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Maximally supersymmetric spacetimes Jos´ e Figueroa-O’Farrill Edinburgh Mathematical Physics Group School of Mathematics Neve Shalom Wahat al-Salam, 27 May 2003

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Page 1: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

Maximally supersymmetric spacetimes

Jose Figueroa-O’Farrill

Edinburgh Mathematical Physics Group

School of Mathematics

Neve Shalom ∼ Wahat al-Salam, 27 May 2003

Page 2: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Maximally symmetric spacetimes

Page 3: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

Page 4: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Maximally symmetric spacetimes

Which are the maximally symmetric spacetimes?

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

Page 5: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 6: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 7: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 8: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 9: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Therefore a Killing vector ξ is uniquely determined by

Page 10: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Therefore a Killing vector ξ is uniquely determined by:

ξ(p)

Page 11: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p)

Page 12: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Therefore a Killing vector ξ is uniquely determined by:

ξ(p) and ∇µξν(p) at any point p.

Page 13: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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.

Page 14: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 15: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 16: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

(ξν

Fνρ

)=(

∇µξν − Fµν

∇µFνρ −Rµνρσξσ

)

Page 17: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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=⇒ (ξµ, Fµν) is parallel

Page 18: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

• ξµ is a Killing vector

Page 19: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

• ξµ is a Killing vector, and

• Fµν = ∇µξν

Page 20: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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=⇒ (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 .

Page 21: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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=⇒ (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.

Page 22: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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=⇒ (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

Page 23: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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=⇒ (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.

Page 24: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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In the riemannian case (and up to local isometry)

Page 25: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

• κ = 0: euclidean space En

Page 26: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

• κ = 0: euclidean space En

• κ > 0: sphere

Sn ⊂ En+1

Page 27: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 28: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 29: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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 : −t21 + x21 + · · ·+ x2

n =−1κ2

Page 30: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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In lorentzian geometry (and up to local isometry)

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In lorentzian geometry (and up to local isometry):

• κ = 0: Minkowski space En−1,1

Page 32: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 33: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 34: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 35: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 36: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

forms with κ 6= 0.

Page 37: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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.

Page 38: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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.

Page 39: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 41: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

Page 42: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

Which are the maximally symmetric spacetimes?

Page 43: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Maximally supersymmetric spacetimes

The supersymmetric analogue of the question:

Which are the maximally symmetric spacetimes?

is the question:

Page 44: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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?

Page 45: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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.

Page 46: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Based on work in collaboration with

Page 47: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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)

Page 48: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 49: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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]

Page 50: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Strategy

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

Page 51: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Strategy

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

• (M, g) a lorentzian spin manifold

Page 52: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Strategy

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

• (M, g) a lorentzian spin manifold

• Φ denotes collectively the other bosonic fields

Page 53: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 54: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 55: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 56: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 57: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 58: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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)

Page 59: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 60: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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There are possibly also algebraic equations

A(g,Φ)ε = 0

Page 61: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 62: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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,...)

Page 63: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 64: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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 =⇒

Page 65: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 66: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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.

Page 67: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

Page 68: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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.

Page 69: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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.

Page 70: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Classifications

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

maximally supersymmetric backgrounds are known:

Page 71: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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Classifications

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

maximally supersymmetric backgrounds are known:

• D = 4 N = 1 [Tod (1984)]

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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)]

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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)]

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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

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14

Eleven-dimensional supergravity

• bosonic fields:

? metric g

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Eleven-dimensional supergravity

• bosonic fields:

? metric g, and

? closed 4-form F

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Eleven-dimensional supergravity

• bosonic fields:

? metric g, and

? closed 4-form F

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

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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

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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).

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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.

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• the gravitino variation defines the connection

Dµ = ∇µ − 1288Fνρστ

(Γνρστ

µ + 8Γνρσδτµ

)

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• 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

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• 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

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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

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

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

The flatness equations are the vanishing of the RµνI.

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

The flatness equations are the vanishing of the RµνI.

Summarising the results:

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(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

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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

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(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

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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

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(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.

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• F obeys the Plucker relations

ιXιY ιZF ∧ F = 0 for all X, Y, Z

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• F obeys the Plucker relations

ιXιY ιZF ∧ F = 0 for all X, Y, Z

or

Fαβγ[µFνρστ ] = 0

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• 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

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• 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

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• 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]

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

spacetime:

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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

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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.

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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.

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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

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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

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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

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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

Page 107: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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

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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

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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

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We now plug these expressions back into the equation which relates

the curvature tensor to F and g

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We now plug these expressions back into the equation which relates

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

• F euclidean

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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)

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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

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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)

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• F null

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• 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

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• 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

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• 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

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• 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.

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• 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.

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Solutions are related by Penrose limits

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21

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

generalised Penrose limit in E2,11

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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

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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

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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

Page 126: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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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]

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(1, 0) D = 6 supergravity

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22

(1, 0) D = 6 supergravity

• bosonic fields

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(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

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(1, 0) D = 6 supergravity

• bosonic fields:

? metric g

? anti-selfdual closed 3-form F

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(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

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(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

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(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.

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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.

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23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

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23

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµ

abΓab

The connection D is actually induced from a metric connection

with torsion

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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

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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

Page 139: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 140: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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.

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24

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

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24

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

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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.

Page 144: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

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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

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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.

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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

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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.

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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.

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Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

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25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras

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25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

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25

Lorentzian Lie algebras

Which Lie algebras have an invariant metric?

• abelian Lie algebras with any metric

• semisimple Lie algebras

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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)

Page 155: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

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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∗

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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

Page 158: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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.

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26

The double extension

• g a Lie algebra with an invariant metric

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26

The double extension

• g a Lie algebra with an invariant metric

• h a Lie algebra acting on g via antisymmetric derivations

Page 161: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

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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

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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

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27

whose dual map

ω : Λ2g → h∗

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27

whose dual map

ω : Λ2g → h∗

is a cocycle

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27

whose dual map

ω : Λ2g → h∗

is a cocycle because h preserves the Lie bracket in g

Page 167: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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∗

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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

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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.

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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

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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∗)

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• the double extension admits an invariant metric

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28

• the double extension admits an invariant metric

Xb Hj Hj

Xa gab 0 0Hi 0 Bij δj

i

Hi 0 δij 0

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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).

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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.

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The structure theorem of Medina and Revoy

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

two orthogonal ideals.

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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)).

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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

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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

Page 180: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 181: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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.

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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.

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Six-dimensional lorentzian Lie algebras

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

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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).

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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

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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)

Page 187: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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.

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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])

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Six-dimensional lorentzian Lie algebras:

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31

Six-dimensional lorentzian Lie algebras:

• R5,1

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31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

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31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

• so(2, 1)⊕ R3

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31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

• so(2, 1)⊕ R3

• so(2, 1)⊕ so(3)

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31

Six-dimensional lorentzian Lie algebras:

• R5,1

• so(3)⊕ R2,1

• so(2, 1)⊕ R3

• so(2, 1)⊕ so(3)

• d(R4, R)

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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)

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32

Antiselfduality of the structure constants narrows the list down to

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32

Antiselfduality of the structure constants narrows the list down to

• R5,1

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32

Antiselfduality of the structure constants narrows the list down to

• R5,1

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

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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

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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.

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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)

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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]

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The vacua are related by Penrose limits:

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33

The vacua are related by Penrose limits:

AdS3×S3

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33

The vacua are related by Penrose limits:

AdS3×S3

NW6

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33

The vacua are related by Penrose limits:

AdS3×S3

NW6

@@

@@

@@R

flat

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33

The vacua are related by Penrose limits:

AdS3×S3

NW6

@@

@@

@@R

flat?

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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]

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34

(2, 0) D = 6 supergravity

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34

(2, 0) D = 6 supergravity

• bosonic fields

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34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

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34

(2, 0) D = 6 supergravity

• bosonic fields:

? metric g

? 5 anti-selfdual closed 3-form F i

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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)

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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

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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

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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.

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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.

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35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

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35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

Maximal supersymmetry =⇒ D is flat

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35

• The gravitino variation yields the connection

Dµ = ∇µ + 18Fµνρ

iγiΓνρ

Maximal supersymmetry =⇒ D is flat =⇒

• F iµνρ = Fµνρv

i

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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

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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

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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.

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36

D = 10 IIB supergravity

• bosonic fields:

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36

D = 10 IIB supergravity

• bosonic fields:

? metric g

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36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ

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36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ ,

? closed complex 3-form H

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36

D = 10 IIB supergravity

• bosonic fields:

? metric g,

? complex scalar τ ,

? closed complex 3-form H, and

? closed selfdual 5-form F

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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

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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

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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

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37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

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37

• the dilatino variation gives rise to an algebraic Killing spinor

equation

Maximal supersymmetry =⇒ τ is constant and H = 0

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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

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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)

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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

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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

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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 .

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Expanding the curvature of D into antisymmetric products of

Γ-matrices

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38

Expanding the curvature of D into antisymmetric products of

Γ-matrices and setting the coefficients to zero

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38

Expanding the curvature of D into antisymmetric products of

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

• F is parallel

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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

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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

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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

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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

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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.

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• F obeys a quadratic identity

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39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0

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39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

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39

• F obeys a quadratic identity:

Fρ ∧ F ρ = 0 or Fµ1µ2µ3ρ[ν1F ρ

ν2ν3ν4ν5] = 0

generalising both the Plucker relations

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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.

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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

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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

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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.

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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)]

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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.)

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n-Lie algebras

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n-Lie algebras

A Lie algebra is a vector space g

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40

n-Lie algebras

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

bilinear map

[ ] : Λ2g → g

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40

n-Lie algebras

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

bilinear map

[ ] : Λ2g → g

satisfying the condition

Page 261: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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 ]

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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 [ ]

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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]]

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An n-Lie algebra

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An n-Lie algebra is a vector space n

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41

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

antisymmetric n-linear map

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41

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

antisymmetric n-linear map

[ ] : Λnn → n

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41

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

antisymmetric n-linear map

[ ] : Λnn → n

satisfying the condition

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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

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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

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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 ]

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42

is a derivation over [ ]

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42

is a derivation over [ ]; that is,

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

i=1

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

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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〉

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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.

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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)])

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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

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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.

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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.

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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

Page 281: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 282: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 283: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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]

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G is decomposable;

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44

G is decomposable; whence, if nonzero

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44

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

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44

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

hence F defines two orthogonal planes.

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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.

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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:

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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

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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

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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

Page 293: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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.

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45

Plugging these expressions back into the relation between the

curvature tensor to F and g

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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)

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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

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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)

)

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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

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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

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46

plane waves:

g = 2dx+dx− − 14µ

28∑

i=1

(xi)2(dx−)2 +8∑

i=1

(dxi)2

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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

)

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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

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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]

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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

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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]

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47

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

quadrics

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47

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

quadrics, and are related by Penrose limits

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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

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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

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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

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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]

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Other theories

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48

Other theories

• D=10: I, heterotic, IIA

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48

Other theories

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

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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.

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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.)

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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]

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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.

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49

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

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49

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

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

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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)

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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

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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

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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)

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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 .

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

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50

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

one-parameter subgroup K:

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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

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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

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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

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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:

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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

Page 333: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 334: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 335: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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

Page 336: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

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, ...

Page 337: Maximally supersymmetric spacetimes - School of …jmf/CV/Seminars/Weizmann.pdf · Maximally supersymmetric spacetimes The supersymmetric analogue of the question: Which are the maximally

51

Thank you.