Post on 30-Sep-2020
Time-dependent string backgrounds viaquotients
Jose Figueroa-O’Farrill
EMPG, School of Mathematics
Erwin Schrodinger Institut, 11 May 2004
1
Based on work in collaboration with Joan Simon (Pennsylvania)
1
Based on work in collaboration with Joan Simon (Pennsylvania),
and Owen Madden and Simon Ross (Durham)
1
Based on work in collaboration with Joan Simon (Pennsylvania),
and Owen Madden and Simon Ross (Durham):
• JHEP 12 (2001) 011, hep-th/0110170
1
Based on work in collaboration with Joan Simon (Pennsylvania),
and Owen Madden and Simon Ross (Durham):
• 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),
and Owen Madden and Simon Ross (Durham):
• 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),
and Owen Madden and Simon Ross (Durham):
• 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 (to appear in Adv. Theor. Math. Phys.)
1
Based on work in collaboration with Joan Simon (Pennsylvania),
and Owen Madden and Simon Ross (Durham):
• 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 (to appear in Adv. Theor. Math. Phys.), and
• hep-th/0402094 (to appear in Phys. Rev. D)
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
General problem
3
General problem
• (M, g, F, ...) a supergravity background
3
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G
3
General problem
• (M, g, F, ...) a supergravity background
• symmetry group G— not just isometries, but also preserving F, ...
3
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
3
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
3
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
3
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
3
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
3
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
3
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
3
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
One-parameter subgroups
4
One-parameter subgroups
• (M, g, F, ...)
4
One-parameter subgroups
• (M, g, F, ...)
• symmetries
4
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M
4
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g
4
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g f∗F = F
4
One-parameter subgroups
• (M, g, F, ...)
• symmetries
f : M∼=−→ M f∗g = g f∗F = F . . .
define a Lie group G
4
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
4
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
4
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}
5
• X ∈ g also defines a Killing vector ξX
5
• X ∈ g also defines a Killing vector ξX:
LξXg = 0
5
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
5
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
5
• X ∈ g also defines a Killing vector ξX:
LξXg = 0 LξX
F = 0 . . .
whose integral curves are the orbits of Γ
• two possible topologies
5
• 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
5
• 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
5
• 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
5
• 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
5
• 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
6
Kaluza–Klein and discrete quotients
6
Kaluza–Klein and discrete quotients
• Γ ∼= S1
6
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
6
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R
6
Kaluza–Klein and discrete quotients
• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction
• Γ ∼= R: quotient performed in two steps:
6
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
6
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
6
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}
6
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
6
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
6
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
6
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
7
• we may stop after the first step
7
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M
7
• we may stop after the first step: obtaining backgrounds M/ΓL
locally isometric to M , but often with very different global
properties
7
• 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
7
• 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
7
• 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
7
• 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
8
Classifying quotients
8
Classifying quotients
• (M, g, F, ...) with symmetry group G
8
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
8
Classifying quotients
• (M, g, F, ...) with symmetry group G, Lie algebra g
• X, X ′ ∈ g generate one-parameter subgroups
Γ = {exp(tX) | t ∈ R}
8
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}
8
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
8
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
8
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′
9
• enough to classify normal forms of X ∈ g under
X ∼ λgXg−1 g ∈ G λ ∈ R×
9
• enough to classify normal forms of X ∈ g under
X ∼ λgXg−1 g ∈ G λ ∈ R×
i.e., projectivised adjoint orbits of g
10
Flat quotients
10
Flat quotients
• (R1,9, F = 0)
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9 ⊂ GL(11,R)
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9 ⊂ GL(11,R):(
A v
0 1
)
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9 ⊂ GL(11,R):(
A v
0 1
)A ∈ O(1, 9)
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9 ⊂ GL(11,R):(
A v
0 1
)A ∈ O(1, 9) v ∈ R
1,9
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9 ⊂ GL(11,R):(
A v
0 1
)A ∈ O(1, 9) v ∈ R
1,9
• Γ ⊂ O(1, 9) n R1,9
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9 ⊂ GL(11,R):(
A v
0 1
)A ∈ O(1, 9) v ∈ R
1,9
• Γ ⊂ O(1, 9) n R1,9, generated by
X = XL + XT ∈ so(1, 9)⊕ R1,9
10
Flat quotients
• (R1,9, F = 0) has symmetry O(1, 9) n R1,9 ⊂ GL(11,R):(
A v
0 1
)A ∈ O(1, 9) v ∈ R
1,9
• Γ ⊂ O(1, 9) n R1,9, generated by
X = XL + XT ∈ so(1, 9)⊕ R1,9 ,
which we need to put in normal form.
11
Normal forms for orthogonal transformations
11
Normal forms for orthogonal transformations
• X ∈ so(p, q)
11
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear
11
Normal forms for orthogonal transformations
• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric
relative to 〈−,−〉 of signature (p, q)
11
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
11
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
11
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
11
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
11
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
11
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
11
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
11
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∗
11
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∗
12
• possible minimal polynomials
12
• possible minimal polynomials:
? λ = 0
12
• possible minimal polynomials:
? λ = 0 µ(x) = xn
12
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R
12
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
12
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
? λ = iϕ ∈ iR
12
• possible minimal polynomials:
? λ = 0 µ(x) = xn
? λ = β ∈ R,
µ(x) = (x2 − β2)n
? λ = iϕ ∈ iR,
µ(x) = (x2 + ϕ2)n
12
• 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
12
• 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
12
• 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
13
Strategy
13
Strategy
• for each µ(x)
13
Strategy
• for each µ(x), write down X in (real) Jordan form
13
Strategy
• for each µ(x), write down X in (real) Jordan form
• determine metric making X skew-symmetric
13
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
13
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
13
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
14
Elementary lorentzian blocks
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1)
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2)
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0)
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1)
14
Elementary lorentzian blocks
Signature Minimal polynomial Type
(0, 1) x trivial
(0, 2) x2 + ϕ2 rotation
(1, 0) x trivial
(1, 1) x2 − β2
14
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
14
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)
14
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
14
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
15
Lorentzian normal forms
15
Lorentzian normal forms
Play with the elementary blocks!
15
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 9)
15
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 9):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
15
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 9):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3)
15
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 9):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3)
• N+2 + R34(ϕ1) + R56(ϕ2) + R78(ϕ3)
15
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 9):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3)
• N+2 + R34(ϕ1) + R56(ϕ2) + R78(ϕ3)
where β > 0
15
Lorentzian normal forms
Play with the elementary blocks!
In signature (1, 9):
• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3)
• N+2 + R34(ϕ1) + R56(ϕ2) + R78(ϕ3)
where β > 0, ϕ1 ≥ ϕ2 ≥ · · · ≥ ϕk−1 ≥ ϕk ≥ 0
16
Normal forms for the Poincare algebra
16
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 9)⊕ R1,9
16
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 9)⊕ R1,9
• conjugate by O(1, 9) to bring λ to normal form
16
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 9)⊕ R1,9
• conjugate by O(1, 9) to bring λ to normal form
• conjugate by R1,9
16
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 9)⊕ R1,9
• conjugate by O(1, 9) to bring λ to normal form
• conjugate by R1,9:
λ + τ 7→ λ + τ − [λ, τ ′]
16
Normal forms for the Poincare algebra
• λ + τ ∈ so(1, 9)⊕ R1,9
• conjugate by O(1, 9) to bring λ to normal form
• conjugate by R1,9:
λ + τ 7→ λ + τ − [λ, τ ′]
to get rid of component of τ in the image of [λ,−]
17
• the subgroups with everywhere spacelike orbits
17
• the subgroups with everywhere spacelike orbits are generated by
either
? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
17
• 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)
17
• 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
17
• 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
17
• 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
17
• 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
18
Adapted coordinates
18
Adapted coordinates
• start with metric in flat coordinates y, z
18
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
18
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
• “undress” the Killing vector
18
Adapted coordinates
• start with metric in flat coordinates y, z
ds2 = 2|dy|2 + dz2
• “undress” the Killing vector
ξ = ∂z + λ
18
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
18
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)
19
• introduce new coordinates
19
• introduce new coordinates
x = U y
19
• introduce new coordinates
x = U y = exp(−zB)y
19
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
19
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
whence ξx = 0
19
• introduce new coordinates
x = U y = exp(−zB)y where λy = By
whence ξx = 0
• rewrite the metric in terms of x
19
• 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
19
• 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
19
• 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
20
• the only data is the matrix
20
• the only data is the matrix
B =
−u
0 −ϕ1 − u
−u ϕ1 + u 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
20
• the only data is the matrix
B =
−u
0 −ϕ1 − u
−u ϕ1 + u 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
where
20
• the only data is the matrix
B =
−u
0 −ϕ1 − u
−u ϕ1 + u 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
where either
? u = 0
20
• the only data is the matrix
B =
−u
0 −ϕ1 − u
−u ϕ1 + u 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
where either
? u = 0 (generalised fluxbranes)
20
• the only data is the matrix
B =
−u
0 −ϕ1 − u
−u ϕ1 + u 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
where either
? u = 0 (generalised fluxbranes); or
? u = 1 and ϕ1 = 0
20
• the only data is the matrix
B =
−u
0 −ϕ1 − u
−u ϕ1 + u 0
0 −ϕ2
ϕ2 0
0 −ϕ3
ϕ3 0
0 −ϕ4
ϕ4 0
where either
? u = 0 (generalised fluxbranes); or
? u = 1 and ϕ1 = 0 (generalised nullbranes)
21
Discrete quotients
21
Discrete quotients
• start with the metric in adapted coordinates
21
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
21
Discrete quotients
• start with the metric in adapted coordinates
ds2 = Λ(dz + A)2 + |dx|2 − ΛA2
and identify z ∼ z + L
21
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
21
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
Λ = 1 + x2+
21
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
Λ = 1 + x2+ and A =
11 + x2
+
(x−dx1 − x1dx−)
21
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
Λ = 1 + x2+ and A =
11 + x2
+
(x−dx1 − x1dx−)
=⇒ half-BPS ten-dimensional nullbrane
22
• the nullbrane is
22
• the nullbrane is
? time-dependent
22
• the nullbrane is
? time-dependent
? smooth
22
• the nullbrane is
? time-dependent
? smooth
? stable
22
• the nullbrane is
? time-dependent
? smooth
? stable
? a smooth transition between Big Crunch and Big Bang
22
• the nullbrane is
? time-dependent
? smooth
? stable
? a smooth transition between Big Crunch and Big Bang
? a resolution of parabolic orbifold [Horowitz–Steif (1991)]
22
• the nullbrane is
? time-dependent
? smooth
? stable
? a smooth transition between Big Crunch and Big Bang
? a resolution of parabolic orbifold [Horowitz–Steif (1991)]
• its conformal field theory is a Z-orbifold of flat space
22
• the nullbrane is
? time-dependent
? smooth
? stable
? a smooth transition between Big Crunch and Big Bang
? a resolution of parabolic orbifold [Horowitz–Steif (1991)]
• its conformal field theory is a Z-orbifold of flat space, and has
been studied [Liu–Moore–Seiberg, hep-th/0206182]
22
• the nullbrane is
? time-dependent
? smooth
? stable
? a smooth transition between Big Crunch and Big Bang
? a resolution of parabolic orbifold [Horowitz–Steif (1991)]
• its conformal field theory is a Z-orbifold of flat space, and has
been studied [Liu–Moore–Seiberg, hep-th/0206182]
• some arithmetic issues remain
23
Supersymmetry
23
Supersymmetry
• (M, g, F, ...) a supersymmetric background
23
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries
23
Supersymmetry
• (M, g, F, ...) a supersymmetric background
• Γ a one-parameter subgroup of symmetries, with Killing vector ξ
23
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?
23
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
23
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
23
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
23
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
23
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
23
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
23
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.
24
“Supersymmetry without supersymmetry”
24
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
24
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example
24
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example: [Duff–Lu–Pope, hep-th/9704186,9803061]
24
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example: [Duff–Lu–Pope, hep-th/9704186,9803061]
AdS5×S5
24
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example: [Duff–Lu–Pope, hep-th/9704186,9803061]
AdS5×S5
((QQQQQQQQQQQQQ
oo // AdS5×CP2 × S1
24
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example: [Duff–Lu–Pope, hep-th/9704186,9803061]
AdS5×S5
((QQQQQQQQQQQQQ
oo // AdS5×CP2 × S1
uukkkkkkkkkkkkkkk
AdS5×CP2
24
“Supersymmetry without supersymmetry”
• T-duality relates backgrounds with different amount of
“supergravitational supersymmetry”
• dramatic example: [Duff–Lu–Pope, hep-th/9704186,9803061]
AdS5×S5
((QQQQQQQQQQQQQ
oo // AdS5×CP2 × S1
uukkkkkkkkkkkkkkk
AdS5×CP2
CP2 is not even spin!
25
Spin structures in quotients
25
Spin structures in quotients
• (M, g) spin
25
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
25
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
25
Spin structures in quotients
• (M, g) spin, Γ a one-parameter subgroup of isometries
Is M/Γ spin?
• if Γ ∼= R
25
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
25
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
• if Γ ∼= S1
25
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
• if Γ ∼= S1 then M/Γ is spin if and only if the action of Γ lifts to
the spin bundle
25
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
• if Γ ∼= S1 then M/Γ is spin if and only if the action of Γ lifts to
the spin bundle
• equivalently, the action of ξ = ξX on spinors has integral weights
26
Supersymmetry of supergravity quotients
26
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
26
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries
26
Supersymmetry of supergravity quotients
• (M, g, F, ...) supersymmetric
• Γ one-parameter group of symmetries, generated by ξ
• Killing spinors of M/Γ
26
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
26
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
26
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., (R1,9)
26
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., (R1,9): Killing spinors are parallel
26
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., (R1,9): Killing spinors are parallel, whence
Lξε = 18∇aξbΓabε
27
• e.g., fluxbranes
27
• e.g., fluxbranes
ξ = ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
27
• e.g., fluxbranes
ξ = ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2(ϕ1Γ12 + ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
28
• for generic ϕi
28
• for generic ϕi, there are no invariant Killing spinors
28
• for generic ϕi, there are no invariant Killing spinors; but there
are hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
28
• 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
29
• e.g., nullbranes
29
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
29
• e.g., nullbranes
ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)
=⇒Lξ = 1
2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)
29
• 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
29
• 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
29
• 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
29
• 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+
29
• 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
30
• for generic ϕi
30
• for generic ϕi, no supersymmetry is preserved
30
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues
30
• for generic ϕi, no supersymmetry is preserved, but there are
hyperplanes (and their intersections) on which Lξ has zero
eigenvalues:
? ϕ2 − ϕ3 − ϕ4 = 0
30
• 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
30
• 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
30
• 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
30
• 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
30
• 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
31
End of first part.
32
Freund–Rubin backgrounds
• purely geometric backgrounds
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h)
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h) and F ∝ dvolg
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h) and F ∝ dvolg
• field equations
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry ⇐⇒ (M, g) and (N,h) admit geometric Killing
spinors
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry ⇐⇒ (M, g) and (N,h) admit geometric Killing
spinors:
∇Xε = λX · ε
32
Freund–Rubin backgrounds
• purely geometric backgrounds, with product geometry
(M ×N, g ⊕ h) and F ∝ dvolg
• field equations ⇐⇒ (M, g) and (N,h) are Einstein
• supersymmetry ⇐⇒ (M, g) and (N,h) admit geometric Killing
spinors:
∇Xε = λX · ε where λ ∈ R×
33
• (M, g) admits geometric Killing spinors
33
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g)
33
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M
33
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g
33
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g ,
admits parallel spinors
33
• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),
M = R+ ×M and g = dr2 + 4λ2r2g ,
admits parallel spinors: ∇ε = 0
33
• (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)]
33
• (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]
34
• (M, g) riemannian
34
• (M, g) riemannian =⇒ (M, g) riemannian
34
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian
34
• (M, g) riemannian =⇒ (M, g) riemannian
• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)
34
• (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
34
• (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,
AdS1+p×Sq
34
• (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,
AdS1+p×Sq
the cones of each factor are flat
34
• (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,
AdS1+p×Sq
the cones of each factor are flat:
? cone of Sq is Rq+1
34
• (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,
AdS1+p×Sq
the cones of each factor are flat:
? cone of Sq is Rq+1
? cone of AdS1+p is (a domain in) R2,p
34
• (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,
AdS1+p×Sq
the cones of each factor are flat:
? cone of Sq is Rq+1
? cone of AdS1+p is (a domain in) R2,p
• again the problem reduces to one of flat spaces!
35
Isometries of AdS1+p
35
Isometries of AdS1+p
• AdS1+p is simply-connected
35
Isometries of AdS1+p
• AdS1+p is simply-connected; it is the universal cover of a quadric
Q1+p ⊂ R2,p
35
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
35
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
35
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
35
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
35
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
35
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
36
• (orientation-preserving) isometries of Q1+p
36
• (orientation-preserving) isometries of Q1+p:
SO(2, p)
36
• (orientation-preserving) isometries of Q1+p:
SO(2, p) ⊂ GL(p + 2,R)
36
• (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
36
• (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?
36
• (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
36
• (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)
36
• (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
36
• (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
36
• (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
36
• (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)
36
• (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)
36
• (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
37
• the central element is the generator of π1Q1+p
37
• the central element is the generator of π1Q1+p
• The bad news
37
• the central element is the generator of π1Q1+p
• The bad news: SO(2, p) is not a matrix group
37
• 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
37
• 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
37
• 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)
37
• 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)
37
• 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
37
• 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)
38
Normal forms for so(2, p)
38
Normal forms for so(2, p)
We play again
38
Normal forms for so(2, p)
We play again but with a bigger set!
38
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks
38
Normal forms for so(2, p)
We play again but with a bigger set!
We can still use the lorentzian elementary blocks:
• (0, 2)
38
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)
38
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
38
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
38
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)
38
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)
38
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
]
39
• (1, 1)
39
• (1, 1), µ(x) = x2 − β2
39
• (1, 1), µ(x) = x2 − β2, boost
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β)
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[0 −β
β 0
]
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[0 −β
β 0
]
• (1, 2)
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[0 −β
β 0
]
• (1, 2) and also (2, 1)
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3, null rotation
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3, null rotation
B(1,2)
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[0 −β
β 0
]
• (1, 2) and also (2, 1), µ(x) = x3, null rotation
B(1,2) = B(2,1)
39
• (1, 1), µ(x) = x2 − β2, boost
B(1,1)(β) =[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
40
But there are also new ones:
• (2, 2)
40
But there are also new ones:
• (2, 2), µ(x) = x2
40
But there are also new ones:
• (2, 2), µ(x) = x2, “rotation” in a totally null plane
40
But there are also new ones:
• (2, 2), µ(x) = x2, “rotation” in a totally null plane
B(2,2)±
40
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
41
• (2, 2)
41
• (2, 2), µ(x) = (x2−β2)2
41
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
41
• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual
boost
B(2,2)± (β > 0)
41
• (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
41
• (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
41
• (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
41
• (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
41
• (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)
41
• (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|
42
• (2, 2)
42
• (2, 2), µ(x) = (x2 + ϕ2)2
42
• (2, 2), µ(x) = (x2 + ϕ2)2, deformation of B(2,2)± by a (anti)self-
dual rotation
42
• (2, 2), µ(x) = (x2 + ϕ2)2, deformation of B(2,2)± by a (anti)self-
dual rotation
B(2,2)± (ϕ)
42
• (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
43
• (2, 2)
43
• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2
43
• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2, self-dual boost + antiself-
dual rotation
43
• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2, self-dual boost + antiself-
dual rotation
B(2,2)± (β > 0, ϕ > 0)
43
• (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
44
• (2, 3)
44
• (2, 3), µ(x) = x5
44
• (2, 3), µ(x) = x5, deformation of B(2,2)+ by a null rotation in a
perpendicular direction
44
• (2, 3), µ(x) = x5, deformation of B(2,2)+ by a null rotation in a
perpendicular direction
B(2,3)
44
• (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
45
• (2, 4)
45
• (2, 4), µ(x) = (x2 + ϕ2)3
45
• (2, 4), µ(x) = (x2 + ϕ2)3, double null rotation + simultaneous
rotation
45
• (2, 4), µ(x) = (x2 + ϕ2)3, double null rotation + simultaneous
rotation
B(2,4)± (ϕ)
45
• (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
45
• (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!
46
Causal properties
46
Causal properties
• Killing vectors on AdS1+p×Sq decompose
46
Causal properties
• Killing vectors on AdS1+p×Sq decompose
ξ = ξA + ξS
46
Causal properties
• Killing vectors on AdS1+p×Sq decompose
ξ = ξA + ξS
whose norms add
‖ξ‖2 = ‖ξA‖2 + ‖ξS‖2
47
• Sq is compact
47
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2
47
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
47
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd
47
• Sq is compact =⇒
R2M2 ≥ ‖ξS‖2 ≥ R2m2
and if q is odd, m2 can be > 0
47
• 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
47
• 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
47
• 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
47
• 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
47
• 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
48
• everywhere non-negative norm
48
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
48
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)
48
• everywhere non-negative norm:
? ⊕iB(0,2)(ϕi)
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|
48
• 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)
48
• 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)
48
• 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)
48
• 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
48
• 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)
48
• 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
48
• 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
48
• 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)
48
• 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
48
• 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
48
• 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!
49
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)
49
? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), unless |β1| = |β2| > 0
49
? 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)
49
? 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)
49
? 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
49
? 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
49
? 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)
49
? 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)
49
? 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)
49
? 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
49
? 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)
49
? 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)
49
? 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)
49
? 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
49
? 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
49
? 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.
50
Discrete quotients with CTCs
50
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1
50
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0
50
Discrete quotients with CTCs
• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike
50
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
50
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
50
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
50
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
50
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)
51
• the “orbifold” of AdS1+p×S2k+1 by ΓL
51
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
51
• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs
• idea of the proof
51
• 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
51
• 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
51
• 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
51
• 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
51
• 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
51
• 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
51
• 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)
51
• 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
51
• 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)
51
• 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
51
• 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
51
• 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
52
? c is uniquely determined by its projections cA onto AdS1+p
52
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
52
? c is uniquely determined by its projections cA onto AdS1+p and
cS onto S2k+1
? cA is the integral curve of ξA
52
? 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
52
? 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
52
? 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‖
52
? 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
52
? 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
52
? 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
52
? 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
52
? 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
52
? 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
53
which is negative for N � 1
53
which is negative for N � 1 where ‖ξA‖2 < 0
53
which is negative for N � 1 where ‖ξA‖2 < 0
• the same argument applies to any Freund–Rubin background
M ×N
53
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
53
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
53
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
53
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
? Bonnet-Myers Theorem =⇒ N has bounded diameter
54
Supersymmetric quotients of AdS3×S3
• yields Freund–Rubin background of IIB
AdS3×S3 ×X4
54
Supersymmetric quotients of AdS3×S3
• yields Freund–Rubin background of IIB
AdS3×S3 ×X4
• equations of motion
54
Supersymmetric quotients of AdS3×S3
• yields Freund–Rubin background of IIB
AdS3×S3 ×X4
• equations of motion =⇒ X Ricci-flat
• supersymmetry
54
Supersymmetric quotients of AdS3×S3
• yields Freund–Rubin background of IIB
AdS3×S3 ×X4
• equations of motion =⇒ X Ricci-flat
• supersymmetry =⇒ X admits parallel spinors
54
Supersymmetric quotients of AdS3×S3
• yields Freund–Rubin background of IIB
AdS3×S3 ×X4
• equations of motion =⇒ X Ricci-flat
• supersymmetry =⇒ X admits parallel spinors
=⇒ X flat or hyperkahler
54
Supersymmetric quotients of AdS3×S3
• yields Freund–Rubin background of IIB
AdS3×S3 ×X4
• equations of motion =⇒ X Ricci-flat
• supersymmetry =⇒ X admits parallel spinors
=⇒ X flat or hyperkahler
55
• for X = R4
55
• for X = R4, Killing spinors are isomorphic to(
∆2,2+ ⊗
[∆4,0
+ ⊗∆0,4+
])⊕
(∆2,2− ⊗
[∆4,0− ⊗∆0,4
+
])
55
• for X = R4, Killing spinors are isomorphic to(
∆2,2+ ⊗
[∆4,0
+ ⊗∆0,4+
])⊕
(∆2,2− ⊗
[∆4,0− ⊗∆0,4
+
])as a representation of Spin(2, 2)× Spin(4)× Spin(4)
55
• for X = R4, Killing spinors are isomorphic to(
∆2,2+ ⊗
[∆4,0
+ ⊗∆0,4+
])⊕
(∆2,2− ⊗
[∆4,0− ⊗∆0,4
+
])as a representation of Spin(2, 2)× Spin(4)× Spin(4)
• here [R] means the underlying real representation of a complex
representation of real type
55
• for X = R4, Killing spinors are isomorphic to(
∆2,2+ ⊗
[∆4,0
+ ⊗∆0,4+
])⊕
(∆2,2− ⊗
[∆4,0− ⊗∆0,4
+
])as a representation of Spin(2, 2)× Spin(4)× Spin(4)
• here [R] means the underlying real representation of a complex
representation of real type; that is,
R = [R]⊗ C
56
Regular one-parameter subgroups
• only consider actions on AdS3×S3
56
Regular one-parameter subgroups
• only consider actions on AdS3×S3
• ξ = ξA + ξS
56
Regular one-parameter subgroups
• only consider actions on AdS3×S3
• ξ = ξA + ξS, with
? ξ spacelike
56
Regular one-parameter subgroups
• only consider actions on AdS3×S3
• ξ = ξA + ξS, with
? ξ spacelike
? smooth quotients
56
Regular one-parameter subgroups
• only consider actions on AdS3×S3
• ξ = ξA + ξS, with
? ξ spacelike
? smooth quotients
? supersymmetric quotients
56
Regular one-parameter subgroups
• only consider actions on AdS3×S3
• ξ = ξA + ξS, with
? ξ spacelike
? smooth quotients
? supersymmetric quotients
• there are two classes
56
Regular one-parameter subgroups
• only consider actions on AdS3×S3
• ξ = ξA + ξS, with
? ξ spacelike
? smooth quotients
? supersymmetric quotients
• there are two classes: having 8 or 4 supersymmetries
57
14-BPS quotients
• ξ = ξS = R12 ±R34
57
14-BPS quotients
• ξ = ξS = R12 ±R34
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ∓R34)
57
14-BPS quotients
• ξ = ξS = R12 ±R34
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ∓R34), θ > 0
57
14-BPS quotients
• ξ = ξS = R12 ±R34
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ∓R34), θ > 0
• ξ = e12 ± e34 + θ(R12 ±R34)
57
14-BPS quotients
• ξ = ξS = R12 ±R34
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ∓R34), θ > 0
• ξ = e12 ± e34 + θ(R12 ±R34), with |θ| > 1
57
14-BPS quotients
• ξ = ξS = R12 ±R34
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ∓R34), θ > 0
• ξ = e12 ± e34 + θ(R12 ±R34), with |θ| > 1
• e13 ± e34 + θ(R12 ±R34)
57
14-BPS quotients
• ξ = ξS = R12 ±R34
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ∓R34), θ > 0
• ξ = e12 ± e34 + θ(R12 ±R34), with |θ| > 1
• e13 ± e34 + θ(R12 ±R34), θ ≥ 0
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34)
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ±R34)
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ±R34), θ > 0
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ±R34), θ > 0
• ξ = ∓e12−e13±e24 +e34 +ϕ(e34∓e12)+ θ(R12±R34)
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ±R34), θ > 0
• ξ = ∓e12−e13±e24 +e34 +ϕ(e34∓e12)+ θ(R12±R34), θ > ϕ
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ±R34), θ > 0
• ξ = ∓e12−e13±e24 +e34 +ϕ(e34∓e12)+ θ(R12±R34), θ > ϕ
• ξ = ∓e12−e13±e24+e34+12(θ1±θ2)(e34∓e12)+θ1R12+θ2R34
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ±R34), θ > 0
• ξ = ∓e12−e13±e24 +e34 +ϕ(e34∓e12)+ θ(R12±R34), θ > ϕ
• ξ = ∓e12−e13±e24+e34+12(θ1±θ2)(e34∓e12)+θ1R12+θ2R34,
θ1 > −θ2 > 0
58
18-BPS quotients
• ξ = 2e34 + R12 ±R34
• ξ = e12 − e13 − e24 + e34 + θ(R12 + R34), θ > 0
• ξ = ∓e12 − e13 ± e24 + e34 + θ(R12 ±R34), θ > 0
• ξ = ∓e12−e13±e24 +e34 +ϕ(e34∓e12)+ θ(R12±R34), θ > ϕ
• ξ = ∓e12−e13±e24+e34+12(θ1±θ2)(e34∓e12)+θ1R12+θ2R34,
θ1 > −θ2 > 0
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34, where 1 ≥ |ϕ|
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34, where 1 ≥ |ϕ|, θ1 ≥ |θ2| > |ϕ|
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34, where 1 ≥ |ϕ|, θ1 ≥ |θ2| > |ϕ|,and 1∓ ϕ = θ1 ∓ θ2
• associated discrete quotients are cyclic orbifolds
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34, where 1 ≥ |ϕ|, θ1 ≥ |θ2| > |ϕ|,and 1∓ ϕ = θ1 ∓ θ2
• associated discrete quotients are cyclic orbifolds (ZN or Z)
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34, where 1 ≥ |ϕ|, θ1 ≥ |θ2| > |ϕ|,and 1∓ ϕ = θ1 ∓ θ2
• associated discrete quotients are cyclic orbifolds (ZN or Z) of a
WZW model with group SL(2,R)× SU(2)
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34, where 1 ≥ |ϕ|, θ1 ≥ |θ2| > |ϕ|,and 1∓ ϕ = θ1 ∓ θ2
• associated discrete quotients are cyclic orbifolds (ZN or Z) of a
WZW model with group SL(2,R)× SU(2)
• most are time-dependent
59
• ξ = e12 + ϕe34 + θ1R12 + θ2R34, where 1 ≥ |ϕ|, θ1 ≥ |θ2| > |ϕ|,and 1∓ ϕ = θ1 ∓ θ2
• associated discrete quotients are cyclic orbifolds (ZN or Z) of a
WZW model with group SL(2,R)× SU(2)
• most are time-dependent, and many have closed timelike curves
60
Thank you.