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Quotienting string backgrounds
Jose Figueroa-O’Farrill
Edinburgh Mathematical Physics Group
School of Mathematics
Supercordas do Noroeste, Oviedo, 4–6 February 2004
1
Based on work in collaboration with Joan Simon (Pennsylvania)
1
Based on work in collaboration with Joan Simon (Pennsylvania):
• JHEP 12 (2001) 011, hep-th/0110170
1
Based on work in collaboration with Joan Simon (Pennsylvania):
• JHEP 12 (2001) 011, hep-th/0110170
• Adv. Theor. Math. Phys. 6 (2003) 703–793, hep-th/0208107
1
Based on work in collaboration with Joan Simon (Pennsylvania):
• JHEP 12 (2001) 011, hep-th/0110170
• Adv. Theor. Math. Phys. 6 (2003) 703–793, hep-th/0208107
• Class. Quant. Grav. 19 (2002) 6147–6174, hep-th/0208108
1
Based on work in collaboration with Joan Simon (Pennsylvania):
• JHEP 12 (2001) 011, hep-th/0110170
• Adv. Theor. Math. Phys. 6 (2003) 703–793, hep-th/0208107
• Class. Quant. Grav. 19 (2002) 6147–6174, hep-th/0208108
• hep-th/0401206
1
Based on work in collaboration with Joan Simon (Pennsylvania):
• JHEP 12 (2001) 011, hep-th/0110170
• Adv. Theor. Math. Phys. 6 (2003) 703–793, hep-th/0208107
• Class. Quant. Grav. 19 (2002) 6147–6174, hep-th/0208108
• hep-th/0401206
and on work in progress with Joan Simon
1
Based on work in collaboration with Joan Simon (Pennsylvania):
• JHEP 12 (2001) 011, hep-th/0110170
• Adv. Theor. Math. Phys. 6 (2003) 703–793, hep-th/0208107
• Class. Quant. Grav. 19 (2002) 6147–6174, hep-th/0208108
• hep-th/0401206
and on work in progress with Joan Simon, Hannu Rajaniemi
(Edinburgh)
1
Based on work in collaboration with Joan Simon (Pennsylvania):
• JHEP 12 (2001) 011, hep-th/0110170
• Adv. Theor. Math. Phys. 6 (2003) 703–793, hep-th/0208107
• Class. Quant. Grav. 19 (2002) 6147–6174, hep-th/0208108
• hep-th/0401206
and on work in progress with Joan Simon, Hannu Rajaniemi
(Edinburgh), Owen Madden and Simon Ross (Durham).
2
Motivation
2
Motivation
• fluxbrane backgrounds in type II string theory
2
Motivation
• fluxbrane backgrounds in type II string theory
• string theory in
? time-dependent backgrounds
2
Motivation
• fluxbrane backgrounds in type II string theory
• string theory in
? time-dependent backgrounds, and
? causally singular backgrounds
2
Motivation
• fluxbrane backgrounds in type II string theory
• string theory in
? time-dependent backgrounds, and
? causally singular backgrounds
• supersymmetric Clifford–Klein space form problem
3
Melvin universe and fluxbranes
3
Melvin universe and fluxbranes
• axisymmetric solution of d=4 Einstein–Maxwell theory
[Melvin (1964)]
3
Melvin universe and fluxbranes
• axisymmetric solution of d=4 Einstein–Maxwell theory
[Melvin (1964)]
• describes a gravitationally stable universe of flux
3
Melvin universe and fluxbranes
• axisymmetric solution of d=4 Einstein–Maxwell theory
[Melvin (1964)]
• describes a gravitationally stable universe of flux
• dilatonic version in supergravity [Gibbons–Maeda (1988)]
3
Melvin universe and fluxbranes
• axisymmetric solution of d=4 Einstein–Maxwell theory
[Melvin (1964)]
• describes a gravitationally stable universe of flux
• dilatonic version in supergravity [Gibbons–Maeda (1988)]
• Kaluza–Klein reduction of a flat five-dimensional spacetime
[F. Dowker et al. (1994)]
3
Melvin universe and fluxbranes
• axisymmetric solution of d=4 Einstein–Maxwell theory
[Melvin (1964)]
• describes a gravitationally stable universe of flux
• dilatonic version in supergravity [Gibbons–Maeda (1988)]
• Kaluza–Klein reduction of a flat five-dimensional spacetime
[F. Dowker et al. (1994)]
• R1,4/Γ, with Γ ∼= R
3
Melvin universe and fluxbranes
• axisymmetric solution of d=4 Einstein–Maxwell theory
[Melvin (1964)]
• describes a gravitationally stable universe of flux
• dilatonic version in supergravity [Gibbons–Maeda (1988)]
• Kaluza–Klein reduction of a flat five-dimensional spacetime
[F. Dowker et al. (1994)]
• R1,4/Γ, with Γ ∼= R, or R1,10/Γ
3
Melvin universe and fluxbranes
• axisymmetric solution of d=4 Einstein–Maxwell theory
[Melvin (1964)]
• describes a gravitationally stable universe of flux
• dilatonic version in supergravity [Gibbons–Maeda (1988)]
• Kaluza–Klein reduction of a flat five-dimensional spacetime
[F. Dowker et al. (1994)]
• R1,4/Γ, with Γ ∼= R, or R1,10/Γ =⇒ IIA fluxbranes
4
General problem
4
General problem
• (M, g, F, ...) a supergravity background
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ, where
Γ ⊂ G is a one-parameter subgroup
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ, where
Γ ⊂ G is a one-parameter subgroup, paying close attention to
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ, where
Γ ⊂ G is a one-parameter subgroup, paying close attention to:
? smoothness
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ, where
Γ ⊂ G is a one-parameter subgroup, paying close attention to:
? smoothness,
? causal regularity
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ, where
Γ ⊂ G is a one-parameter subgroup, paying close attention to:
? smoothness,
? causal regularity,
? spin structure
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ, where
Γ ⊂ G is a one-parameter subgroup, paying close attention to:
? smoothness,
? causal regularity,
? spin structure,
? supersymmetry
4
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
• determine all quotient supergravity backgrounds M/Γ, where
Γ ⊂ G is a one-parameter subgroup, paying close attention to:
? smoothness,
? causal regularity,
? spin structure,
? supersymmetry,...
5
One-parameter subgroups
5
One-parameter subgroups
• (M, g, F, ...)
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g f∗F = F
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g f∗F = F . . .
define a Lie group G
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g f∗F = F . . .
define a Lie group G, with Lie algebra g
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g f∗F = F . . .
define a Lie group G, with Lie algebra g
• X ∈ g defines a one-parameter subgroup
5
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g f∗F = F . . .
define a Lie group G, with Lie algebra g
• X ∈ g defines a one-parameter subgroup
Γ = {exp(tX) | t ∈ R}
6
• X ∈ g also defines a Killing vector ξX
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
• two possible topologies
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
• two possible topologies:
? Γ ∼= S1
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
• two possible topologies:
? Γ ∼= S1, if and only if ∃T > 0 such that exp(TX) = 1
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
• two possible topologies:
? Γ ∼= S1, if and only if ∃T > 0 such that exp(TX) = 1? Γ ∼= R
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
• two possible topologies:
? Γ ∼= S1, if and only if ∃T > 0 such that exp(TX) = 1? Γ ∼= R, otherwise
6
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
• two possible topologies:
? Γ ∼= S1, if and only if ∃T > 0 such that exp(TX) = 1? Γ ∼= R, otherwise
• we are interested in the orbit space M/Γ
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Kaluza–Klein and discrete quotients
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
? discrete quotient M/ΓL, where
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
? discrete quotient M/ΓL, where L > 0
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
? discrete quotient M/ΓL, where L > 0 and
ΓL = {exp(nLX) | n ∈ Z}
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
? discrete quotient M/ΓL, where L > 0 and
ΓL = {exp(nLX) | n ∈ Z} ∼= Z
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
? discrete quotient M/ΓL, where L > 0 and
ΓL = {exp(nLX) | n ∈ Z} ∼= Z
? Kaluza–Klein reduction by Γ/ΓL
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
? discrete quotient M/ΓL, where L > 0 and
ΓL = {exp(nLX) | n ∈ Z} ∼= Z
? Kaluza–Klein reduction by Γ/ΓL∼= R/Z
7
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
? discrete quotient M/ΓL, where L > 0 and
ΓL = {exp(nLX) | n ∈ Z} ∼= Z
? Kaluza–Klein reduction by Γ/ΓL∼= R/Z ∼= S1
8
• we may stop after the first step
8
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M
8
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M , but often with very different global
properties
8
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M , but often with very different global
properties, e.g.,
? M static, but M/ΓL time-dependent
8
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M , but often with very different global
properties, e.g.,
? M static, but M/ΓL time-dependent
? M causally regular, but M/ΓL causally singular
8
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M , but often with very different global
properties, e.g.,
? M static, but M/ΓL time-dependent
? M causally regular, but M/ΓL causally singular
? M spin, but M/ΓL not spin
8
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M , but often with very different global
properties, e.g.,
? M static, but M/ΓL time-dependent
? M causally regular, but M/ΓL causally singular
? M spin, but M/ΓL not spin
? M supersymmetric, but M/ΓL breaking all supersymmetry
9
Classifying quotients
9
Classifying quotients
• (M, g, F, ...) with symmetry group G
9
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
9
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
• X, X ′ ∈ g generate one-parameter subgroups
Γ = {exp(tX) | t ∈ R}
9
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
• X, X ′ ∈ g generate one-parameter subgroups
Γ = {exp(tX) | t ∈ R} Γ′ = {exp(tX ′) | t ∈ R}
9
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
• X, X ′ ∈ g generate one-parameter subgroups
Γ = {exp(tX) | t ∈ R} Γ′ = {exp(tX ′) | t ∈ R}
• if X ′ = λX, λ 6= 0, then Γ′ = Γ
9
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
• X, X ′ ∈ g generate one-parameter subgroups
Γ = {exp(tX) | t ∈ R} Γ′ = {exp(tX ′) | t ∈ R}
• if X ′ = λX, λ 6= 0, then Γ′ = Γ
• if X ′ = gXg−1, then Γ′ = gΓg−1
9
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
• X, X ′ ∈ g generate one-parameter subgroups
Γ = {exp(tX) | t ∈ R} Γ′ = {exp(tX ′) | t ∈ R}
• if X ′ = λX, λ 6= 0, then Γ′ = Γ
• if X ′ = gXg−1, then Γ′ = gΓg−1, and moreover M/Γ ∼= M/Γ′
10
• enough to classify normal forms of X ∈ g under
X ∼ λgXg−1 g ∈ G λ ∈ R×
10
• enough to classify normal forms of X ∈ g under
X ∼ λgXg−1 g ∈ G λ ∈ R×
i.e., projectivised adjoint orbits of g
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Eleven-dimensional supergravity backgrounds
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Eleven-dimensional supergravity backgrounds
• (M, g): lorentzian spin 11-dimensional connected manifold
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Eleven-dimensional supergravity backgrounds
• (M, g): lorentzian spin 11-dimensional connected manifold
• F a closed 4-form
11
Eleven-dimensional supergravity backgrounds
• (M, g): lorentzian spin 11-dimensional connected manifold
• F a closed 4-form
• (M, g, F ) is supersymmetric
11
Eleven-dimensional supergravity backgrounds
• (M, g): lorentzian spin 11-dimensional connected manifold
• F a closed 4-form
• (M, g, F ) is supersymmetric ⇐⇒ admits nonzero Killing spinors
11
Eleven-dimensional supergravity backgrounds
• (M, g): lorentzian spin 11-dimensional connected manifold
• F a closed 4-form
• (M, g, F ) is supersymmetric ⇐⇒ admits nonzero Killing spinors,
parallel with respect to
DX = ∇X + 16ιXF − 1
12X[ ∧ F
12
• (M, g, F ) is ν-BPS
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
∗ Minkowski
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
∗ Minkowski
∗ Freund–Rubin
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
∗ Minkowski
∗ Freund–Rubin
∗ Kowalski-Glikman wave
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
∗ Minkowski
∗ Freund–Rubin
∗ Kowalski-Glikman wave
? ν = 12 backgrounds
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
∗ Minkowski
∗ Freund–Rubin
∗ Kowalski-Glikman wave
? ν = 12 backgrounds
∗ M2 and M5 branes
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
∗ Minkowski
∗ Freund–Rubin
∗ Kowalski-Glikman wave
? ν = 12 backgrounds
∗ M2 and M5 branes
∗ Kaluza–Klein monopole
12
• (M, g, F ) is ν-BPS ⇐⇒
dim {Killing spinors} = 32ν
e.g.,
? ν = 1 backgrounds
∗ Minkowski
∗ Freund–Rubin
∗ Kowalski-Glikman wave
? ν = 12 backgrounds
∗ M2 and M5 branes
∗ Kaluza–Klein monopole
∗ M-wave
13
Minkowski vacuum
13
Minkowski vacuum
• (R1,10, F = 0)
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R)
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R):(
A v
0 1
)
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R):(
A v
0 1
)A ∈ O(1, 10)
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R):(
A v
0 1
)A ∈ O(1, 10) v ∈ R
1,10
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R):(
A v
0 1
)A ∈ O(1, 10) v ∈ R
1,10
• Γ ⊂ O(1, 10) n R1,10
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R):(
A v
0 1
)A ∈ O(1, 10) v ∈ R
1,10
• Γ ⊂ O(1, 10) n R1,10, generated by
X = XL + XT ∈ so(1, 10)⊕ R1,10
13
Minkowski vacuum
• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R):(
A v
0 1
)A ∈ O(1, 10) v ∈ R
1,10
• Γ ⊂ O(1, 10) n R1,10, generated by
X = XL + XT ∈ so(1, 10)⊕ R1,10 ,
which we need to put in normal form
14
Normal forms for orthogonal transformations
14
Normal forms for orthogonal transformations
• X ∈ so(p, q)
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
Rp+q =
⊕i
Vi
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
Rp+q =
⊕i
Vi with Vi indecomposable
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
Rp+q =
⊕i
Vi with Vi indecomposable
• for each indecomposable block
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
Rp+q =
⊕i
Vi with Vi indecomposable
• for each indecomposable block, if λ is an eigenvalue
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
Rp+q =
⊕i
Vi with Vi indecomposable
• for each indecomposable block, if λ is an eigenvalue, then so are
−λ
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
Rp+q =
⊕i
Vi with Vi indecomposable
• for each indecomposable block, if λ is an eigenvalue, then so are
−λ, λ∗
14
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
• X =∑
i Xi relative to an orthogonal decomposition
Rp+q =
⊕i
Vi with Vi indecomposable
• for each indecomposable block, if λ is an eigenvalue, then so are
−λ, λ∗, and −λ∗
15
• possible minimal polynomials
15
• possible minimal polynomials:
? λ = 0
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
? λ = iϕ ∈ iR
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
? λ = iϕ ∈ iR,
µ(x) = (x2 + ϕ2)n
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
? λ = iϕ ∈ iR,
µ(x) = (x2 + ϕ2)n
? λ = β + iϕ
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
? λ = iϕ ∈ iR,
µ(x) = (x2 + ϕ2)n
? λ = β + iϕ, βϕ 6= 0
15
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
? λ = iϕ ∈ iR,
µ(x) = (x2 + ϕ2)n
? λ = β + iϕ, βϕ 6= 0,
µ(x) =((
x2 + β2 + ϕ2)2 − 4β2x2
)n
16
Strategy
16
Strategy
• for each µ(x)
16
Strategy
• for each µ(x), write down X in (real) Jordan form
16
Strategy
• for each µ(x), write down X in (real) Jordan form
• determine metric making X skew-symmetric
16
Strategy
• for each µ(x), write down X in (real) Jordan form
• determine metric making X skew-symmetric, using automorphism
of Jordan form if necessary to bring the metric to standard form
16
Strategy
• for each µ(x), write down X in (real) Jordan form
• determine metric making X skew-symmetric, using automorphism
of Jordan form if necessary to bring the metric to standard form
• keep only those blocks with appropriate signature
16
Strategy
• for each µ(x), write down X in (real) Jordan form
• determine metric making X skew-symmetric, using automorphism
of Jordan form if necessary to bring the metric to standard form
• keep only those blocks with appropriate signature
• example: µ(x) = x3
17
Elementary lorentzian blocks
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1)
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2)
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0)
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1)
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1) x2 − β2
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1) x2 − β2 boost
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1) x2 − β2 boost
(1, 2)
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1) x2 − β2 boost
(1, 2) x3
17
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1) x2 − β2 boost
(1, 2) x3 null rotation
18
Lorentzian normal forms
18
Lorentzian normal forms
Play with the elementary blocks!
18
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 10)
18
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 10):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4) + R9\(ϕ5)
18
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 10):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4) + R9\(ϕ5)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)
18
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 10):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4) + R9\(ϕ5)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)
• N+2 + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)
18
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 10):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4) + R9\(ϕ5)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)
• N+2 + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)
where β > 0
18
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 10):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4) + R9\(ϕ5)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)
• N+2 + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)
where β > 0, ϕ1 ≥ ϕ2 ≥ · · · ≥ ϕk−1 ≥ ϕk ≥ 0
19
Normal forms for the Poincare algebra
19
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 10)⊕ R1,10
19
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 10)⊕ R1,10
• conjugate by O(1, 10) to bring λ to normal form
19
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 10)⊕ R1,10
• conjugate by O(1, 10) to bring λ to normal form
• conjugate by R1,10
19
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 10)⊕ R1,10
• conjugate by O(1, 10) to bring λ to normal form
• conjugate by R1,10:
λ + τ 7→ λ + τ − [λ, τ ′]
19
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 10)⊕ R1,10
• conjugate by O(1, 10) to bring λ to normal form
• conjugate by R1,10:
λ + τ 7→ λ + τ − [λ, τ ′]
to get rid of component of τ in the image of [λ,−]
20
• the subgroups with everywhere spacelike orbits
20
• the subgroups with everywhere spacelike orbits are generated by
either
? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
20
• the subgroups with everywhere spacelike orbits are generated by
either
? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4); or
? ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
20
• the subgroups with everywhere spacelike orbits are generated by
either
? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4); or
? ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4),
where ϕ1 ≥ ϕ2 ≥ ϕ3 ≥ ϕ4 ≥ 0
20
• the subgroups with everywhere spacelike orbits are generated by
either
? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4); or
? ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4),
where ϕ1 ≥ ϕ2 ≥ ϕ3 ≥ ϕ4 ≥ 0
• both are ∼= R
20
• the subgroups with everywhere spacelike orbits are generated by
either
? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4); or
? ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4),
where ϕ1 ≥ ϕ2 ≥ ϕ3 ≥ ϕ4 ≥ 0
• both are ∼= R
• the former gives rise to fluxbranes
20
• the subgroups with everywhere spacelike orbits are generated by
either
? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4); or
? ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4),
where ϕ1 ≥ ϕ2 ≥ ϕ3 ≥ ϕ4 ≥ 0
• both are ∼= R
• the former gives rise to fluxbranes and the latter to nullbranes
21
Adapted coordinates
21
Adapted coordinates
• start with metric in flat coordinates y, z
21
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
21
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
• “undress” the Killing vector
21
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
• “undress” the Killing vector
ξ = ∂z + λ
21
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
• “undress” the Killing vector
ξ = ∂z + λ = U ∂z U−1
21
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
• “undress” the Killing vector
ξ = ∂z + λ = U ∂z U−1 with U = exp(−zλ)
22
• introduce new coordinates
22
• introduce new coordinates
x = U y
22
• introduce new coordinates
x = U y = exp(−zB)y
22
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
22
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
whence ξx = 0
22
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
whence ξx = 0
• rewrite the metric in terms of x
22
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
whence ξx = 0
• rewrite the metric in terms of x:
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
22
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
whence ξx = 0
• rewrite the metric in terms of x:
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
where
? Λ = 1 + |Bx|2
22
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
whence ξx = 0
• rewrite the metric in terms of x:
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
where
? Λ = 1 + |Bx|2? A = Λ−1 Bx · dx
23
• comparing with the Kaluza–Klein “ansatz”
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
we read off the IIA fields
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
we read off the IIA fields
? dilaton
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
we read off the IIA fields
? dilaton: Φ = 34 log(1 + |Bx|2)
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
we read off the IIA fields
? dilaton: Φ = 34 log(1 + |Bx|2)
? RR 1-form potential
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
we read off the IIA fields
? dilaton: Φ = 34 log(1 + |Bx|2)
? RR 1-form potential:
A =Bx · dx
1 + |Bx|2
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
we read off the IIA fields
? dilaton: Φ = 34 log(1 + |Bx|2)
? RR 1-form potential:
A =Bx · dx
1 + |Bx|2
? string frame metric
23
• comparing with the Kaluza–Klein “ansatz”
ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h
we read off the IIA fields
? dilaton: Φ = 34 log(1 + |Bx|2)
? RR 1-form potential:
A =Bx · dx
1 + |Bx|2
? string frame metric:
h = Λ1/2|dx|2 − Λ−1/2(Bx · dx)2
24
• the only data is the matrix
24
• the only data is the matrix
B =
0 −ϕ1 0 u
ϕ1 0 0 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
−u 0
0 0
24
• the only data is the matrix
B =
0 −ϕ1 0 u
ϕ1 0 0 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
−u 0
0 0
where
24
• the only data is the matrix
B =
0 −ϕ1 0 u
ϕ1 0 0 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
−u 0
0 0
where either
? u = 0
24
• the only data is the matrix
B =
0 −ϕ1 0 u
ϕ1 0 0 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
−u 0
0 0
where either
? u = 0 (generalised fluxbranes)
24
• the only data is the matrix
B =
0 −ϕ1 0 u
ϕ1 0 0 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
−u 0
0 0
where either
? u = 0 (generalised fluxbranes); or
? u = 1 and ϕ1 = 0
24
• the only data is the matrix
B =
0 −ϕ1 0 u
ϕ1 0 0 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
−u 0
0 0
where either
? u = 0 (generalised fluxbranes); or
? u = 1 and ϕ1 = 0 (generalised nullbranes)
25
e.g.,
? u = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
? u = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
? u = 0, ϕ1 = ϕ2
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ 14-BPS flux-string
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ 14-BPS flux-string
? u = 1
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ 14-BPS flux-string
? u = 1, ϕi = 0
25
e.g.,
? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane
[Costa–Gutperle, hep-th/0012072]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane
[Gutperle–Strominger, hep-th/0104136]
? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane
? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string
[Uranga, hep-th/0108196]
? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ 14-BPS flux-string
? u = 1, ϕi = 0 =⇒ half-BPS nullbrane
26
Discrete quotients
26
Discrete quotients
• start with the metric in adapted coordinates
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B
=⇒ half-BPS eleven-dimensional nullbrane
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B
=⇒ half-BPS eleven-dimensional nullbrane:
? time-dependent
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B
=⇒ half-BPS eleven-dimensional nullbrane:
? time-dependent
? smooth
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B
=⇒ half-BPS eleven-dimensional nullbrane:
? time-dependent
? smooth
? stable
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B
=⇒ half-BPS eleven-dimensional nullbrane:
? time-dependent
? smooth
? stable
? smooth transition between Big Crunch and Big Bang
26
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B
=⇒ half-BPS eleven-dimensional nullbrane:
? time-dependent
? smooth
? stable
? smooth transition between Big Crunch and Big Bang
? resolution of parabolic orbifold [Horowitz–Steif (1991)]
27
End of first lecture
28
Supersymmetry
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
In supergravity
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
In supergravity: Γ-invariant Killing spinors
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
In supergravity: Γ-invariant Killing spinors:
Lξε
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
In supergravity: Γ-invariant Killing spinors:
Lξε = ∇ξε + 18∇aξbΓabε
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
In supergravity: Γ-invariant Killing spinors:
Lξε = ∇ξε + 18∇aξbΓabε = 0
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
In supergravity: Γ-invariant Killing spinors:
Lξε = ∇ξε + 18∇aξbΓabε = 0
In string/M-theory
28
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
How much supersymmetry will the quotient M/Γ preserve?
In supergravity: Γ-invariant Killing spinors:
Lξε = ∇ξε + 18∇aξbΓabε = 0
In string/M-theory this cannot be the end of the story.
29
“Supersymmetry without supersymmetry”
29
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
29
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example
29
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example:
AdS5×S5
29
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example:
AdS5×S5
((QQQQQQQQQQQQQ
oo // AdS5×CP2 × S1
29
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example:
AdS5×S5
((QQQQQQQQQQQQQ
oo // AdS5×CP2 × S1
uukkkkkkkkkkkkkkk
AdS5×CP2
29
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example:
AdS5×S5
((QQQQQQQQQQQQQ
oo // AdS5×CP2 × S1
uukkkkkkkkkkkkkkk
AdS5×CP2
CP2 is not even spin! [Duff–Lu–Pope, hep-th/9704186,9803061]
30
Spin structures in quotients
30
Spin structures in quotients
• (M, g) spin
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
• so let Γ ∼= S1
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
• so let Γ ∼= S1 with (normalised) Killing vector ξ = ξX
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
• so let Γ ∼= S1 with (normalised) Killing vector ξ = ξX
• Lξ has integral weights on tensors
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
• so let Γ ∼= S1 with (normalised) Killing vector ξ = ξX
• Lξ has integral weights on tensors , e.g.,
LξT
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
• so let Γ ∼= S1 with (normalised) Killing vector ξ = ξX
• Lξ has integral weights on tensors , e.g.,
LξT = inT
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
• so let Γ ∼= S1 with (normalised) Killing vector ξ = ξX
• Lξ has integral weights on tensors , e.g.,
LξT = inT for n ∈ Z
30
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R, then M/Γ is always spin
• so let Γ ∼= S1 with (normalised) Killing vector ξ = ξX
• Lξ has integral weights on tensors , e.g.,
LξT = inT for n ∈ Z
so that exp(2πX) does act like 1 on tensors
31
• ξ also acts on spinors via Lξ
31
• ξ also acts on spinors via Lξ; but two things may happen
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral:
Lξε = i(n + 12)ε
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral:
Lξε = i(n + 12)ε
• if Γ acts on spinors
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral:
Lξε = i(n + 12)ε
• if Γ acts on spinors, M/Γ is spin
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral:
Lξε = i(n + 12)ε
• if Γ acts on spinors, M/Γ is spin;
• if only Γ does
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral:
Lξε = i(n + 12)ε
• if Γ acts on spinors, M/Γ is spin;
• if only Γ does, M/Γ is not spin
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral:
Lξε = i(n + 12)ε
• if Γ acts on spinors, M/Γ is spin;
• if only Γ does, M/Γ is not spin, but spinc
31
• ξ also acts on spinors via Lξ; but two things may happen:
? Lξ integrates to an action of Γ: weights are again integral; or
? Lξ integrates to an action of a double cover Γ: weights are
now half-integral:
Lξε = i(n + 12)ε
• if Γ acts on spinors, M/Γ is spin;
• if only Γ does, M/Γ is not spin, but spinc
• it suffices to check this on Killing spinors
32
• e.g., the Hopf fibration S5 → CP2
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1 ,
and S1-action
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1 ,
and S1-action
(z1, z2, z3)
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1 ,
and S1-action
(z1, z2, z3) 7→ (eitz1, eitz2, e
itz3)
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1 ,
and S1-action
(z1, z2, z3) 7→ (eitz1, eitz2, e
itz3)
• Killing spinors on S5
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1 ,
and S1-action
(z1, z2, z3) 7→ (eitz1, eitz2, e
itz3)
• Killing spinors on S5 are restrictions of parallel spinors on C3
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1 ,
and S1-action
(z1, z2, z3) 7→ (eitz1, eitz2, e
itz3)
• Killing spinors on S5 are restrictions of parallel spinors on C3
• Lξ acts as
Lξε
32
• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C
3 as the quadric
|z1|2 + |z2|2 + |z3|2 = 1 ,
and S1-action
(z1, z2, z3) 7→ (eitz1, eitz2, e
itz3)
• Killing spinors on S5 are restrictions of parallel spinors on C3
• Lξ acts as
Lξε = 12(Γ12 + Γ34 + Γ56)ε
33
• Γ2ij = −1
33
• Γ2ij = −1: its eigenvalues are ±i
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1)
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1) and ±12 (with multiplicity 3)
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1) and ±12 (with multiplicity 3)
• weights are half-integral
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1) and ±12 (with multiplicity 3)
• weights are half-integral =⇒ CP2 has no spin structure
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1) and ±12 (with multiplicity 3)
• weights are half-integral =⇒ CP2 has no spin structure
• more generally
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1) and ±12 (with multiplicity 3)
• weights are half-integral =⇒ CP2 has no spin structure
• more generally, M → M/Γ is a principal circle bundle
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1) and ±12 (with multiplicity 3)
• weights are half-integral =⇒ CP2 has no spin structure
• more generally, M → M/Γ is a principal circle bundle
• let L = M ×Γ C be the complex line bundle on M/Γ
33
• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3
2
(with multiplicity 1) and ±12 (with multiplicity 3)
• weights are half-integral =⇒ CP2 has no spin structure
• more generally, M → M/Γ is a principal circle bundle
• let L = M ×Γ C be the complex line bundle on M/Γ associated
to the representation with charge 1
34
• a spinor ε on M
34
• a spinor ε on M such that
Lξε = inε
34
• a spinor ε on M such that
Lξε = inε
defines a section of S ⊗ Ln over M/Γ
34
• a spinor ε on M such that
Lξε = inε
defines a section of S ⊗ Ln over M/Γ, where S → M/Γ is the
bundle of spinors
34
• a spinor ε on M such that
Lξε = inε
defines a section of S ⊗ Ln over M/Γ, where S → M/Γ is the
bundle of spinors
• a spinor ε on M
34
• a spinor ε on M such that
Lξε = inε
defines a section of S ⊗ Ln over M/Γ, where S → M/Γ is the
bundle of spinors
• a spinor ε on M such that
Lξε = i(n + 1/2)ε
34
• a spinor ε on M such that
Lξε = inε
defines a section of S ⊗ Ln over M/Γ, where S → M/Γ is the
bundle of spinors
• a spinor ε on M such that
Lξε = i(n + 1/2)ε
would define a section of
S ⊗ Ln+1/2
34
• a spinor ε on M such that
Lξε = inε
defines a section of S ⊗ Ln over M/Γ, where S → M/Γ is the
bundle of spinors
• a spinor ε on M such that
Lξε = i(n + 1/2)ε
would define a section of
S ⊗ Ln+1/2 ∼= (S ⊗ L1/2)
34
• a spinor ε on M such that
Lξε = inε
defines a section of S ⊗ Ln over M/Γ, where S → M/Γ is the
bundle of spinors
• a spinor ε on M such that
Lξε = i(n + 1/2)ε
would define a section of
S ⊗ Ln+1/2 ∼= (S ⊗ L1/2)⊗ Ln
35
• however neither S
35
• however neither S nor L1/2 exist as bundles
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
• L carries a natural connection
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
• L carries a natural connection: the RR 1-form
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
• L carries a natural connection: the RR 1-form
• sections through Ln carry n units of RR charge
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
• L carries a natural connection: the RR 1-form
• sections through Ln carry n units of RR charge
• spinc spinors carry fractional RR charge
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
• L carries a natural connection: the RR 1-form
• sections through Ln carry n units of RR charge
• spinc spinors carry fractional RR charge
• RR charge
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
• L carries a natural connection: the RR 1-form
• sections through Ln carry n units of RR charge
• spinc spinors carry fractional RR charge
• RR charge ↔ momentum along the compact direction given by ξ
35
• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2
does: it is the bundle of spinc spinors
• L carries a natural connection: the RR 1-form
• sections through Ln carry n units of RR charge
• spinc spinors carry fractional RR charge
• RR charge ↔ momentum along the compact direction given by ξ
• RR charged states are T-dual to winding states
36
• e.g., Killing spinors in AdS5×S5
36
• e.g., Killing spinors in AdS5×S5 are T-dual to winding states in
AdS5×CP2 × S1
36
• e.g., Killing spinors in AdS5×S5 are T-dual to winding states in
AdS5×CP2 × S1, which the supergravity does not see
36
• e.g., Killing spinors in AdS5×S5 are T-dual to winding states in
AdS5×CP2 × S1, which the supergravity does not see, or does
it?
[e.g., Hull hep-th/0305039]
37
Supersymmetry of supergravity quotients
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries, generated by ξ
• Killing spinors of M/Γ
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries, generated by ξ
• Killing spinors of M/Γ ⇐⇒ Γ-invariant Killing spinors of M
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries, generated by ξ
• Killing spinors of M/Γ ⇐⇒ Γ-invariant Killing spinors of M
• it suffices to determine zero weights of Lξ on Killing spinors
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries, generated by ξ
• Killing spinors of M/Γ ⇐⇒ Γ-invariant Killing spinors of M
• it suffices to determine zero weights of Lξ on Killing spinors
• e.g., (R1,10)
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries, generated by ξ
• Killing spinors of M/Γ ⇐⇒ Γ-invariant Killing spinors of M
• it suffices to determine zero weights of Lξ on Killing spinors
• e.g., (R1,10): Killing spinors are parallel
37
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries, generated by ξ
• Killing spinors of M/Γ ⇐⇒ Γ-invariant Killing spinors of M
• it suffices to determine zero weights of Lξ on Killing spinors
• e.g., (R1,10): Killing spinors are parallel, whence
Lξε = 18∇aξbΓabε
38
• e.g., fluxbranes
38
• e.g., fluxbranes
ξ = ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
38
• e.g., fluxbranes
ξ = ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2(ϕ1Γ12 + ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
39
• for generic ϕi
39
• for generic ϕi, there are no invariant Killing spinors
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14
? ϕ1 = ϕ2 = ϕ3 = ϕ4
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14
? ϕ1 = ϕ2 = ϕ3 = ϕ4 =⇒ ν = 38
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14
? ϕ1 = ϕ2 = ϕ3 = ϕ4 =⇒ ν = 38
? ϕ1 = ϕ2, ϕ3 = ϕ4 = 0
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14
? ϕ1 = ϕ2 = ϕ3 = ϕ4 =⇒ ν = 38
? ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ ν = 12
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14
? ϕ1 = ϕ2 = ϕ3 = ϕ4 =⇒ ν = 38
? ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ ν = 12
? ϕ1 = ϕ2 = ϕ3 = ϕ4 = 0
39
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18
? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14
? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14
? ϕ1 = ϕ2 = ϕ3 = ϕ4 =⇒ ν = 38
? ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ ν = 12
? ϕ1 = ϕ2 = ϕ3 = ϕ4 = 0 =⇒ ν = 1
40
• e.g., nullbranes
40
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
40
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
40
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
• N+2 is nilpotent
40
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
• N+2 is nilpotent, whereas 12(ϕ2Γ34+ϕ3Γ56+ϕ4Γ78) is semisimple
and commutes with it
40
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
• N+2 is nilpotent, whereas 12(ϕ2Γ34+ϕ3Γ56+ϕ4Γ78) is semisimple
and commutes with it; whence invariant spinors are annihilated
by both
40
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
• N+2 is nilpotent, whereas 12(ϕ2Γ34+ϕ3Γ56+ϕ4Γ78) is semisimple
and commutes with it; whence invariant spinors are annihilated
by both
• ker N+2 = ker Γ+
40
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
• N+2 is nilpotent, whereas 12(ϕ2Γ34+ϕ3Γ56+ϕ4Γ78) is semisimple
and commutes with it; whence invariant spinors are annihilated
by both
• ker N+2 = ker Γ+, and this simply halves the number of
supersymmetries
41
• for generic ϕi
41
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues
41
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ2 − ϕ3 − ϕ4 = 0
41
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18
41
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18
? ϕ2 = ϕ3, ϕ4 = 0
41
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18
? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14
41
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18
? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14
? ϕ2 = ϕ3 = ϕ4 = 0
41
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18
? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14
? ϕ2 = ϕ3 = ϕ4 = 0 =⇒ ν = 12
42
Brane quotients
42
Brane quotients
• typical electric p-brane solution
42
Brane quotients
• typical electric p-brane solution:
g = e2A(r)ds2(R1,p)
42
Brane quotients
• typical electric p-brane solution:
g = e2A(r)ds2(R1,p) + e2B(r)ds2(RD−1−p)
42
Brane quotients
• typical electric p-brane solution:
g = e2A(r)ds2(R1,p) + e2B(r)ds2(RD−1−p)
Fp+2 = dvol(R1,p) ∧ dC(r)
42
Brane quotients
• typical electric p-brane solution:
g = e2A(r)ds2(R1,p) + e2B(r)ds2(RD−1−p)
Fp+2 = dvol(R1,p) ∧ dC(r)
with r the transverse radius
42
Brane quotients
• typical electric p-brane solution:
g = e2A(r)ds2(R1,p) + e2B(r)ds2(RD−1−p)
Fp+2 = dvol(R1,p) ∧ dC(r)
with r the transverse radius
• A(r), B(r), C(r) → 0 as r → ∞
42
Brane quotients
• typical electric p-brane solution:
g = e2A(r)ds2(R1,p) + e2B(r)ds2(RD−1−p)
Fp+2 = dvol(R1,p) ∧ dC(r)
with r the transverse radius
• A(r), B(r), C(r) → 0 as r → ∞ =⇒ asymptotic to
(R1,D−1, F = 0)
43
• symmetry group
43
• symmetry group
G =(SO(1, p) n R
1,p)
43
• symmetry group
G =(SO(1, p) n R
1,p)×O(D− p− 1)
43
• symmetry group
G =(SO(1, p) n R
1,p)×O(D− p− 1) ⊂ O(1, D − 1) n R
1,D−1
43
• symmetry group
G =(SO(1, p) n R
1,p)×O(D− p− 1) ⊂ O(1, D − 1) n R
1,D−1
• Killing spinors
43
• symmetry group
G =(SO(1, p) n R
1,p)×O(D− p− 1) ⊂ O(1, D − 1) n R
1,D−1
• Killing spinors
ε = eD(r)ε∞
where ε∞ is a Killing spinor of the asymptotic background
43
• symmetry group
G =(SO(1, p) n R
1,p)×O(D− p− 1) ⊂ O(1, D − 1) n R
1,D−1
• Killing spinors
ε = eD(r)ε∞
where ε∞ is a Killing spinor of the asymptotic background,
satisfying
dvol(R1,p) · ε∞ = ε∞
44
• e.g., M2-brane
44
• e.g., M2-brane
g = V −2/3ds2(R1,2) + V 1/3ds2(R8)
44
• e.g., M2-brane
g = V −2/3ds2(R1,2) + V 1/3ds2(R8)
F = dvol(R1,2) ∧ dV −1
44
• e.g., M2-brane
g = V −2/3ds2(R1,2) + V 1/3ds2(R8)
F = dvol(R1,2) ∧ dV −1
ε = V −1/6ε∞
44
• e.g., M2-brane
g = V −2/3ds2(R1,2) + V 1/3ds2(R8)
F = dvol(R1,2) ∧ dV −1
ε = V −1/6ε∞
with V (r) = 1 + Q/r6
44
• e.g., M2-brane
g = V −2/3ds2(R1,2) + V 1/3ds2(R8)
F = dvol(R1,2) ∧ dV −1
ε = V −1/6ε∞
with V (r) = 1 + Q/r6, and
dvol(R1,2) · ε∞ = ε∞
44
• e.g., M2-brane
g = V −2/3ds2(R1,2) + V 1/3ds2(R8)
F = dvol(R1,2) ∧ dV −1
ε = V −1/6ε∞
with V (r) = 1 + Q/r6, and
dvol(R1,2) · ε∞ = ε∞
=⇒ the M2-brane is half-BPS
45
• action of G on Killing spinors also induced from asymptotic limit
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε = eD(r)ε′∞
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε = eD(r)ε′∞
but also
Lξε = LξeD(r)ε∞
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε = eD(r)ε′∞
but also
Lξε = LξeD(r)ε∞ = eD(r)Lξε∞
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε = eD(r)ε′∞
but also
Lξε = LξeD(r)ε∞ = eD(r)Lξε∞
whence
Lξε∞ = ε′∞
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε = eD(r)ε′∞
but also
Lξε = LξeD(r)ε∞ = eD(r)Lξε∞
whence
Lξε∞ = ε′∞
(independent of r)
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε = eD(r)ε′∞
but also
Lξε = LξeD(r)ε∞ = eD(r)Lξε∞
whence
Lξε∞ = ε′∞
(independent of r)
? compute it at r →∞
45
• action of G on Killing spinors also induced from asymptotic limit:
? ξ a Killing vector
? ε a Killing spinor =⇒ so is Lξε, whence
Lξε = eD(r)ε′∞
but also
Lξε = LξeD(r)ε∞ = eD(r)Lξε∞
whence
Lξε∞ = ε′∞
(independent of r)
? compute it at r →∞
46
=⇒ the classification problem reduces to the flat case
46
=⇒ the classification problem reduces to the flat case, with two
important differences:
? G is not the full asymptotic symmetry group
46
=⇒ the classification problem reduces to the flat case, with two
important differences:
? G is not the full asymptotic symmetry group, and
? brane metric is not flat
46
=⇒ the classification problem reduces to the flat case, with two
important differences:
? G is not the full asymptotic symmetry group, and
? brane metric is not flat, so causal properties of Killing vectors
may differ from R1,D−1
46
=⇒ the classification problem reduces to the flat case, with two
important differences:
? G is not the full asymptotic symmetry group, and
? brane metric is not flat, so causal properties of Killing vectors
may differ from R1,D−1
• e.g., ξ may be everywhere spacelike in the brane metric
46
=⇒ the classification problem reduces to the flat case, with two
important differences:
? G is not the full asymptotic symmetry group, and
? brane metric is not flat, so causal properties of Killing vectors
may differ from R1,D−1
• e.g., ξ may be everywhere spacelike in the brane metric, but
its asymptotic value may not be everywhere spacelike in the flat
metric
47
Example: M2-brane
47
Example: M2-brane
• metric
47
Example: M2-brane
• metric:
V −2/3ds2(R1,2) + V 1/3ds2(R8)
47
Example: M2-brane
• metric:
V −2/3ds2(R1,2) + V 1/3ds2(R8)
with V = 1 + |Q|/r6
47
Example: M2-brane
• metric:
V −2/3ds2(R1,2) + V 1/3ds2(R8)
with V = 1 + |Q|/r6
• Killing vector:
ξ
47
Example: M2-brane
• metric:
V −2/3ds2(R1,2) + V 1/3ds2(R8)
with V = 1 + |Q|/r6
• Killing vector:
ξ = τ‖
47
Example: M2-brane
• metric:
V −2/3ds2(R1,2) + V 1/3ds2(R8)
with V = 1 + |Q|/r6
• Killing vector:
ξ = τ‖ + ρ‖
47
Example: M2-brane
• metric:
V −2/3ds2(R1,2) + V 1/3ds2(R8)
with V = 1 + |Q|/r6
• Killing vector:
ξ = τ‖ + ρ‖ + ρ⊥
47
Example: M2-brane
• metric:
V −2/3ds2(R1,2) + V 1/3ds2(R8)
with V = 1 + |Q|/r6
• Killing vector:
ξ = τ‖ + ρ‖ + ρ⊥
mutually orthogonal
48
• norm
48
• norm:
‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3|ρ⊥|2
48
• norm:
‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3|ρ⊥|2
= V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3r2|ρ⊥|2S
48
• norm:
‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3|ρ⊥|2
= V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3r2|ρ⊥|2S
≥ V −2/3|τ |2 + V 1/3r2m2
48
• norm:
‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3|ρ⊥|2
= V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3r2|ρ⊥|2S
≥ V −2/3|τ |2 + V 1/3r2m2
where |ρ⊥|2S ≥ m2
48
• norm:
‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3|ρ⊥|2
= V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3r2|ρ⊥|2S
≥ V −2/3|τ |2 + V 1/3r2m2
where |ρ⊥|2S ≥ m2, which defines
f(r) := V (r)−2/3|τ |2 + V (r)1/3r2m2
48
• norm:
‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3|ρ⊥|2
= V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3r2|ρ⊥|2S
≥ V −2/3|τ |2 + V 1/3r2m2
where |ρ⊥|2S ≥ m2, which defines
f(r) := V (r)−2/3|τ |2 + V (r)1/3r2m2
• odd-dimensional spheres can be combed
48
• norm:
‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3|ρ⊥|2
= V −2/3(|τ |2 + |ρ‖|2
)+ V 1/3r2|ρ⊥|2S
≥ V −2/3|τ |2 + V 1/3r2m2
where |ρ⊥|2S ≥ m2, which defines
f(r) := V (r)−2/3|τ |2 + V (r)1/3r2m2
• odd-dimensional spheres can be combed =⇒ m2 can be > 0
49
• f(r) has a minimum at r0 > 0
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
? if |τ |2 ≥ 0
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
? if |τ |2 ≥ 0, there is no such r0 > 0
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
? if |τ |2 ≥ 0, there is no such r0 > 0? if |τ |2 < 0
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
? if |τ |2 ≥ 0, there is no such r0 > 0? if |τ |2 < 0, there is
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
? if |τ |2 ≥ 0, there is no such r0 > 0? if |τ |2 < 0, there is, and
‖ξ‖2 ≥ V (r0)−2/3|τ |2 + V (r0)1/3r20m
2
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
? if |τ |2 ≥ 0, there is no such r0 > 0? if |τ |2 < 0, there is, and
‖ξ‖2 ≥ V (r0)−2/3|τ |2 + V (r0)1/3r20m
2
which is > 0
49
• f(r) has a minimum at r0 > 0, where
V (r0)r80 = −2|Q|
m2|τ |2
? if |τ |2 ≥ 0, there is no such r0 > 0? if |τ |2 < 0, there is, and
‖ξ‖2 ≥ V (r0)−2/3|τ |2 + V (r0)1/3r20m
2
which is > 0 provided that
|τ |2 > −32m
2(2|Q|)1/3
50
• such ξ are not everywhere spacelike in R1,10
50
• such ξ are not everywhere spacelike in R1,10, so the resulting
quotient does not have a straight-forward physical interpretation
50
• such ξ are not everywhere spacelike in R1,10, so the resulting
quotient does not have a straight-forward physical interpretation
• moreover they contain closed timelike curves
[Maoz–Simon, hep-th/0310255]
50
• such ξ are not everywhere spacelike in R1,10, so the resulting
quotient does not have a straight-forward physical interpretation
• moreover they contain closed timelike curves
[Maoz–Simon, hep-th/0310255]
• so do the associated discrete quotients
50
• such ξ are not everywhere spacelike in R1,10, so the resulting
quotient does not have a straight-forward physical interpretation
• moreover they contain closed timelike curves
[Maoz–Simon, hep-th/0310255]
• so do the associated discrete quotients
• similar results hold for delocalised branes
50
• such ξ are not everywhere spacelike in R1,10, so the resulting
quotient does not have a straight-forward physical interpretation
• moreover they contain closed timelike curves
[Maoz–Simon, hep-th/0310255]
• so do the associated discrete quotients
• similar results hold for delocalised branes, intersecting branes
50
• such ξ are not everywhere spacelike in R1,10, so the resulting
quotient does not have a straight-forward physical interpretation
• moreover they contain closed timelike curves
[Maoz–Simon, hep-th/0310255]
• so do the associated discrete quotients
• similar results hold for delocalised branes, intersecting branes,...
51
End of second lecture
52
Freund–Rubin backgrounds
• purely geometric backgrounds
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h)
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h) and F ∝ dvolg
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h) and F ∝ dvolg
• field equations
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry ⇐⇒ (M, g) and (N,h) admit geometric Killing
spinors
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry ⇐⇒ (M, g) and (N,h) admit geometric Killing
spinors:
∇aε = λΓaε
52
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M4 ×N7, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry ⇐⇒ (M, g) and (N,h) admit geometric Killing
spinors:
∇aε = λΓaε where λ ∈ R×
53
• (M, g) admits geometric Killing spinors
53
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g)
53
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M
53
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g
53
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g ,
admits parallel spinors
53
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g ,
admits parallel spinors: ∇ε = 0
53
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g ,
admits parallel spinors: ∇ε = 0[Bar (1993), Kath (1999)]
53
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g ,
admits parallel spinors: ∇ε = 0[Bar (1993), Kath (1999)]
• equivariant under the isometry group G of (M, g)[hep-th/9902066]
54
• (M, g) riemannian
54
• (M, g) riemannian =⇒ (M, g) riemannian
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
• for the maximally supersymmetric Freund–Rubin backgrounds
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
• for the maximally supersymmetric Freund–Rubin backgrounds,
AdS4×S7
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
• for the maximally supersymmetric Freund–Rubin backgrounds,
AdS4×S7 and S4 ×AdS7 ,
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
• for the maximally supersymmetric Freund–Rubin backgrounds,
AdS4×S7 and S4 ×AdS7 ,
the cones of each factor are flat
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
• for the maximally supersymmetric Freund–Rubin backgrounds,
AdS4×S7 and S4 ×AdS7 ,
the cones of each factor are flat:
? cone of Sn is Rn+1
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
• for the maximally supersymmetric Freund–Rubin backgrounds,
AdS4×S7 and S4 ×AdS7 ,
the cones of each factor are flat:
? cone of Sn is Rn+1
? cone of AdS1+p is (a domain in) R2,p
54
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
• for the maximally supersymmetric Freund–Rubin backgrounds,
AdS4×S7 and S4 ×AdS7 ,
the cones of each factor are flat:
? cone of Sn is Rn+1
? cone of AdS1+p is (a domain in) R2,p
• again the problem reduces to one of flat spaces!
55
Isometries of AdS1+p
55
Isometries of AdS1+p
• AdS1+p is simply-connected
55
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p
55
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p, given by
−x21 − x2
2 + x23 + · · ·+ x2
p+2 = −R2
55
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p, given by
−x21 − x2
2 + x23 + · · ·+ x2
p+2 = −R2
• For p > 2
55
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p, given by
−x21 − x2
2 + x23 + · · ·+ x2
p+2 = −R2
• For p > 2, π1Q1+p∼= Z
55
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p, given by
−x21 − x2
2 + x23 + · · ·+ x2
p+2 = −R2
• For p > 2, π1Q1+p∼= Z, generated by (topological) CTCs
55
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p, given by
−x21 − x2
2 + x23 + · · ·+ x2
p+2 = −R2
• For p > 2, π1Q1+p∼= Z, generated by (topological) CTCs
x1(t) + ix2(t) = reit
55
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p, given by
−x21 − x2
2 + x23 + · · ·+ x2
p+2 = −R2
• For p > 2, π1Q1+p∼= Z, generated by (topological) CTCs
x1(t) + ix2(t) = reit with r2 = R2 + x23 + · · ·+ x2
p+2
56
• (orientation-preserving) isometries of Q1+p
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p)
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why?
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
? SO(2, p) has maximal compact subgroup SO(2)× SO(p)
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
? SO(2, p) has maximal compact subgroup SO(2)× SO(p)? the orbits of SO(2) are the CTCs above
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
? SO(2, p) has maximal compact subgroup SO(2)× SO(p)? the orbits of SO(2) are the CTCs above
? these curves are not closed in AdS1+p
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
? SO(2, p) has maximal compact subgroup SO(2)× SO(p)? the orbits of SO(2) are the CTCs above
? these curves are not closed in AdS1+p
? in AdS1+p, x1∂2 − x2∂1 does not generate SO(2) but R
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
? SO(2, p) has maximal compact subgroup SO(2)× SO(p)? the orbits of SO(2) are the CTCs above
? these curves are not closed in AdS1+p
? in AdS1+p, x1∂2 − x2∂1 does not generate SO(2) but R
• the (orientation-preserving) isometry group of AdS1+p is an
infinite cover SO(2, p)
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
? SO(2, p) has maximal compact subgroup SO(2)× SO(p)? the orbits of SO(2) are the CTCs above
? these curves are not closed in AdS1+p
? in AdS1+p, x1∂2 − x2∂1 does not generate SO(2) but R
• the (orientation-preserving) isometry group of AdS1+p is an
infinite cover SO(2, p), a central extension of SO(2, p)
56
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
• SO(2, p) is not the (orientation-preserving) isometry group of
AdS1+p. Why? Because...
? SO(2, p) has maximal compact subgroup SO(2)× SO(p)? the orbits of SO(2) are the CTCs above
? these curves are not closed in AdS1+p
? in AdS1+p, x1∂2 − x2∂1 does not generate SO(2) but R
• the (orientation-preserving) isometry group of AdS1+p is an
infinite cover SO(2, p), a central extension of SO(2, p) by Z
57
• the central element is the generator of π1Q1+p
57
• the central element is the generator of π1Q1+p
• The bad news
57
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group
57
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group; it has no finite-
dimensional matrix representations
57
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group; it has no finite-
dimensional matrix representations
• The good news
57
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group; it has no finite-
dimensional matrix representations
• The good news:
? the Lie algebra of SO(2, p) is still so(2, p)
57
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group; it has no finite-
dimensional matrix representations
• The good news:
? the Lie algebra of SO(2, p) is still so(2, p); and
? adjoint group is again SO(2, p)
57
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group; it has no finite-
dimensional matrix representations
• The good news:
? the Lie algebra of SO(2, p) is still so(2, p); and
? adjoint group is again SO(2, p)
whence
• one-parameter subgroups
57
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group; it has no finite-
dimensional matrix representations
• The good news:
? the Lie algebra of SO(2, p) is still so(2, p); and
? adjoint group is again SO(2, p)
whence
• one-parameter subgroups↔ projectivised adjoint orbits of so(2, p)under SO(2, p)
58
Normal forms for so(2, p)
58
Normal forms for so(2, p)
We play again
58
Normal forms for so(2, p)
We play again but with a bigger set!
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2)
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2) and also (2, 0)
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2) and also (2, 0), µ(x) = x2 + ϕ2
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2) and also (2, 0), µ(x) = x2 + ϕ2, rotation
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2) and also (2, 0), µ(x) = x2 + ϕ2, rotation
B(0,2)(ϕ)
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2) and also (2, 0), µ(x) = x2 + ϕ2, rotation
B(0,2)(ϕ) = B(2,0)(ϕ)
58
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2) and also (2, 0), µ(x) = x2 + ϕ2, rotation
B(0,2)(ϕ) = B(2,0)(ϕ) =[
0 ϕ
−ϕ 0
]
59
• (1, 1)
59
• (1, 1), µ(x) = x2 − β2
59
• (1, 1), µ(x) = x2 − β2, boost
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
• (1, 2)
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
• (1, 2) and also (2, 1)
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3, null rotation
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3, null rotation
B(1,2)
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3, null rotation
B(1,2) = B(2,1)
59
• (1, 1), µ(x) = x2 − β2, boost
B(1,1) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3, null rotation
B(1,2) = B(2,1) =
0 −1 01 0 −10 1 0
60
But there are also new ones:
• (2, 2)
60
But there are also new ones:
• (2, 2), µ(x) = x2
60
But there are also new ones:
• (2, 2), µ(x) = x2, “rotation” in a totally null plane
60
But there are also new ones:
• (2, 2), µ(x) = x2, “rotation” in a totally null plane
B(2,2)±
60
But there are also new ones:
• (2, 2), µ(x) = x2, “rotation” in a totally null plane
B(2,2)± =
0 ∓1 1 0±1 0 0 ∓1−1 0 0 10 ±1 −1 0
61
• (2, 2)
61
• (2, 2), µ(x) = (x2−β2)2
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0)
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0) =
0 ∓1 1 −β
±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0) =
0 ∓1 1 −β
±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0
The associated discrete quotient of AdS3
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0) =
0 ∓1 1 −β
±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0
The associated discrete quotient of AdS3 yields the extremal
BTZ black hole
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0) =
0 ∓1 1 −β
±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0
The associated discrete quotient of AdS3 yields the extremal
BTZ black hole; the non-extremal black hole
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0) =
0 ∓1 1 −β
±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0
The associated discrete quotient of AdS3 yields the extremal
BTZ black hole; the non-extremal black hole is obtained from
B(1,1)(β1)⊕B(1,1)(β2)
61
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0) =
0 ∓1 1 −β
±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0
The associated discrete quotient of AdS3 yields the extremal
BTZ black hole; the non-extremal black hole is obtained from
B(1,1)(β1)⊕B(1,1)(β2), for |β1| 6= |β2|
62
• (2, 2)
62
• (2, 2), µ(x) = (x2 + ϕ2)2
62
• (2, 2), µ(x) = (x2 + ϕ2)2, deformation of B(2,2)± by a (anti)self-
dual rotation
62
• (2, 2), µ(x) = (x2 + ϕ2)2, deformation of B(2,2)± by a (anti)self-
dual rotation
B(2,2)± (ϕ)
62
• (2, 2), µ(x) = (x2 + ϕ2)2, deformation of B(2,2)± by a (anti)self-
dual rotation
B(2,2)± (ϕ) =
0 ∓1± ϕ 1 0
±1∓ ϕ 0 0 ∓1−1 0 0 1 + ϕ
0 ±1 −1− ϕ 0
63
• (2, 2)
63
• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2
63
• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2, self-dual boost + antiself-
dual rotation
63
• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2, self-dual boost + antiself-
dual rotation
B(2,2)± (β > 0, ϕ > 0)
63
• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2, self-dual boost + antiself-
dual rotation
B(2,2)± (β > 0, ϕ > 0) =
0 ±ϕ 0 −β
∓ϕ 0 ±β 00 ∓β 0 −ϕ
β 0 ϕ 0
64
• (2, 3)
64
• (2, 3), µ(x) = x5
64
• (2, 3), µ(x) = x5, deformation of B(2,2)+ by a null rotation in a
perpendicular direction
64
• (2, 3), µ(x) = x5, deformation of B(2,2)+ by a null rotation in a
perpendicular direction
B(2,3)
64
• (2, 3), µ(x) = x5, deformation of B(2,2)+ by a null rotation in a
perpendicular direction
B(2,3) =
0 1 −1 0 −1−1 0 0 1 01 0 0 −1 00 −1 1 0 −11 0 0 1 0
65
• (2, 4)
65
• (2, 4), µ(x) = (x2 + ϕ2)3
65
• (2, 4), µ(x) = (x2 + ϕ2)3, double null rotation + simultaneous
rotation
65
• (2, 4), µ(x) = (x2 + ϕ2)3, double null rotation + simultaneous
rotation
B(2,4)± (ϕ)
65
• (2, 4), µ(x) = (x2 + ϕ2)3, double null rotation + simultaneous
rotation
B(2,4)± (ϕ) =
0 ∓ϕ 0 0 −1 0±ϕ 0 0 0 0 ∓10 0 0 ϕ −1 00 0 −ϕ 0 0 −11 0 1 0 0 ϕ
0 ±1 0 1 −ϕ 0
65
• (2, 4), µ(x) = (x2 + ϕ2)3, double null rotation + simultaneous
rotation
B(2,4)± (ϕ) =
0 ∓ϕ 0 0 −1 0±ϕ 0 0 0 0 ∓10 0 0 ϕ −1 00 0 −ϕ 0 0 −11 0 1 0 0 ϕ
0 ±1 0 1 −ϕ 0
• and that’s all!
66
Causal properties
66
Causal properties
• Killing vectors on AdS1+p×Sq decompose
66
Causal properties
• Killing vectors on AdS1+p×Sq decompose
ξ = ξA + ξS
66
Causal properties
• Killing vectors on AdS1+p×Sq decompose
ξ = ξA + ξS
whose norms add
‖ξ‖2 = ‖ξA‖2 + ‖ξS‖2
67
• Sq is compact
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd, m2 can be > 0
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd, m2 can be > 0
• ξ can be everywhere spacelike on AdS1+p×S2k+1
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd, m2 can be > 0
• ξ can be everywhere spacelike on AdS1+p×S2k+1, even if ξA is
not spacelike everywhere
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd, m2 can be > 0
• ξ can be everywhere spacelike on AdS1+p×S2k+1, even if ξA is
not spacelike everywhere, provided that ‖ξA‖2 is bounded below
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd, m2 can be > 0
• ξ can be everywhere spacelike on AdS1+p×S2k+1, even if ξA is
not spacelike everywhere, provided that ‖ξA‖2 is bounded below
and ξS has no zeroes
67
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd, m2 can be > 0
• ξ can be everywhere spacelike on AdS1+p×S2k+1, even if ξA is
not spacelike everywhere, provided that ‖ξA‖2 is bounded below
and ξS has no zeroes
• it is convenient to distinguish Killing vectors according to norm
68
• everywhere non-negative norm
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below:
? B(2,0)(ϕ)⊕i B(0,2)(ϕi)
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below:
? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below:
? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even and |ϕi| ≥ ϕ > 0 for all i
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below:
? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even and |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi)
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below:
? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even and |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), if |ϕi| ≥ |ϕ| ≥ 0 for all i
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below:
? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even and |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), if |ϕi| ≥ |ϕ| ≥ 0 for all i
• arbitrarily negative norm
68
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|? B(1,2) ⊕i B(0,2)(ϕi)? B(1,2) ⊕B(1,2) ⊕i B(0,2)(ϕi)? B
(2,2)± ⊕i B(0,2)(ϕi)
• norm bounded below:
? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even and |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), if |ϕi| ≥ |ϕ| ≥ 0 for all i
• arbitrarily negative norm: the rest!
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), unless |ϕi| ≥ ϕ > 0 for all i
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), unless |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), unless |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)
? B(2,3) ⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), unless |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)
? B(2,3) ⊕i B(0,2)(ϕi)? B
(2,4)± (ϕ)⊕i B(0,2)(ϕi)
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), unless |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)
? B(2,3) ⊕i B(0,2)(ϕi)? B
(2,4)± (ϕ)⊕i B(0,2)(ϕi)
Some of these give rise to higher-dimensional BTZ-like black
holes
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), unless |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)
? B(2,3) ⊕i B(0,2)(ϕi)? B
(2,4)± (ϕ)⊕i B(0,2)(ϕi)
Some of these give rise to higher-dimensional BTZ-like black
holes: quotient only a part of AdS
69
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0? B(1,2) ⊕B(1,1)(β)⊕i B(0,2)(ϕi)? B(2,0)(ϕ)⊕i B(0,2)(ϕi), unless p is even and |ϕi| ≥ |ϕ| for all i
? B(2,1) ⊕i B(0,2)(ϕi)? B
(2,2)± (β)⊕i B(0,2)(ϕi)
? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), unless |ϕi| ≥ ϕ > 0 for all i
? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)
? B(2,3) ⊕i B(0,2)(ϕi)? B
(2,4)± (ϕ)⊕i B(0,2)(ϕi)
Some of these give rise to higher-dimensional BTZ-like black
holes: quotient only a part of AdS and check that the boundary
thus introduced lies behind a horizon.
70
Discrete quotients with CTCs
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike
• the corresponding one-parameter subgroup Γ
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike
• the corresponding one-parameter subgroup Γ ∼= R
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike
• the corresponding one-parameter subgroup Γ ∼= R
• pick L > 0 and consider the cyclic subgroup ΓL
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike
• the corresponding one-parameter subgroup Γ ∼= R
• pick L > 0 and consider the cyclic subgroup ΓL∼= Z
70
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike
• the corresponding one-parameter subgroup Γ ∼= R
• pick L > 0 and consider the cyclic subgroup ΓL∼= Z generated
by
γ = exp(LX)
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point and γN · x
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS)
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS) its
image under γN
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS) its
image under γN
? we will construct a timelike curve c(t)
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS) its
image under γN
? we will construct a timelike curve c(t) between c(0) = x
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS) its
image under γN
? we will construct a timelike curve c(t) between c(0) = x and
c(NL) = γN · x
71
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof: find a timelike curve which connects a point x
to its image γNx for N � 1
• e.g., a Z-quotient of a lorentzian cylinder
• general case:
? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS) its
image under γN
? we will construct a timelike curve c(t) between c(0) = x and
c(NL) = γN · x for N � 1
72
? c is uniquely determined by its projections cA onto AdS1+p
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt = NL‖cS‖
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt = NL‖cS‖ ≤ D
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt = NL‖cS‖ ≤ D
? therefore ‖cS‖ ≤ D/NL
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt = NL‖cS‖ ≤ D
? therefore ‖cS‖ ≤ D/NL, and
‖c‖2
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt = NL‖cS‖ ≤ D
? therefore ‖cS‖ ≤ D/NL, and
‖c‖2 = ‖cA‖2 + ‖cS‖2
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt = NL‖cS‖ ≤ D
? therefore ‖cS‖ ≤ D/NL, and
‖c‖2 = ‖cA‖2 + ‖cS‖2 ≤ ‖ξA‖2 +D2
N2L2
72
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
? cS is a length-minimising geodesic between xS and γN · xS,
whose arclength
∫ NL
0
‖cS(t)‖dt = NL‖cS‖ ≤ D
? therefore ‖cS‖ ≤ D/NL, and
‖c‖2 = ‖cA‖2 + ‖cS‖2 ≤ ‖ξA‖2 +D2
N2L2
73
which is negative for N � 1
73
which is negative for N � 1 where ‖ξA‖2 < 0
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete:
? N is Einstein with positive cosmological constant
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete:
? N is Einstein with positive cosmological constant and Einstein
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete:
? N is Einstein with positive cosmological constant and Einstein
? Bonnet-Myers Theorem =⇒ N is compact
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete:
? N is Einstein with positive cosmological constant and Einstein
? Bonnet-Myers Theorem =⇒ N is compact =⇒ has bounded
diameter
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete:
? N is Einstein with positive cosmological constant and Einstein
? Bonnet-Myers Theorem =⇒ N is compact =⇒ has bounded
diameter
• geometrical CTCs are also natural in certain kinds of
supersymmetric Freund–Rubin backgrounds M × N
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete:
? N is Einstein with positive cosmological constant and Einstein
? Bonnet-Myers Theorem =⇒ N is compact =⇒ has bounded
diameter
• geometrical CTCs are also natural in certain kinds of
supersymmetric Freund–Rubin backgrounds M × N , where M
is lorentzian Einstein–Sasaki
73
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N , where M is lorentzian admitting such isometries and N
is complete:
? N is Einstein with positive cosmological constant and Einstein
? Bonnet-Myers Theorem =⇒ N is compact =⇒ has bounded
diameter
• geometrical CTCs are also natural in certain kinds of
supersymmetric Freund–Rubin backgrounds M × N , where M
is lorentzian Einstein–Sasaki: timelike circle bundles over Kahler
manifolds
74
Some interesting reductions
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3 × R4.
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3 × R4.
• 34-BPS AdS4×CP3 background of IIA
[Duff–Lu–Pope, hep-th/9704186]
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Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3 × R4.
• 34-BPS AdS4×CP3 background of IIA
[Duff–Lu–Pope, hep-th/9704186]
• 916-BPS IIA backgrounds
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3 × R4.
• 34-BPS AdS4×CP3 background of IIA
[Duff–Lu–Pope, hep-th/9704186]
• 916-BPS IIA backgrounds: reductions of AdS4×S7
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3 × R4.
• 34-BPS AdS4×CP3 background of IIA
[Duff–Lu–Pope, hep-th/9704186]
• 916-BPS IIA backgrounds: reductions of AdS4×S7 by
B(2,2)+
74
Some interesting reductions
• there are many families of smooth supersymmetric reductions of
AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3 × R4.
• 34-BPS AdS4×CP3 background of IIA
[Duff–Lu–Pope, hep-th/9704186]
• 916-BPS IIA backgrounds: reductions of AdS4×S7 by
B(2,2)+ ⊕ ϕ(R12 + R34 + R56 −R78)
75
• a half-BPS IIA background
75
• a half-BPS IIA background: reduction of S4 ×AdS7
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
• a family of half-BPS IIA backgrounds
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
• a family of half-BPS IIA backgrounds: reductions of S4 × AdS7
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
• a family of half-BPS IIA backgrounds: reductions of S4 × AdS7
by
B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
• a family of half-BPS IIA backgrounds: reductions of S4 × AdS7
by
B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)
Both these half-BPS quotients are of the form S4 × (AdS7 /Γ)
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
• a family of half-BPS IIA backgrounds: reductions of S4 × AdS7
by
B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)
Both these half-BPS quotients are of the form S4 × (AdS7 /Γ)
• a number of maximally supersymmetric reductions of AdS3×S3
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
• a family of half-BPS IIA backgrounds: reductions of S4 × AdS7
by
B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)
Both these half-BPS quotients are of the form S4 × (AdS7 /Γ)
• a number of maximally supersymmetric reductions of AdS3×S3:
near-horizon geometries of the supersymmetric rotating black
holes
75
• a half-BPS IIA background: reduction of S4 ×AdS7 by a double
null rotation
B(1,2) ⊕B(1,2)
• a family of half-BPS IIA backgrounds: reductions of S4 × AdS7
by
B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)
Both these half-BPS quotients are of the form S4 × (AdS7 /Γ)
• a number of maximally supersymmetric reductions of AdS3×S3:
near-horizon geometries of the supersymmetric rotating black
holes, including over-rotating cases
76
• reductions which break no supersymmetry are rare
76
• reductions which break no supersymmetry are rare: this is
intimately linked to the fact that AdS3×S3 is a lorentzian Lie
group
76
• reductions which break no supersymmetry are rare: this is
intimately linked to the fact that AdS3×S3 is a lorentzian Lie
group, and the Killing vectors are left-invariant
76
• reductions which break no supersymmetry are rare: this is
intimately linked to the fact that AdS3×S3 is a lorentzian Lie
group, and the Killing vectors are left-invariant
[Chamseddine–FO–Sabra, hep-th/0306278]
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Thank you.