Third Frontiers Lecture at Texas A&M p. 1/17 Conformal geometry and twistor theory Higher symmetries of the Laplacian Michael Eastwood Australian National University
Third Frontiers Lecture at Texas A&M p. 2/17 References ME, Higher symmetries of the Laplacian, Ann. Math. 161 (2005) 1645 1665. ME, Petr Somberg, and Vladimír Souček, Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras, Jour. Geom. Phys. 57 (2007) 2539 2546. ME and Thomas Leistner, Higher symmetries of the square of the Laplacian, IMA Volumes 144, Springer Verlag 2007, pp. 319 338. Thanks: Erik van den Ban, David Calderbank, Andreas Čap, Rod Gover, Robin Graham, Keith Hannabuss, Bertram Kostant, Toshio Oshima, Paul Tod, Misha Vasiliev, Nolan Wallach, Edward Witten, and Joseph Wolf.
Third Frontiers Lecture at Texas A&M p. 3/17 A simple question on R n, n 3 Question: Which linear differential operators preserve harmonic functions? Answer on R 3 : Zeroth order f constant f First order 1 = / x 1 2 = / x 2 3 = / x 3 x 1 2 x 2 1 &c. x 1 1 + x 2 2 + x 3 3 +1/2 (x 2 1 x 2 2 x 2 3 ) 1 + 2x 1 x 2 2 + 2x 1 x 3 3 + x 1 &c. Dimensions............................................... [D 1, D 2 ] D 1 D 2 D 2 D 1 Lie Algebra = so(4, 1) = conformal algebra NB! 1 3 3 1 3 10
Third Frontiers Lecture at Texas A&M p. 4/17 Second order Boyer-Kalnins-Miller (1976) Extras: Laplacian (f h f for any smooth h) plus a 35-dim l family of new ones! {D 1, D 2 } D 1 D 2 + D 2 D 1 2 so(4, 1) =? dim = 10 11/2 = 55 = R 55 = 35 + 14 + 1 + 5 Separation of variables (Bôcher, Bateman,... ). Third order...?
Third Frontiers Lecture at Texas A&M p. 5/17 Conformal geometry S n R n Action of SO(n + 1, 1) on S n by conformal transformations stereographic projection {generators} S n flat model
Third Frontiers Lecture at Texas A&M p. 6/17 Conformal Laplacian Dirac 1935 r x 1 2 + x n 2 + x n+1 2 x n+2 2 2 x + + 2 2 1 x + 2 2 n x 2 2 n+1 x 2 n+2 f on null cone R n+2 homogeneous of degree w ambiently extend to f of degree w freedom f f + rg for g of degree w 2 calculate: (rg) = r g + 2(n + 2w 2)g w = 1 n/2 f ( f) r=0 is invariantly defined. On R n it s On S n it s n 2 4(n 1) R AdS/CFT Fefferman-Graham ambient metric
Third Frontiers Lecture at Texas A&M p. 7/17 Symmetries of D a symmetry D = δ for some δ. trivial example: D = P for any P equivalence: D 1 D 2 D 1 D 2 = P R n A n algebra of symmetries under composition up to equivalence Write D = V bc d b c d + lower order terms symbol normalise w.l.g. to be trace-free
Third Frontiers Lecture at Texas A&M p. 8/17 Theorems D a symmetry trace-free part of (a V bc d) = 0 Easy On R n, such a conformal Killing tensor V bc d D V Not So Easy D V is a canonically associated symmetry of the form D V = V bc d b c d + lower order terms. E.g. First order E.g. Second order D V f = V a a f + n 2 2n ( av a )f D V f = V ab a b f + n n+2 ( av ab ) b f + n(n 2) 4(n+2)(n+1) ( a b V ab )f
Third Frontiers Lecture at Texas A&M p. 9/17 Ingredients of proof We can solve the conformal Killing tensor equation (a V bc d) = g (ab λ c d) on R n by prolongation and/or BGG machinery: w.r.t. so(n + 1, 1). }{{} # of columns = # of indices on V bc d E.g. V b = s b + m bc x c + λx b + r c x c x b 1 2 xc x c r b = translation + rotation + dilation + inversion. Use ambient methods to construct D V.
Third Frontiers Lecture at Texas A&M p. 10/17 Corollary As a vector space A n = s=0 }{{} s Question: What about the algebra structure? Cf.: let g be a Lie algebra. As a vector space U(g) = s=0 s g but the algebra structure is opaque viewed this way.
Third Frontiers Lecture at Texas A&M p. 11/17 The algebra structure U(g) = Theorem = g (X Y Y X [X,Y ]) g (X Y X Y 1 2 [X,Y ]) so(n + 1, 1) A n = (X Y X Y 1 n 2 2 [X,Y ] + 4n(n+1) X,Y ) Cartan Lie Killing Equivalently, A n = U(so(n + 1, 1))/Joseph Ideal.
Proof of algebra structure Calculate by ambient means that D X D Y = D X Y + 1 2 D [X,Y ] n 2 4n(n+1) D X,Y and use properties of Cartan product (due to Kostant). Remark: simple Lie algebra = g sl(2, C) dim g (X Y X Y 1 = 2 [X,Y ] λ X,Y ) for precisely one value of λ (Braverman and Joseph) graded algebra s=0 s g. Third Frontiers Lecture at Texas A&M p. 12/17
Third Frontiers Lecture at Texas A&M p. 13/17 Curved analogues For any vector field V a, f V a a f + n 2 2n ( av a )f is conformally invariant. For any trace-free symmetric tensor field V ab, f V ab a b f + n n+2 ( av ab ) b f + n(n 2) 4(n+2)(n+1) ( a b V ab )f is conformally invariant n+2 4(n+1) R abv ab f &c. &c. = curvature correction terms
Third Frontiers Lecture at Texas A&M p. 14/17 Curved symmetries? V a is a conformal Killing vector D V f V a a f + n 2 2n ( av a )f is symmetry of the conformal Laplacian. V ab is a conformal Killing tensor =? D V f V ab a b f + n n+2 ( av ab ) b f + n(n 2) 4(n+2)(n+1) ( a b V ab )f n+2 4(n+1) R abv ab f is a symmetry of the conformal Laplacian. Unknown!
Third Frontiers Lecture at Texas A&M p. 15/17 Another operator In even dimensions, there is the Dirac operator D : S + S. A symmetry of D is an operator D : S + S + s.t. D S + S + D D δ S S commutes for some differential operator δ : S S. The symbol of D satisfies a conformally invariant overdetermined system of equations. First order symmetries: Benn and Kress. Higher order symmetries in the flat case: E, Somberg, and Souček.
Third Frontiers Lecture at Texas A&M p. 16/17 Yet another operator For the square of the Laplacian (E and Leistner) symmetry algebra = so(n + 1, 1) ( X Y X Y X Y 1 (n 4)(n+4) [X,Y ] + X,Y ) 2 4n(n+1)(n+2) and some fourth order elements with graded counterpart new different s=0 }{{} s s=2 }{{} s
Third Frontiers Lecture at Texas A&M p. 17/17 THANK YOU THE END