9 / 2 HR ∼ E − 1 / 2 So solutions are parametrically under control 13
Further comments on solutions • SUSY solutions: all e i have same sign other signs: non-SUSY controlled solutions ∼ skew-whiffing (Duff/Nillson/Pope) • Can check B -mode stability SUSY solutions: all modes stable non-SUSY solutions: BF-allowed tachyons • Can stabilize blow-up modes with additional F 4 fluxes can choose in regime where blow-up modes ≪ R • Number of vacua with R ≤ R ∗ goes as ( R ∗ ) 4 cutoff dominated • Expect similar results for other models some axions not stabilized, fix anomalies (CFI) 14
4. Synthesis: non-geometric vacua Upshot so far: IIA, IIB vacua seem very different But mirror symmetry: IIA ↔ IIB?? Reconciliation: non-geometric fluxes Example: Consider T 3 with B xy = Nz ⇒ H xyz = N flux T-duality T x : “geometric flux” f x yz ds 2 = ( dx + f x yz zdy ) 2 + dy 2 + dz 2 (twisted tori: Scherk/Schwarz, Kaloper/Myers, . . . ; SU(3) structure: Hitchin, Gurrieri/Louis/Micu/Waldram, . . . ) T y : “non-geometric flux” Q xy z Locally geometric T 2 bundle over T 1 , duality twist in BC’s 1 ds 2 = dx 2 + dy 2 � + dz 2 � 1 + N 2 z 2 Nz B xy = 1 + N 2 z 2 . (Dabholkar/Hull, Hellerman/McGreevy/Williams, Flourney/Wecht/Williams, . . . ) T z : more non-geometric flux R xyz ; not yet understood 15
T-duality rules for NS-NS fluxes T a T b T c → f a → Q ab → R abc H abc ← ← ← bc c Like T-duality rules for R-R fluxes T x F xα 1 ··· α p ← → F α 1 ··· α p Generalize Buscher rules to include 0-forms Example: T 6 = ( T 2 ) 3 in IIA, IIB • Duality ⇒ superpotential, constraints • Demonstrates consistency of NG fluxes moduli IIB IIA τ CS K¨ ahler S axiodilaton axiodilaton U K¨ ahler CS Previously known flux superpotentials IIB: W = P (3) ( τ ) + SP (3) ( τ ) 1 2 (geometric, coefficients F, H ) IIA: W = P (3) ( τ ) + SP (1) ( τ ) + UP (1) ( τ ) 1 2 3 (w/ geometric flux; Villadoro/Zwirner, Camara/Font/Ibanez ) 16
Claim: full IIA/IIB superpotential is W = P (3) ( τ ) + SP (3) ( τ ) + UP (3) ( τ ) 1 2 3 coefficients: NS-NS fluxes H abc , f a bc , Q ab c , R abc Explicit construction (O6 on α, β, γ ) Term IIA flux IIB flux ¯ ¯ 1 F αiβjγk F ijk ¯ ¯ τ F αiβj F ijγ ¯ ¯ τ 2 F αi F iβγ ¯ τ 3 F (0) F αβγ ¯ ¯ S H ijk H ijk ¯ Q αβ U H αβk k ¯ f α Sτ H αjk jk f j βγ Q αj k , Q iβ kα , f i βk , f α k , Q βγ Uτ α ¯ Q αβ Sτ 2 H iβγ k Q γi γ , Q ij γ , Q γi β , Q ij Uτ 2 β , Q iβ Q iβ k k ¯ Sτ 3 R αβγ H αβγ Uτ 3 R ijγ Q ij γ Black: already known; Blue: T-dual of black Green: rotation of blue; Purple: T-dual of Green 17
Use T-duality to find (Bianchi/tadpole) constraints � NS-NS constraints ( ∼ dH = 0 ) H x [ ab f x ¯ cd ] = 0 f a x [ b f x cd ] + ¯ H x [ bc Q ax d ] = 0 H x [ cd ] R [ ab ] x = 0 [ cd ] − 4 f [ a x [ c Q b ] x d ] + ¯ Q [ ab ] f x x xd R bc ] x = 0 x Q c ] x + f [ a Q [ ab d x R cd ] x = 0 . Q [ ab � R-R constraints ( ∼ ( d + H ) F = 0 ) F [ abc ¯ ¯ H def ] = 0 F x [ abc f x ¯ de ] − ¯ F [ ab ¯ H cde ] = 0 F xy [ abc Q xy ¯ d ] − 3 ¯ F x [ ab f x cd ] − 2 ¯ F [ a ¯ H bcd ] = 0 F xyz [ abc ] R xyz − 9 ¯ ¯ F xy [ ab Q xy c ] bc ] + 6 F (0) ¯ − 18 ¯ F x [ a f x H [ abc ] = 0 F xyz [ ab ] R xyz + 6 ¯ ¯ F xy [ a Q xy b ] − 6 ¯ F x f x [ ab ] = 0 F xyza R xyz − 3 ¯ ¯ F xy Q xy = 0 a F xyz R xyz = 0 . ¯ 18
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