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Maximally supersymmetric spacetimes

Jose Figueroa-O’Farrill

Edinburgh Mathematical Physics Group

School of Mathematics

Neve Shalom ∼ Wahat al-Salam, 27 May 2003

1

Maximally symmetric spacetimes

1

Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

1

Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

Locally, those which admit the maximum number of Killing vectors.

1

Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

Locally, those which admit the maximum number of Killing vectors.

Recall: ξµ is Killing ⇐⇒ ∇µξν = −∇νξµ

1

Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

Locally, those which admit the maximum number of Killing vectors.

Recall: ξµ is Killing ⇐⇒ ∇µξν = −∇νξµ

Differentiating

1

Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

Locally, those which admit the maximum number of Killing vectors.

Recall: ξµ is Killing ⇐⇒ ∇µξν = −∇νξµ

Differentiating =⇒ Killing’s identity

1

Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

Locally, those which admit the maximum number of Killing vectors.

Recall: ξµ is Killing ⇐⇒ ∇µξν = −∇νξµ

Differentiating =⇒ Killing’s identity:

∇µ∇νξρ = Rµνρσξσ

2

Therefore a Killing vector ξ is uniquely determined by

2

Therefore a Killing vector ξ is uniquely determined by:

ξ(p)

2

Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p)

2

Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p) at any point p.

2

Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p) at any point p.

This is suggestive of parallel sections of a vector bundle.

2

Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p) at any point p.

This is suggestive of parallel sections of a vector bundle.

Indeed on the bundle

E(M) = TM ⊕ Λ2T ∗M

2

Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p) at any point p.

This is suggestive of parallel sections of a vector bundle.

Indeed on the bundle

E(M) = TM ⊕ Λ2T ∗M

there is a connection D defined on a section (ξµ, Fµν) of E(M) by

2

Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p) at any point p.

This is suggestive of parallel sections of a vector bundle.

Indeed on the bundle

E(M) = TM ⊕ Λ2T ∗M

there is a connection D defined on a section (ξµ, Fµν) of E(M) by

Dµ

(ξν

Fνρ

)=(

∇µξν − Fµν

∇µFνρ −Rµνρσξσ

)

3

=⇒ (ξµ, Fµν) is parallel

3

=⇒ (ξµ, Fµν) is parallel if and only if

• ξµ is a Killing vector

3

=⇒ (ξµ, Fµν) is parallel if and only if

• ξµ is a Killing vector, and

• Fµν = ∇µξν

3

=⇒ (ξµ, Fµν) is parallel if and only if

• ξµ is a Killing vector, and

• Fµν = ∇µξν

E(M) has rank n(n + 1)/2 for an n-dimensional M .

3

=⇒ (ξµ, Fµν) is parallel if and only if

• ξµ is a Killing vector, and

• Fµν = ∇µξν

E(M) has rank n(n + 1)/2 for an n-dimensional M .

=⇒ ∃ ≤ n(n + 1)/2 linearly independent Killing vectors.

3

=⇒ (ξµ, Fµν) is parallel if and only if

• ξµ is a Killing vector, and

• Fµν = ∇µξν

E(M) has rank n(n + 1)/2 for an n-dimensional M .

=⇒ ∃ ≤ n(n + 1)/2 linearly independent Killing vectors.

Maximal symmetry =⇒ E(M) is flat

3

=⇒ (ξµ, Fµν) is parallel if and only if

• ξµ is a Killing vector, and

• Fµν = ∇µξν

E(M) has rank n(n + 1)/2 for an n-dimensional M .

=⇒ ∃ ≤ n(n + 1)/2 linearly independent Killing vectors.

Maximal symmetry =⇒ E(M) is flat

=⇒ M has constant sectional curvature κ.

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In the riemannian case (and up to local isometry)

4

In the riemannian case (and up to local isometry):

• κ = 0: euclidean space En

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In the riemannian case (and up to local isometry):

• κ = 0: euclidean space En

• κ > 0: sphere

Sn ⊂ En+1

4

In the riemannian case (and up to local isometry):

• κ = 0: euclidean space En

• κ > 0: sphere

Sn ⊂ En+1 : x21 + x2

2 + · · ·+ x2n+1 =

1κ2

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In the riemannian case (and up to local isometry):

• κ = 0: euclidean space En

• κ > 0: sphere

Sn ⊂ En+1 : x21 + x2

2 + · · ·+ x2n+1 =

1κ2

• κ < 0: hyperbolic space

Hn ⊂ E1,n

4

In the riemannian case (and up to local isometry):

• κ = 0: euclidean space En

• κ > 0: sphere

Sn ⊂ En+1 : x21 + x2

2 + · · ·+ x2n+1 =

1κ2

• κ < 0: hyperbolic space

Hn ⊂ E1,n : −t21 + x21 + · · ·+ x2

n =−1κ2

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In lorentzian geometry (and up to local isometry)

5

In lorentzian geometry (and up to local isometry):

• κ = 0: Minkowski space En−1,1

5

In lorentzian geometry (and up to local isometry):

• κ = 0: Minkowski space En−1,1

• κ > 0: de Sitter space

dSn ⊂ E1,n

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In lorentzian geometry (and up to local isometry):

• κ = 0: Minkowski space En−1,1

• κ > 0: de Sitter space

dSn ⊂ E1,n : −t21 + x21 + x2

2 + · · ·+ x2n =

1κ2

5

In lorentzian geometry (and up to local isometry):

• κ = 0: Minkowski space En−1,1

• κ > 0: de Sitter space

dSn ⊂ E1,n : −t21 + x21 + x2

2 + · · ·+ x2n =

1κ2

• κ < 0: anti de Sitter space

AdSn ⊂ E2,n−1

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In lorentzian geometry (and up to local isometry):

• κ = 0: Minkowski space En−1,1

• κ > 0: de Sitter space

dSn ⊂ E1,n : −t21 + x21 + x2

2 + · · ·+ x2n =

1κ2

• κ < 0: anti de Sitter space

AdSn ⊂ E2,n−1 : −t21 − t22 + x21 + · · ·+ x2

n−1 =−1κ2

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Note: the space form with κ = 0 is a geometric limit of the space

forms with κ 6= 0.

6

Note: the space form with κ = 0 is a geometric limit of the space

forms with κ 6= 0.

It remains to classify smooth discrete quotients of the universal

covers of the above spaces.

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Note: the space form with κ = 0 is a geometric limit of the space

forms with κ 6= 0.

It remains to classify smooth discrete quotients of the universal

covers of the above spaces.

This is the Clifford–Klein space form problem, first posed by Killing

in 1891 and reformulated in these terms by Hopf in 1925.

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Note: the space form with κ = 0 is a geometric limit of the space

forms with κ 6= 0.

It remains to classify smooth discrete quotients of the universal

covers of the above spaces.

This is the Clifford–Klein space form problem, first posed by Killing

in 1891 and reformulated in these terms by Hopf in 1925.

The flat and spherical cases are completely solved (culminating in

the work of Wolf in the 1970s), but the possible quotients of

hyperbolic and lorentzian cases are still unclassified.

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Maximally supersymmetric spacetimes

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Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

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Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

Which are the maximally symmetric spacetimes?

7

Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

Which are the maximally symmetric spacetimes?

is the question:

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Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

Which are the maximally symmetric spacetimes?

is the question:

Which are the maximally supersymmetric backgrounds of

supergravity theories?

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Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

Which are the maximally symmetric spacetimes?

is the question:

Which are the maximally supersymmetric backgrounds of

supergravity theories?

In this talk I will report on progress towards answering this

question.

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Based on work in collaboration with

8

Based on work in collaboration with

• George Papadopoulos (King’s College, London)

? hep-th/0211089 (JHEP 03 (2003) 048)

? math.AG/0211170 (J Geom Phys to appear)

8

Based on work in collaboration with

• George Papadopoulos (King’s College, London)

? hep-th/0211089 (JHEP 03 (2003) 048)

? math.AG/0211170 (J Geom Phys to appear)

• Ali Chamseddine and Wafic Sabra (CAMS, Beirut)

? in preparation

9

Supergravities

32 24 20 16 12 8 4

11 M

10 IIA IIB I

9 N = 2 N = 1

8 N = 2 N = 1

7 N = 4 N = 2

6 (2, 2) (3,1) (4,0) (2, 1) (3,0) (1, 1) (2, 0) (1, 0)

5 N = 8 N = 6 N = 4 N = 2

4 N = 8 N = 6 N = 5 N = 4 N = 3 N = 2 N = 1

[Van Proeyen, hep-th/0301005]

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Strategy

Let (M, g,Φ, S) be a supergravity background

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Strategy

Let (M, g,Φ, S) be a supergravity background:

• (M, g) a lorentzian spin manifold

10

Strategy

Let (M, g,Φ, S) be a supergravity background:

• (M, g) a lorentzian spin manifold

• Φ denotes collectively the other bosonic fields

10

Strategy

Let (M, g,Φ, S) be a supergravity background:

• (M, g) a lorentzian spin manifold

• Φ denotes collectively the other bosonic fields

• S a real vector bundle of spinors

10

Strategy

Let (M, g,Φ, S) be a supergravity background:

• (M, g) a lorentzian spin manifold

• Φ denotes collectively the other bosonic fields

• S a real vector bundle of spinors

• fermions have been put to zero

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(M, g,Φ, S) is supersymmetric if it admits Killing spinors

11

(M, g,Φ, S) is supersymmetric if it admits Killing spinors; that is,

sections ε of S such that

Dµε = 0

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(M, g,Φ, S) is supersymmetric if it admits Killing spinors; that is,

sections ε of S such that

Dµε = 0

where D is the connection on S

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(M, g,Φ, S) is supersymmetric if it admits Killing spinors; that is,

sections ε of S such that

Dµε = 0

where D is the connection on S

Dµ = ∇µ + Ωµ(g,Φ)

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(M, g,Φ, S) is supersymmetric if it admits Killing spinors; that is,

sections ε of S such that

Dµε = 0

where D is the connection on S

Dµ = ∇µ + Ωµ(g,Φ)

defined by the supersymmetric variation of the gravitino:

δεΨµ = Dµε

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There are possibly also algebraic equations

A(g,Φ)ε = 0

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

δεχ = Aε

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

δεχ = Aε

Maximal supersymmetry =⇒

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

δεχ = Aε

Maximal supersymmetry =⇒ D is flat

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

δεχ = Aε

Maximal supersymmetry =⇒ D is flat and A = 0.

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

δεχ = Aε

Maximal supersymmetry =⇒ D is flat and A = 0.

Typically A = 0 sets some gauge field strengths to zero

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

δεχ = Aε

Maximal supersymmetry =⇒ D is flat and A = 0.

Typically A = 0 sets some gauge field strengths to zero, and the

flatness of D constrains the geometry and any remaining field

strengths.

12

There are possibly also algebraic equations

A(g,Φ)ε = 0

where A is the algebraic operator defined by the supersymmetric

variation of any other fermionic fields (dilatinos, gauginos,...)

δεχ = Aε

Maximal supersymmetry =⇒ D is flat and A = 0.

Typically A = 0 sets some gauge field strengths to zero, and the

flatness of D constrains the geometry and any remaining field

strengths. The strategy is therefore to study the flatness equations

for D.

13

Classifications

In the table we have highlighted the “top” theories whose

maximally supersymmetric backgrounds are known:

13

Classifications

In the table we have highlighted the “top” theories whose

maximally supersymmetric backgrounds are known:

• D = 4 N = 1 [Tod (1984)]

13

Classifications

In the table we have highlighted the “top” theories whose

maximally supersymmetric backgrounds are known:

• D = 4 N = 1 [Tod (1984)]

• D = 6 (1, 0), (2, 0) [Chamseddine–FO–Sabra (2003)]

13

Classifications

In the table we have highlighted the “top” theories whose

maximally supersymmetric backgrounds are known:

• D = 4 N = 1 [Tod (1984)]

• D = 6 (1, 0), (2, 0) [Chamseddine–FO–Sabra (2003)]

• D = 10 IIB and I [FO–Papadopoulos (2002)]

13

Classifications

In the table we have highlighted the “top” theories whose

maximally supersymmetric backgrounds are known:

• D = 4 N = 1 [Tod (1984)]

• D = 6 (1, 0), (2, 0) [Chamseddine–FO–Sabra (2003)]

• D = 10 IIB and I [FO–Papadopoulos (2002)]

• D = 11 M [FO–Papadopoulos (2001)]

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Eleven-dimensional supergravity

• bosonic fields

14

Eleven-dimensional supergravity

• bosonic fields:

? metric g

14

Eleven-dimensional supergravity

• bosonic fields:

? metric g, and

? closed 4-form F

14

Eleven-dimensional supergravity

• bosonic fields:

? metric g, and

? closed 4-form F

for a total of 44 + 84 = 128 bosonic physical degrees of freedom.

14

Eleven-dimensional supergravity

• bosonic fields:

? metric g, and

? closed 4-form F

for a total of 44 + 84 = 128 bosonic physical degrees of freedom.

• spinors are Majorana

14

Eleven-dimensional supergravity

• bosonic fields:

? metric g, and

? closed 4-form F

for a total of 44 + 84 = 128 bosonic physical degrees of freedom.

• spinors are Majorana; that is, associated to one of the two

irreducible real 32-dimensional representations of C`(1, 10).

14

Eleven-dimensional supergravity

• bosonic fields:

? metric g, and

? closed 4-form F

for a total of 44 + 84 = 128 bosonic physical degrees of freedom.

• spinors are Majorana; that is, associated to one of the two

irreducible real 32-dimensional representations of C`(1, 10).Therefore the gravitino also has 128 physical degrees of freedom.

15

• the gravitino variation defines the connection

Dµ = ∇µ − 1288Fνρστ

(Γνρστ

µ + 8Γνρσδτµ

)

15

• the gravitino variation defines the connection

Dµ = ∇µ − 1288Fνρστ

(Γνρστ

µ + 8Γνρσδτµ

)

For fixed µ, ν, the curvature Rµν of D can be expanded in terms of

antisymmetric products of Γ matrices

15

• the gravitino variation defines the connection

Dµ = ∇µ − 1288Fνρστ

(Γνρστ

µ + 8Γνρσδτµ

)

For fixed µ, ν, the curvature Rµν of D can be expanded in terms of

antisymmetric products of Γ matrices

[Dµ, Dν] = RµνIΓI

15

• the gravitino variation defines the connection

Dµ = ∇µ − 1288Fνρστ

(Γνρστ

µ + 8Γνρσδτµ

)

For fixed µ, ν, the curvature Rµν of D can be expanded in terms of

antisymmetric products of Γ matrices

[Dµ, Dν] = RµνIΓI

where I is an index labeling the following elements

Γa Γab Γabc Γabcd Γabcde

16

(We have used that Γ01···9\ = −1 in this representation.)

16

(We have used that Γ01···9\ = −1 in this representation.)

The flatness equations are the vanishing of the RµνI.

16

(We have used that Γ01···9\ = −1 in this representation.)

The flatness equations are the vanishing of the RµνI.

Summarising the results:

16

(We have used that Γ01···9\ = −1 in this representation.)

The flatness equations are the vanishing of the RµνI.

Summarising the results:

• F is parallel: ∇F = 0

16

(We have used that Γ01···9\ = −1 in this representation.)

The flatness equations are the vanishing of the RµνI.

Summarising the results:

• F is parallel: ∇F = 0

• the Riemann curvature tensor is determined algebraically in terms

of F and g

16

(We have used that Γ01···9\ = −1 in this representation.)

The flatness equations are the vanishing of the RµνI.

Summarising the results:

• F is parallel: ∇F = 0

• the Riemann curvature tensor is determined algebraically in terms

of F and g:

Rµνρσ = Tµνρσ(F, g)

with T quadratic in F

16

(We have used that Γ01···9\ = −1 in this representation.)

The flatness equations are the vanishing of the RµνI.

Summarising the results:

• F is parallel: ∇F = 0

• the Riemann curvature tensor is determined algebraically in terms

of F and g:

Rµνρσ = Tµνρσ(F, g)

with T quadratic in F . This means that Rµνρσ is parallel

16

(We have used that Γ01···9\ = −1 in this representation.)

The flatness equations are the vanishing of the RµνI.

Summarising the results:

• F is parallel: ∇F = 0

• the Riemann curvature tensor is determined algebraically in terms

of F and g:

Rµνρσ = Tµνρσ(F, g)

with T quadratic in F . This means that Rµνρσ is parallel;

equivalently, that g is locally symmetric.

17

• F obeys the Plucker relations

ιXιY ιZF ∧ F = 0 for all X, Y, Z

17

• F obeys the Plucker relations

ιXιY ιZF ∧ F = 0 for all X, Y, Z

or

Fαβγ[µFνρστ ] = 0

17

• F obeys the Plucker relations

ιXιY ιZF ∧ F = 0 for all X, Y, Z

or

Fαβγ[µFνρστ ] = 0

The solution is that F is decomposable into a wedge product of

four 1-forms

17

• F obeys the Plucker relations

ιXιY ιZF ∧ F = 0 for all X, Y, Z

or

Fαβγ[µFνρστ ] = 0

The solution is that F is decomposable into a wedge product of

four 1-forms:

F = θ1 ∧ θ2 ∧ θ3 ∧ θ4

17

• F obeys the Plucker relations

ιXιY ιZF ∧ F = 0 for all X, Y, Z

or

Fαβγ[µFνρστ ] = 0

The solution is that F is decomposable into a wedge product of

four 1-forms:

F = θ1 ∧ θ2 ∧ θ3 ∧ θ4 or Fµνρσ = θ1[µθ2

νθ3ρθ

4σ]

18

We can restrict to the tangent space at any one point in the

spacetime:

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat.

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat. Otherwise

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat. Otherwise:

• if the plane is euclidean

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat. Otherwise:

• if the plane is euclidean, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F1234

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat. Otherwise:

• if the plane is euclidean, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F1234

• if the plane is lorentzian

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat. Otherwise:

• if the plane is euclidean, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F1234

• if the plane is lorentzian, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F0123

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat. Otherwise:

• if the plane is euclidean, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F1234

• if the plane is lorentzian, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F0123

• if the plane is null

18

We can restrict to the tangent space at any one point in the

spacetime: the metric g defines a lorentzian inner product and F is

either zero or defines a 4-plane: the plane spanned by the θi.

If F is zero, then the solution is flat. Otherwise:

• if the plane is euclidean, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F1234

• if the plane is lorentzian, we can choose a pseudo-orthonormal

frame in which the only nonzero component of F is F0123

• if the plane is null, we can choose a pseudo-orthonormal frame in

which the only nonzero component of F is F−123

19

We now plug these expressions back into the equation which relates

the curvature tensor to F and g

19

We now plug these expressions back into the equation which relates

the curvature tensor to F and g finding the following solutions:

• F euclidean

19

We now plug these expressions back into the equation which relates

the curvature tensor to F and g finding the following solutions:

• F euclidean: a one parameter R > 0 family of vacua

AdS7(−7R)× S4(8R) F =√

6R dvol(S4)

19

We now plug these expressions back into the equation which relates

the curvature tensor to F and g finding the following solutions:

• F euclidean: a one parameter R > 0 family of vacua

AdS7(−7R)× S4(8R) F =√

6R dvol(S4)

• F lorentzian

19

We now plug these expressions back into the equation which relates

the curvature tensor to F and g finding the following solutions:

• F euclidean: a one parameter R > 0 family of vacua

AdS7(−7R)× S4(8R) F =√

6R dvol(S4)

• F lorentzian: a one parameter R < 0 family of vacua

AdS4(8R)× S7(−7R) F =√−6R dvol(AdS4)

20

• F null

20

• F null: a one parameter µ ∈ R family of symmetric plane waves:

g = 2dx+dx− − 136µ

2

(4

3∑i=1

(xi)2 +9∑

i=4

(xi)2)

(dx−)2 +9∑

i=1

(dxi)2

20

• F null: a one parameter µ ∈ R family of symmetric plane waves:

g = 2dx+dx− − 136µ

2

(4

3∑i=1

(xi)2 +9∑

i=4

(xi)2)

(dx−)2 +9∑

i=1

(dxi)2

F = µdx− ∧ dx1 ∧ dx2 ∧ dx3

Notice that for µ = 0 we recover the flat space solution

20

• F null: a one parameter µ ∈ R family of symmetric plane waves:

g = 2dx+dx− − 136µ

2

(4

3∑i=1

(xi)2 +9∑

i=4

(xi)2)

(dx−)2 +9∑

i=1

(dxi)2

F = µdx− ∧ dx1 ∧ dx2 ∧ dx3

Notice that for µ = 0 we recover the flat space solution; whereas

for µ 6= 0 all solutions are equivalent

20

• F null: a one parameter µ ∈ R family of symmetric plane waves:

g = 2dx+dx− − 136µ

2

(4

3∑i=1

(xi)2 +9∑

i=4

(xi)2)

(dx−)2 +9∑

i=1

(dxi)2

F = µdx− ∧ dx1 ∧ dx2 ∧ dx3

Notice that for µ = 0 we recover the flat space solution; whereas

for µ 6= 0 all solutions are equivalent and coincide with the

eleven-dimensional vacuum discovered by Kowalski-Glikman in

1984.

20

• F null: a one parameter µ ∈ R family of symmetric plane waves:

g = 2dx+dx− − 136µ

2

(4

3∑i=1

(xi)2 +9∑

i=4

(xi)2)

(dx−)2 +9∑

i=1

(dxi)2

F = µdx− ∧ dx1 ∧ dx2 ∧ dx3

Notice that for µ = 0 we recover the flat space solution; whereas

for µ 6= 0 all solutions are equivalent and coincide with the

eleven-dimensional vacuum discovered by Kowalski-Glikman in

1984.

All vacua embed isometrically in E2,11 as the intersections of two

quadrics.

21

Solutions are related by Penrose limits

21

Solutions are related by Penrose limits, which are induced by a

generalised Penrose limit in E2,11

21

Solutions are related by Penrose limits, which are induced by a

generalised Penrose limit in E2,11:

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Blau–FO–Papadopoulos hep-th/0202111]

AdS4×S7 S4 ×AdS7

21

Solutions are related by Penrose limits, which are induced by a

generalised Penrose limit in E2,11:

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Blau–FO–Papadopoulos hep-th/0202111]

AdS4×S7 S4 ×AdS7

@@

@@

@@I

KG

21

Solutions are related by Penrose limits, which are induced by a

generalised Penrose limit in E2,11:

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Blau–FO–Papadopoulos hep-th/0202111]

AdS4×S7 S4 ×AdS7

@@

@@

@@I

KG

@@

@@

@@R

flat

21

Solutions are related by Penrose limits, which are induced by a

generalised Penrose limit in E2,11:

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Blau–FO–Papadopoulos hep-th/0202111]

AdS4×S7 S4 ×AdS7

@@

@@

@@I

KG

@@

@@

@@R

flat?

[Back]

22

(1, 0) D = 6 supergravity

22

(1, 0) D = 6 supergravity

• bosonic fields

22

(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

22

(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

? anti-selfdual closed 3-form F

22

(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

? anti-selfdual closed 3-form F

for a total of 9 + 3 = 12 physical bosonic degrees of freedom

22

(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

? anti-selfdual closed 3-form F

for a total of 9 + 3 = 12 physical bosonic degrees of freedom

• spinors are positive-chirality symplectic Majorana–Weyl

22

(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

? anti-selfdual closed 3-form F

for a total of 9 + 3 = 12 physical bosonic degrees of freedom

• spinors are positive-chirality symplectic Majorana–Weyl; i.e.,

associated to the 8-dimensional real representation of Spin(1, 5)×Sp(1) having positive six-dimensional chirality.

22

(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

? anti-selfdual closed 3-form F

for a total of 9 + 3 = 12 physical bosonic degrees of freedom

• spinors are positive-chirality symplectic Majorana–Weyl; i.e.,

associated to the 8-dimensional real representation of Spin(1, 5)×Sp(1) having positive six-dimensional chirality.

The gravitino has therefore also 12 physical degrees of freedom.

23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

The connection D is actually induced from a metric connection

with torsion

23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

The connection D is actually induced from a metric connection

with torsion; i.e., Dg = 0

23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

The connection D is actually induced from a metric connection

with torsion; i.e., Dg = 0 and

Dµ∂ν = Γµνρ∂ρ

23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

The connection D is actually induced from a metric connection

with torsion; i.e., Dg = 0 and

Dµ∂ν = Γµνρ∂ρ with Γµν

ρ = Γµνρ + Fµν

ρ

23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

The connection D is actually induced from a metric connection

with torsion; i.e., Dg = 0 and

Dµ∂ν = Γµνρ∂ρ with Γµν

ρ = Γµνρ + Fµν

ρ

Maximal supersymmetry =⇒ D is flat.

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion is locally isometric to a parallelised Lie group with bi-invariant metric.

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion is locally isometric to a parallelised Lie group with bi-invariant metric.

As a corollary, vacua of (1, 0) D = 6 supergravity are locally

isometric to six-dimensional Lie groups

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion is locally isometric to a parallelised Lie group with bi-invariant metric.

As a corollary, vacua of (1, 0) D = 6 supergravity are locally

isometric to six-dimensional Lie groups admitting a bi-invariant

lorentzian metric

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion is locally isometric to a parallelised Lie group with bi-invariant metric.

As a corollary, vacua of (1, 0) D = 6 supergravity are locally

isometric to six-dimensional Lie groups admitting a bi-invariant

lorentzian metric and whose parallelizing torsion is anti-self-dual.

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion is locally isometric to a parallelised Lie group with bi-invariant metric.

As a corollary, vacua of (1, 0) D = 6 supergravity are locally

isometric to six-dimensional Lie groups admitting a bi-invariant

lorentzian metric and whose parallelizing torsion is anti-self-dual.

Equivalently, they are in one-to-one correspondence with

six-dimensional Lie algebras with an invariant lorentzian metric

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion is locally isometric to a parallelised Lie group with bi-invariant metric.

As a corollary, vacua of (1, 0) D = 6 supergravity are locally

isometric to six-dimensional Lie groups admitting a bi-invariant

lorentzian metric and whose parallelizing torsion is anti-self-dual.

Equivalently, they are in one-to-one correspondence with

six-dimensional Lie algebras with an invariant lorentzian metric and

with anti-selfdual structure constants fabc.

24

Theorem (Cartan–Schouten (1926), Wolf (1971/2),Cahen–Parker (1977)).A lorentzian manifold admitting a flat metric connection withtorsion is locally isometric to a parallelised Lie group with bi-invariant metric.

As a corollary, vacua of (1, 0) D = 6 supergravity are locally

isometric to six-dimensional Lie groups admitting a bi-invariant

lorentzian metric and whose parallelizing torsion is anti-self-dual.

Equivalently, they are in one-to-one correspondence with

six-dimensional Lie algebras with an invariant lorentzian metric and

with anti-selfdual structure constants fabc.

The solution to this problem is known.

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

• semisimple Lie algebras

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

• semisimple Lie algebras with the Killing form (Cartan’s criterion)

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

• semisimple Lie algebras with the Killing form (Cartan’s criterion)

• reductive Lie algebras = semisimple ⊕ abelian

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

• semisimple Lie algebras with the Killing form (Cartan’s criterion)

• reductive Lie algebras = semisimple ⊕ abelian

• classical doubles h n h∗

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

• semisimple Lie algebras with the Killing form (Cartan’s criterion)

• reductive Lie algebras = semisimple ⊕ abelian

• classical doubles h n h∗ with the dual pairing

25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

• semisimple Lie algebras with the Killing form (Cartan’s criterion)

• reductive Lie algebras = semisimple ⊕ abelian

• classical doubles h n h∗ with the dual pairing

But there is a more general construction.

26

The double extension

• g a Lie algebra with an invariant metric

26

The double extension

• g a Lie algebra with an invariant metric

• h a Lie algebra acting on g via antisymmetric derivations

26

The double extension

• g a Lie algebra with an invariant metric

• h a Lie algebra acting on g via antisymmetric derivations; i.e.,

? preserving the Lie bracket of g

26

The double extension

• g a Lie algebra with an invariant metric

• h a Lie algebra acting on g via antisymmetric derivations; i.e.,

? preserving the Lie bracket of g, and

? preserving the metric

26

The double extension

• g a Lie algebra with an invariant metric

• h a Lie algebra acting on g via antisymmetric derivations; i.e.,

? preserving the Lie bracket of g, and

? preserving the metric

• since h preserves the metric on g, there is a linear map

h → Λ2g

27

whose dual map

ω : Λ2g → h∗

27

whose dual map

ω : Λ2g → h∗

is a cocycle

27

whose dual map

ω : Λ2g → h∗

is a cocycle because h preserves the Lie bracket in g

27

whose dual map

ω : Λ2g → h∗

is a cocycle because h preserves the Lie bracket in g

• so we build the central extension g×ω h∗

27

whose dual map

ω : Λ2g → h∗

is a cocycle because h preserves the Lie bracket in g

• so we build the central extension g×ω h∗; i.e.,

[Xa, Xb] = fabcXc + ωab iH

i

27

whose dual map

ω : Λ2g → h∗

is a cocycle because h preserves the Lie bracket in g

• so we build the central extension g×ω h∗; i.e.,

[Xa, Xb] = fabcXc + ωab iH

i

relative to bases Xa, Hi and Hi for g, h and h∗, respectively.

27

whose dual map

ω : Λ2g → h∗

is a cocycle because h preserves the Lie bracket in g

• so we build the central extension g×ω h∗; i.e.,

[Xa, Xb] = fabcXc + ωab iH

i

relative to bases Xa, Hi and Hi for g, h and h∗, respectively.

• h acts on g×ω h∗ preserving the Lie bracket

27

whose dual map

ω : Λ2g → h∗

is a cocycle because h preserves the Lie bracket in g

• so we build the central extension g×ω h∗; i.e.,

[Xa, Xb] = fabcXc + ωab iH

i

relative to bases Xa, Hi and Hi for g, h and h∗, respectively.

• h acts on g×ω h∗ preserving the Lie bracket, so we can form the

double extension

d(g, h) = h n (g×ω h∗)

28

• the double extension admits an invariant metric

28

• the double extension admits an invariant metric

Xb Hj Hj

Xa gab 0 0Hi 0 Bij δj

i

Hi 0 δij 0

28

• the double extension admits an invariant metric

Xb Hj Hj

Xa gab 0 0Hi 0 Bij δj

i

Hi 0 δij 0

where Bij is any invariant symmetric bilinear form on h (not

necessarily nondegenerate).

28

• the double extension admits an invariant metric

Xb Hj Hj

Xa gab 0 0Hi 0 Bij δj

i

Hi 0 δij 0

where Bij is any invariant symmetric bilinear form on h (not

necessarily nondegenerate).

This construction is due to Medina and Revoy who proved an

important structure theorem.

29

The structure theorem of Medina and Revoy

A metric Lie algebra is indecomposable if it is not the direct sum of

two orthogonal ideals.

29

The structure theorem of Medina and Revoy

A metric Lie algebra is indecomposable if it is not the direct sum of

two orthogonal ideals.

Theorem (Medina–Revoy (1985)).

29

The structure theorem of Medina and Revoy

A metric Lie algebra is indecomposable if it is not the direct sum of

two orthogonal ideals.

Theorem (Medina–Revoy (1985)).An indecomposable metric Lie algebra is either

29

The structure theorem of Medina and Revoy

A metric Lie algebra is indecomposable if it is not the direct sum of

two orthogonal ideals.

Theorem (Medina–Revoy (1985)).An indecomposable metric Lie algebra is either simple

29

The structure theorem of Medina and Revoy

A metric Lie algebra is indecomposable if it is not the direct sum of

two orthogonal ideals.

Theorem (Medina–Revoy (1985)).An indecomposable metric Lie algebra is either simple, one-dimensional

29

The structure theorem of Medina and Revoy

A metric Lie algebra is indecomposable if it is not the direct sum of

two orthogonal ideals.

Theorem (Medina–Revoy (1985)).An indecomposable metric Lie algebra is either simple, one-dimensional, or a double extension d(g, h) where h is either simpleor one-dimensional.

29

The structure theorem of Medina and Revoy

A metric Lie algebra is indecomposable if it is not the direct sum of

two orthogonal ideals.

Theorem (Medina–Revoy (1985)).An indecomposable metric Lie algebra is either simple, one-dimensional, or a double extension d(g, h) where h is either simpleor one-dimensional.Every metric Lie algebra is obtained as an orthogonal direct sumof indecomposables.

30

Six-dimensional lorentzian Lie algebras

It is now easy to list all six-dimensional lorentzian Lie algebras.

30

Six-dimensional lorentzian Lie algebras

It is now easy to list all six-dimensional lorentzian Lie algebras.

Notice that if the metric on g has signature (p, q) and h is

r-dimensional, the metric on d(g, h) has signature (p + r, q + r).

30

Six-dimensional lorentzian Lie algebras

It is now easy to list all six-dimensional lorentzian Lie algebras.

Notice that if the metric on g has signature (p, q) and h is

r-dimensional, the metric on d(g, h) has signature (p + r, q + r).

Therefore a lorentzian Lie algebra takes the general form

30

Six-dimensional lorentzian Lie algebras

It is now easy to list all six-dimensional lorentzian Lie algebras.

Notice that if the metric on g has signature (p, q) and h is

r-dimensional, the metric on d(g, h) has signature (p + r, q + r).

Therefore a lorentzian Lie algebra takes the general form

reductive⊕ d(a, h)

30

Six-dimensional lorentzian Lie algebras

It is now easy to list all six-dimensional lorentzian Lie algebras.

Notice that if the metric on g has signature (p, q) and h is

r-dimensional, the metric on d(g, h) has signature (p + r, q + r).

Therefore a lorentzian Lie algebra takes the general form

reductive⊕ d(a, h)

where a is abelian with euclidean metric and h is one-dimensional.

30

Six-dimensional lorentzian Lie algebras

It is now easy to list all six-dimensional lorentzian Lie algebras.

Notice that if the metric on g has signature (p, q) and h is

r-dimensional, the metric on d(g, h) has signature (p + r, q + r).

Therefore a lorentzian Lie algebra takes the general form

reductive⊕ d(a, h)

where a is abelian with euclidean metric and h is one-dimensional.

(Any semisimple factors in a factor out of the double extension.

[FO–Stanciu hep-th/9402035])

31

Six-dimensional lorentzian Lie algebras:

31

Six-dimensional lorentzian Lie algebras:

• R5,1

31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

• so(2, 1)⊕ R3

31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

• so(2, 1)⊕ R3

• so(2, 1)⊕ so(3)

31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

• so(2, 1)⊕ R3

• so(2, 1)⊕ so(3)

• d(R4, R)

31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

• so(2, 1)⊕ R3

• so(2, 1)⊕ so(3)

• d(R4, R), actually a family of Lie algebras parametrised by

homomorphisms

R → Λ2R4 ∼= so(4)

32

Antiselfduality of the structure constants narrows the list down to

32

Antiselfduality of the structure constants narrows the list down to

• R5,1

32

Antiselfduality of the structure constants narrows the list down to

• R5,1

• so(2, 1)⊕ so(3) with “commensurate” metrics

32

Antiselfduality of the structure constants narrows the list down to

• R5,1

• so(2, 1)⊕ so(3) with “commensurate” metrics, and

• d(R4, R) with the image of R → Λ2R4 self-dual

32

Antiselfduality of the structure constants narrows the list down to

• R5,1

• so(2, 1)⊕ so(3) with “commensurate” metrics, and

• d(R4, R) with the image of R → Λ2R4 self-dual

The first case corresponds to the flat vacuum.

32

Antiselfduality of the structure constants narrows the list down to

• R5,1

• so(2, 1)⊕ so(3) with “commensurate” metrics, and

• d(R4, R) with the image of R → Λ2R4 self-dual

The first case corresponds to the flat vacuum. The second case

corresponds to AdS3×S3 with equal radii of curvature and

F ∝ dvol(AdS3) + dvol(S3)

32

Antiselfduality of the structure constants narrows the list down to

• R5,1

• so(2, 1)⊕ so(3) with “commensurate” metrics, and

• d(R4, R) with the image of R → Λ2R4 self-dual

The first case corresponds to the flat vacuum. The second case

corresponds to AdS3×S3 with equal radii of curvature and

F ∝ dvol(AdS3) + dvol(S3)

The third case is a six-dimensional version of the Nappi-Witten

spacetime, NW6, discovered by Meessen. [Meessen hep-th/0111031]

33

The vacua are related by Penrose limits:

33

The vacua are related by Penrose limits:

AdS3×S3

33

The vacua are related by Penrose limits:

AdS3×S3

NW6

33

The vacua are related by Penrose limits:

AdS3×S3

NW6

@@

@@

@@R

flat

33

The vacua are related by Penrose limits:

AdS3×S3

NW6

@@

@@

@@R

flat?

33

The vacua are related by Penrose limits:

AdS3×S3

NW6

@@

@@

@@R

flat?

which in this case are group contractions a la Inonu–Wigner.

[Stanciu–FO hep-th/0303212, Olive–Rabinovici–Schwimmer hep-th/9311081]

[Back]

34

(2, 0) D = 6 supergravity

34

(2, 0) D = 6 supergravity

• bosonic fields

34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

? 5 anti-selfdual closed 3-form F i

34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

? 5 anti-selfdual closed 3-form F i transforming in the 5-

dimensional representation of Sp(2) ∼= Spin(5)

34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

? 5 anti-selfdual closed 3-form F i transforming in the 5-

dimensional representation of Sp(2) ∼= Spin(5)

for a total of 9 + 15 = 24 physical bosonic degrees of freedom

34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

? 5 anti-selfdual closed 3-form F i transforming in the 5-

dimensional representation of Sp(2) ∼= Spin(5)

for a total of 9 + 15 = 24 physical bosonic degrees of freedom

• spinors are positive-chirality symplectic Majorana–Weyl

34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

? 5 anti-selfdual closed 3-form F i transforming in the 5-

dimensional representation of Sp(2) ∼= Spin(5)

for a total of 9 + 15 = 24 physical bosonic degrees of freedom

• spinors are positive-chirality symplectic Majorana–Weyl; i.e.,

associated to the 16-dimensional real representation of

Spin(1, 5)× Sp(2) having positive six-dimensional chirality.

34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

? 5 anti-selfdual closed 3-form F i transforming in the 5-

dimensional representation of Sp(2) ∼= Spin(5)

for a total of 9 + 15 = 24 physical bosonic degrees of freedom

• spinors are positive-chirality symplectic Majorana–Weyl; i.e.,

associated to the 16-dimensional real representation of

Spin(1, 5)× Sp(2) having positive six-dimensional chirality.

The gravitino has therefore also 24 physical degrees of freedom.

35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

Maximal supersymmetry =⇒ D is flat

35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

Maximal supersymmetry =⇒ D is flat =⇒

• F iµνρ = Fµνρv

i

35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

Maximal supersymmetry =⇒ D is flat =⇒

• F iµνρ = Fµνρv

i, where F is an antiself-dual 3-form and v some

constant unit vector

35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

Maximal supersymmetry =⇒ D is flat =⇒

• F iµνρ = Fµνρv

i, where F is an antiself-dual 3-form and v some

constant unit vector

• (g, F ) is a vacuum solution of the (1, 0) theory

35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

Maximal supersymmetry =⇒ D is flat =⇒

• F iµνρ = Fµνρv

i, where F is an antiself-dual 3-form and v some

constant unit vector

• (g, F ) is a vacuum solution of the (1, 0) theory

=⇒ (1, 0) vacua ↔ (2, 0) vacua up to Sp(2) R-symmetry.

36

D = 10 IIB supergravity

• bosonic fields:

36

D = 10 IIB supergravity

• bosonic fields:

? metric g

36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ

36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ ,

? closed complex 3-form H

36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ ,

? closed complex 3-form H, and

? closed selfdual 5-form F

36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ ,

? closed complex 3-form H, and

? closed selfdual 5-form F

=⇒ 35+2+56+35 = 128 bosonic physical degrees of freedom

36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ ,

? closed complex 3-form H, and

? closed selfdual 5-form F

=⇒ 35+2+56+35 = 128 bosonic physical degrees of freedom

• spinors are positive-chirality Majorana–Weyl spinors.

There are two gravitini and two dilatini

36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ ,

? closed complex 3-form H, and

? closed selfdual 5-form F

=⇒ 35+2+56+35 = 128 bosonic physical degrees of freedom

• spinors are positive-chirality Majorana–Weyl spinors.

There are two gravitini and two dilatini

=⇒ 112 + 16 = 128 fermionic physical degrees of freedom

37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

Maximal supersymmetry =⇒ τ is constant and H = 0

37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

Maximal supersymmetry =⇒ τ is constant and H = 0

• the gravitino variation defines the connection

37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

Maximal supersymmetry =⇒ τ is constant and H = 0

• the gravitino variation defines the connection

(with H = 0 and τ constant)

37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

Maximal supersymmetry =⇒ τ is constant and H = 0

• the gravitino variation defines the connection

(with H = 0 and τ constant)

Dµ = ∇µ + iαFν1ν2ν3ν4ν5Γν1ν2ν3ν4ν5Γµ

37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

Maximal supersymmetry =⇒ τ is constant and H = 0

• the gravitino variation defines the connection

(with H = 0 and τ constant)

Dµ = ∇µ + iαFν1ν2ν3ν4ν5Γν1ν2ν3ν4ν5Γµ

where we have written the two real spinors as a complex spinor

37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

Maximal supersymmetry =⇒ τ is constant and H = 0

• the gravitino variation defines the connection

(with H = 0 and τ constant)

Dµ = ∇µ + iαFν1ν2ν3ν4ν5Γν1ν2ν3ν4ν5Γµ

where we have written the two real spinors as a complex spinor,

and α depends on the constant value of τ .

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero, we find

• F is parallel

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero, we find

• F is parallel: ∇F = 0

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero, we find

• F is parallel: ∇F = 0

• the Riemann curvature tensor is again determined algebraically in

terms of F and g

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero, we find

• F is parallel: ∇F = 0

• the Riemann curvature tensor is again determined algebraically in

terms of F and g:

Rµνρσ = Tµνρσ(F, g)

with T quadratic in F

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero, we find

• F is parallel: ∇F = 0

• the Riemann curvature tensor is again determined algebraically in

terms of F and g:

Rµνρσ = Tµνρσ(F, g)

with T quadratic in F . Again this means that Rµνρσ is parallel

38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero, we find

• F is parallel: ∇F = 0

• the Riemann curvature tensor is again determined algebraically in

terms of F and g:

Rµνρσ = Tµνρσ(F, g)

with T quadratic in F . Again this means that Rµνρσ is parallel,

so that g is locally symmetric.

39

• F obeys a quadratic identity

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations and the Jacobi identity.

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations and the Jacobi identity.

Again we can work in the tangent space at a point

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations and the Jacobi identity.

Again we can work in the tangent space at a point, where g gives

rise to a lorentzian innner product

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations and the Jacobi identity.

Again we can work in the tangent space at a point, where g gives

rise to a lorentzian innner product and F defines a self-dual 5-form

obeying a quadratic equation.

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations and the Jacobi identity.

Again we can work in the tangent space at a point, where g gives

rise to a lorentzian innner product and F defines a self-dual 5-form

obeying a quadratic equation.

This equation defines a generalisation of a Lie algebra known as a

4-Lie algebra (with an invariant metric). [Filippov (1985)]

39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations and the Jacobi identity.

Again we can work in the tangent space at a point, where g gives

rise to a lorentzian innner product and F defines a self-dual 5-form

obeying a quadratic equation.

This equation defines a generalisation of a Lie algebra known as a

4-Lie algebra (with an invariant metric). [Filippov (1985)]

(Unfortunate notation: a 2-Lie algebra is a Lie algebra.)

40

n-Lie algebras

40

n-Lie algebras

A Lie algebra is a vector space g

40

n-Lie algebras

A Lie algebra is a vector space g together with an antisymmetric

bilinear map

[ ] : Λ2g → g

40

n-Lie algebras

A Lie algebra is a vector space g together with an antisymmetric

bilinear map

[ ] : Λ2g → g

satisfying the condition

40

n-Lie algebras

A Lie algebra is a vector space g together with an antisymmetric

bilinear map

[ ] : Λ2g → g

satisfying the condition: for all X ∈ g the map

adX : g → g defined by adX Y = [X, Y ]

40

n-Lie algebras

A Lie algebra is a vector space g together with an antisymmetric

bilinear map

[ ] : Λ2g → g

satisfying the condition: for all X ∈ g the map

adX : g → g defined by adX Y = [X, Y ]

is a derivation over [ ]

40

n-Lie algebras

A Lie algebra is a vector space g together with an antisymmetric

bilinear map

[ ] : Λ2g → g

satisfying the condition: for all X ∈ g the map

adX : g → g defined by adX Y = [X, Y ]

is a derivation over [ ]; that is,

[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]]

41

An n-Lie algebra

41

An n-Lie algebra is a vector space n

41

An n-Lie algebra is a vector space n together with an

antisymmetric n-linear map

41

An n-Lie algebra is a vector space n together with an

antisymmetric n-linear map

[ ] : Λnn → n

41

An n-Lie algebra is a vector space n together with an

antisymmetric n-linear map

[ ] : Λnn → n

satisfying the condition

41

An n-Lie algebra is a vector space n together with an

antisymmetric n-linear map

[ ] : Λnn → n

satisfying the condition: for all X1, . . . , Xn−1 ∈ n

41

An n-Lie algebra is a vector space n together with an

antisymmetric n-linear map

[ ] : Λnn → n

satisfying the condition: for all X1, . . . , Xn−1 ∈ n, the map

adX1,...,Xn−1 : n → n

41

An n-Lie algebra is a vector space n together with an

antisymmetric n-linear map

[ ] : Λnn → n

satisfying the condition: for all X1, . . . , Xn−1 ∈ n, the map

adX1,...,Xn−1 : n → n

defined by

adX1,...,Xn−1 Y = [X1, . . . , Xn−1, Y ]

42

is a derivation over [ ]

42

is a derivation over [ ]; that is,

[X1, . . . , Xn−1, [Y1, . . . , Yn]] =n∑

i=1

[Y1, . . . , [X1, . . . , Xn−1, Yi], . . . , Yn]

42

is a derivation over [ ]; that is,

[X1, . . . , Xn−1, [Y1, . . . , Yn]] =n∑

i=1

[Y1, . . . , [X1, . . . , Xn−1, Yi], . . . , Yn]

If 〈−,−〉 is a metric on n, we can define F by

F (X1, . . . , Xn+1) = 〈[X1, . . . , Xn], Xn+1〉

42

is a derivation over [ ]; that is,

[X1, . . . , Xn−1, [Y1, . . . , Yn]] =n∑

i=1

[Y1, . . . , [X1, . . . , Xn−1, Yi], . . . , Yn]

If 〈−,−〉 is a metric on n, we can define F by

F (X1, . . . , Xn+1) = 〈[X1, . . . , Xn], Xn+1〉

If F is totally antisymmetric then 〈−,−〉 is an invariant metric.

42

is a derivation over [ ]; that is,

[X1, . . . , Xn−1, [Y1, . . . , Yn]] =n∑

i=1

[Y1, . . . , [X1, . . . , Xn−1, Yi], . . . , Yn]

If 〈−,−〉 is a metric on n, we can define F by

F (X1, . . . , Xn+1) = 〈[X1, . . . , Xn], Xn+1〉

If F is totally antisymmetric then 〈−,−〉 is an invariant metric.

(n-Lie algebras also appear naturally in the context of Nambu

dynamics. [Nambu (1973)])

43

Ten-dimensional lorentzian 4-Lie algebras

In this language, IIB vacua are in one-to-one correspondence with

ten-dimensional selfdual lorentzian 4-Lie algebras

43

Ten-dimensional lorentzian 4-Lie algebras

In this language, IIB vacua are in one-to-one correspondence with

ten-dimensional selfdual lorentzian 4-Lie algebras; but this is not

particularly helpful since the theory of n-Lie algebras is still largely

undeveloped.

43

Ten-dimensional lorentzian 4-Lie algebras

In this language, IIB vacua are in one-to-one correspondence with

ten-dimensional selfdual lorentzian 4-Lie algebras; but this is not

particularly helpful since the theory of n-Lie algebras is still largely

undeveloped.

One is forced to solve the equations.

43

Ten-dimensional lorentzian 4-Lie algebras

In this language, IIB vacua are in one-to-one correspondence with

ten-dimensional selfdual lorentzian 4-Lie algebras; but this is not

particularly helpful since the theory of n-Lie algebras is still largely

undeveloped.

One is forced to solve the equations. After a lot of work, we

found that a selfdual 5-form obeys the equation if and only if

43

Ten-dimensional lorentzian 4-Lie algebras

In this language, IIB vacua are in one-to-one correspondence with

ten-dimensional selfdual lorentzian 4-Lie algebras; but this is not

particularly helpful since the theory of n-Lie algebras is still largely

undeveloped.

One is forced to solve the equations. After a lot of work, we

found that a selfdual 5-form obeys the equation if and only if

F = G + ?G

43

Ten-dimensional lorentzian 4-Lie algebras

In this language, IIB vacua are in one-to-one correspondence with

ten-dimensional selfdual lorentzian 4-Lie algebras; but this is not

particularly helpful since the theory of n-Lie algebras is still largely

undeveloped.

One is forced to solve the equations. After a lot of work, we

found that a selfdual 5-form obeys the equation if and only if

F = G + ?G where G = θ1 ∧ θ2 ∧ θ3 ∧ θ4 ∧ θ5

43

Ten-dimensional lorentzian 4-Lie algebras

In this language, IIB vacua are in one-to-one correspondence with

ten-dimensional selfdual lorentzian 4-Lie algebras; but this is not

particularly helpful since the theory of n-Lie algebras is still largely

undeveloped.

One is forced to solve the equations. After a lot of work, we

found that a selfdual 5-form obeys the equation if and only if

F = G + ?G where G = θ1 ∧ θ2 ∧ θ3 ∧ θ4 ∧ θ5

[FO–Papadopoulos math.AG/0211170]

44

G is decomposable;

44

G is decomposable; whence, if nonzero

44

G is decomposable; whence, if nonzero, it defines a 5-plane

44

G is decomposable; whence, if nonzero, it defines a 5-plane, and

hence F defines two orthogonal planes.

44

G is decomposable; whence, if nonzero, it defines a 5-plane, and

hence F defines two orthogonal planes.

If F = 0 we recover the flat vacuum.

44

G is decomposable; whence, if nonzero, it defines a 5-plane, and

hence F defines two orthogonal planes.

If F = 0 we recover the flat vacuum. Otherwise there are two

possibilities:

44

G is decomposable; whence, if nonzero, it defines a 5-plane, and

hence F defines two orthogonal planes.

If F = 0 we recover the flat vacuum. Otherwise there are two

possibilities:

• one plane is lorentzian and the other euclidean

44

G is decomposable; whence, if nonzero, it defines a 5-plane, and

hence F defines two orthogonal planes.

If F = 0 we recover the flat vacuum. Otherwise there are two

possibilities:

• one plane is lorentzian and the other euclidean: we can choose a

pseudo-orthonormal frame in which the only nonzero components

of F are F01234 = F56789

44

G is decomposable; whence, if nonzero, it defines a 5-plane, and

hence F defines two orthogonal planes.

If F = 0 we recover the flat vacuum. Otherwise there are two

possibilities:

• one plane is lorentzian and the other euclidean: we can choose a

pseudo-orthonormal frame in which the only nonzero components

of F are F01234 = F56789, or

• both planes are null

44

G is decomposable; whence, if nonzero, it defines a 5-plane, and

hence F defines two orthogonal planes.

If F = 0 we recover the flat vacuum. Otherwise there are two

possibilities:

• one plane is lorentzian and the other euclidean: we can choose a

pseudo-orthonormal frame in which the only nonzero components

of F are F01234 = F56789, or

• both planes are null: we can choose a pseudo-orthonormal frame

in which the only nonzero components of F are F−1234 = F−5678.

45

Plugging these expressions back into the relation between the

curvature tensor to F and g

45

Plugging these expressions back into the relation between the

curvature tensor to F and g, one finds the following backgrounds

(up to local isometry)

45

Plugging these expressions back into the relation between the

curvature tensor to F and g, one finds the following backgrounds

(up to local isometry):

• F non-degenerate case

45

Plugging these expressions back into the relation between the

curvature tensor to F and g, one finds the following backgrounds

(up to local isometry):

• F non-degenerate case: a one-parameter (R > 0) family of vacua

AdS5(−R)× S5(R) F =

√4R

5(dvol(AdS5)− dvol(S5)

)

45

Plugging these expressions back into the relation between the

curvature tensor to F and g, one finds the following backgrounds

(up to local isometry):

• F non-degenerate case: a one-parameter (R > 0) family of vacua

AdS5(−R)× S5(R) F =

√4R

5(dvol(AdS5)− dvol(S5)

)

• F degenerate

45

Plugging these expressions back into the relation between the

curvature tensor to F and g, one finds the following backgrounds

(up to local isometry):

• F non-degenerate case: a one-parameter (R > 0) family of vacua

AdS5(−R)× S5(R) F =

√4R

5(dvol(AdS5)− dvol(S5)

)

• F degenerate: a one-parameter (µ ∈ R) family of symmetric

46

plane waves:

g = 2dx+dx− − 14µ

28∑

i=1

(xi)2(dx−)2 +8∑

i=1

(dxi)2

46

plane waves:

g = 2dx+dx− − 14µ

28∑

i=1

(xi)2(dx−)2 +8∑

i=1

(dxi)2

F = 12µdx− ∧

(dx1 ∧ dx2 ∧ dx3 ∧ dx4 + dx5 ∧ dx6 ∧ dx7 ∧ dx8

)

46

plane waves:

g = 2dx+dx− − 14µ

28∑

i=1

(xi)2(dx−)2 +8∑

i=1

(dxi)2

F = 12µdx− ∧

(dx1 ∧ dx2 ∧ dx3 ∧ dx4 + dx5 ∧ dx6 ∧ dx7 ∧ dx8

)Again for µ = 0 we recover the flat space solution

46

plane waves:

g = 2dx+dx− − 14µ

28∑

i=1

(xi)2(dx−)2 +8∑

i=1

(dxi)2

F = 12µdx− ∧

(dx1 ∧ dx2 ∧ dx3 ∧ dx4 + dx5 ∧ dx6 ∧ dx7 ∧ dx8

)Again for µ = 0 we recover the flat space solution; whereas for

µ 6= 0 all solutions are isometric to the same plane wave

[Blau–FO–Hull–Papadopoulos hep-th/0110242]

46

plane waves:

g = 2dx+dx− − 14µ

28∑

i=1

(xi)2(dx−)2 +8∑

i=1

(dxi)2

F = 12µdx− ∧

(dx1 ∧ dx2 ∧ dx3 ∧ dx4 + dx5 ∧ dx6 ∧ dx7 ∧ dx8

)Again for µ = 0 we recover the flat space solution; whereas for

µ 6= 0 all solutions are isometric to the same plane wave

[Blau–FO–Hull–Papadopoulos hep-th/0110242]

Notice that g is a bi-invariant metric on a Lie group

46

plane waves:

g = 2dx+dx− − 14µ

28∑

i=1

(xi)2(dx−)2 +8∑

i=1

(dxi)2

F = 12µdx− ∧

(dx1 ∧ dx2 ∧ dx3 ∧ dx4 + dx5 ∧ dx6 ∧ dx7 ∧ dx8

)Again for µ = 0 we recover the flat space solution; whereas for

µ 6= 0 all solutions are isometric to the same plane wave

[Blau–FO–Hull–Papadopoulos hep-th/0110242]

Notice that g is a bi-invariant metric on a Lie group: a

ten-dimensional version of the Nappi–Witten spacetime.

[Stanciu–FO hep-th/0303212]

47

These vacua again embed isometrically in E2,10 as intersections of

quadrics

47

These vacua again embed isometrically in E2,10 as intersections of

quadrics, and are related by Penrose limits

47

These vacua again embed isometrically in E2,10 as intersections of

quadrics, and are related by Penrose limits

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Berenstein–Maldacena–Nastase hep-th/0202021]

AdS5×S5

47

These vacua again embed isometrically in E2,10 as intersections of

quadrics, and are related by Penrose limits

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Berenstein–Maldacena–Nastase hep-th/0202021]

AdS5×S5

BFHP

47

These vacua again embed isometrically in E2,10 as intersections of

quadrics, and are related by Penrose limits

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Berenstein–Maldacena–Nastase hep-th/0202021]

AdS5×S5

BFHP

@@

@@

@@R

flat

47

These vacua again embed isometrically in E2,10 as intersections of

quadrics, and are related by Penrose limits

[Blau–FO–Hull–Papadopoulos hep-th/0201081]

[Berenstein–Maldacena–Nastase hep-th/0202021]

AdS5×S5

BFHP

@@

@@

@@R

flat?

[Back]

48

Other theories

48

Other theories

• D=10: I, heterotic, IIA

48

Other theories

• D=10: I, heterotic, IIA only have the flat vacuum.

48

Other theories

• D=10: I, heterotic, IIA only have the flat vacuum. The same is

true for any theory lower in the corresponding columns.

48

Other theories

• D=10: I, heterotic, IIA only have the flat vacuum. The same is

true for any theory lower in the corresponding columns. (Roman’s

massive supergravity has not vacua at all.)

48

Other theories

• D=10: I, heterotic, IIA only have the flat vacuum. The same is

true for any theory lower in the corresponding columns. (Roman’s

massive supergravity has not vacua at all.)

• D=6: (1, 0) vacua do have reductions preserving all

supersymmetry.

[Gauntlett–Gutowsky–Hull–Pakis–Reall hep-th/0209114]

[Lozano-Tellechea–Meessen–Ortın hep-th/0206200]

48

Other theories

• D=10: I, heterotic, IIA only have the flat vacuum. The same is

true for any theory lower in the corresponding columns. (Roman’s

massive supergravity has not vacua at all.)

• D=6: (1, 0) vacua do have reductions preserving all

supersymmetry.

[Gauntlett–Gutowsky–Hull–Pakis–Reall hep-th/0209114]

[Lozano-Tellechea–Meessen–Ortın hep-th/0206200]

This now has a natural explanation.

49

D=5 N=2 from D=6 (1, 0)

49

D=5 N=2 from D=6 (1, 0)

D=6 (1, 0) D=5 N=2 coupled to a vector multiplet

49

D=5 N=2 from D=6 (1, 0)

D=6 (1, 0) D=5 N=2 coupled to a vector multiplet:

(g, F3) (h, φ, F2, G2)

49

D=5 N=2 from D=6 (1, 0)

D=6 (1, 0) D=5 N=2 coupled to a vector multiplet:

(g, F3) (h, φ, F2, G2)

Minimal D=5 N=2 requires a truncation

49

D=5 N=2 from D=6 (1, 0)

D=6 (1, 0) D=5 N=2 coupled to a vector multiplet:

(g, F3) (h, φ, F2, G2)

Minimal D=5 N=2 requires a truncation

dφ = 0 F2 = G2

49

D=5 N=2 from D=6 (1, 0)

D=6 (1, 0) D=5 N=2 coupled to a vector multiplet:

(g, F3) (h, φ, F2, G2)

Minimal D=5 N=2 requires a truncation

dφ = 0 F2 = G2

Let (M, g, F ) be a vacuum solution of D=6 (1, 0)

49

D=5 N=2 from D=6 (1, 0)

D=6 (1, 0) D=5 N=2 coupled to a vector multiplet:

(g, F3) (h, φ, F2, G2)

Minimal D=5 N=2 requires a truncation

dφ = 0 F2 = G2

Let (M, g, F ) be a vacuum solution of D=6 (1, 0): (M, g) a

parallelised (anti-selfdual, lorentzian) Lie group with torsion F .

50

Let ξ be a (spacelike) left-invariant vector field

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

• truncation is automatic

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

• truncation is automatic

• all vacua of minimal D=5 N=2 supergravity arise in this way:

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

• truncation is automatic

• all vacua of minimal D=5 N=2 supergravity arise in this way:

AdS2×S3 ↔ AdS3×S2

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

• truncation is automatic

• all vacua of minimal D=5 N=2 supergravity arise in this way:

AdS2×S3 ↔ AdS3×S2, E1,4

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

• truncation is automatic

• all vacua of minimal D=5 N=2 supergravity arise in this way:

AdS2×S3 ↔ AdS3×S2, E1,4, KG5

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

• truncation is automatic

• all vacua of minimal D=5 N=2 supergravity arise in this way:

AdS2×S3 ↔ AdS3×S2, E1,4, KG5, Godel

50

Let ξ be a (spacelike) left-invariant vector field, with corresponding

one-parameter subgroup K:

• Kaluza–Klein reduction yields space M/K of right K-cosets

• ξ leaves invariant all Killing spinors

• truncation is automatic

• all vacua of minimal D=5 N=2 supergravity arise in this way:

AdS2×S3 ↔ AdS3×S2, E1,4, KG5, Godel, ...

51

Thank you.