Classifying Highly Supersymmetric Solutions Jan Gutowski Kings - PowerPoint PPT Presentation

Classifying Highly Supersymmetric Solutions Jan Gutowski King’s College London with U. Gran, G. Papadopoulos and D. Roest arXiv:????.????, arXiv:0902.3642, arXiv:0710.1829, hep-th/0610331, hep-th/0606049 15-th European Workshop on String Theory

Outline IIB Supergravity and Killing Spinors

Outline IIB Supergravity and Killing Spinors Analysis of Solutions with 28 < N < 32 supersymmetries

Outline IIB Supergravity and Killing Spinors Analysis of Solutions with 28 < N < 32 supersymmetries No assumptions about the spacetime geometry/fluxes are made - the analysis is completely general.

Outline IIB Supergravity and Killing Spinors Analysis of Solutions with 28 < N < 32 supersymmetries No assumptions about the spacetime geometry/fluxes are made - the analysis is completely general. All maximally supersymmetric solutions, i.e. those with 32 linearly independent Killing spinors, are completely classified [Figueroa O’Farrill, Papadopoulos]

Outline IIB Supergravity and Killing Spinors Analysis of Solutions with 28 < N < 32 supersymmetries No assumptions about the spacetime geometry/fluxes are made - the analysis is completely general. All maximally supersymmetric solutions, i.e. those with 32 linearly independent Killing spinors, are completely classified [Figueroa O’Farrill, Papadopoulos] One finds: R 9 , 1 , AdS 5 × S 5 and a maximally supersymmetric plane wave solution. Conclusions

IIB Supergravity and Killing Spinors The bosonic fields of IIB supergravity are the spacetime metric g , the axion σ and dilaton φ , two three-form field strengths G α = dA α ( α = 1 , 2 ), and a self-dual five-form field strength F The axion and dilaton give rise to a complex 1-form P [Schwarz]. The 3-forms are combined to give a complex 3-form G . To achieve this, introduce a SU (1 , 1) matrix U = ( V α + , V α − ) , α = 1 , 2 such that − ) ∗ = V 2 − ) ∗ = V 1 + = ǫ αβ , − V β + − V β ( V 1 ( V 2 V α − V α + , + ǫ 12 = 1 = ǫ 12 .

The V α ± are related to the axion and dilaton by V 2 = 1 + i ( σ + ie − φ ) − 1 − i ( σ + ie − φ ) . V 1 − Then P and G are defined by + ∂ M V β + G β P M = − ǫ αβ V α G MNR = − ǫ αβ V α + , MNR

The gravitino Killing spinor equation is 48Γ N 1 ...N 4 ǫF N 1 ...N 4 M − 1 ∇ M ǫ + i ˜ N 1 N 2 N 3 G N 1 N 2 N 3 96(Γ M − 9Γ N 1 N 2 G MN 1 N 2 )( Cǫ ) ∗ = 0 where 2 Q M + 1 ∇ M = ∂ M − i ˜ 4Ω M,AB Γ AB is the standard covariant derivative twisted with U (1) connection Q M , given in terms of the SU (1 , 1) scalars by − ∂ M V β Q M = − iǫ αβ V α + and Ω is the spin connection.

There is also an algebraic constraint P M Γ M ( Cǫ ) ∗ + 1 24 G N 1 N 2 N 3 Γ N 1 N 2 N 3 ǫ = 0 The Killing spinor ǫ is a complex Weyl spinor constructed from two copies of the same Majorana-Weyl representation ∆ + 16 : ǫ = ψ 1 + iψ 2 Majorana-Weyl spinors ψ satisfy ψ = C ( ψ ∗ ) C is the charge conjugation matrix.

Spinors as Forms Let e 1 , . . . , e 5 be a locally defined orthonormal basis of R 5 . Take U to be the span over R of e 1 , . . . , e 5 . The space of Dirac spinors is ∆ c = Λ ∗ ( U ⊗ C ) (the complexified space of all forms on U ). ∆ c decomposes into even forms ∆ + c and odd forms ∆ − c , which are the complex Weyl representations of Spin (9 , 1) .

The gamma matrices are represented on ∆ c as Γ 0 η = − e 5 ∧ η + e 5 � η Γ 5 η = e 5 ∧ η + e 5 � η Γ i η = e i ∧ η + e i � η i = 1 , . . . , 4 Γ 5+ i η = ie i ∧ η − ie i � η i = 1 , . . . , 4 Γ j for j = 1 , . . . , 9 are hermitian and Γ 0 is anti-hermitian with respect to the inner product 5 ( z a ) ∗ w a , � < z a e a , w b e b > = a =1 This inner product can be extended from U ⊗ C to ∆ c .

There is a Spin (9 , 1) invariant inner product defined on ∆ c defined by B ( ǫ 1 , ǫ 2 ) = < Γ 0 C ( ǫ 1 ) ∗ , ǫ 2 > B is skew-symmetric in ǫ 1 , ǫ 2 . B vanishes when restricted to ∆ + c or ∆ − c . This defines a non-degenerate pairing B : ∆ + c ⊗ ∆ − c → R given by B ( ǫ, ξ ) = Re B ( ǫ, ξ )

Canonical forms of spinors We wish to write a spinor ν = ν 1 + iν 2 , where ν i ∈ ∆ − 16 in a simple canonical form.

Canonical forms of spinors We wish to write a spinor ν = ν 1 + iν 2 , where ν i ∈ ∆ − 16 in a simple canonical form. Spin (9 , 1) has one type of orbit with stability subgroup Spin (7) ⋉ R 8 in ∆ − 16 [Figueroa-O’Farrill, Bryant]. 16 = R < e 5 + e 12345 > +Λ 1 ( R 7 ) + ∆ 8 , ∆ −

Canonical forms of spinors We wish to write a spinor ν = ν 1 + iν 2 , where ν i ∈ ∆ − 16 in a simple canonical form. Spin (9 , 1) has one type of orbit with stability subgroup Spin (7) ⋉ R 8 in ∆ − 16 [Figueroa-O’Farrill, Bryant]. 16 = R < e 5 + e 12345 > +Λ 1 ( R 7 ) + ∆ 8 , ∆ − R < e 5 + e 12345 > is the singlet generated by e 5 + e 12345

Canonical forms of spinors We wish to write a spinor ν = ν 1 + iν 2 , where ν i ∈ ∆ − 16 in a simple canonical form. Spin (9 , 1) has one type of orbit with stability subgroup Spin (7) ⋉ R 8 in ∆ − 16 [Figueroa-O’Farrill, Bryant]. 16 = R < e 5 + e 12345 > +Λ 1 ( R 7 ) + ∆ 8 , ∆ − R < e 5 + e 12345 > is the singlet generated by e 5 + e 12345 Λ 1 ( R 7 ) is the vector representation of Spin (7) spanned by (j,k=1,...,4) e jk 5 − 1 2 ǫ jkmn e mn 5 , i ( e jk 5 + 1 2 ǫ jkmn e mn 5 ) and i ( e 5 − e 12345 ) .

Canonical forms of spinors We wish to write a spinor ν = ν 1 + iν 2 , where ν i ∈ ∆ − 16 in a simple canonical form. Spin (9 , 1) has one type of orbit with stability subgroup Spin (7) ⋉ R 8 in ∆ − 16 [Figueroa-O’Farrill, Bryant]. 16 = R < e 5 + e 12345 > +Λ 1 ( R 7 ) + ∆ 8 , ∆ − R < e 5 + e 12345 > is the singlet generated by e 5 + e 12345 Λ 1 ( R 7 ) is the vector representation of Spin (7) spanned by (j,k=1,...,4) e jk 5 − 1 2 ǫ jkmn e mn 5 , i ( e jk 5 + 1 2 ǫ jkmn e mn 5 ) and i ( e 5 − e 12345 ) . ∆ 8 is the spin representation of Spin (7) spanned by e j + 1 6 ǫ jq 1 q 2 q 3 e q 1 q 2 q 3 , i ( e j − 1 6 ǫ jq 1 q 2 q 3 e q 1 q 2 q 3 ) .

Spin (7) acts transitively on the S 7 in ∆ 8 , with stability subgroup G 2 , and G 2 acts transitively on the S 6 in Λ 1 ( R 7 ) with stability subgroup SU (3) [Salamon] Using these transitive actions, any ν 1 ∈ ∆ − 16 can be written as ν 1 = a 1 ( e 5 + e 12345 ) + ia 2 ( e 5 − e 12345 ) + a 3 ( e 1 + e 234 ) For all possible choices of (real) a 1 , a 2 , a 3 , there exist Spin (9 , 1) transformations which set ν 1 = e 5 + e 12345 .

Spin (7) acts transitively on the S 7 in ∆ 8 , with stability subgroup G 2 , and G 2 acts transitively on the S 6 in Λ 1 ( R 7 ) with stability subgroup SU (3) [Salamon] Using these transitive actions, any ν 1 ∈ ∆ − 16 can be written as ν 1 = a 1 ( e 5 + e 12345 ) + ia 2 ( e 5 − e 12345 ) + a 3 ( e 1 + e 234 ) For all possible choices of (real) a 1 , a 2 , a 3 , there exist Spin (9 , 1) transformations which set ν 1 = e 5 + e 12345 . This spinor is Spin (7) ⋉ R 8 invariant.

Spin (7) acts transitively on the S 7 in ∆ 8 , with stability subgroup G 2 , and G 2 acts transitively on the S 6 in Λ 1 ( R 7 ) with stability subgroup SU (3) [Salamon] Using these transitive actions, any ν 1 ∈ ∆ − 16 can be written as ν 1 = a 1 ( e 5 + e 12345 ) + ia 2 ( e 5 − e 12345 ) + a 3 ( e 1 + e 234 ) For all possible choices of (real) a 1 , a 2 , a 3 , there exist Spin (9 , 1) transformations which set ν 1 = e 5 + e 12345 . This spinor is Spin (7) ⋉ R 8 invariant. Having fixed ν 1 , it remains to consider ν 2 : By using Spin (7) gauge transformations, which leave ν 1 invariant, one can write ν 2 = b 1 ( e 5 + e 12345 ) + ib 2 ( e 5 − e 12345 ) + b 3 ( e 1 + e 234 )

There are various cases i) b 3 � = 0 . Then using Spin (7) ⋉ R 8 gauge transformations one can take ν 2 = g ( e 1 + e 234 ) The stability subgroup of Spin (9 , 1) which leaves ν 1 and ν 2 invariant is G 2 . ii) If b 3 = 0 then ν 2 = g 1 ( e 5 + e 12345 ) + ig 2 ( e 5 − e 12345 ) and the stability subgroup is SU (4) ⋉ R 8 iii) If b 2 = b 3 = 0 then ν 2 = g ( e 5 + e 12345 ) and the stability subgroup is Spin (7) ⋉ R 8 .

N = 31 Solutions: Algebraic Constraints Suppose that there exists a solution with exactly (and no more than) 31 linearly independent Killing spinors over R . Consider the algebraic constraint P M Γ M ( Cǫ r ) ∗ + 1 24 G N 1 N 2 N 3 Γ N 1 N 2 N 3 ǫ r = 0 where ǫ r are Killing spinors for r = 1 , . . . , 31 .

N = 31 Solutions: Algebraic Constraints Suppose that there exists a solution with exactly (and no more than) 31 linearly independent Killing spinors over R . Consider the algebraic constraint P M Γ M ( Cǫ r ) ∗ + 1 24 G N 1 N 2 N 3 Γ N 1 N 2 N 3 ǫ r = 0 where ǫ r are Killing spinors for r = 1 , . . . , 31 . The space of Killing spinors is orthogonal to a single normal spinor , ν ∈ ∆ − c with respect to the Spin (9 , 1) invariant inner product B . Using Spin (9 , 1) gauge transformations, this normal spinor can be brought into one of 3 canonical forms: Spin (7) ⋉ R 8 : ν = ( n + im )( e 5 + e 12345 ) , SU (4) ⋉ R 8 : ν = ( n − ℓ + im ) e 5 + ( n + ℓ + im ) e 12345 , G 2 : ν = n ( e 5 + e 12345 ) + im ( e 1 + e 234 ) ,

In general, one can write 32 ǫ r = � f r i η i i =1 where f ri are real, η p for p = 1 , . . . , 16 is a basis for ∆ + 16 and η 16+ p = iη p . The matrix with components f ri is of rank 31.