Pseudo-supersymmetry: a tale of alternate realities Jan Rosseel - PowerPoint PPT Presentation

Pseudo-supersymmetry: a tale of alternate realities Jan Rosseel (ITF, K. U. Leuven) Work in progress by: E. Bergshoeff, J. Hartong, A. Ploegh, D. Van den Bleeken, J.R. Firenze, april 4th 2007

Outline 1. Introduction and motivation Goal Domain-wall vs. cosmology correspondence Variant supergravities The superalgebra 2. The strategy 3. More generally 4. The domain-wall cosmology correspondence 5. Summary and discussion

Introduction and motivation Goal ◮ Goal : Construct different supergravity actions from one ’complex’ action by taking different real slices. ◮ Motivation : 1. The domain-wall vs. cosmology correspondence ( Townsend, Skenderis ) suggests that this can be done. Explicit realisation of this correspondence in a supergravity setting. 2. ’Variant’ supergravities in 10 and 11 dimensions have been considered by looking at time-like T-duality, e.g. the so-called *-theories. ( Hull, Bergshoeff, Van Proeyen, Vaula ). Can we construct these explicitly?

Introduction and motivation Goal ◮ Goal : Construct different supergravity actions from one ’complex’ action by taking different real slices. ◮ Motivation : 1. The domain-wall vs. cosmology correspondence ( Townsend, Skenderis ) suggests that this can be done. Explicit realisation of this correspondence in a supergravity setting. 2. ’Variant’ supergravities in 10 and 11 dimensions have been considered by looking at time-like T-duality, e.g. the so-called *-theories. ( Hull, Bergshoeff, Van Proeyen, Vaula ). Can we construct these explicitly?

Introduction and motivation Goal ◮ Goal : Construct different supergravity actions from one ’complex’ action by taking different real slices. ◮ Motivation : 1. The domain-wall vs. cosmology correspondence ( Townsend, Skenderis ) suggests that this can be done. Explicit realisation of this correspondence in a supergravity setting. 2. ’Variant’ supergravities in 10 and 11 dimensions have been considered by looking at time-like T-duality, e.g. the so-called *-theories. ( Hull, Bergshoeff, Van Proeyen, Vaula ). Can we construct these explicitly?

Introduction and motivation Domain-walls vs. cosmologies There is a correspondence between domain-walls and cosmologies ( Townsend, Skenderis ). ◮ Domain wall metric d τ 2 d s 2 = dz 2 + e 2 βϕ � 1 + k τ 2 + τ 2 ( d ψ 2 + sinh 2 ψ d Ω 2 � − d − 2 ) . where k = 0 , ± 1, ϕ = ϕ ( z ) . ◮ FLRW cosmology d r 2 d s 2 = − dt 2 + e 2 βφ � 1 − kr 2 + r 2 ( d θ 2 + sin 2 θ d Ω 2 � d − 2 ) . where k = 0 , ± 1, φ = φ ( t ) . Related via analytical continuation : ( t , r , θ ) = − i ( z , τ, ψ ) and φ ( t ) = ϕ ( i t ) .

Introduction and motivation Domain-walls vs. cosmologies There is a correspondence between domain-walls and cosmologies ( Townsend, Skenderis ). ◮ Domain wall metric d τ 2 d s 2 = dz 2 + e 2 βϕ � 1 + k τ 2 + τ 2 ( d ψ 2 + sinh 2 ψ d Ω 2 � − d − 2 ) . where k = 0 , ± 1, ϕ = ϕ ( z ) . ◮ FLRW cosmology d r 2 d s 2 = − dt 2 + e 2 βφ � 1 − kr 2 + r 2 ( d θ 2 + sin 2 θ d Ω 2 � d − 2 ) . where k = 0 , ± 1, φ = φ ( t ) . Related via analytical continuation : ( t , r , θ ) = − i ( z , τ, ψ ) and φ ( t ) = ϕ ( i t ) .

Introduction and motivation Domain-walls vs. cosmologies There is a correspondence between domain-walls and cosmologies ( Townsend, Skenderis ). ◮ Domain wall metric d τ 2 d s 2 = dz 2 + e 2 βϕ � 1 + k τ 2 + τ 2 ( d ψ 2 + sinh 2 ψ d Ω 2 � − d − 2 ) . where k = 0 , ± 1, ϕ = ϕ ( z ) . ◮ FLRW cosmology d r 2 d s 2 = − dt 2 + e 2 βφ � 1 − kr 2 + r 2 ( d θ 2 + sin 2 θ d Ω 2 � d − 2 ) . where k = 0 , ± 1, φ = φ ( t ) . Related via analytical continuation : ( t , r , θ ) = − i ( z , τ, ψ ) and φ ( t ) = ϕ ( i t ) .

Introduction and motivation Domain-walls vs. cosmologies There is a correspondence between domain-walls and cosmologies ( Townsend, Skenderis ). ◮ Domain wall metric d τ 2 d s 2 = dz 2 + e 2 βϕ � 1 + k τ 2 + τ 2 ( d ψ 2 + sinh 2 ψ d Ω 2 � − d − 2 ) . where k = 0 , ± 1, ϕ = ϕ ( z ) . ◮ FLRW cosmology d r 2 d s 2 = − dt 2 + e 2 βφ � 1 − kr 2 + r 2 ( d θ 2 + sin 2 θ d Ω 2 � d − 2 ) . where k = 0 , ± 1, φ = φ ( t ) . Related via analytical continuation : ( t , r , θ ) = − i ( z , τ, ψ ) and φ ( t ) = ϕ ( i t ) .

Introduction and motivation Domain-walls vs. cosmologies ◮ Considering gravity coupled to scalars: L = √− g R − 1 2 ( ∂σ ) 2 − η V ( σ ) � � , η = ± 1 . DW for ( η = 1 , k = ± 1 or 0 ) → cosmology for ( η = − 1 , k = ∓ 1 or 0 ) . ◮ For the DW (fake supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = 2 and ( D µ − αβ W Γ µ ) ǫ = 0 ◮ For the cosmology (fake pseudo-supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = − 2 and ( D µ − i αβ W Γ µ ) ǫ = 0 ◮ Γ µ D µ ǫ = M ǫ : 1. susy : M hermitian 2. pseudo-susy : M anti-hermitian.

Introduction and motivation Domain-walls vs. cosmologies ◮ Considering gravity coupled to scalars: L = √− g R − 1 2 ( ∂σ ) 2 − η V ( σ ) � � , η = ± 1 . DW for ( η = 1 , k = ± 1 or 0 ) → cosmology for ( η = − 1 , k = ∓ 1 or 0 ) . ◮ For the DW (fake supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = 2 and ( D µ − αβ W Γ µ ) ǫ = 0 ◮ For the cosmology (fake pseudo-supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = − 2 and ( D µ − i αβ W Γ µ ) ǫ = 0 ◮ Γ µ D µ ǫ = M ǫ : 1. susy : M hermitian 2. pseudo-susy : M anti-hermitian.

Introduction and motivation Domain-walls vs. cosmologies ◮ Considering gravity coupled to scalars: L = √− g R − 1 2 ( ∂σ ) 2 − η V ( σ ) � � , η = ± 1 . DW for ( η = 1 , k = ± 1 or 0 ) → cosmology for ( η = − 1 , k = ∓ 1 or 0 ) . ◮ For the DW (fake supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = 2 and ( D µ − αβ W Γ µ ) ǫ = 0 ◮ For the cosmology (fake pseudo-supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = − 2 and ( D µ − i αβ W Γ µ ) ǫ = 0 ◮ Γ µ D µ ǫ = M ǫ : 1. susy : M hermitian 2. pseudo-susy : M anti-hermitian.

Introduction and motivation Domain-walls vs. cosmologies ◮ Considering gravity coupled to scalars: L = √− g R − 1 2 ( ∂σ ) 2 − η V ( σ ) � � , η = ± 1 . DW for ( η = 1 , k = ± 1 or 0 ) → cosmology for ( η = − 1 , k = ∓ 1 or 0 ) . ◮ For the DW (fake supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = 2 and ( D µ − αβ W Γ µ ) ǫ = 0 ◮ For the cosmology (fake pseudo-supersymmetry) | W ′ | 2 − α 2 | W | 2 � � V = − 2 and ( D µ − i αβ W Γ µ ) ǫ = 0 ◮ Γ µ D µ ǫ = M ǫ : 1. susy : M hermitian 2. pseudo-susy : M anti-hermitian.

Domain-walls vs. cosmologies ◮ From a supergravity point of view this correspondence looks rather strange: • Supersymmetric domain walls can be generically found, supersymmetric cosmologies not. • V → − V , W → i W ? • In real supergravity you do care about reality of fermions ↔ fake supergravity. ◮ Is there a way of realizing this in a supergravity context, i.e. see the Killing spinor conditions as arising from δ ǫ ψ µ = 0? ◮ Strategy : 1. Look at ’complex’ supergravity theories. 2. Impose reality conditions, i.e. take real slices 3. See how many slices per signature are possible and what the implications of this are.

Domain-walls vs. cosmologies ◮ From a supergravity point of view this correspondence looks rather strange: • Supersymmetric domain walls can be generically found, supersymmetric cosmologies not. • V → − V , W → i W ? • In real supergravity you do care about reality of fermions ↔ fake supergravity. ◮ Is there a way of realizing this in a supergravity context, i.e. see the Killing spinor conditions as arising from δ ǫ ψ µ = 0? ◮ Strategy : 1. Look at ’complex’ supergravity theories. 2. Impose reality conditions, i.e. take real slices 3. See how many slices per signature are possible and what the implications of this are.

Domain-walls vs. cosmologies ◮ From a supergravity point of view this correspondence looks rather strange: • Supersymmetric domain walls can be generically found, supersymmetric cosmologies not. • V → − V , W → i W ? • In real supergravity you do care about reality of fermions ↔ fake supergravity. ◮ Is there a way of realizing this in a supergravity context, i.e. see the Killing spinor conditions as arising from δ ǫ ψ µ = 0? ◮ Strategy : 1. Look at ’complex’ supergravity theories. 2. Impose reality conditions, i.e. take real slices 3. See how many slices per signature are possible and what the implications of this are.

Domain-walls vs. cosmologies ◮ From a supergravity point of view this correspondence looks rather strange: • Supersymmetric domain walls can be generically found, supersymmetric cosmologies not. • V → − V , W → i W ? • In real supergravity you do care about reality of fermions ↔ fake supergravity. ◮ Is there a way of realizing this in a supergravity context, i.e. see the Killing spinor conditions as arising from δ ǫ ψ µ = 0? ◮ Strategy : 1. Look at ’complex’ supergravity theories. 2. Impose reality conditions, i.e. take real slices 3. See how many slices per signature are possible and what the implications of this are.