· 1 based on work in collaboration with joan sim´on (pennsylvania), and owen madden and simon...

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


and Owen Madden and Simon Ross (Durham):

• 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 (to appear in Adv. Theor. Math. Phys.)

1



• JHEP 12 (2001) 011, hep-th/0110170



• 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


• 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

General problem

3

General problem

• (M, g, F, ...) a supergravity background

3

General problem


• symmetry group G

3

General problem


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

3

General problem



• determine all quotient supergravity backgrounds M/Γ

3

General problem



• determine all quotient supergravity backgrounds M/Γ, where

Γ ⊂ G is a one-parameter subgroup

3

General problem




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

3

General problem




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

? smoothness

3

General problem





? smoothness,

? causal regularity

3

General problem





? smoothness,

? causal regularity,

? spin structure

3

General problem





? smoothness,


? spin structure,

? supersymmetry

3

General problem





? smoothness,


? spin structure,

? supersymmetry,...

4

One-parameter subgroups

4


• (M, g, F, ...)

4


• (M, g, F, ...)

• symmetries

4


• (M, g, F, ...)

• symmetries

f : M∼=−→ M

4


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g

4


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F

4


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .

define a Lie group G

4


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .

define a Lie group G, with Lie algebra g

4


• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .


• X ∈ g defines a one-parameter subgroup

4


• (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}

5

• X ∈ g also defines a Killing vector ξX

5

• X ∈ g also defines a Killing vector ξX:

LξXg = 0

5


LξXg = 0 LξX

F = 0 . . .

5


LξXg = 0 LξX

F = 0 . . .

whose integral curves are the orbits of Γ

5


LξXg = 0 LξX

F = 0 . . .


• two possible topologies

5


LξXg = 0 LξX

F = 0 . . .


• two possible topologies:

? Γ ∼= S1

5


LξXg = 0 LξX

F = 0 . . .



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

5


LξXg = 0 LξX

F = 0 . . .



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

5


LξXg = 0 LξX

F = 0 . . .



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

5


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

6

Kaluza–Klein and discrete quotients

6


• Γ ∼= S1

6


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

6



• Γ ∼= R

6



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

6




? discrete quotient M/ΓL, where

6




? discrete quotient M/ΓL, where L > 0

6




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

ΓL = {exp(nLX) | n ∈ Z}

6





ΓL = {exp(nLX) | n ∈ Z} ∼= Z

6





ΓL = {exp(nLX) | n ∈ Z} ∼= Z

? Kaluza–Klein reduction by Γ/ΓL

6





ΓL = {exp(nLX) | n ∈ Z} ∼= Z

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

6





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


locally isometric to M , but often with very different global

properties

7



properties, e.g.,

? M static, but M/ΓL time-dependent

7



properties, e.g.,


? M causally regular, but M/ΓL causally singular

7



properties, e.g.,



? M spin, but M/ΓL not spin

7



properties, e.g.,



? M spin, but M/ΓL not spin

? M supersymmetric, but M/ΓL breaking all supersymmetry

8

Classifying quotients

8


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

8


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

8



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

Γ = {exp(tX) | t ∈ R}

8




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

8





• if X ′ = λX, λ 6= 0, then Γ′ = Γ

8





• if X ′ = λX, λ 6= 0, then Γ′ = Γ

• if X ′ = gXg−1, then Γ′ = gΓg−1

8





• 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


A v

0 1

)A ∈ O(1, 9)

10

Flat quotients


A v

0 1

)A ∈ O(1, 9) v ∈ R

1,9

10

Flat quotients


A v

0 1

)A ∈ O(1, 9) v ∈ R

1,9

• Γ ⊂ O(1, 9) n R1,9

10

Flat quotients


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


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


• X ∈ so(p, q)

11


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

11


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

relative to 〈−,−〉 of signature (p, q)

11




• X =∑

i Xi

11




• X =∑

i Xi relative to an orthogonal decomposition

11




• X =∑


Rp+q =

⊕i

Vi

11




• X =∑


Rp+q =

⊕i

Vi with Vi indecomposable

11




• X =∑


Rp+q =

⊕i


• for each indecomposable block

11




• X =∑


Rp+q =

⊕i


• for each indecomposable block, if λ is an eigenvalue

11




• X =∑


Rp+q =

⊕i


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

−λ

11




• X =∑


Rp+q =

⊕i



−λ, λ∗

11




• X =∑


Rp+q =

⊕i



−λ, λ∗, and −λ∗

12

• possible minimal polynomials

12

• possible minimal polynomials:

? λ = 0

12


? λ = 0 µ(x) = xn

12


? λ = 0 µ(x) = xn

? λ = β ∈ R

12


? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

12


? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = iϕ ∈ iR

12


? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = iϕ ∈ iR,

µ(x) = (x2 + ϕ2)n

12


? 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


? 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


? 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


• determine metric making X skew-symmetric

13

Strategy


• determine metric making X skew-symmetric, using automorphism

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

13

Strategy


• 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


Signature Minimal polynomial Type

14



(0, 1)

14



(0, 1) x

14



(0, 1) x trivial

14



(0, 1) x trivial

(0, 2)

14



(0, 1) x trivial

(0, 2) x2 + ϕ2

14



(0, 1) x trivial

(0, 2) x2 + ϕ2 rotation

14



(0, 1) x trivial


(1, 0)

14



(0, 1) x trivial


(1, 0) x

14



(0, 1) x trivial


(1, 0) x trivial

14



(0, 1) x trivial


(1, 0) x trivial

(1, 1)

14



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2

14



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

14



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2)

14



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2) x3

14



(0, 1) x trivial


(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2) x3 null rotation

15

Lorentzian normal forms

15


Play with the elementary blocks!

15



In signature (1, 9)

15



In signature (1, 9):

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

15




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

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

15




• 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




• 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




• 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


• λ + τ ∈ so(1, 9)⊕ R1,9

16


• λ + τ ∈ so(1, 9)⊕ R1,9

• conjugate by O(1, 9) to bring λ to normal form

16


• λ + τ ∈ so(1, 9)⊕ R1,9


• conjugate by R1,9

16


• λ + τ ∈ so(1, 9)⊕ R1,9


• conjugate by R1,9:

λ + τ 7→ λ + τ − [λ, τ ′]

16


• λ + τ ∈ so(1, 9)⊕ R1,9


• 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


either

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

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

17


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


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


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


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


ds2 = 2|dy|2 + dz2

18

Adapted coordinates


ds2 = 2|dy|2 + dz2

• “undress” the Killing vector

18

Adapted coordinates


ds2 = 2|dy|2 + dz2


ξ = ∂z + λ

18

Adapted coordinates


ds2 = 2|dy|2 + dz2


ξ = ∂z + λ = U ∂z U−1

18

Adapted coordinates


ds2 = 2|dy|2 + dz2


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

19

• introduce new coordinates

19


x = U y

19


x = U y = exp(−zB)y

19


x = U y = exp(−zB)y where λy = By

19



whence ξx = 0

19



whence ξx = 0

• rewrite the metric in terms of x

19



whence ξx = 0

• rewrite the metric in terms of x:

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

19



whence ξx = 0


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

where

? Λ = 1 + |Bx|2

19



whence ξx = 0


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


B =

−u

0 −ϕ1 − u

−u ϕ1 + u 0

0 −ϕ2

ϕ2 0

0 −ϕ3

ϕ3 0

0 −ϕ4

ϕ4 0

20


B =

−u

0 −ϕ1 − u

−u ϕ1 + u 0

0 −ϕ2

ϕ2 0

0 −ϕ3

ϕ3 0

0 −ϕ4

ϕ4 0

where

20


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


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


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


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


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

21

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

and identify z ∼ z + L

21

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

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

21

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2


Λ = 1 + x2+

21

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2


Λ = 1 + x2+ and A =

11 + x2

+

(x−dx1 − x1dx−)

21

Discrete quotients


ds2 = Λ(dz + A)2 + |dx|2 − ΛA2


Λ = 1 + x2+ and A =

11 + x2

+

(x−dx1 − x1dx−)

=⇒ half-BPS ten-dimensional nullbrane

22

• the nullbrane is

22


? time-dependent

22


? time-dependent

? smooth

22


? time-dependent

? smooth

? stable

22


? time-dependent

? smooth

? stable

? a smooth transition between Big Crunch and Big Bang

22


? time-dependent

? smooth

? stable


? a resolution of parabolic orbifold [Horowitz–Steif (1991)]

22


? time-dependent

? smooth

? stable



• its conformal field theory is a Z-orbifold of flat space

22


? time-dependent

? smooth

? stable



• its conformal field theory is a Z-orbifold of flat space, and has

been studied [Liu–Moore–Seiberg, hep-th/0206182]

22


? time-dependent

? smooth

? stable



• 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


• Γ a one-parameter subgroup of symmetries

23

Supersymmetry


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

23

Supersymmetry



How much supersymmetry will the quotient M/Γ preserve?

23

Supersymmetry




In supergravity

23

Supersymmetry




In supergravity: Γ-invariant Killing spinors

23

Supersymmetry




In supergravity: Γ-invariant Killing spinors:

Lξε

23

Supersymmetry





Lξε = ∇ξε + 18∇aξbΓabε

23

Supersymmetry





Lξε = ∇ξε + 18∇aξbΓabε = 0

23

Supersymmetry






In string/M-theory

23

Supersymmetry






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

24

“Supersymmetry without supersymmetry”

24


• T-duality relates backgrounds with different amount of

“supergravitational supersymmetry”

24




• dramatic example

24




• dramatic example: [Duff–Lu–Pope, hep-th/9704186,9803061]

24





AdS5×S5

24





AdS5×S5

((QQQQQQQQQQQQQ

oo // AdS5×CP2 × S1

24





AdS5×S5

((QQQQQQQQQQQQQ


uukkkkkkkkkkkkkkk

AdS5×CP2

24





AdS5×S5

((QQQQQQQQQQQQQ


uukkkkkkkkkkkkkkk

AdS5×CP2

CP2 is not even spin!

25

Spin structures in quotients

25


• (M, g) spin

25


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

25



Is M/Γ spin?

25



Is M/Γ spin?

• if Γ ∼= R

25



Is M/Γ spin?

• if Γ ∼= R, then M/Γ is always spin

25



Is M/Γ spin?


• if Γ ∼= S1

25



Is M/Γ spin?


• if Γ ∼= S1 then M/Γ is spin if and only if the action of Γ lifts to

the spin bundle

25



Is M/Γ 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


• (M, g, F, ...) supersymmetric

26



• Γ one-parameter group of symmetries

26



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

• Killing spinors of M/Γ

26




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

26





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

26






• e.g., (R1,9)

26






• e.g., (R1,9): Killing spinors are parallel

26






• e.g., (R1,9): Killing spinors are parallel, whence

Lξε = 18∇aξbΓabε

27

• e.g., fluxbranes

27


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

27


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



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0

28



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

28



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4

28



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

28



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4

28



eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14

28



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



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



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



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



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



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


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

29


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

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)

29


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

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)

• N+2 is nilpotent

29


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


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

29


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+

29


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

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



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0

30



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

30



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0

30



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14

30



eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14

? ϕ2 = ϕ3 = ϕ4 = 0

30



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


• purely geometric backgrounds, with product geometry

(M ×N, g ⊕ h)

32



(M ×N, g ⊕ h) and F ∝ dvolg

32




• field equations

32




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

32





• supersymmetry

32





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

spinors

32






spinors:

∇Xε = λX · ε

32






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 = R+ ×M and g = dr2 + 4λ2r2g

33


M = R+ ×M and g = dr2 + 4λ2r2g ,

admits parallel spinors

33


M = R+ ×M and g = dr2 + 4λ2r2g ,

admits parallel spinors: ∇ε = 0

33


M = R+ ×M and g = dr2 + 4λ2r2g ,

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

33


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


• (M1,n−1, g) lorentzian

34


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

34



• for the maximally supersymmetric Freund–Rubin backgrounds

34



• for the maximally supersymmetric Freund–Rubin backgrounds,

AdS1+p×Sq

34




AdS1+p×Sq

the cones of each factor are flat

34




AdS1+p×Sq

the cones of each factor are flat:

? cone of Sq is Rq+1

34




AdS1+p×Sq



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

34




AdS1+p×Sq



? 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


• AdS1+p is simply-connected

35


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

Q1+p ⊂ R2,p

35



Q1+p ⊂ R2,p, given by

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

35




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

• For p > 2

35




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

• For p > 2, π1Q1+p∼= Z

35




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

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

35




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2


x1(t) + ix2(t) = reit

35




−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2


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


SO(2, p) ⊂ GL(p + 2,R)

36


SO(2, p) ⊂ GL(p + 2,R)

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

AdS1+p

36


SO(2, p) ⊂ GL(p + 2,R)


AdS1+p. Why?

36


SO(2, p) ⊂ GL(p + 2,R)


AdS1+p. Why? Because

36


SO(2, p) ⊂ GL(p + 2,R)



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

36


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

36


SO(2, p) ⊂ GL(p + 2,R)




? these curves are not closed in AdS1+p

36


SO(2, p) ⊂ GL(p + 2,R)





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

36


SO(2, p) ⊂ GL(p + 2,R)






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

infinite cover SO(2, p)

36


SO(2, p) ⊂ GL(p + 2,R)







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

36


SO(2, p) ⊂ GL(p + 2,R)







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 bad news

37


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

37


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

dimensional matrix representations

37




• The good news

37




• The good news:

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

37




• 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 good news:



whence

• one-parameter subgroups

37




• The good news:



whence

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

38

Normal forms for so(2, p)

38


We play again

38


We play again but with a bigger set!

38



We can still use the lorentzian elementary blocks

38



We can still use the lorentzian elementary blocks:

• (0, 2)

38




• (0, 2) and also (2, 0)

38




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

38




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

38





B(0,2)(ϕ)

38





B(0,2)(ϕ) = B(2,0)(ϕ)

38





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

]


B(1,2)

39

• (1, 1), µ(x) = x2 − β2, boost

B(1,1)(β) =[0 −β

β 0

]


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

39

• (1, 1), µ(x) = x2 − β2, boost

B(1,1)(β) =[0 −β

β 0

]


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

0 −1 01 0 −10 1 0

40

But there are also new ones:

• (2, 2)

40


• (2, 2), µ(x) = x2

40


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

40



B(2,2)±

40



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


boost

B(2,2)± (β > 0)

41


boost

B(2,2)± (β > 0) =

0 ∓1 1 −β

±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0

41


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


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


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

41


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)

41


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|

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


dual rotation

B(2,2)± (ϕ)

42


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


dual rotation

B(2,2)± (β > 0, ϕ > 0)

43


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



B(2,3)

44



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


rotation

B(2,4)± (ϕ)

45


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


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


ξ = ξA + ξS

46

Causal properties


ξ = ξA + ξS

whose norms add

‖ξ‖2 = ‖ξA‖2 + ‖ξS‖2

47

• Sq is compact

47

• Sq is compact =⇒

R2M2 ≥ ‖ξS‖2

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2

and if q is odd

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2

and if q is odd, m2 can be > 0

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2


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

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2


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

not spacelike everywhere

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2



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

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2




and ξS has no zeroes

47


R2M2 ≥ ‖ξS‖2 ≥ R2m2




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


? ⊕iB(0,2)(ϕi)

? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)

48


? ⊕iB(0,2)(ϕi)

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

48


? ⊕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


? ⊕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


? ⊕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


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)

• norm bounded below

48


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)

• norm bounded below:

? B(2,0)(ϕ)⊕i B(0,2)(ϕi)

48


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)


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

48


? ⊕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

48


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)



? B(2,2)± (ϕ)⊕i B(0,2)(ϕi)

48


? ⊕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

48


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)




• arbitrarily negative norm

48


? ⊕iB(0,2)(ϕi)


(2,2)± ⊕i B(0,2)(ϕi)




• 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(2,1) ⊕i B(0,2)(ϕi)

49


? B(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)

49


? 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(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(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)

49


? 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)

49


? 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)

49


? 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

49


? 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

49


? 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.

50

Discrete quotients with CTCs

50


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

50


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

50


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

50



• the corresponding one-parameter subgroup Γ

50



• the corresponding one-parameter subgroup Γ ∼= R

50




• pick L > 0 and consider the cyclic subgroup ΓL

50




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

50




• 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


• idea of the proof

51


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

to its image γNx

51



to its image γNx for N � 1

51




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

51





• general case

51





• general case:

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

51





• general case:

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

51





• general case:

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

51





• general case:

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

image under γN

51





• general case:


image under γN

? we will construct a timelike curve c(t)

51





• general case:


image under γN

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

51





• general case:


image under γN

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

c(NL) = γN · x

51





• 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

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


cS onto S2k+1

? cA is the integral curve of ξA

52


cS onto S2k+1


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

52


cS onto S2k+1


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

whose arclength

∫ NL

0

‖cS(t)‖dt

52


cS onto S2k+1



whose arclength

∫ NL

0

‖cS(t)‖dt = NL‖cS‖

52


cS onto S2k+1



whose arclength

∫ NL

0

‖cS(t)‖dt = NL‖cS‖ ≤ D

52


cS onto S2k+1



whose arclength

∫ NL

0


? therefore ‖cS‖ ≤ D/NL

52


cS onto S2k+1



whose arclength

∫ NL

0


? therefore ‖cS‖ ≤ D/NL, and

‖c‖2

52


cS onto S2k+1



whose arclength

∫ NL

0



‖c‖2 = ‖cA‖2 + ‖cS‖2

52


cS onto S2k+1



whose arclength

∫ NL

0



‖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


• the same argument applies to any Freund–Rubin background

M ×N

53



M ×N , where M is lorentzian admitting such isometries

53



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

is complete

53




is complete:

? N is Einstein with positive cosmological constant

53




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



AdS3×S3 ×X4

• equations of motion

54



AdS3×S3 ×X4

• equations of motion =⇒ X Ricci-flat

• supersymmetry

54



AdS3×S3 ×X4


• supersymmetry =⇒ X admits parallel spinors

54



AdS3×S3 ×X4


• 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


∆2,2+ ⊗

[∆4,0

+ ⊗∆0,4+

])⊕

(∆2,2− ⊗

[∆4,0− ⊗∆0,4

+

])as a representation of Spin(2, 2)× Spin(4)× Spin(4)

55


∆2,2+ ⊗

[∆4,0

+ ⊗∆0,4+

])⊕

(∆2,2− ⊗

[∆4,0− ⊗∆0,4

+


• here [R] means the underlying real representation of a complex

representation of real type

55


∆2,2+ ⊗

[∆4,0

+ ⊗∆0,4+

])⊕

(∆2,2− ⊗

[∆4,0− ⊗∆0,4

+


• 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



• ξ = ξA + ξS

56



• ξ = ξA + ξS, with

? ξ spacelike

56




? ξ spacelike

? smooth quotients

56




? ξ spacelike

? smooth quotients

? supersymmetric quotients

56




? ξ spacelike

? smooth quotients


• there are two classes

56




? ξ spacelike

? smooth 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

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


• associated discrete quotients are cyclic orbifolds (ZN or Z)

59


• associated discrete quotients are cyclic orbifolds (ZN or Z) of a

WZW model with group SL(2,R)× SU(2)

59




• most are time-dependent

59




• most are time-dependent, and many have closed timelike curves

60

Thank you.

· 1 based on work in collaboration with joan sim´on (pennsylvania), and owen madden and simon...

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