Conformally invariant differential operators

Transcription

1 Spring Lecture Two at the University of Arkansas p. 1/15 Conformal differential geometry and its interaction with representation theory Conformally invariant differential operators Michael Eastwood Australian National University

2 Spring Lecture Two at the University of Arkansas p. 2/15 Examples Maxwell in dimension four Laplacian in dimension two ( = 4 2 / z z) Exterior derivative d : Λ p Λ p+1 (in all dimensions) Λ m = Λ m + Λm in dimension 2m Conformal Killing X a (a X b) trace Dirac operator Rarita-Schwinger operator Yamabe operator (aka conformal Laplacian, + ) Paneitz operator ( 2 + ), etc etc etc

3 Spring Lecture Two at the University of Arkansas p. 3/15 Meaning? WARNING! 8 possible formulations! But... Invariance as operator on conformal manifold Invariance on S n under action of SO(n + 1, 1) or Spin Naïve verification g ab ĝ ab = Ω 2 g ab a ˆ a ˆ a X b = a X b + Υ a X b Υ b X a + Υ c X c b δ a where Υ a ( a Ω)/Ω. ˆ a X b = a X b + Υ a X b Υ b X a + Υ c X c g ab Hence ˆ (a X b) = (a X b) + Υ c X c g ab Conformal Killing

4 Spring Lecture Two at the University of Arkansas p. 4/15 Yamabe operator Conformal densities of weight w f ˆf = Ω w f ˆ a ω b = a ω b + (w 1)Υ a ω b Υ b ω a + Υ c ω c g ab ˆ a ˆ b f = ˆ a ( b f + wυ b f) = a ( b f + wυ b f) + (w 1)Υ a ( b f + wυ b f) Υ b ( a f + wυ a f) + Υ c ( c f + wυ c f)g ab ˆ f = f + (n + 2w 2)Υ a a f + w( a Υ a + (n + w 2)Υ a Υ a )f

5 Spring Lecture Two at the University of Arkansas p. 5/15 Yamabe operator cont d ˆ f = f + (n + 2w 2)Υ a a f + w( a Υ a + (n + w 2)Υ a Υ a )f If w = 1 n/2 ˆ f = f 1 4 (n 2)(2 a Υ a + (n 2)Υ a Υ a )f But ˆR = R (n 1)(2 a Υ a + (n 2)Υ a Υ a ) ˆ n 2 4(n 1) ˆR = n 2 4(n 1) R invariant L n 2 4(n 1) R : Λ0 [1 n/2] Λ 0 [ 1 n/2]

6 Spring Lecture Two at the University of Arkansas p. 6/15 SO(n+1,1)-invariance Recall S n = SO(n + 1, 1)/P = G/P, where p = } rotations {{ dilations } inversions Levi subalgebra SO(n) {λ > 0} Irreducible homogeneous vector bundles on S n irreducible SO(n)-module λ λ w irreducible Riemannian tensor bundle V [w] conformal weight w EG: Λ p, b Λ1,,,,... (or ±-part thereof )

7 Spring Lecture Two at the University of Arkansas p. 7/15 Operators on the three-sphere A complete list of SO(4, 1)-invariant linear differential operators between irreducible tensor bundles Standard (with suitable conformal weights) b a+1 Λ1 a+b+1 Λ 1 2b+1 a+b+1 Λ 1 a+1 b for a,b Z 0 (a = b = 0 de Rham complex) Λ1 Non-standard b Λ1 [a + 2b] 2a+2b+3 b Λ1 [ a 3] for a + 1/2,b Z 0 (a = 1/2,b = 0 Laplacian) Proof is by algebra (Lie theory and Verma modules) Theorem All these operators have conformally invariant curved analogues.

8 Spring Lecture Two at the University of Arkansas p. 8/15 Conformal-Einstein operator Let P ab 1 n 2 ( ) R ab 1 2(n 1) Rg ab. Then σ D trace-free part of( a b σ + P ab σ) is conformally invariant, σ weight 1 (a = 1, b = 0) Geometric meaning where σ 0 (LeBrun 1985) Dσ = 0 σ 2 g ab is an Einstein metric Prolong a σ µ a = 0 Dσ = 0 a µ b + P ab σ + g ab ρ = 0 a ρ P b a µ b = 0 Curved translation principle Cartan connection

9 Spring Lecture Two at the University of Arkansas p. 9/15 Representation theory a b c d SO(9) a,b,c Z 0, d 2Z EG: a a b c d Spin(9) EG: Spin(10) = S + 0 = R = Λ 3 R = a R9 a,b,c,d Z 0 = Λ 2 R 9 = Λ 4 R 9 = = S, basic spin representation

10 Spring Lecture Two at the University of Arkansas p. 10/15 Representation theory cont d On a Riemannian 9-manifold Λ (a ω b) trace [a ω b] a ω a = On a 9-dimensional spin manifold twistor operator = Dirac operator

11 Spring Lecture Two at the University of Arkansas p. 11/15 Some invariant operators S 9 = Spin(10, 1)/P = Killing 7/ / Yamabe Conformal-Einstein Twistor 9/ / Dirac 11/ / Rarita-Schwinger

12 Spring Lecture Two at the University of Arkansas p. 12/15 BGG complexes on 3-sphere Conformal a b a 2 2a + b + 2 a b 3 2a + b + 2 a b 3 Contact projective a b a 2 a + b + 1 a 2b 4 a + b + 1 CR a b a 2 a + b + 1 a + b + 1 b 2 a b 3 a b a b 3 b a 2b 4 b b 2 a 2

13 Spring Lecture Two at the University of Arkansas p. 13/15 Beware the four-sphere Pattern ր ց ց ր cf. de Rham Maxwell Theorem Most of these operators have conformally invariant curved analogues. Standard Non-standard Bateman, Yamabe, et alia 2 Paneitz, et alia, GJMS 3 Graham 4 Gover & Hirachi general cf. Eastwood & Slovák

14 C.R. Graham, R. Jenne, L.J. Mason, and G.A.J. Sparling, Conformally invariant powers of the Laplacian, I: existence, Jour. LMS 46 (1992) Spring Lecture Two at the University of Arkansas p. 14/15 References A. Čap & J. Slovák, Parabolic Geometries 1, AMS P.A.M. Dirac, The electron wave equation in de-sitter space, Ann. Math. 36 (1935) M.G. Eastwood & J.W. Rice, Conformally invariant differential operators on Minkowski space and their curved analogues, Commun. Math. Phys. 109 (1987) & Erratum 144 (1992) 213. M.G. Eastwood & J. Slovák, Semi-holonomic Verma modules, Jour. Alg. 197 (1997) A.R. Gover & K. Hirachi, Conformally invariant powers of the Laplacian a complete non-existence theorem, Jour. AMS 17 (2004) C.R. Graham, Conformally invariant powers of the Laplacian, II: nonexistence, Jour. LMS 46 (1992)

15 Spring Lecture Two at the University of Arkansas p. 15/15 THANK YOU END OF PART TWO