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The twistor equation and its consequences Michael Eastwood University of Adelaide New Horizons in Twistor Theory, University of Oxford – p. 1/15
Transcript of The twistor equation and its consequences - Peoplepeople.maths.ox.ac.uk/lmason/New...
The twistor equation and itsconsequences
Michael Eastwood
University of Adelaide
New Horizons in Twistor Theory, University of Oxford – p. 1/15
Spinors in four dimensions
NB: 4 = 2 × 2 ∧1M = S ⊗ S′
φa = φAA′
Example and ‘flat model’ M = Gr2(T)↑
4-dimensional
Π
TΠGr2(T) = Hom(Π, T/Π)
= T/Π ⊗ Π∗ S∗ ⊗ S′∗
⊙2∧
1 = [⊙2S ⊗
⊙2S′] ⊕ [∧2S ⊗ ∧2S′]
neutral signature. . . conformal structure
gab = ǫABǫA′B′
New Horizons in Twistor Theory, University of Oxford – p. 2/15
The twistor equation
∧1 = S ⊗ S′
(and ∧
2S = ∧2S′ ≡ E [−1]
)
Pick a metric gab Levi-Civita connection ∇a
S∗ ∇a−−→ ∧1 ⊗ S∗ = S ⊗ S′ ⊗ S∗ ր
ց
S′
S ⊗S∗ ⊗ S′
Dirac
Twistor
Dirac ωA 7−→ ∇AA′ωA
Twistor ωB 7−→ ∇AA′ωB − 12δA
B∇CA′ωC
Conformal change gab 7→ gab = Ω2gab
∇AA′ωB = ∇AA′ωB + δABΥCA′ωC where Υa ≡ Ω−1∇aΩ
Therefore, the twistor equation is conformally invariant!
New Horizons in Twistor Theory, University of Oxford – p. 3/15
The twistor equation prolonged
The flat model M = Gr2(T) ∋ Π 0 → Π → T → T/Π → 0
0 → S′ → Gr2(T) × T → S∗ → 0
T → Γ(M, S∗) T = ωB ∈ Γ(S∗) | (∇AA′ωB) = 0
‖ ︸ ︷︷ ︸
twistor space local twistors
(∇AA′ωB) = 0 ⇔ ∇AA′ωB + δABπA′ = 0
∇AA′πB′ = 0 in flat space∇AA′πB′ = PABA′B′ωB
︸ ︷︷ ︸in general
where Pab = 12Rab −
112Rgab is the Schouten tensor
Closed
New Horizons in Twistor Theory, University of Oxford – p. 4/15
Local twistors
Twistor equation ⇔ ∇AA′ωB + δABπA′ = 0
∇AA′πB′ − PABA′B′ωB = 0
Connection
[ωB
πB′
]∇a7−→
[∇AA′ωB + δA
BπA′
∇AA′πB′ − PABA′B′ωB
] Localtwistortransport
[ωB
πB′
]=
[ωB
πB′ − ΥBB′ωB
]conformally invariant
(∇a∇b −∇b∇a)
[ωC
πC′
]=
[−ΨAB
CDωD
(∇CC′ΨABC
D)ωD
]ǫA′B′ +
[0
ΨA′B′D′
C′πD′
]ǫAB
curvature
where ΨABC
D ↔ anti-self-dual Weyl curvature &c
New Horizons in Twistor Theory, University of Oxford – p. 5/15
Consequences: equivalence problem
Twistor bundle 0 → S′ → T → S∗ → 0 with connection
∇a : T → ∧1 ⊗ T
flat ⇔ Weyl curvature Wabcd = 0 locally ∼= Gr2(T).
Solves the ‘equivalence problem’ cf. Cartan connection
Twistors T = S∗+ S′ Tractors ∧
2T = ∧2S∗+ S∗⊗ S′ + ∧
2S′
‖ ‖ ‖
E [1] ∧1[1] E [−1]
σ
µb
ρ
∇a7−→
∇aσ − µb
∇aµb + gabρ + Pabσ
∇aρ − Pabµb
OK in alldimensions > 2
New Horizons in Twistor Theory, University of Oxford – p. 6/15
Consequences: invariant operators
E(ABC)[−1] ∋ φABC 7−→ ∇CA′φABC ∈ E(AB)A′ [−2]
is conformally invariant (cf. Dirac operator). Couple:
T(ABC)[−1]∇−→ T(AB)A′ [−2]
‖ ‖E(ABCD) ⊕ E(AB)[−1] E(ABC)A′ [−1] ⊕ EAA′ [−2]
+ +E(ABC)A′ [−1] E(AB)(A′B′)[−2] ⊕ E(AB)[−3]
s
E(ABCD) ∋ ΨABCD 7→ (∇C(A′∇
DB′) + P CD
A′B′)ΨABCD︸ ︷︷ ︸Bach tensor
Tran
slat
ion
prin
cipl
e
‘Classification’ ofconformally invariant linear differential operators
New Horizons in Twistor Theory, University of Oxford – p. 7/15
Back to tractors
conformal geometryTracey Thomas
vector tensor spinortractor∗ twistor
∗A.P. Hodges1991
T =
E [1]+
∧1[1]+
E [−1]
∋
σ
µb
ρ
∇a7−→
∇aσ − µb
∇aµb + gabρ + Pabσ
∇aρ − Pabµb
∈ ∧
1 ⊗ T
Prolongation of conformal to Einstein equation
(∇a∇b + Pab)σ = 0 ⇔ σ−2gab is Einstein
Structure group = SO(n + 1, 1)
‖(σ, µb, ρ)‖2 ≡ 2σρ + µbµb is preserved
New Horizons in Twistor Theory, University of Oxford – p. 8/15
Consequences: conformal geodesics
Consider γ → (M, [g]) a smooth curve,t : γ → R a smooth parameterisation.
Let ∂/∂t ≡ V a∇a and v ≡√
V aVa ∈ Γ(E [1]).
(0, 0, v−1) ∈ Γ(γ,T ) A ≡∂
∂t
∂
∂t(0, 0, v−1) ∈ Γ(γ,T )
‖A‖2 = 0∂
∂tA = 0
conformallyinvariantconditions
thirdorderODE
cf.S
chw
arzi
an
projective structure on γ
conformal geodesics
circles on the round sphere
New Horizons in Twistor Theory, University of Oxford – p. 9/15
Puncture repair
M
∈ ⊂
p M \ p
^ compact Riemannian manifold_ N⊂
U≃ conformally diffeomorphic
Then M ∼= N &c: p must bereplacedLocal
• ∼= U
N
A A A know nothing about ∂U ⊂ N
OK in 2 dimensions (> 2 C. Frances, Comm. Math. Helv. 2014)Greene-Wu: harmonic functions N → R separate points.But bounded harmonic functions Disc \ p → R extend.
New Horizons in Twistor Theory, University of Oxford – p. 10/15
Puncture repair in four dimensions
A A A harmonic functions not conformally invariant
• Yamabe f 7→ (∇a∇a −16R)f
gab 7→ gab = Ω2gab ⇒ (∇a∇a −16R)Ω−1f = Ω−3(∇a∇a −
16R)f
• Paneitz f 7−→ ∇a
(∇a∇b + 2Rab − 2
3Rgab)∇bf ?
• Four-Laplacian f 7→ ∇a(‖∇f‖2∇af) promising?
• Preferred metric?
‖W‖2 ≡ W abcdWabcd ‖W‖2 = Ω−4‖W‖2
‖W‖gab is conformally invariant. OK if Wabcd(p) 6= 0.
New Horizons in Twistor Theory, University of Oxford – p. 11/15
Consequences: metric propagationw/ A.R. Gover
Recall ‖A‖2 = 0 (and write ˙ = ∂/∂t)
⇔ VaVa =
3
VbV b(VaV
a)2 −3
2VaV
a − VcVcPabV
aV b
Better (K.P. Tod) Pick a metric gab Ua unit tangent vector
and write Ca ≡ ∂Ua, where ∂ ≡ Ua∇a.
2(∂3t)∂t − 3(∂2t)2 = (2PabUaU b + CaC
a)(∂t)2
Even better, for τ ∈ Γ(γ, E [1/2]),
∂2τ + 12(PabU
aU b + 12CaC
a)τ = 0
τ−4gab, a preferred metric along γ (as long as τ 6= 0).
New Horizons in Twistor Theory, University of Oxford – p. 12/15
Consequences: preferred metricsw/ A.R. Gover
Conformal geodesics (following K.P. Tod)
∂Ca = PbaU b − (PbcU
bU c + CbCb)Ua
OK as long as Ca remains finite.
Theorem ‘Conformal injectivity radius’ OK.
•• p
q
Theorem Punctures are repaired.
New Horizons in Twistor Theory, University of Oxford – p. 13/15
Consequences: new and old horizons
Parabolic geometry (BGG machinery) Flat models
Gr2(R4) = SL(4,R)/
8
>
<
>
:
2
6
4
∗ ∗ ∗ ∗∗ ∗ ∗ ∗0 0 ∗ ∗0 0 ∗ ∗
3
7
5
9
>
=
>
;
Sn = SO(n + 1, 1)/P
M = SO(4, 2)/P = SU(2, 2)/P
RP 1967Conformal
S3 = SL(4,R)/
8
>
<
>
:
2
6
4
λ ∗ ∗ ∗0 ∗ ∗ ∗0 ∗ ∗ ∗0 ∗ ∗ ∗
3
7
5| λ > 0
9
>
=
>
;
Projective geometry(⊂ Riemannian geometry)
F1,2(R3) = SL(3,R)/
[∗ ∗ ∗0 ∗ ∗0 0 ∗
]S3 = SU(2, 1)/P CR
G2/P (2, 3, 5) geometry (É. Cartan 1910)
CPn = SL(n + 1, C)/P C-projective geometry. . .
New Horizons in Twistor Theory, University of Oxford – p. 14/15
THANK YOU
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ROGER AND NICK!¥ ¥ ¥ ¥ ¥ ¥ ¥
New Horizons in Twistor Theory, University of Oxford – p. 15/15
