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

Based on work in collaboration with Joan Simon (Pennsylvania)

Based on work in collaboration with Joan Simon (Pennsylvania):

• JHEP 12 (2001) 011, hep-th/0110170

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

• JHEP 12 (2001) 011, hep-th/0110170

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

• JHEP 12 (2001) 011, hep-th/0110170

• hep-th/0401206

• JHEP 12 (2001) 011, hep-th/0110170

• hep-th/0401206

and on work in progress with Joan Simon

• JHEP 12 (2001) 011, hep-th/0110170

• hep-th/0401206

and on work in progress with Joan Simon, Hannu Rajaniemi

(Edinburgh)

• 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).

Motivation

• fluxbrane backgrounds in type II string theory

Motivation

• string theory in

? time-dependent backgrounds

Motivation

? time-dependent backgrounds, and

? causally singular backgrounds

Motivation

? time-dependent backgrounds, and

? causally singular backgrounds

• supersymmetric Clifford–Klein space form problem

Melvin universe and fluxbranes

• axisymmetric solution of d=4 Einstein–Maxwell theory

[Melvin (1964)]

• describes a gravitationally stable universe of flux

[Melvin (1964)]

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

[Melvin (1964)]

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

[F. Dowker et al. (1994)]

[Melvin (1964)]

• R1,4/Γ, with Γ ∼= R

[Melvin (1964)]

• R1,4/Γ, with Γ ∼= R, or R1,10/Γ

[Melvin (1964)]

• R1,4/Γ, with Γ ∼= R, or R1,10/Γ =⇒ IIA fluxbranes

General problem

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

General problem

• symmetry group G

General problem

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

General problem

• determine all quotient supergravity backgrounds M/Γ

General problem

• determine all quotient supergravity backgrounds M/Γ, where

Γ ⊂ G is a one-parameter subgroup

General problem

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

General problem

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

? smoothness

General problem

? smoothness,

? causal regularity

General problem

? smoothness,

? causal regularity,

? spin structure

General problem

? smoothness,

? spin structure,

? supersymmetry

General problem

? smoothness,

? spin structure,

? supersymmetry,...

One-parameter subgroups

• (M, g, F, ...)

• symmetries

• (M, g, F, ...)

• symmetries

f : M∼=−→ M

• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g

• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F

• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .

define a Lie group G

• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .

define a Lie group G, with Lie algebra g

• (M, g, F, ...)

• symmetries

f : M∼=−→ M f∗g = g f∗F = F . . .

• X ∈ g defines a one-parameter subgroup

• (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}

• X ∈ g also defines a Killing vector ξX

• X ∈ g also defines a Killing vector ξX:

LξXg = 0

LξXg = 0 LξX

F = 0 . . .

LξXg = 0 LξX

F = 0 . . .

whose integral curves are the orbits of Γ

LξXg = 0 LξX

F = 0 . . .

• two possible topologies

LξXg = 0 LξX

F = 0 . . .

• two possible topologies:

? Γ ∼= S1

LξXg = 0 LξX

F = 0 . . .

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

LξXg = 0 LξX

F = 0 . . .

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

LξXg = 0 LξX

F = 0 . . .

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

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

Kaluza–Klein and discrete quotients

• Γ ∼= S1

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

• Γ ∼= R

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

? discrete quotient M/ΓL, where

? discrete quotient M/ΓL, where L > 0

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

ΓL = {exp(nLX) | n ∈ Z}

ΓL = {exp(nLX) | n ∈ Z} ∼= Z

? Kaluza–Klein reduction by Γ/ΓL

ΓL = {exp(nLX) | n ∈ Z} ∼= Z

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

ΓL = {exp(nLX) | n ∈ Z} ∼= Z

? Kaluza–Klein reduction by Γ/ΓL∼= R/Z ∼= S1

• we may stop after the first step

• we may stop after the first step: obtaining backgrounds M/ΓL

locally isometric to M

locally isometric to M , but often with very different global

properties

properties, e.g.,

? M static, but M/ΓL time-dependent

properties, e.g.,

? M causally regular, but M/ΓL causally singular

properties, e.g.,

? M spin, but M/ΓL not spin

properties, e.g.,

? M spin, but M/ΓL not spin

? M supersymmetric, but M/ΓL breaking all supersymmetry

Classifying quotients

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

• (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} Γ′ = {exp(tX ′) | t ∈ R}

• if X ′ = λX, λ 6= 0, then Γ′ = Γ

• if X ′ = gXg−1, then Γ′ = gΓg−1

• if X ′ = λX, λ 6= 0, then Γ′ = Γ

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

• enough to classify normal forms of X ∈ g under

X ∼ λgXg−1 g ∈ G λ ∈ R×

• enough to classify normal forms of X ∈ g under

X ∼ λgXg−1 g ∈ G λ ∈ R×

i.e., projectivised adjoint orbits of g

Eleven-dimensional supergravity backgrounds

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

• F a closed 4-form

• (M, g, F ) is supersymmetric

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

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

parallel with respect to

DX = ∇X + 16ιXF − 1

12X[ ∧ F

• (M, g, F ) is ν-BPS

• (M, g, F ) is ν-BPS ⇐⇒

dim {Killing spinors} = 32ν

• (M, g, F ) is ν-BPS ⇐⇒

? ν = 1 backgrounds

• (M, g, F ) is ν-BPS ⇐⇒

∗ Minkowski

• (M, g, F ) is ν-BPS ⇐⇒

∗ Minkowski

∗ Freund–Rubin

• (M, g, F ) is ν-BPS ⇐⇒

∗ Minkowski

∗ Freund–Rubin

∗ Kowalski-Glikman wave

• (M, g, F ) is ν-BPS ⇐⇒

∗ Minkowski

∗ Freund–Rubin

• (M, g, F ) is ν-BPS ⇐⇒

∗ Minkowski

∗ Freund–Rubin

∗ M2 and M5 branes

• (M, g, F ) is ν-BPS ⇐⇒

∗ Minkowski

∗ Freund–Rubin

∗ Kaluza–Klein monopole

• (M, g, F ) is ν-BPS ⇐⇒

∗ Minkowski

∗ Freund–Rubin

∗ Kaluza–Klein monopole

∗ M-wave

Minkowski vacuum

• (R1,10, F = 0)

Minkowski vacuum

• (R1,10, F = 0) has symmetry O(1, 10) n R1,10

Minkowski vacuum

• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R)

Minkowski vacuum

• (R1,10, F = 0) has symmetry O(1, 10) n R1,10 ⊂ GL(12,R):(

Minkowski vacuum

)A ∈ O(1, 10)

Minkowski vacuum

)A ∈ O(1, 10) v ∈ R

Minkowski vacuum

)A ∈ O(1, 10) v ∈ R

• Γ ⊂ O(1, 10) n R1,10

Minkowski vacuum

)A ∈ O(1, 10) v ∈ R

• Γ ⊂ O(1, 10) n R1,10, generated by

X = XL + XT ∈ so(1, 10)⊕ R1,10

Minkowski vacuum

)A ∈ O(1, 10) v ∈ R

• 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

Normal forms for orthogonal transformations

• X ∈ so(p, q)

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

• 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

• X =∑

Rp+q =

• X =∑

Rp+q =

Vi with Vi indecomposable

• X =∑

Rp+q =

• for each indecomposable block

• X =∑

Rp+q =

• for each indecomposable block, if λ is an eigenvalue

• X =∑

Rp+q =

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

• X =∑

Rp+q =

−λ, λ∗

• X =∑

Rp+q =

−λ, λ∗, and −λ∗

• possible minimal polynomials

• possible minimal polynomials:

? λ = 0

? λ = 0 µ(x) = xn

? λ = β ∈ R

? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = iϕ ∈ iR

? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = iϕ ∈ iR,

µ(x) = (x2 + ϕ2)n

? λ = 0 µ(x) = xn

? λ = β ∈ R,

µ(x) = (x2 − β2)n

? λ = iϕ ∈ iR,

µ(x) = (x2 + ϕ2)n

? λ = β + iϕ

? 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

? 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

Strategy

• for each µ(x)

Strategy

• for each µ(x), write down X in (real) Jordan form

Strategy

• determine metric making X skew-symmetric

Strategy

• determine metric making X skew-symmetric, using automorphism

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

Strategy

• keep only those blocks with appropriate signature

Strategy

• keep only those blocks with appropriate signature

• example: µ(x) = x3

Elementary lorentzian blocks

Signature Minimal polynomial Type

(0, 1)

(0, 1) x

(0, 1) x trivial

(0, 2)

(0, 1) x trivial

(0, 2) x2 + ϕ2

(0, 1) x trivial

(0, 2) x2 + ϕ2 rotation

(0, 1) x trivial

(1, 0)

(0, 1) x trivial

(1, 0) x

(0, 1) x trivial

(1, 0) x trivial

(0, 1) x trivial

(1, 0) x trivial

(1, 1)

(0, 1) x trivial

(1, 0) x trivial

(1, 1) x2 − β2

(0, 1) x trivial

(1, 0) x trivial

(1, 1) x2 − β2 boost

(0, 1) x trivial

(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2)

(0, 1) x trivial

(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2) x3

(0, 1) x trivial

(1, 0) x trivial

(1, 1) x2 − β2 boost

(1, 2) x3 null rotation

Lorentzian normal forms

Play with the elementary blocks!

In signature (1, 10)

In signature (1, 10):

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

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

• 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)

• 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

• 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

Normal forms for the Poincare algebra

• λ + τ ∈ so(1, 10)⊕ R1,10

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

• λ + τ ∈ so(1, 10)⊕ R1,10

• conjugate by R1,10

• λ + τ ∈ so(1, 10)⊕ R1,10

• conjugate by R1,10:

λ + τ 7→ λ + τ − [λ, τ ′]

• 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,−]

• the subgroups with everywhere spacelike orbits

• the subgroups with everywhere spacelike orbits are generated by

either

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

either

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

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

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

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

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

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

Adapted coordinates

• start with metric in flat coordinates y, z

Adapted coordinates

ds2 = 2|dy|2 + dz2

Adapted coordinates

ds2 = 2|dy|2 + dz2

• “undress” the Killing vector

Adapted coordinates

ds2 = 2|dy|2 + dz2

ξ = ∂z + λ

Adapted coordinates

ds2 = 2|dy|2 + dz2

ξ = ∂z + λ = U ∂z U−1

Adapted coordinates

ds2 = 2|dy|2 + dz2

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

• introduce new coordinates

x = U y

x = U y = exp(−zB)y

x = U y = exp(−zB)y where λy = By

whence ξx = 0

• rewrite the metric in terms of x

whence ξx = 0

• rewrite the metric in terms of x:

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

whence ξx = 0

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

? Λ = 1 + |Bx|2

whence ξx = 0

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

? Λ = 1 + |Bx|2? A = Λ−1 Bx · dx

• comparing with the Kaluza–Klein “ansatz”

ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

we read off the IIA fields

ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

? dilaton

ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

? dilaton: Φ = 34 log(1 + |Bx|2)

ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

? RR 1-form potential

ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

? RR 1-form potential:

A =Bx · dx

1 + |Bx|2

ds2 = e4Φ/3(dz + A)2 + e−2Φ/3h

A =Bx · dx

1 + |Bx|2

? string frame metric

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

• the only data is the matrix

0 −ϕ1 0 u

ϕ1 0 0 0

0 −ϕ2

0 −ϕ3

0 −ϕ4

−u 0

0 −ϕ1 0 u

ϕ1 0 0 0

0 −ϕ2

0 −ϕ3

0 −ϕ4

−u 0

0 −ϕ1 0 u

ϕ1 0 0 0

0 −ϕ2

0 −ϕ3

0 −ϕ4

−u 0

where either

? u = 0

0 −ϕ1 0 u

ϕ1 0 0 0

0 −ϕ2

0 −ϕ3

0 −ϕ4

−u 0

where either

? u = 0 (generalised fluxbranes)

0 −ϕ1 0 u

ϕ1 0 0 0

0 −ϕ2

0 −ϕ3

0 −ϕ4

−u 0

where either

? u = 0 (generalised fluxbranes); or

? u = 1 and ϕ1 = 0

0 −ϕ1 0 u

ϕ1 0 0 0

0 −ϕ2

0 −ϕ3

0 −ϕ4

−u 0

where either

? u = 0 (generalised fluxbranes); or

? u = 1 and ϕ1 = 0 (generalised nullbranes)

? u = 0

? u = 0, ϕ2 = ϕ3 = ϕ4 = 0

? u = 0, ϕ2 = ϕ3 = ϕ4 = 0 =⇒ flux-sevenbrane

[Costa–Gutperle, hep-th/0012072]

? u = 0

? u = 0, ϕ1 = ϕ2

? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4 = 0

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

[Gutperle–Strominger, hep-th/0104136]

? u = 0

? u = 0, ϕ1 = ϕ2 + ϕ3

? u = 0, ϕ1 = ϕ2 + ϕ3, ϕ4 = 0

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

? u = 0

? u = 0, ϕ1 − ϕ2 = ϕ3 ± ϕ4

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

[Uranga, hep-th/0108196]

? u = 0

? u = 0, ϕ1 = ϕ2

? u = 0, ϕ1 = ϕ2, ϕ3 = ϕ4

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

? u = 1

? u = 1, ϕi = 0

? u = 1, ϕi = 0 =⇒ half-BPS nullbrane

Discrete quotients

• start with the metric in adapted coordinates

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

and identify z ∼ z + L

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

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

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

=⇒ half-BPS eleven-dimensional nullbrane

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

=⇒ half-BPS eleven-dimensional nullbrane:

? time-dependent

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

? time-dependent

? smooth

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

? time-dependent

? smooth

? stable

Discrete quotients

ds2 = Λ(dz + A)2 + |dx|2 − ΛA2

? time-dependent

? smooth

? stable

? smooth transition between Big Crunch and Big Bang

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)]

End of first lecture

Supersymmetry

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

Supersymmetry

• Γ a one-parameter subgroup of symmetries

Supersymmetry

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

Supersymmetry

How much supersymmetry will the quotient M/Γ preserve?

Supersymmetry

In supergravity

Supersymmetry

In supergravity: Γ-invariant Killing spinors

Supersymmetry

In supergravity: Γ-invariant Killing spinors:

Supersymmetry

Lξε = ∇ξε + 18∇aξbΓabε

Supersymmetry

Lξε = ∇ξε + 18∇aξbΓabε = 0

Supersymmetry

In string/M-theory

Supersymmetry

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

“Supersymmetry without supersymmetry”

• T-duality relates backgrounds with different amount of

“supergravitational supersymmetry”

• dramatic example

• dramatic example:

AdS5×S5

((QQQQQQQQQQQQQ

oo // AdS5×CP2 × S1

AdS5×S5

((QQQQQQQQQQQQQ

uukkkkkkkkkkkkkkk

AdS5×CP2

AdS5×S5

((QQQQQQQQQQQQQ

uukkkkkkkkkkkkkkk

AdS5×CP2

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

Spin structures in quotients

• (M, g) spin

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

Is M/Γ spin?

• if Γ ∼= R

Is M/Γ spin?

• if Γ ∼= R, then M/Γ is always spin

Is M/Γ spin?

• so let Γ ∼= S1

Is M/Γ spin?

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

Is M/Γ spin?

• Lξ has integral weights on tensors

Is M/Γ spin?

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

Is M/Γ spin?

LξT = inT

Is M/Γ spin?

LξT = inT for n ∈ Z

Is M/Γ spin?

LξT = inT for n ∈ Z

so that exp(2πX) does act like 1 on tensors

• ξ also acts on spinors via Lξ

• ξ also acts on spinors via Lξ; but two things may happen

• ξ also acts on spinors via Lξ; but two things may happen:

? Lξ integrates to an action of Γ

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

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

? Lξ integrates to an action of a double cover Γ

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

now half-integral

now half-integral:

Lξε = i(n + 12)ε

now half-integral:

Lξε = i(n + 12)ε

• if Γ acts on spinors

now half-integral:

Lξε = i(n + 12)ε

• if Γ acts on spinors, M/Γ is spin

now half-integral:

Lξε = i(n + 12)ε

• if Γ acts on spinors, M/Γ is spin;

• if only Γ does

now half-integral:

Lξε = i(n + 12)ε

• if only Γ does, M/Γ is not spin

now half-integral:

Lξε = i(n + 12)ε

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

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

• e.g., the Hopf fibration S5 → CP2

• e.g., the Hopf fibration S5 → CP2; with S5 ⊂ C

3 as the quadric

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

3 as the quadric

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

and S1-action

3 as the quadric

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

and S1-action

(z1, z2, z3)

3 as the quadric

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

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

3 as the quadric

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

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

• Killing spinors on S5

3 as the quadric

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

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

• Killing spinors on S5 are restrictions of parallel spinors on C3

3 as the quadric

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

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

• Lξ acts as

3 as the quadric

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

and S1-action

(z1, z2, z3) 7→ (eitz1, eitz2, e

• Lξ acts as

Lξε = 12(Γ12 + Γ34 + Γ56)ε

• Γ2ij = −1

• Γ2ij = −1: its eigenvalues are ±i

• Γ2ij = −1: its eigenvalues are ±i, hence weights of Lξ are ±3

(with multiplicity 1)

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

• weights are half-integral

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

• more generally

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

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

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

to the representation with charge 1

• a spinor ε on M

• a spinor ε on M such that

Lξε = inε

defines a section of S ⊗ Ln over M/Γ

Lξε = inε

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

bundle of spinors

Lξε = inε

bundle of spinors

• a spinor ε on M

Lξε = inε

bundle of spinors

Lξε = i(n + 1/2)ε

Lξε = inε

bundle of spinors

Lξε = i(n + 1/2)ε

would define a section of

S ⊗ Ln+1/2

Lξε = inε

bundle of spinors

Lξε = i(n + 1/2)ε

S ⊗ Ln+1/2 ∼= (S ⊗ L1/2)

Lξε = inε

bundle of spinors

Lξε = i(n + 1/2)ε

S ⊗ Ln+1/2 ∼= (S ⊗ L1/2)⊗ Ln

• however neither S

• however neither S nor L1/2 exist as bundles

• however neither S nor L1/2 exist as bundles; although S ⊗ L1/2

does: it is the bundle of spinc spinors

• L carries a natural connection

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

• sections through Ln carry n units of RR charge

• spinc spinors carry fractional RR charge

• RR charge

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

• RR charged states are T-dual to winding states

• e.g., Killing spinors in AdS5×S5

• e.g., Killing spinors in AdS5×S5 are T-dual to winding states in

AdS5×CP2 × S1

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

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

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

Supersymmetry of supergravity quotients

• (M, g, F, ...) supersymmetric

• Γ one-parameter group of symmetries

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

• Killing spinors of M/Γ

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

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

• e.g., (R1,10)

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

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

Lξε = 18∇aξbΓabε

• e.g., fluxbranes

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

• e.g., fluxbranes

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

=⇒Lξ = 1

2(ϕ1Γ12 + ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)

• for generic ϕi

• for generic ϕi, there are no invariant Killing spinors

• 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

eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4

eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4

eigenvalues:

? ϕ1 − ϕ2 − ϕ3 ± ϕ4 = 0 =⇒ ν = 18

? ϕ1 = ϕ2, ϕ3 = ϕ4 =⇒ ν = 14

? ϕ1 − ϕ2 − ϕ3 = 0 = ϕ4 =⇒ ν = 14

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

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

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

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

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

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

• e.g., nullbranes

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

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)

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

=⇒Lξ = 1

2Γ+2 + 12(ϕ2Γ34 + ϕ3Γ56 + ϕ4Γ78)

• N+2 is nilpotent

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

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

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+

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

• for generic ϕi

• for generic ϕi, no supersymmetry is preserved, but there are

hyperplanes (and their intersections) on which Lξ has zero

eigenvalues

eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0

eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0

eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14

eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14

? ϕ2 = ϕ3 = ϕ4 = 0

eigenvalues:

? ϕ2 − ϕ3 − ϕ4 = 0 =⇒ ν = 18

? ϕ2 = ϕ3, ϕ4 = 0 =⇒ ν = 14

? ϕ2 = ϕ3 = ϕ4 = 0 =⇒ ν = 12

Brane quotients

• typical electric p-brane solution

Brane quotients

• typical electric p-brane solution:

g = e2A(r)ds2(R1,p)

Brane quotients

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

Brane quotients

Fp+2 = dvol(R1,p) ∧ dC(r)

Brane quotients

with r the transverse radius

Brane quotients

• A(r), B(r), C(r) → 0 as r → ∞

Brane quotients

• A(r), B(r), C(r) → 0 as r → ∞ =⇒ asymptotic to

(R1,D−1, F = 0)

• symmetry group

G =(SO(1, p) n R

• symmetry group

G =(SO(1, p) n R

1,p)×O(D− p− 1)

• symmetry group

G =(SO(1, p) n R

1,p)×O(D− p− 1) ⊂ O(1, D − 1) n R

1,D−1

• 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

• 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

• 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∞

• e.g., M2-brane

g = V −2/3ds2(R1,2) + V 1/3ds2(R8)

• e.g., M2-brane

g = V −2/3ds2(R1,2) + V 1/3ds2(R8)

F = dvol(R1,2) ∧ dV −1

• e.g., M2-brane

g = V −2/3ds2(R1,2) + V 1/3ds2(R8)

F = dvol(R1,2) ∧ dV −1

ε = V −1/6ε∞

• 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

• 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∞

• 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

• action of G on Killing spinors also induced from asymptotic limit

• action of G on Killing spinors also induced from asymptotic limit:

? ξ a Killing vector

? ε a Killing spinor

? ε a Killing spinor =⇒ so is Lξε

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

Lξε = eD(r)ε′∞

but also

Lξε = LξeD(r)ε∞

but also

Lξε = LξeD(r)ε∞ = eD(r)Lξε∞

but also

whence

Lξε∞ = ε′∞

but also

whence

Lξε∞ = ε′∞

(independent of r)

but also

whence

Lξε∞ = ε′∞

(independent of r)

? compute it at r →∞

but also

whence

Lξε∞ = ε′∞

(independent of r)

? compute it at r →∞

=⇒ the classification problem reduces to the flat case

=⇒ the classification problem reduces to the flat case, with two

important differences:

? G is not the full asymptotic symmetry group

? G is not the full asymptotic symmetry group, and

? brane metric is not flat

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

may differ from R1,D−1

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

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

its asymptotic value may not be everywhere spacelike in the flat

metric

Example: M2-brane

• metric

Example: M2-brane

• metric:

V −2/3ds2(R1,2) + V 1/3ds2(R8)

Example: M2-brane

• metric:

V −2/3ds2(R1,2) + V 1/3ds2(R8)

with V = 1 + |Q|/r6

Example: M2-brane

• metric:

V −2/3ds2(R1,2) + V 1/3ds2(R8)

with V = 1 + |Q|/r6

• Killing vector:

Example: M2-brane

• metric:

V −2/3ds2(R1,2) + V 1/3ds2(R8)

with V = 1 + |Q|/r6

• Killing vector:

ξ = τ‖

Example: M2-brane

• metric:

V −2/3ds2(R1,2) + V 1/3ds2(R8)

with V = 1 + |Q|/r6

• Killing vector:

ξ = τ‖ + ρ‖

Example: M2-brane

• metric:

V −2/3ds2(R1,2) + V 1/3ds2(R8)

with V = 1 + |Q|/r6

• Killing vector:

ξ = τ‖ + ρ‖ + ρ⊥

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

• norm

• norm:

‖ξ‖2 = V −2/3(|τ |2 + |ρ‖|2

)+ V 1/3|ρ⊥|2

• 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

• 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

• 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

• 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

• 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

• 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

• f(r) has a minimum at r0 > 0

• f(r) has a minimum at r0 > 0, where

V (r0)r80 = −2|Q|

m2|τ |2

V (r0)r80 = −2|Q|

m2|τ |2

? if |τ |2 ≥ 0

V (r0)r80 = −2|Q|

m2|τ |2

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

V (r0)r80 = −2|Q|

m2|τ |2

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

V (r0)r80 = −2|Q|

m2|τ |2

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

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

V (r0)r80 = −2|Q|

m2|τ |2

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

which is > 0

V (r0)r80 = −2|Q|

m2|τ |2

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

which is > 0 provided that

|τ |2 > −32m

2(2|Q|)1/3

• such ξ are not everywhere spacelike in R1,10

• such ξ are not everywhere spacelike in R1,10, so the resulting

quotient does not have a straight-forward physical interpretation

• moreover they contain closed timelike curves

[Maoz–Simon, hep-th/0310255]

• so do the associated discrete quotients

• similar results hold for delocalised branes

• similar results hold for delocalised branes, intersecting branes

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

End of second lecture

Freund–Rubin backgrounds

• purely geometric backgrounds

• purely geometric backgrounds, with product geometry

(M4 ×N7, g ⊕ h)

(M4 ×N7, g ⊕ h) and F ∝ dvolg

• field equations

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

• supersymmetry

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

spinors

spinors:

∇aε = λΓaε

spinors:

∇aε = λΓaε where λ ∈ R×

• (M, g) admits geometric Killing spinors

• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g)

• (M, g) admits geometric Killing spinors ⇐⇒ the cone (M, g),

M = R+ ×M

M = R+ ×M and g = dr2 + 4λ2r2g

M = R+ ×M and g = dr2 + 4λ2r2g ,

admits parallel spinors

M = R+ ×M and g = dr2 + 4λ2r2g ,

admits parallel spinors: ∇ε = 0

M = R+ ×M and g = dr2 + 4λ2r2g ,

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

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]

• (M, g) riemannian

• (M, g) riemannian =⇒ (M, g) riemannian

• (M1,n−1, g) lorentzian

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

• for the maximally supersymmetric Freund–Rubin backgrounds

• for the maximally supersymmetric Freund–Rubin backgrounds,

AdS4×S7

AdS4×S7 and S4 ×AdS7 ,

the cones of each factor are flat

the cones of each factor are flat:

? cone of Sn is Rn+1

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

• again the problem reduces to one of flat spaces!

Isometries of AdS1+p

• AdS1+p is simply-connected

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

Q1+p ⊂ R2,p

Q1+p ⊂ R2,p, given by

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

• For p > 2

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

• For p > 2, π1Q1+p∼= Z

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

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

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

x1(t) + ix2(t) = reit

−x21 − x2

2 + x23 + · · ·+ x2

p+2 = −R2

x1(t) + ix2(t) = reit with r2 = R2 + x23 + · · ·+ x2

• (orientation-preserving) isometries of Q1+p

• (orientation-preserving) isometries of Q1+p:

SO(2, p)

SO(2, p) ⊂ GL(p + 2,R)

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

AdS1+p

SO(2, p) ⊂ GL(p + 2,R)

AdS1+p. Why?

SO(2, p) ⊂ GL(p + 2,R)

AdS1+p. Why? Because...

SO(2, p) ⊂ GL(p + 2,R)

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

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

SO(2, p) ⊂ GL(p + 2,R)

? these curves are not closed in AdS1+p

SO(2, p) ⊂ GL(p + 2,R)

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

SO(2, p) ⊂ GL(p + 2,R)

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

infinite cover SO(2, p)

SO(2, p) ⊂ GL(p + 2,R)

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

SO(2, p) ⊂ GL(p + 2,R)

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

• the central element is the generator of π1Q1+p

• The bad news

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

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

dimensional matrix representations

• The good news

• The good news:

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

• The good news:

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

? adjoint group is again SO(2, p)

• The good news:

whence

• one-parameter subgroups

• The good news:

whence

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

Normal forms for so(2, p)

We play again

We play again but with a bigger set!

We can still use the lorentzian elementary blocks

We can still use the lorentzian elementary blocks:

• (0, 2)

• (0, 2) and also (2, 0)

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

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

B(0,2)(ϕ)

B(0,2)(ϕ) = B(2,0)(ϕ)

B(0,2)(ϕ) = B(2,0)(ϕ) =[

−ϕ 0

• (1, 1)

• (1, 1), µ(x) = x2 − β2

• (1, 1), µ(x) = x2 − β2, boost

B(1,1)

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

• (1, 2)

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

• (1, 2) and also (2, 1)

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

• (1, 2) and also (2, 1), µ(x) = x3

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

• (1, 2) and also (2, 1), µ(x) = x3, null rotation

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

B(1,2)

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

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

• (1, 1), µ(x) = x2 − β2, boost

B(1,1) =[0 −β

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

0 −1 01 0 −10 1 0

But there are also new ones:

• (2, 2)

• (2, 2), µ(x) = x2

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

B(2,2)±

B(2,2)± =

0 ∓1 1 0±1 0 0 ∓1−1 0 0 10 ±1 −1 0

• (2, 2)

• (2, 2), µ(x) = (x2−β2)2

• (2, 2), µ(x) = (x2−β2)2, deformation of B(2,2)± by a (anti)selfdual

B(2,2)± (β > 0)

B(2,2)± (β > 0) =

0 ∓1 1 −β

±1 0 ±β ∓1−1 ∓β 0 1β ±1 −1 0

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

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

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

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)

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|

• (2, 2)

• (2, 2), µ(x) = (x2 + ϕ2)2

• (2, 2), µ(x) = (x2 + ϕ2)2, deformation of B(2,2)± by a (anti)self-

dual rotation

B(2,2)± (ϕ)

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

• (2, 2)

• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2

• (2, 2), µ(x) = (x2 +β2 +ϕ2)− 4β2x2, self-dual boost + antiself-

dual rotation

B(2,2)± (β > 0, ϕ > 0)

dual rotation

B(2,2)± (β > 0, ϕ > 0) =

0 ±ϕ 0 −β

∓ϕ 0 ±β 00 ∓β 0 −ϕ

β 0 ϕ 0

• (2, 3)

• (2, 3), µ(x) = x5

• (2, 3), µ(x) = x5, deformation of B(2,2)+ by a null rotation in a

perpendicular direction

B(2,3)

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

• (2, 4)

• (2, 4), µ(x) = (x2 + ϕ2)3

• (2, 4), µ(x) = (x2 + ϕ2)3, double null rotation + simultaneous

rotation

B(2,4)± (ϕ)

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

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!

Causal properties

• Killing vectors on AdS1+p×Sq decompose

Causal properties

ξ = ξA + ξS

Causal properties

ξ = ξA + ξS

whose norms add

‖ξ‖2 = ‖ξA‖2 + ‖ξS‖2

• Sq is compact

• Sq is compact =⇒

R2M2 ≥ ‖ξS‖2

R2M2 ≥ ‖ξS‖2 ≥ R2m2

and if q is odd

R2M2 ≥ ‖ξS‖2 ≥ R2m2

and if q is odd, m2 can be > 0

R2M2 ≥ ‖ξS‖2 ≥ R2m2

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

R2M2 ≥ ‖ξS‖2 ≥ R2m2

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

not spacelike everywhere

R2M2 ≥ ‖ξS‖2 ≥ R2m2

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

R2M2 ≥ ‖ξS‖2 ≥ R2m2

and ξS has no zeroes

R2M2 ≥ ‖ξS‖2 ≥ R2m2

and ξS has no zeroes

• it is convenient to distinguish Killing vectors according to norm

• everywhere non-negative norm

• everywhere non-negative norm:

? ⊕iB(0,2)(ϕi)

? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)

? ⊕iB(0,2)(ϕi)

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

? ⊕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)

? ⊕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)

? ⊕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)

? ⊕iB(0,2)(ϕi)

(2,2)± ⊕i B(0,2)(ϕi)

• norm bounded below

? ⊕iB(0,2)(ϕi)

(2,2)± ⊕i B(0,2)(ϕi)

• norm bounded below:

? B(2,0)(ϕ)⊕i B(0,2)(ϕi)

? ⊕iB(0,2)(ϕi)

(2,2)± ⊕i B(0,2)(ϕi)

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

? ⊕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

? ⊕iB(0,2)(ϕi)

(2,2)± ⊕i B(0,2)(ϕi)

? B(2,2)± (ϕ)⊕i B(0,2)(ϕi)

? ⊕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

? ⊕iB(0,2)(ϕi)

(2,2)± ⊕i B(0,2)(ϕi)

• arbitrarily negative norm

? ⊕iB(0,2)(ϕi)

(2,2)± ⊕i B(0,2)(ϕi)

• arbitrarily negative norm: the rest!

? B(1,1)(β1)⊕B(1,1)(β2)⊕i B(0,2)(ϕi)

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

? 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(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)

? 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

? 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(2,1) ⊕i B(0,2)(ϕi)? B

(2,2)± (β)⊕i B(0,2)(ϕi)

? 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)

? 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(2,1) ⊕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(2,1) ⊕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(2,1) ⊕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)

? 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

? 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

? 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.

Discrete quotients with CTCs

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

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

• 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

• the corresponding one-parameter subgroup Γ ∼= R

• pick L > 0 and consider the cyclic subgroup ΓL

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

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

γ = exp(LX)

• the “orbifold” of AdS1+p×S2k+1 by ΓL

• the “orbifold” of AdS1+p×S2k+1 by ΓL contains CTCs

• idea of the proof

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

to its image γNx

to its image γNx for N � 1

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

• general case

• general case:

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

• general case:

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

• general case:

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

• general case:

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

image under γN

• general case:

image under γN

? we will construct a timelike curve c(t)

• general case:

image under γN

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

• general case:

image under γN

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

c(NL) = γN · x

• 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

? c is uniquely determined by its projections cA onto AdS1+p

? c is uniquely determined by its projections cA onto AdS1+p and

cS onto S2k+1

? cA is the integral curve of ξA

cS onto S2k+1

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

cS onto S2k+1

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

whose arclength

∫ NL

‖cS(t)‖dt

cS onto S2k+1

whose arclength

∫ NL

‖cS(t)‖dt = NL‖cS‖

cS onto S2k+1

whose arclength

∫ NL

‖cS(t)‖dt = NL‖cS‖ ≤ D

cS onto S2k+1

whose arclength

∫ NL

? therefore ‖cS‖ ≤ D/NL

cS onto S2k+1

whose arclength

∫ NL

? therefore ‖cS‖ ≤ D/NL, and

‖c‖2

cS onto S2k+1

whose arclength

∫ NL

‖c‖2 = ‖cA‖2 + ‖cS‖2

cS onto S2k+1

whose arclength

∫ NL

‖c‖2 = ‖cA‖2 + ‖cS‖2 ≤ ‖ξA‖2 +D2

cS onto S2k+1

whose arclength

∫ NL

‖c‖2 = ‖cA‖2 + ‖cS‖2 ≤ ‖ξA‖2 +D2

which is negative for N � 1

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

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

is complete

is complete:

? N is Einstein with positive cosmological constant

is complete:

? N is Einstein with positive cosmological constant and Einstein

is complete:

? Bonnet-Myers Theorem =⇒ N is compact

is complete:

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

diameter

is complete:

diameter

• geometrical CTCs are also natural in certain kinds of

supersymmetric Freund–Rubin backgrounds M × N

is complete:

diameter

supersymmetric Freund–Rubin backgrounds M × N , where M

is lorentzian Einstein–Sasaki

is complete:

diameter

supersymmetric Freund–Rubin backgrounds M × N , where M

is lorentzian Einstein–Sasaki: timelike circle bundles over Kahler

manifolds

Some interesting reductions

• there are many families of smooth supersymmetric reductions of

AdS4×S7

AdS4×S7, S4 ×AdS7

AdS4×S7, S4 ×AdS7, AdS5×S5

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

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

• 34-BPS AdS4×CP3 background of IIA

[Duff–Lu–Pope, hep-th/9704186]

• 916-BPS IIA backgrounds

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

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

B(2,2)+

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

B(2,2)+ ⊕ ϕ(R12 + R34 + R56 −R78)

• a half-BPS IIA background

• a half-BPS IIA background: reduction of S4 ×AdS7

• a half-BPS IIA background: reduction of S4 ×AdS7 by a double

null rotation

B(1,2) ⊕B(1,2)

null rotation

B(1,2) ⊕B(1,2)

• a family of half-BPS IIA backgrounds

null rotation

B(1,2) ⊕B(1,2)

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

null rotation

B(1,2) ⊕B(1,2)

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

null rotation

B(1,2) ⊕B(1,2)

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)

null rotation

B(1,2) ⊕B(1,2)

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

null rotation

B(1,2) ⊕B(1,2)

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

null rotation

B(1,2) ⊕B(1,2)

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

• reductions which break no supersymmetry are rare

• reductions which break no supersymmetry are rare: this is

intimately linked to the fact that AdS3×S3 is a lorentzian Lie

group, and the Killing vectors are left-invariant

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

Thank you.

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

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