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Conformal differential geometry
and its interaction with representation theory
Introduction to conformaldifferential geometry
Michael Eastwood
Australian National University
Spring Lecture One at the University of Arkansas – p. 1/13
Motivation from physics• GR: null geodesics are conformally invariant• Maxwell’s equations are conformally invariant
gab = (pseudo-)metric onM, a smoothn-manifoldgab = Ω
2gab = conformally related metric(angles OK)
g2 : T ∗M → R dg2 Xg2 geodesic spray
g2 = Ω2g2 dg2 = Ω2dg2 + g2dΩ2
∴ Xg2|g=0 ∝ Xg2|g=0
gab ǫab···de volume form (e.g.ǫab···deǫab···de = n!)∴ gab = Ω
2gab =⇒ ǫab···de = Ωnǫab···de
∴ ǫabcd = ǫab
cd whenn = 4∴ Fab 7→ ∗Fab ≡ ǫab
cdFcd is invariantdF = 0d∗F = 0
Spring Lecture One at the University of Arkansas – p. 2/13
Motivation from geometry
stereographic projectionS n
Rn
SS
SS
SS
SS
SS•
•
BB
BBB
conformal
Q
•
•
trough
Rn ∋ x 7→ 1
‖x‖2+4
4x‖x‖2 − 4
∈ S n
Spring Lecture One at the University of Arkansas – p. 3/13
Motivation from navigation•Mercator(Cartographer) 1569
•Wright (Mathematician) 1599
S 2 \ polesstereographic−−−−−−−−−→ R2 \ 0 = C \ 0
log−−→ C
Jac=
ux uy
vx vy
=
c -s
s c
⇐⇒
ux = vy
vx = -uy
Cauchy-Riemann
Spring Lecture One at the University of Arkansas – p. 4/13
Euclidean symmetriesX =vector field
X = X1 ∂
∂x1+ X2 ∂
∂x2+ · · · + Xn ∂
∂xn= Xa∇a
Infinitesimal Euclidean symmetry: LXδab︸︷︷︸
Lie derivative
= 0.
Compute
LXδab = Xc∇cδab + δcb∇aXc + δac∇bXc
= ∇aXb + ∇bXa
∴ LXδab = 0 ⇐⇒ ∇(aXb) = 0 Killing field
Spring Lecture One at the University of Arkansas – p. 5/13
Killing fields by prolongationKilling operator: Xa 7→ ∇(aXb)
Kernelin flat space:Kab ≡ ∇aXb is skew.
Claim: ∇aKbc = 0. ∇aKbc = ∇cKba − ∇bKca
= ∇c∇bXa − ∇b∇cXa
= 0, as required.
Hence,∇(aXb) = 0 ⇐⇒∇aXb = Kab
∇aKbc = 0Closed!
Conclusion: Xa = sa + mabxb wheremab = −mba.
translations
6
rotations
6
Spring Lecture One at the University of Arkansas – p. 6/13
Conformal symmetries
trace-free part∇(aXb) = 0 conformal Killing field
Rewrite as ∇aXb = Kab + Λδab whereKab is skew.
∇aKbc = ∇cKba − ∇bKca
= ∇c∇bXa − ∇b∇cXa − δab∇cΛ + δac∇bΛ so
∇aKbc = δabQc − δacQb where ∇aΛ = −Qc but
0 = δab(∇d∇aKbc − ∇a∇dKbc)= δab(δab∇dQc − δac∇dQb − δdb∇aQc + δdc∇aQb)= (n − 2)∇dQc + δdc∇
aQa whence
∇aQb = 0 if n ≥ 3 Closed!!
Spring Lecture One at the University of Arkansas – p. 7/13
Conformal symmetries cont’d
Solve ∇aXb = Kab + Λδab
∇aKbc = δabQc − δacQb
∇aΛ = −Qc
∇aQb = 0
Qb = −rb Λ = λ + rbxb Kbc = rbxc − rcxb − mbc
Xa = sa + mabxb + λxa + rbxbxa −12raxbxb
translation+ rotation+ dilation+ inversion
Integrate the inversions
xa 7−→xa − 1
2ra‖x‖2
1− raxa + 14‖r‖
2‖x‖2
Spring Lecture One at the University of Arkansas – p. 8/13
Conformal group
SO(n + 1,1) acts onS n
by conformal transformations
AA
AA
AA
AA
AA S n
CCCCCCCCCCCO generators
S n = SO(n + 1,1)/P
flat model of conformal differential geometry
Rn = (SO(n) ⋉ Rn)/SO(n)
flat model of Riemannian differential geometry
semisimpleHHj parabolic
Spring Lecture One at the University of Arkansas – p. 9/13
A simple question onRn, n ≥ 3Question: Which linear differential operators preserveharmonic functions? Answer onR3:–Zeroth order f 7→ constant× fFirst order∇1 = ∂/∂x1 ∇2 = ∂/∂x2 ∇3 = ∂/∂x3
x1∇2 − x2∇1 &c.x1∇1 + x2∇2 + x3∇3 +1/2
(x12 − x2
2 − x32)∇1 + 2x1x2∇2 + 2x1x3∇3 + x1
&c.Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3313
10
1
[D1,D2] ≡ D1D2 −D2D1
Lie Algebra so(4,1) = conformal algebra← NB!
Spring Lecture One at the University of Arkansas – p. 10/13
Surroundings
Spring Lecture One at the University of Arkansas – p. 11/13
Next four talks• What about conformally invariant operators?
(Beyond Maxwell)• What about higher order operators preserving
harmonic functions? (Beyond first order)• Bateman’s formula and twistor theory• The X-ray transform
Further Reading• M.G. Eastwood, Notes on conformal differential geometry,
Suppl. Rendi. Circ. Mat. Palermo43 (1996) 57–76.
• R. Penrose and W. Rindler, Spinors and space-time, vols 1
and 2, Cambridge University Press 1984 and 1986.
Spring Lecture One at the University of Arkansas – p. 12/13
THANK YOU
END OF PART ONE
Spring Lecture One at the University of Arkansas – p. 13/13