quotienting string backgrounds - school of mathematicsjmf/cv/seminars/oviedo.pdf · 2014. 11....

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


• JHEP 12 (2001) 011, hep-th/0110170

• Adv. Theor. Math. Phys. 6 (2003) 703–793, hep-th/0208107

1


• JHEP 12 (2001) 011, hep-th/0110170


• Class. Quant. Grav. 19 (2002) 6147–6174, hep-th/0208108

1


• JHEP 12 (2001) 011, hep-th/0110170



• hep-th/0401206

1


• JHEP 12 (2001) 011, hep-th/0110170



• hep-th/0401206

and on work in progress with Joan Simon

1


• JHEP 12 (2001) 011, hep-th/0110170



• hep-th/0401206

and on work in progress with Joan Simon, Hannu Rajaniemi

(Edinburgh)

1


• JHEP 12 (2001) 011, hep-th/0110170



• 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


• string theory in

? time-dependent backgrounds

2

Motivation



? time-dependent backgrounds, and

? causally singular backgrounds

2

Motivation



? time-dependent backgrounds, and

? causally singular backgrounds

• supersymmetric Clifford–Klein space form problem

3

Melvin universe and fluxbranes

3


• axisymmetric solution of d=4 Einstein–Maxwell theory

[Melvin (1964)]

3



[Melvin (1964)]

• describes a gravitationally stable universe of flux

3



[Melvin (1964)]


• dilatonic version in supergravity [Gibbons–Maeda (1988)]

3



[Melvin (1964)]



• Kaluza–Klein reduction of a flat five-dimensional spacetime

[F. Dowker et al. (1994)]

3



[Melvin (1964)]





• R1,4/Γ, with Γ ∼= R

3



[Melvin (1964)]





• R1,4/Γ, with Γ ∼= R, or R1,10/Γ

3



[Melvin (1964)]





• 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


• symmetry group G

4

General problem


• symmetry group G— not just isometries, but also preserving F, ...

4

General problem



• determine all quotient supergravity backgrounds M/Γ

4

General problem



• determine all quotient supergravity backgrounds M/Γ, where

Γ ⊂ G is a one-parameter subgroup

4

General problem




Γ ⊂ G is a one-parameter subgroup, paying close attention to

4

General problem




Γ ⊂ G is a one-parameter subgroup, paying close attention to:

? smoothness

4

General problem





? smoothness,

? causal regularity

4

General problem





? smoothness,

? causal regularity,

? spin structure

4

General problem





? smoothness,


? spin structure,

? supersymmetry

4

General problem





? smoothness,


? spin structure,

? supersymmetry,...

5

One-parameter subgroups

5


• (M, g, F, ...)

5


• (M, g, F, ...)

• symmetries

5


• (M, g, F, ...)

• symmetries

f : M∼=−→ M

5


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g

5


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F

5


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .

define a Lie group G

5


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .

define a Lie group G, with Lie algebra g

5


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .


• X ∈ g defines a one-parameter subgroup

5


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .


• 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


LξXg = 0 LξX

F = 0 . . .

6


LξXg = 0 LξX

F = 0 . . .

whose integral curves are the orbits of Γ

6


LξXg = 0 LξX

F = 0 . . .


• two possible topologies

6


LξXg = 0 LξX

F = 0 . . .


• two possible topologies:

? Γ ∼= S1

6


LξXg = 0 LξX

F = 0 . . .



? Γ ∼= S1, if and only if ∃T > 0 such that exp(TX) = 1

6


LξXg = 0 LξX

F = 0 . . .



? Γ ∼= S1, if and only if ∃T > 0 such that exp(TX) = 1? Γ ∼= R

6


LξXg = 0 LξX

F = 0 . . .



? Γ ∼= S1, if and only if ∃T > 0 such that exp(TX) = 1? Γ ∼= R, otherwise

6


LξXg = 0 LξX

F = 0 . . .



? 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

7

Kaluza–Klein and discrete quotients

7


• Γ ∼= S1

7


• Γ ∼= S1: M/Γ is standard Kaluza–Klein reduction

7



• Γ ∼= R

7



• Γ ∼= R: quotient performed in two steps:

7




? discrete quotient M/ΓL, where

7




? discrete quotient M/ΓL, where L > 0

7




? discrete quotient M/ΓL, where L > 0 and

ΓL = {exp(nLX) | n ∈ Z}

7





ΓL = {exp(nLX) | n ∈ Z} ∼= Z

7





ΓL = {exp(nLX) | n ∈ Z} ∼= Z

? Kaluza–Klein reduction by Γ/ΓL

7





ΓL = {exp(nLX) | n ∈ Z} ∼= Z

? Kaluza–Klein reduction by Γ/ΓL∼= R/Z

7





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


locally isometric to M , but often with very different global

properties

8



properties, e.g.,

? M static, but M/ΓL time-dependent

8



properties, e.g.,


? M causally regular, but M/ΓL causally singular

8



properties, e.g.,



? M spin, but M/ΓL not spin

8



properties, e.g.,



? M spin, but M/ΓL not spin

? M supersymmetric, but M/ΓL breaking all supersymmetry

9

Classifying quotients

9


• (M, g, F, ...) with symmetry group G

9


• (M, g, F, ...) with symmetry group G, Lie algebra g

9



• X, X ′ ∈ g generate one-parameter subgroups

Γ = {exp(tX) | t ∈ R}

9




Γ = {exp(tX) | t ∈ R} Γ′ = {exp(tX ′) | t ∈ R}

9





• if X ′ = λX, λ 6= 0, then Γ′ = Γ

9





• if X ′ = λX, λ 6= 0, then Γ′ = Γ

• if X ′ = gXg−1, then Γ′ = gΓg−1

9





• if X ′ = λX, λ 6= 0, then Γ′ = Γ

• if X ′ = gXg−1, then Γ′ = gΓg−1, and moreover M/Γ ∼= M/Γ′

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• 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

11

Eleven-dimensional supergravity backgrounds

11


• (M, g): lorentzian spin 11-dimensional connected manifold

11



• F a closed 4-form

11




• (M, g, F ) is supersymmetric

11




• (M, g, F ) is supersymmetric ⇐⇒ admits nonzero Killing spinors

11




• (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 ⇐⇒


e.g.,

? ν = 1 backgrounds

12

• (M, g, F ) is ν-BPS ⇐⇒


e.g.,


∗ Minkowski

12

• (M, g, F ) is ν-BPS ⇐⇒


e.g.,


∗ Minkowski

∗ Freund–Rubin

12

• (M, g, F ) is ν-BPS ⇐⇒


e.g.,


∗ Minkowski

∗ Freund–Rubin

∗ Kowalski-Glikman wave

12

• (M, g, F ) is ν-BPS ⇐⇒


e.g.,


∗ Minkowski

∗ Freund–Rubin



12

• (M, g, F ) is ν-BPS ⇐⇒


e.g.,


∗ Minkowski

∗ Freund–Rubin



∗ M2 and M5 branes

12

• (M, g, F ) is ν-BPS ⇐⇒


e.g.,


∗ Minkowski

∗ Freund–Rubin




∗ Kaluza–Klein monopole

12

• (M, g, F ) is ν-BPS ⇐⇒


e.g.,


∗ Minkowski

∗ Freund–Rubin




∗ 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


A v

0 1

)A ∈ O(1, 10)

13

Minkowski vacuum


A v

0 1

)A ∈ O(1, 10) v ∈ R

1,10

13

Minkowski vacuum


A v

0 1

)A ∈ O(1, 10) v ∈ R

1,10

• Γ ⊂ O(1, 10) n R1,10

13

Minkowski vacuum


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


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


• X ∈ so(p, q)

14


• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear

14


• X ∈ so(p, q) ⇐⇒ X : Rp+q → Rp+q linear, skew-symmetric

relative to 〈−,−〉 of signature (p, q)

14




• X =∑

i Xi

14




• X =∑

i Xi relative to an orthogonal decomposition

14




• X =∑


Rp+q =

⊕i

Vi

14




• X =∑


Rp+q =

⊕i

Vi with Vi indecomposable

14




• X =∑


Rp+q =

⊕i


• for each indecomposable block

14




• X =∑


Rp+q =

⊕i


• for each indecomposable block, if λ is an eigenvalue

14




• X =∑


Rp+q =

⊕i


• for each indecomposable block, if λ is an eigenvalue, then so are

−λ

14




• X =∑


Rp+q =

⊕i



−λ, λ∗

14




• X =∑


Rp+q =

⊕i



−λ, λ∗, and −λ∗

15

• possible minimal polynomials

15

• possible minimal polynomials:

? λ = 0

15


? λ = 0 µ(x) = xn

15


? λ = 0 µ(x) = xn

? λ = β ∈ R

15


? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

15


? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = iϕ ∈ iR

15


? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = iϕ ∈ iR,

µ(x) = (x2 + ϕ2)n

15


? 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


? 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


? 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


• determine metric making X skew-symmetric

16

Strategy


• determine metric making X skew-symmetric, using automorphism

of Jordan form if necessary to bring the metric to standard form

16

Strategy




• keep only those blocks with appropriate signature

16

Strategy




• keep only those blocks with appropriate signature

• example: µ(x) = x3

17

Elementary lorentzian blocks

17


Signature Minimal polynomial Type

17



(0, 1)

17



(0, 1) x

17



(0, 1) x trivial

17



(0, 1) x trivial

(0, 2)

17



(0, 1) x trivial

(0, 2) x2 + ϕ2

17



(0, 1) x trivial

(0, 2) x2 + ϕ2 rotation

17



(0, 1) x trivial


(1, 0)

17



(0, 1) x trivial


(1, 0) x

17



(0, 1) x trivial


(1, 0) x trivial

17



(0, 1) x trivial


(1, 0) x trivial

(1, 1)

17



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2

17



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

17



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2)

17



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2) x3

17



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2) x3 null rotation

18

Lorentzian normal forms

18


Play with the elementary blocks!

18



In signature (1, 10)

18



In signature (1, 10):

• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4) + R9\(ϕ5)

18




• R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4) + R9\(ϕ5)

• B02(β) + R34(ϕ1) + R56(ϕ2) + R78(ϕ3) + R9\(ϕ4)

18




• 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




• 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




• 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


• λ + τ ∈ so(1, 10)⊕ R1,10

19


• λ + τ ∈ so(1, 10)⊕ R1,10

• conjugate by O(1, 10) to bring λ to normal form

19


• λ + τ ∈ so(1, 10)⊕ R1,10


• conjugate by R1,10

19


• λ + τ ∈ so(1, 10)⊕ R1,10


• conjugate by R1,10:

λ + τ 7→ λ + τ − [λ, τ ′]

19


• λ + τ ∈ so(1, 10)⊕ R1,10


• 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


either

? ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4); or

? ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

20


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


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


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


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


ds2 = 2|dy|2 + dz2

21

Adapted coordinates


ds2 = 2|dy|2 + dz2

• “undress” the Killing vector

21

Adapted coordinates


ds2 = 2|dy|2 + dz2


ξ = ∂z + λ

21

Adapted coordinates


ds2 = 2|dy|2 + dz2


ξ = ∂z + λ = U ∂z U−1

21

Adapted coordinates


ds2 = 2|dy|2 + dz2


ξ = ∂z + λ = U ∂z U−1 with U = exp(−zλ)

22

• introduce new coordinates

22


x = U y

22


x = U y = exp(−zB)y

22


x = U y = exp(−zB)y where λy = By

22



whence ξx = 0

22



whence ξx = 0

• rewrite the metric in terms of x

22



whence ξx = 0

• rewrite the metric in terms of x:

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

22



whence ξx = 0


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

where

? Λ = 1 + |Bx|2

22



whence ξx = 0


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


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

23


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

we read off the IIA fields

23


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h


? dilaton

23


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h


? dilaton: Φ = 34 log(1 + |Bx|2)

23


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h



? RR 1-form potential

23


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h



? RR 1-form potential:

A =Bx · dx

1 + |Bx|2

23


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h




A =Bx · dx

1 + |Bx|2

? string frame metric

23


ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h




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


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


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


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


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


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


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

25

e.g.,



? u = 0, ϕ1 = ϕ2

25

e.g.,



? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0

25

e.g.,



? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0 =⇒ half-BPS flux-fivebrane

[Gutperle–Strominger, hep-th/0104136]

25

e.g.,





? u = 0

25

e.g.,





? u = 0, ϕ1 = ϕ2 + ϕ3

25

e.g.,





? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0

25

e.g.,





? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0 =⇒ 14-BPS flux-threebrane

25

e.g.,






? u = 0

25

e.g.,






? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4

25

e.g.,






? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4 =⇒ 18-BPS flux-string

[Uranga, hep-th/0108196]

25

e.g.,








? u = 0

25

e.g.,








? u = 0, ϕ1 = ϕ2

25

e.g.,








? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4

25

e.g.,








? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ 14-BPS flux-string

25

e.g.,









? u = 1

25

e.g.,









? u = 1, ϕi = 0

25

e.g.,









? u = 1, ϕi = 0 =⇒ half-BPS nullbrane

26

Discrete quotients

26

Discrete quotients

• start with the metric in adapted coordinates

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

and identify z ∼ z + L

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

and identify z ∼ z + L; e.g., u = 1, ϕi = 0 in B

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2


=⇒ half-BPS eleven-dimensional nullbrane

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2


=⇒ half-BPS eleven-dimensional nullbrane:

? time-dependent

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2



? time-dependent

? smooth

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2



? time-dependent

? smooth

? stable

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2



? time-dependent

? smooth

? stable

? smooth transition between Big Crunch and Big Bang

26

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2



? 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


• Γ a one-parameter subgroup of symmetries

28

Supersymmetry


• Γ a one-parameter subgroup of symmetries, with Killing vector ξ

28

Supersymmetry



How much supersymmetry will the quotient M/Γ preserve?

28

Supersymmetry




In supergravity

28

Supersymmetry




In supergravity: Γ-invariant Killing spinors

28

Supersymmetry




In supergravity: Γ-invariant Killing spinors:

Lξε

28

Supersymmetry





Lξε = ∇ξε + 18∇aξbΓabε

28

Supersymmetry





Lξε = ∇ξε + 18∇aξbΓabε = 0

28

Supersymmetry






In string/M-theory

28

Supersymmetry






In string/M-theory this cannot be the end of the story.

29

“Supersymmetry without supersymmetry”

29


• T-duality relates backgrounds with different amount of

“supergravitational supersymmetry”

29




• dramatic example

29




• dramatic example:

AdS5×S5

29





AdS5×S5

((QQQQQQQQQQQQQ

oo // AdS5×CP2 × S1

29





AdS5×S5

((QQQQQQQQQQQQQ


uukkkkkkkkkkkkkkk

AdS5×CP2

29





AdS5×S5

((QQQQQQQQQQQQQ


uukkkkkkkkkkkkkkk

AdS5×CP2

CP2 is not even spin! [Duff–Lu–Pope, hep-th/9704186,9803061]

30

Spin structures in quotients

30


• (M, g) spin

30


• (M, g) spin, Γ a one-parameter subgroup of isometries

30



Is M/Γ spin?

30



Is M/Γ spin?

• if Γ ∼= R

30



Is M/Γ spin?

• if Γ ∼= R, then M/Γ is always spin

30



Is M/Γ spin?


• so let Γ ∼= S1

30



Is M/Γ spin?


• so let Γ ∼= S1 with (normalised) Killing vector ξ = ξX

30



Is M/Γ spin?



• Lξ has integral weights on tensors

30



Is M/Γ spin?



• Lξ has integral weights on tensors , e.g.,

LξT

30



Is M/Γ spin?




LξT = inT

30



Is M/Γ spin?




LξT = inT for n ∈ Z

30



Is M/Γ spin?




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


? Lξ integrates to an action of Γ: weights are again integral

31


? Lξ integrates to an action of Γ: weights are again integral; or

? Lξ integrates to an action of a double cover Γ

31



? Lξ integrates to an action of a double cover Γ: weights are

now half-integral

31




now half-integral:

Lξε = i(n + 12)ε

31




now half-integral:

Lξε = i(n + 12)ε

• if Γ acts on spinors

31




now half-integral:

Lξε = i(n + 12)ε

• if Γ acts on spinors, M/Γ is spin

31




now half-integral:

Lξε = i(n + 12)ε

• if Γ acts on spinors, M/Γ is spin;

• if only Γ does

31




now half-integral:

Lξε = i(n + 12)ε


• if only Γ does, M/Γ is not spin

31




now half-integral:

Lξε = i(n + 12)ε


• if only Γ does, M/Γ is not spin, but spinc

31




now half-integral:

Lξε = i(n + 12)ε


• 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


3 as the quadric

|z1|2 + |z2|2 + |z3|2 = 1 ,

and S1-action

32


3 as the quadric

|z1|2 + |z2|2 + |z3|2 = 1 ,

and S1-action

(z1, z2, z3)

32


3 as the quadric

|z1|2 + |z2|2 + |z3|2 = 1 ,

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

itz3)

32


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


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


3 as the quadric

|z1|2 + |z2|2 + |z3|2 = 1 ,

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

itz3)


• Lξ acts as

Lξε

32


3 as the quadric

|z1|2 + |z2|2 + |z3|2 = 1 ,

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

itz3)


• 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


2

(with multiplicity 1) and ±12 (with multiplicity 3)

33


2


• weights are half-integral

33


2


• weights are half-integral =⇒ CP2 has no spin structure

33


2



• more generally

33


2



• more generally, M → M/Γ is a principal circle bundle

33


2




• let L = M ×Γ C be the complex line bundle on M/Γ

33


2




• 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


Lξε = inε

defines a section of S ⊗ Ln over M/Γ

34


Lξε = inε

defines a section of S ⊗ Ln over M/Γ, where S → M/Γ is the

bundle of spinors

34


Lξε = inε


bundle of spinors

• a spinor ε on M

34


Lξε = inε


bundle of spinors


Lξε = i(n + 1/2)ε

34


Lξε = inε


bundle of spinors


Lξε = i(n + 1/2)ε

would define a section of

S ⊗ Ln+1/2

34


Lξε = inε


bundle of spinors


Lξε = i(n + 1/2)ε


S ⊗ Ln+1/2 ∼= (S ⊗ L1/2)

34


Lξε = inε


bundle of spinors


Lξε = i(n + 1/2)ε


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


does: it is the bundle of spinc spinors

35



• L carries a natural connection

35



• L carries a natural connection: the RR 1-form

35




• sections through Ln carry n units of RR charge

35





• spinc spinors carry fractional RR charge

35






• RR charge

35






• RR charge ↔ momentum along the compact direction given by ξ

35






• 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


AdS5×CP2 × S1, which the supergravity does not see

36


AdS5×CP2 × S1, which the supergravity does not see, or does

it?

[e.g., Hull hep-th/0305039]

37

Supersymmetry of supergravity quotients

37


• (M, g, F, ...) supersymmetric

37



• Γ one-parameter group of symmetries

37



• Γ one-parameter group of symmetries, generated by ξ

• Killing spinors of M/Γ

37




• Killing spinors of M/Γ ⇐⇒ Γ-invariant Killing spinors of M

37





• it suffices to determine zero weights of Lξ on Killing spinors

37






• e.g., (R1,10)

37






• e.g., (R1,10): Killing spinors are parallel

37






• e.g., (R1,10): Killing spinors are parallel, whence

Lξε = 18∇aξbΓabε

38

• e.g., fluxbranes

38


ξ = ∂z + R12(ϕ1) + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

38


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



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0

39



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

39



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4

39



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

39



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4

39



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14

39



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



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



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



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



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



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


ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

40


ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)

40


ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)

• N+2 is nilpotent

40


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


ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)


and commutes with it; whence invariant spinors are annihilated

by both

40


ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)



by both

• ker N+2 = ker Γ+

40


ξ = ∂z + N+2 + R34(ϕ2) + R56(ϕ3) + R78(ϕ4)

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)



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



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0

41



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

41



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0

41



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14

41



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14

? ϕ2 = ϕ3 = ϕ4 = 0

41



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


g = e2A(r)ds2(R1,p) + e2B(r)ds2(RD−1−p)

42

Brane quotients



Fp+2 = dvol(R1,p) ∧ dC(r)

42

Brane quotients




with r the transverse radius

42

Brane quotients





• A(r), B(r), C(r) → 0 as r → ∞

42

Brane quotients





• 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



? ε a Killing spinor

45



? ε a Killing spinor =⇒ so is Lξε

45



? ε a Killing spinor =⇒ so is Lξε, whence

Lξε

45




Lξε = eD(r)ε′∞

45





but also

Lξε = LξeD(r)ε∞

45





but also

Lξε = LξeD(r)ε∞ = eD(r)Lξε∞

45





but also


whence

Lξε∞ = ε′∞

45





but also


whence

Lξε∞ = ε′∞

(independent of r)

45





but also


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



? G is not the full asymptotic symmetry group, and

? brane metric is not flat

46




? brane metric is not flat, so causal properties of Killing vectors

may differ from R1,D−1

46






• e.g., ξ may be everywhere spacelike in the brane metric

46






• 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


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


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


V (r0)r80 = −2|Q|

m2|τ |2

? if |τ |2 ≥ 0

49


V (r0)r80 = −2|Q|

m2|τ |2

? if |τ |2 ≥ 0, there is no such r0 > 0

49


V (r0)r80 = −2|Q|

m2|τ |2

? if |τ |2 ≥ 0, there is no such r0 > 0? if |τ |2 < 0

49


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


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


V (r0)r80 = −2|Q|

m2|τ |2


‖ξ‖2 ≥ V (r0)−2/3|τ |2 + V (r0)1/3r20m

2

which is > 0

49


V (r0)r80 = −2|Q|

m2|τ |2


‖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



• moreover they contain closed timelike curves

[Maoz–Simon, hep-th/0310255]

50





• so do the associated discrete quotients

50






• similar results hold for delocalised branes

50






• similar results hold for delocalised branes, intersecting branes

50






• similar results hold for delocalised branes, intersecting branes,...

51

End of second lecture

52

Freund–Rubin backgrounds

• purely geometric backgrounds

52


• purely geometric backgrounds, with product geometry

(M4 ×N7, g ⊕ h)

52



(M4 ×N7, g ⊕ h) and F ∝ dvolg

52




• field equations

52




• field equations ⇐⇒ (M, g) and (N,h) are Einstein

52





• supersymmetry

52





• supersymmetry ⇐⇒ (M, g) and (N,h) admit geometric Killing

spinors

52






spinors:

∇aε = λΓaε

52






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 = R+ ×M and g = dr2 + 4λ2r2g

53


M = R+ ×M and g = dr2 + 4λ2r2g ,

admits parallel spinors

53


M = R+ ×M and g = dr2 + 4λ2r2g ,

admits parallel spinors: ∇ε = 0

53


M = R+ ×M and g = dr2 + 4λ2r2g ,

admits parallel spinors: ∇ε = 0[Bar (1993), Kath (1999)]

53


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


• (M1,n−1, g) lorentzian

54


• (M1,n−1, g) lorentzian =⇒ (M, g) has signature (2, n− 1)

54



• for the maximally supersymmetric Freund–Rubin backgrounds

54



• for the maximally supersymmetric Freund–Rubin backgrounds,

AdS4×S7

54




AdS4×S7 and S4 ×AdS7 ,

54





the cones of each factor are flat

54





the cones of each factor are flat:

? cone of Sn is Rn+1

54







? cone of AdS1+p is (a domain in) R2,p

54







? 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


• AdS1+p is simply-connected

55


• AdS1+p is simply-connected; it is the universal cover of a quadric

Q1+p ⊂ R2,p

55



Q1+p ⊂ R2,p, given by

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

55




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

• For p > 2

55




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

• For p > 2, π1Q1+p∼= Z

55




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

• For p > 2, π1Q1+p∼= Z, generated by (topological) CTCs

55




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2


x1(t) + ix2(t) = reit

55




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2


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


SO(2, p) ⊂ GL(p + 2,R)

56


SO(2, p) ⊂ GL(p + 2,R)

• SO(2, p) is not the (orientation-preserving) isometry group of

AdS1+p

56


SO(2, p) ⊂ GL(p + 2,R)


AdS1+p. Why?

56


SO(2, p) ⊂ GL(p + 2,R)


AdS1+p. Why? Because...

56


SO(2, p) ⊂ GL(p + 2,R)



? SO(2, p) has maximal compact subgroup SO(2)× SO(p)

56


SO(2, p) ⊂ GL(p + 2,R)



? SO(2, p) has maximal compact subgroup SO(2)× SO(p)? the orbits of SO(2) are the CTCs above

56


SO(2, p) ⊂ GL(p + 2,R)




? these curves are not closed in AdS1+p

56


SO(2, p) ⊂ GL(p + 2,R)





? in AdS1+p, x1∂2 − x2∂1 does not generate SO(2) but R

56


SO(2, p) ⊂ GL(p + 2,R)






• the (orientation-preserving) isometry group of AdS1+p is an

infinite cover SO(2, p)

56


SO(2, p) ⊂ GL(p + 2,R)







infinite cover SO(2, p), a central extension of SO(2, p)

56


SO(2, p) ⊂ GL(p + 2,R)







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 bad news

57


• The bad news: SO(2, p) is not a matrix group

57


• The bad news: SO(2, p) is not a matrix group; it has no finite-

dimensional matrix representations

57




• The good news

57




• The good news:

? the Lie algebra of SO(2, p) is still so(2, p)

57




• 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 good news:



whence

• one-parameter subgroups

57




• The good news:



whence

• one-parameter subgroups↔ projectivised adjoint orbits of so(2, p)under SO(2, p)

58

Normal forms for so(2, p)

58


We play again

58


We play again but with a bigger set!

58



We can still use the lorentzian elementary blocks

58



We can still use the lorentzian elementary blocks:

• (0, 2)

58




• (0, 2) and also (2, 0)

58




• (0, 2) and also (2, 0), µ(x) = x2 + ϕ2

58




• (0, 2) and also (2, 0), µ(x) = x2 + ϕ2, rotation

58





B(0,2)(ϕ)

58





B(0,2)(ϕ) = B(2,0)(ϕ)

58





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

]


B(1,2)

59

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

β 0

]


B(1,2) = B(2,1)

59

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

β 0

]


B(1,2) = B(2,1) =

0 −1 01 0 −10 1 0

60

But there are also new ones:

• (2, 2)

60


• (2, 2), µ(x) = x2

60


• (2, 2), µ(x) = x2, “rotation” in a totally null plane

60



B(2,2)±

60



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


boost

B(2,2)± (β > 0)

61


boost

B(2,2)± (β > 0) =

0 ∓1 1 −β

±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0

61


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


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


boost

B(2,2)± (β > 0) =

0 ∓1 1 −β

±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0


BTZ black hole; the non-extremal black hole

61


boost

B(2,2)± (β > 0) =

0 ∓1 1 −β

±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0


BTZ black hole; the non-extremal black hole is obtained from

B(1,1)(β1)⊕B(1,1)(β2)

61


boost

B(2,2)± (β > 0) =

0 ∓1 1 −β

±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0


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


dual rotation

B(2,2)± (ϕ)

62


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


dual rotation

B(2,2)± (β > 0, ϕ > 0)

63


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



B(2,3)

64



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


rotation

B(2,4)± (ϕ)

65


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


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


ξ = ξA + ξS

66

Causal properties


ξ = ξA + ξS

whose norms add

‖ξ‖2 = ‖ξA‖2 + ‖ξS‖2

67

• Sq is compact

67

• Sq is compact =⇒

R2M2 ≥ ‖ξS‖2

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2

and if q is odd

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2

and if q is odd, m2 can be > 0

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2


• ξ can be everywhere spacelike on AdS1+p×S2k+1

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2


• ξ can be everywhere spacelike on AdS1+p×S2k+1, even if ξA is

not spacelike everywhere

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2



not spacelike everywhere, provided that ‖ξA‖2 is bounded below

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2




and ξS has no zeroes

67


R2M2 ≥ ‖ξS‖2 ≥ R2m2




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


? ⊕iB(0,2)(ϕi)

? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)

68


? ⊕iB(0,2)(ϕi)

? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi), if |β1| = |β2|

68


? ⊕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


? ⊕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


? ⊕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


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)

• norm bounded below

68


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)

• norm bounded below:

? B(2,0)(ϕ)⊕i B(0,2)(ϕi)

68


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)


? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even

68


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)


? B(2,0)(ϕ)⊕i B(0,2)(ϕi), if p is even and |ϕi| ≥ ϕ > 0 for all i

68


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)



? B(2,2)± (ϕ)⊕i B(0,2)(ϕi)

68


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)



? B(2,2)± (ϕ)⊕i B(0,2)(ϕi), if |ϕi| ≥ |ϕ| ≥ 0 for all i

68


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)




• arbitrarily negative norm

68


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)




• 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(2,1) ⊕i B(0,2)(ϕi)

69


? B(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)

69


? 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(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(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)


? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)

69


? B(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)


? B(2,2)± (β, ϕ)⊕i B(0,2)(ϕi)

? B(2,3) ⊕i B(0,2)(ϕi)

69


? B(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)


? 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(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)


? 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(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)


? 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)


holes: quotient only a part of AdS

69


? B(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)


? 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)


holes: quotient only a part of AdS and check that the boundary

thus introduced lies behind a horizon.

70

Discrete quotients with CTCs

70


• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1

70


• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0

70


• ξ = ξA + ξS a Killing vector in AdS1+p×S2k+1, with ‖ξ‖2 > 0but ‖ξA‖ not everywhere spacelike

70



• the corresponding one-parameter subgroup Γ

70



• the corresponding one-parameter subgroup Γ ∼= R

70




• pick L > 0 and consider the cyclic subgroup ΓL

70




• pick L > 0 and consider the cyclic subgroup ΓL∼= Z

70




• 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


• idea of the proof

71


• idea of the proof: find a timelike curve which connects a point x

to its image γNx

71



to its image γNx for N � 1

71




• e.g., a Z-quotient of a lorentzian cylinder

71





• general case

71





• general case:

? let x = (xA, xS) be a point

71





• general case:

? let x = (xA, xS) be a point and γN · x

71





• general case:

? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS)

71





• general case:

? let x = (xA, xS) be a point and γN · x = (γN · xA, γN · xS) its

image under γN

71





• general case:


image under γN

? we will construct a timelike curve c(t)

71





• general case:


image under γN

? we will construct a timelike curve c(t) between c(0) = x

71





• general case:


image under γN

? we will construct a timelike curve c(t) between c(0) = x and

c(NL) = γN · x

71





• general case:


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


cS onto S2k+1

? cA is the integral curve of ξA

72


cS onto S2k+1


? cS is a length-minimising geodesic between xS and γN · xS

72


cS onto S2k+1


? cS is a length-minimising geodesic between xS and γN · xS,

whose arclength

∫ NL

0

‖cS(t)‖dt

72


cS onto S2k+1



whose arclength

∫ NL

0

‖cS(t)‖dt = NL‖cS‖

72


cS onto S2k+1



whose arclength

∫ NL

0

‖cS(t)‖dt = NL‖cS‖ ≤ D

72


cS onto S2k+1



whose arclength

∫ NL

0


? therefore ‖cS‖ ≤ D/NL

72


cS onto S2k+1



whose arclength

∫ NL

0


? therefore ‖cS‖ ≤ D/NL, and

‖c‖2

72


cS onto S2k+1



whose arclength

∫ NL

0



‖c‖2 = ‖cA‖2 + ‖cS‖2

72


cS onto S2k+1



whose arclength

∫ NL

0



‖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


• the same argument applies to any Freund–Rubin background

M ×N

73



M ×N , where M is lorentzian admitting such isometries

73



M ×N , where M is lorentzian admitting such isometries and N

is complete

73




is complete:

? N is Einstein with positive cosmological constant

73




is complete:

? N is Einstein with positive cosmological constant and Einstein

73




is complete:


? Bonnet-Myers Theorem =⇒ N is compact

73




is complete:


? Bonnet-Myers Theorem =⇒ N is compact =⇒ has bounded

diameter

73




is complete:



diameter

• geometrical CTCs are also natural in certain kinds of

supersymmetric Freund–Rubin backgrounds M × N

73




is complete:



diameter


supersymmetric Freund–Rubin backgrounds M × N , where M

is lorentzian Einstein–Sasaki

73




is complete:



diameter


supersymmetric Freund–Rubin backgrounds M × N , where M

is lorentzian Einstein–Sasaki: timelike circle bundles over Kahler

manifolds

74

Some interesting reductions

74


• there are many families of smooth supersymmetric reductions of

AdS4×S7

74



AdS4×S7, S4 ×AdS7

74



AdS4×S7, S4 ×AdS7, AdS5×S5

74



AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3

74



AdS4×S7, S4 ×AdS7, AdS5×S5, and AdS3×S3 × R4.

74




• 34-BPS AdS4×CP3 background of IIA

[Duff–Lu–Pope, hep-th/9704186]

74






• 916-BPS IIA backgrounds

74






• 916-BPS IIA backgrounds: reductions of AdS4×S7

74






• 916-BPS IIA backgrounds: reductions of AdS4×S7 by

B(2,2)+

74






• 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


null rotation

B(1,2) ⊕B(1,2)

75


null rotation

B(1,2) ⊕B(1,2)

• a family of half-BPS IIA backgrounds

75


null rotation

B(1,2) ⊕B(1,2)

• a family of half-BPS IIA backgrounds: reductions of S4 × AdS7

75


null rotation

B(1,2) ⊕B(1,2)


by

B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)

75


null rotation

B(1,2) ⊕B(1,2)


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


null rotation

B(1,2) ⊕B(1,2)


by

B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)


• a number of maximally supersymmetric reductions of AdS3×S3

75


null rotation

B(1,2) ⊕B(1,2)


by

B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)


• a number of maximally supersymmetric reductions of AdS3×S3:

near-horizon geometries of the supersymmetric rotating black

holes

75


null rotation

B(1,2) ⊕B(1,2)


by

B(1,1)(β)⊕B(1,1)(β)⊕B(0,2)(ϕ)⊕B(0,2)(−ϕ)


• 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



group, and the Killing vectors are left-invariant

76



group, and the Killing vectors are left-invariant

[Chamseddine–FO–Sabra, hep-th/0306278]

77

Thank you.

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