Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential...

13
Conformal differential geometry and its interaction with representation theory Introduction to conformal differential geometry Michael Eastwood Australian National University Spring Lecture One at the University of Arkansas – p. 1/13

Transcript of Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential...

Page 1: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 2: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 3: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

Motivation from geometry

stereographic projectionS n

Rn

SS

SS

SS

SS

SS•

BB

BBB

conformal

QQ

QQ

QQ

QQ

QQ

QQ

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

Page 4: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 5: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 6: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 7: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 8: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 9: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 10: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 11: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

Surroundings

Spring Lecture One at the University of Arkansas – p. 11/13

Page 12: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

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

Page 13: Introduction to conformal differential geometryeastwood/fayetteville1.pdf · Conformal differential geometry and its interaction with representation theory Introduction to conformal

THANK YOU

END OF PART ONE

Spring Lecture One at the University of Arkansas – p. 13/13