Gauged SDFT and stable de Sitter in d = 7 Jose J. Fern - PowerPoint PPT Presentation

Gauged SDFT and stable de Sitter in d = 7 Jose J. Fern´ andez-Melgarejo Harvard University Based on 1505.01301: W. Cho, J.J. FM, I. Jeon, J.-H. Park (JHEP 2015) 1506.01294: G. Dibitetto, J.J. FM, D. Marqu´ es (sub. JHEP 2015) Duality symmetries in String and M-Theories August 10, 2015

Our goals Goal 1: DFT from relaxed (SC) gauge principle. Uniqueness of the action upon what symmetries? Cho,FM,Jeon,Park’15 Generalised diffeomorphisms Double Lorentz transformations O(D,D) ... SUSY? Goal 2: Half-maximal d = 7: natural continuation of previous results Dibitetto,FM,Marqu´ es’15 Classification of deformations Vacua Mass spectrum

Outline 1 Twisted supersymmetric DFT (SDFT) [Cho,Jeon,FM,Park] Relaxation approaches Untwisted SDFT Twisting 2 Stable de Sitter in half-maximal d = 7 [Dibitetto,FM,Marqu´ es] 3 Conclusions

Outline 1 Twisted supersymmetric DFT (SDFT) [Cho,Jeon,FM,Park] Relaxation approaches Untwisted SDFT Twisting 2 Stable de Sitter in half-maximal d = 7 [Dibitetto,FM,Marqu´ es] 3 Conclusions

Geometric formulation Jeon...’11, Hohm...’11, Hohm...’12 S DFT ( H AB , d ) as a curvature: S ABCD Non-vanishing scalar curvatures � J AC J BD � H AC H BD , S ABCD Upon SC: S ABCD J AC J BD = − S ABCD H AC H BD [DFT + SC] lacks some O ( d , d ) gaugings Aldazabal. . . ’11,Geissb¨ uhler’11 Relaxation of the SC?

Relaxation of the SC Aldazabal. . . ’11, Geissb¨ uhler’11, Gra˜ na. . . ’12, Geissb¨ uhler. . . ’13, Berman. . . ’13 Lower-dim SC stronger than quadratic constraints in half-maximal SUGRA S DFT +SC-terms G ABCD = [ S + ∆] ABCD ∆ = ∆( E AM ) Geometric approaches Weitzenb¨ ock connection Generalised flux formulation G ABCD ( α H AC H BD + β J AC J BD ) such that half-max SUGRA DFT action fixed by first principles?

Supersymmetric theories N = 1 SDFT Siegel’93, Hohm. . . ’12, Jeon. . . ’12, Berman. . . ’13 � ρ α , � ¯ ψ α d , V Ap , V A ¯ p , ¯ p N = 2 SDFT Jeon. . . ’12 � ρ α , α , � ¯ C α ¯ ρ ′ ¯ ¯ ψ α ψ ′ α d , V Ap , V A ¯ p , α , p , ¯ p Symmetries: O(10,10) Generalised diffeo’s Spin(1,9) × Spin(9,1) SUSY

SDFT - { V , ¯ V } Double vielbein formalism Siegel, Hohm. . . , Jeon. . . V Ap V Aq = η pq , ¯ p ¯ V A ¯ q = ¯ η ¯ q , V A ¯ p ¯ V Ap V Bp + ¯ p = J AB . V Ap ¯ V A ¯ p ¯ V B ¯ q = 0 , V A ¯ p ¯ P AB = ¯ ¯ P AB = V Ap V Bp V A ¯ Projectors V B ¯ p � � D = ∂ + Γ + Φ + ¯ ¯ Φ D V Ap , p , d , J AB = 0 V A ¯ Connections Γ CAB ( V , ¯ V , d ) Christoffel Φ Apq ( V , ¯ V , d ) Spin(1,9) ¯ q ( V , ¯ Φ A ¯ V , d ) Spin(9,1) p ¯

SDFT - { V , ¯ V } Field strengths � � S ABCD = 1 R ABCD + R CDAB − Γ E AB Γ ECD 2 E Φ | B ] ED F ABCD = 2 ∇ [ A Φ B ] CD − 2Φ [ A | C E ¯ F ABCD = 2 ∇ [ A ¯ ¯ Φ B ] CD − 2¯ Φ [ A | C Φ | B ] ED � � ( F + ¯ F ) ABCD + ( F + ¯ F ) CDAB + (Φ + ¯ AB (Φ + ¯ G ABCD = 1 Φ) E Φ) ECD 2 q ∂ E ¯ p ∂ E V Bp + ¯ p ∂ E ¯ q ∂ E V Dq + ¯ = S ABCD + 1 ¯ ¯ 2 ( V A V A V B ¯ p )( V C V C V D ¯ q ) ( δ X − L X ) G ABCD ∼ ( δ X − L X ) S ABCD ∼ 0 qr q ¯ r Non-vanishing Ricci G pr ¯ G p ¯ r ¯ Non-vanishing scalar curvatures G pqpq q ¯ p ¯ q G ¯ p ¯

SDFT - RR sector O(10,10) scalar bispinor C α ¯ F := ¯ ¯ C − 1 + ( F ) T C + − → F := D + C β where D ± T := γ p D p T ± γ (11) D ¯ γ ¯ p p T ¯ Gauge invariance upon nilpotency of D + δ F = ( D + ) 2 Λ ∼ 0 δ C = D + Λ − → Nilpotency upon strong constraint ( D ± ) 2 T ∼ 0

Twisting Twisted fields ˙ ˙ T A 1 ··· A n ( X , Y ) = e − 2 ωλ ( Y ) U A 1 A n ( Y ) ˙ A 1 ( Y ) · · · U A n A n ( X ) T ˙ A 1 ··· ˙ Twisting derivatives ˙ ˙ A n ˙ ˙ A 1 · · · U A n C ˙ ∂ C T A 1 ··· A n = e − 2 ωλ U C C U A 1 D ˙ T ˙ A 1 ··· ˙ A n where n � B ˙ ˙ ˙ C ˙ A n := ˙ C ˙ A n − 2 ω ˙ C λ ˙ ∂ ˙ ∂ ˙ A n + Ω ˙ D ˙ T ˙ T ˙ T ˙ T ˙ A 1 ··· ˙ A 1 ··· ˙ A 1 ··· ˙ C ˙ A 1 ··· ˙ B ··· ˙ A i A n i =1 � � B : = ˙ U − 1 ˙ ˙ ˙ C : = ( U − 1 ) ˙ C ∂ C B ∂ ˙ Ω ˙ ∂ ˙ C U C ˙ C A ˙ A We define ˙ B ˙ A − 2 ˙ f ˙ C := 3Ω [ ˙ f ˙ A := Ω ∂ ˙ A λ A ˙ B ˙ A ˙ B ˙ B ˙ C ]

Consistency/twistability conditions Twisted ˙ L ˙ X ˙ X ˙ A 1 · · · ( U − 1 ) ˙ A n L X T A 1 ··· A n A n = e 2 ωλ ( U − 1 ) ˙ L ˙ T ˙ A 1 ··· ˙ A 1 A n B ˙ B ˙ ˙ ˙ = ˙ B ˙ A n + ω ˙ B ˙ X D ˙ T ˙ D ˙ X T ˙ A 1 ··· ˙ A 1 ··· ˙ A n n � ˙ ( ˙ A i ˙ B − ˙ B ˙ A i ) ˙ B ˙ + D ˙ X ˙ D ˙ X ˙ T ˙ A 1 ··· ˙ A i +1 ··· ˙ A i − 1 A n i =1 Twisted C-bracket ˙ [ ˙ X , ˙ C = ( U − 1 ) ˙ A A [ X , Y ] A Y ] ˙ C A B ˙ ˙ A − ˙ ˙ B ˙ ˙ A + 1 ˙ B ˙ ˙ A ˙ ˙ B ˙ ˙ A ˙ ˙ = ˙ B ˙ B ˙ 2 ˙ B − 1 2 ˙ X D ˙ Y Y D ˙ X Y D X ˙ X D Y ˙ B Closure of the algebra � � [ ˙ X , ˙ Y ] − ˙ ˙ L ˙ L ˙ L [ ˙ A n � = 0 T ˙ X , ˙ A 1 ··· ˙ Y ] ˙ C

Closure conditions Gra˜ na-Marqu´ es’12 SC for all the dotted twisted fields, M ≡ 0 ˙ ˙ M ˙ ∂ ˙ ∂ Orthogonality between connection and derivatives ˙ M ˙ G ˙ Ω ∂ ˙ M ≡ 0 F ˙ Constancy of f ˙ A ˙ B ˙ C ˙ ∂ ˙ C ≡ 0 E f ˙ A ˙ B ˙ Jacobi identities ˙ E f ˙ f [ ˙ E ≡ 0 A ˙ C ] ˙ D ˙ B Triviality of f ˙ A ˙ A − 2 ˙ A − 2 ˙ C ˙ A λ = ∂ C U C ˙ A = Ω ∂ ˙ ∂ ˙ A λ ≡ 0 f ˙ C ˙

Twisted connections and field strengths Connections ( ∂ → D ) � � � � Φ , ˙ Γ , Φ , ¯ Γ , ˙ ˙ ¯ Φ → Φ (Semi-)covariant generalised curvature P + ˙ P + ˙ X − ˆ X ) ˙ D ≡ ˙ A ( ˙ ¯ D + ˙ C ( ˙ ¯ ( δ ˙ L ˙ G ˙ D [ ˙ P ) ˙ D [ ˙ P ) ˙ A ˙ B ˙ C ˙ B ] ˙ C ˙ D ] ˙ A ˙ B Natural curvature G ABCD q ] T p ≡ ˙ 2 [ ˙ D p , ˙ 1 qr T p D ¯ G pr ¯ q ≡ − ˙ 2 [ ˙ D p , ˙ 1 q ] T ¯ q ¯ r T ¯ q D ¯ G p ¯ r ¯

N = 1 twisted SDFT N = 1 d � Twisted SDFT = e − 2 ˙ G pqpq + i 1 ργ p ˙ ˙ 4 ˙ 1 L Half − maximal 2 ¯ D p ρ � p ˙ p γ q ˙ − i ¯ ψ ¯ 2 ¯ ψ ¯ p ρ − i 1 D ¯ D q ψ ¯ p Twisted transformations for fermions δ ε ρ = − γ p ˙ p = ˙ D p ε , δ ε ψ ¯ D ¯ p ε . C breaks Z 2 symmetry C f ˙ A ˙ B ˙ f ˙ A ˙ B ˙ G pqpq + ˙ q = 1 C f ˙ A ˙ B ˙ ˙ q ¯ p ¯ C G ¯ 6 f ˙ p ¯ A ˙ B ˙ where Aldazabal...’11, Dibitetto...’12 A ˙ ˙ B ˙ C = − 3 ∂ D U A A ∂ D � ˙ U − 1 � A − 24 ∂ D λ ∂ D λ f ˙ C f A ˙ B ˙ ˙ A

N = 2 twisted SDFT N = 2 d � G pqpq − ˙ F ¯ Twisted SDFT = e − 2 ˙ ˙ 8 ( ˙ 2 Tr ( ˙ ˙ 1 q ¯ p ¯ q ) + 1 L Maximal G ¯ F ) p ¯ ργ p ˙ p ˙ + i 1 D p ρ − i 1 ρ ′ ¯ γ ¯ p ρ ′ 2 ¯ 2 ¯ D ¯ p γ q ˙ q ˙ 2 ¯ 2 ¯ − i 1 ψ ¯ p + i 1 ψ ′ p ¯ γ ¯ q ψ ′ D q ψ ¯ D ¯ p D p ρ ′ � F ρ ′ + i ¯ p ψ ′ q − i ¯ p ˙ ψ ′ p ˙ ρ ˙ p γ q ˙ p ρ + i ¯ γ ¯ ψ ¯ − i ¯ ψ ¯ F ¯ D ¯ Twisted SUSY transformations for fermions δ ε ρ = − γ p ˙ δ ε ρ ′ = − ¯ p ˙ γ ¯ p ε ′ D p ε D ¯ D p ε ′ + ¯ p = ˙ p ε + ˙ p = ˙ ˙ p ε ′ δ ε ψ ′ δ ε ψ ¯ D ¯ F ¯ γ ¯ F γ p ε R-R sector C Λ ! C f ˙ A ˙ B ˙ δ C = ˙ δ ˙ F = ( ˙ D + ) 2 Λ ≡ − 1 D + Λ − → = 0 24 f ˙ A ˙ B ˙ G pqpq + ˙ q = 0 ˙ q ¯ p ¯ G ¯ p ¯

Fixing the actions N = 1 − → Z 2 transformation B (2) → − B (2) J → −J ⇔ ˙ Twisted SDFT ( d , V Ap , ¯ p , ρ ′ ¯ α , ψ ′ α ) ¯ L Half − maximal V A ¯ p N = 2 − → Z 2 unbroken G pqpq = −G ¯ q ¯ p ¯ q p ¯ SC relaxation genuinely non-geometric configurations (deformations) stringy origin? Level-matching conditions

Outline 1 Twisted supersymmetric DFT (SDFT) [Cho,Jeon,FM,Park] Relaxation approaches Untwisted SDFT Twisting 2 Stable de Sitter in half-maximal d = 7 [Dibitetto,FM,Marqu´ es] 3 Conclusions

Half-maximal d = 7 SUGRA R + × SL (4) � β � α ˆ e µ a , A µ [ mn ] , B µν , Σ , V m α ˆ α , ψ µα , χ α , λ α ˆ Field content Embedding tensor 10 ′ Θ ∈ ⊕ ⊕ ⊕ 1 ( − 4) 10 (+1) 6 (+1) (+1) � �� � � �� � � �� � � �� � θ ˜ ξ [ mn ] Q ( mn ) Q ( mn ) Previous work: 10 ⊕ 10 ′ Dibitetto,FM,Marqu´ es,Roest’12 Scalar potential V = 1 4 Q mn Q pq Σ − 2 (2 M mp M nq − M mn M pq ) + 1 Q mn ˜ Q pq Σ − 2 (2 M mp M nq − M mn M pq ) + Q mn ˜ Q mn Σ − 2 ˜ 4 � � Σ 3 + 3 + θ 2 Σ 8 − θ Q mn M mn − ˜ 2 ξ mn ξ pq Σ − 2 M mp M nq Q mn M mn