Gauged SUGRAs in light of double field theory Jose Juan Fern - PowerPoint PPT Presentation

Gauged SUGRAs in light of double field theory Jose Juan Fern´ andez-Melgarejo KIAS February 24, 2014

Outline 1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

Outline 1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

Supergravity essentials Quantum field theory Field content Action / eom’s Supersymmetry transformations Global symmetry group

Examples: N = 2 A D = 10 supergravity Field content � µ , χ ± � g µν , φ , B µν , C (3) µνρ , C (1) , ψ ± . µ Lagrangian of the bosonic sector e − 1 L 2A = R − 1 2 ( ∂φ ) 2 � 2 e − φ | H | 2 − 1 − 1 e (4 − d ) φ/ 2 | G ( d +1) | 2 2 d =1 , 3 � � dC (3) ∧ dC (3) ∧ B − 1 2 ⋆ Global symmetry group R + × R +

Examples: N = 2 B D = 10 supergravity Field content � � g µν , B µν , φ , C (0) , C (2) , C (4) µνρσ SD , ψ I µα , λ I α Lagrangian of the bosonic sector e − 1 L 2B = R − 1 2 ( ∂φ ) 2 � 2 e − φ | H | 2 − 1 − 1 | G ( d +1) | 2 2 d =0 , 2 , 4 � � C (4) ∧ dC (2) ∧ B − 1 2 ⋆ Global symmetry group SL (2 , R ) × R + Self-duality relation G (5) = ⋆ G (5) . (1)

Examples: D = 11 supergravity Field content { e µ a , C µνρ , ψ µ } Lagrangian � 1 ψ µ γ µνρ D ν ψ ρ − 1 d 11 xe [ e a µ e b ν R µν ab − ¯ 24 F µνρσ F µνρσ S = 2 κ 2 √ � γ αβγδνρ + 12 γ αβ g γν g δρ � 2 ¯ ψ ρ ( F αβγδ + ˜ − ψ ν F alpha βγδ ) 192 √ − 2 2 (144) 2 e − 1 ǫ α ′ β ′ γ ′ δ ′ αβγδµνρ F α ′ β ′ γ ′ δ ′ F αβγδ C µνρ ] Global symmetry group: R + (trombone symmetry) N = 8 D = 4 → perturbatively UV finite up to 3 loops Bern, Carrasco, Dixon, Johansson, Kosower, Roiban’07

Massive supergravities Massive SUGRAs Deform a theory such that SUSY is preserved L = L 0 + L def ( m ) Gauged SUGRA: deformed theory in which vector fields gauge a Yang-Mills subgroup of the global symmetry group Relation between RR ( p + 1)-form potentials in d = 10 N = 2 A / 2 B D -branes in string theory Polchinski’95 How to find higher-rank fields? (extended field content) Democratic formulations Bergshoeff et al. ’01 E 11 + U-duality arguments Bergshoeff,Riccioni’10’11’12 Gauged supergravities

Massive supergravity There is not a general procedure Dimensional reduction on tori T n : Abelian gaugings on twisted tori: non-Abelian gaugings Scherk-Schwarz dimensional reductions: non-Abelian gaugings Embedding tensor formalism: systematic procedure to scan all the possible gaugings Tensor hierarchy + embedding tensor: including all possible deformations

Outline 1 Introduction 2 Gauged supergravities 3 Double field theory 4 Conclusions

Scherk-Schwarz dimensional reduction Scherk-Schwarz reduction of the D-dim NSNS sector of the string effective action { g ij , b ij , φ } to d = ( D − n ) dimensions � � � � 1 R − 4( ∂φ ) 2 + | g | e − 2 φ 2 · 3! H µνρ H µνρ d D x S =

Scherk-Schwarz dimensional reduction Scherk-Schwarz reduction of the D-dim NSNS sector of the string effective action { g ij , b ij , φ } to d = ( D − n ) dimensions � � g µν + g pq A p µ A q ν A p µ g pn g ij = g mp A p ν g mn � � b µν − 1 2 ( A p µ V p ν − A p ν V p µ ) + A p µ A q ν b pq V n µ − b np A p µ b ij = − V m ν + b mp A p ν b mn b µν = � g µν = � g µν ( x ) , b µν ( x ) m ( y ) � m ( y ) � A m A a V m µ = u a µ = u a µ ( x ) , V a µ ( x ) , n ( y ) � g mn = u a m ( y ) u b b mn = u a m ( y ) u b n ( y ) � g ab ( x ) , b ab ( x ) + v mn ( y ) . � µ = ( � V a µ , � A A A a µ ) � � g ac � g ab � − � b cb � M AB = � g ab − � g cd � g cb b ac � � b ac � b db

Scherk-Schwarz dimensional reduction Scherk-Schwarz reduction of the D-dim NSNS sector of the string effective action { g ij , b ij , φ } to d = ( D − n ) dimensions � � � φ − 1 R + 4 ∂ µ � ge − 2 � φ∂ µ � � d d x φ M AB F A µν F B S = � µν 4 � − 1 12 G µνρ G µνρ + 1 M AB D µ � M AB + V 8 D µ � µν = ∂ µ � ν − ∂ ν � A � µ � F A A A A A A B A C µ − f BC ν G µρλ = 3 ∂ [ µ � b ρλ ] − f ABC � µ � ρ � λ + 3 ∂ [ µ � ρ � A A A B A C A A A λ ] A C � C � D µ � M AB = ∂ µ � µ � µ � A D A D M AB − f AD M CB − f BD M AC V = − 1 M AB − 1 M EF − 1 C f CB D � E f BD F � M AB � M CD � 6 f ABC f ABC 4 f DA 12 f AC f abc = 0 d v c ] d ) f abc = 3( ∂ [ a v bc ] + f [ ab f abc = u am ∂ m u bn u cn − u bm ∂ m u an u cn f abc = 0

Embedding tensor formalism ϑ I A Systematic study of the most general gaugings Cordaro et al. ’98 / Nicolai,Samtleben’01 / deWit et al. ’02 / deWit et al. ’03 Promote subgroup G 0 ⊂ G to be local X I ⋄⋄ = ϑ I A ( t A ) ⋄⋄ ( n v × dim G ) matrix ϑ I A ⇒ Constraints ϑ K A X IJ K − ϑ J B X IB A = 0 quadratic (gauge) linear (SUSY) g ⊗ V = θ 1 ⊕ θ 2 ⊕ . . . ⊕ θ n St¨ uckelberg couplings ⇒ tensor hierarchy  field strengths  scalar potential  Deformation consequences  fermionic SUSY transf  . . .

Embedding tensor formalism ϑ I A Systematic study of the most general gaugings Cordaro et al. ’98 / Nicolai,Samtleben’01 / deWit et al. ’02 / deWit et al. ’03 Promote subgroup G 0 ⊂ G to be local X I ⋄⋄ = ϑ I A ( t A ) ⋄⋄ ( n v × dim G ) matrix ϑ I A ⇒ Constraints ϑ K A ϑ I B ( t B ) J K − ϑ J B ϑ I C f CB A = 0 quadratic (gauge) linear (SUSY) g ⊗ V = θ 1 ⊕ θ 2 ⊕ . . . ⊕ θ n St¨ uckelberg couplings ⇒ tensor hierarchy  field strengths  scalar potential  Deformation consequences  fermionic SUSY transf  . . .

N = 2 d = 9 supergravity One undeformed maximal theory Gates’86 Global symmetry: SL (2 , R ) × ( R + ) 2 Field content Vielbein: e µ a Scalar fields: ϕ , τ ≡ χ + ie − φ p -form potentials: A µ 0 , A µ 1 , A µ 2 , B µν 1 , B µν 2 , C µνρ fermions: ψ µ , ˜ λ, λ

Magnetic duals p -forms F (7) A (6) ˜ ˜ A (1) I F (2) I ∼ ⋆ F I → ⇒ ⇒ I I H (6) B (5) ˜ ˜ B (2) i H (3) i ∼ ⋆ H i → ⇒ ⇒ i i G (5) ∼ ⋆ G ˜ ˜ C (3) G (4) C (4) → ⇒ ⇒ � magnetic � � � electric eom’s fields ≡ Bianchi’s fields electric magnetic 7-, 8- and 9-forms? ( d − 2) = G AB ⋆ j B ˜ J A ≡ d ˜ A A A A ⇒ Noether currents ( d − 2) � A ♯ A ♯ ˜ ♯ dm ♯ ∧ ˜ ⇒ Deformation parameters ( d − 1) ( d − 1) � ˜ ♭ Q ♭ ˜ A ♭ A ♭ ⇒ Quadratic constraints ( d ) ( d )

Deformation recipe 1 Deform supersymmetric transformations Deformation ingredients of fermionic fields 1 embedding tensor ϑ I A (3 × 5) St¨ uckelberg couplings δ ǫ ψ µ = δ 0 ψ µ + f γ µ ǫ + k γ µ ǫ ∗ Z •⋄ δ ǫ ˜ λ = δ 0 ˜ g ǫ + ˜ h ǫ ∗ λ + ˜ Fermion shifts δ ǫ λ = δ 0 λ + g ǫ + h ǫ ∗ g , ˜ f , k , ˜ h , g , h Other ingredients � � � 1 undeformed theory 2 e φ D 5 µ χ + A I µ ϑ I m P m ∇ µ + i 1 D µ ǫ ≡ 2 Gauge generators 14 γ µ � A I ϑ I 4 � X I ⋄• = ϑ I A t A ⋄• + 9 ǫ Covariant derivatives D µ = ∂ µ + X I A I µ Gauge parameters Λ ( p )

Deformation recipe 2 Covariant derivatives and gauge Deformation ingredients transformations of the 0-forms 1 embedding tensor ϑ I A (3 × 5) St¨ uckelberg couplings D ϕ = d ϕ + A I ϑ I A k A ϕ Z •⋄ D τ = d τ + A I ϑ I A k A τ Fermion shifts g , ˜ f , k , ˜ h , g , h δ Λ A I = − D Λ I + Z I i Λ i Other ingredients 1 undeformed theory Gauge generators X I ⋄• = ϑ I A t A ⋄• Covariant derivatives D µ = ∂ µ + X I A I µ Gauge parameters Λ ( p )

Deformation recipe 3 Closure of the algebra of the 0-forms Deformation ingredients 1 embedding tensor ϑ I A (3 × 5) [ δ ǫ 1 , δ ǫ 2 ] ϕ = ξ µ D µ ϕ + ℜ e (˜ h ) b − ℑ m (˜ g ) c + ℜ e (˜ g ) d St¨ uckelberg couplings [ δ ǫ 1 , δ ǫ 2 ] τ = ξ µ D µ τ + e − φ [ g ( c − id ) − ihb ] Z •⋄ Fermion shifts [ δ ǫ 1 , δ ǫ 2 ] ϕ = L ξ ϕ + Λ I ϑ I A k A ϕ g , ˜ f , k , ˜ h , g , h [ δ ǫ 1 , δ ǫ 2 ] τ = L ξ τ + Λ I ϑ I A k A τ Other ingredients 1 undeformed theory g ) d = (Λ I − a I ) ϑ I ℜ e (˜ A k A ϕ h ) b − ℑ m (˜ g ) c + ℜ e (˜ Gauge generators g ( c − id ) − ihb = e φ (Λ I − a I ) ϑ I X I ⋄• = ϑ I A t A ⋄• A k A τ Covariant derivatives D µ = ∂ µ + X I A I µ Gauge parameters Λ ( p )

Deformation recipe 4 Deformed field strength of the 1-forms Deformation ingredients 1 embedding tensor F I = dA I + 1 I A J ∧ A K + Z I ϑ I A (3 × 5) i B i 2 X JK St¨ uckelberg couplings i � 2 A I ∧ δ Λ A J � δ Λ B i = − D Λ i − 2 h IJ Λ I F J + 1 Z •⋄ + Z i Λ Fermion shifts g , ˜ f , k , ˜ h , g , h I + Z I i = 0 X ( JK ) i h JK Other ingredients i Z i = 0 Z I 1 undeformed theory Gauge generators X I ⋄• = ϑ I A t A ⋄• j − 2 X I ( J i = 0 i h JK L h K ) L X I j Covariant derivatives X I Z i − X I j i Z j = 0 D µ = ∂ µ + X I A I µ Gauge parameters Λ ( p )