Higher symmetries of the system /D
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1 Higher symmetries of the system /D Jean-Philippe Michel (Université de Liège) joint work with Josef ilhan (Masaryk University) 34 rd Winter school in Geometry and Physics
2 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Overview on higher symmetries Let E and F be two vector bundles over the smooth manifold M. Denition Let D D(M; E, F ) be a dierential operator. A higher symmetry of D is a di. op. A D(M, E) such that D A = B D for some di. op. B D(M, F ). Higher symmetries (HS) preserve the kernel of D. The space of HS is a subalgebra of D(M, E), i.e., an associative and non-commutative ltered algebra.
3 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 On R n, the algebras of HS are classied for: the Laplacian [Eastwwod, Ann. Math. 2005], its powers k [Gover- ilhan, JMP 2012], the CR-subLaplacian [Vlasáková, 2012], the Dirac operator /D [Eastwood-Somberg-Sou ek], the Schrödinger operator [Bekaert-Meunier-Moroz, JHEP 2012], the superlaplacian [Coulembier-Somberg-Sou ek, IMRN 2013]. They all read as U(g)/J, with g Vect(R n ) acting by Lie derivative on Γ(E).
4 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 On R n, the algebras of HS are classied for: the Laplacian [Eastwwod, Ann. Math. 2005], its powers k [Gover- ilhan, JMP 2012], the CR-subLaplacian [Vlasáková, 2012], the Dirac operator /D [Eastwood-Somberg-Sou ek], the Schrödinger operator [Bekaert-Meunier-Moroz, JHEP 2012], the superlaplacian [Coulembier-Somberg-Sou ek, IMRN 2013]. They all read as U(g)/J, with g Vect(R n ) acting by Lie derivative on Γ(E). For the Laplacian = 2 x x 2 2 x 2 p p+1 E := C (R n, C): 2 x 2 p+q, acting on second order symmetries allow to classify separating coordinates for the Laplace equation f = 0; g = o(n + 2, C) and J is the Joseph ideal, so that ker is the minimal representation of O(p + 1, q + 1); U(g)/J is an algebra of symmetries in higher spin eld theories.
5 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Conjecture Let E, F be two irreducible homogeneous vector bundles over M = G/P. Assume D D(M; E, F ) is a G-invariant di. op. Conjecture The algebra of HS of D is the quotient U(g)/J, with J the annihilator of ker D. What about HS of systems of invariant dierential operators?
6 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Problematic Let S be the spinor bundle over the pseudo-euclidean space (R n, ě). Determine the algebra of HS of the system of dierential operators E[ n 2 2 n 1 n+1 n+2 ] S[ 2 ] S[ 2 ] E[ ( ) ) f φ ( f /Dφ 2 ]
7 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Problematic Let S be the spinor bundle over the pseudo-euclidean space (R n, ě). Determine the algebra of HS of the system of dierential operators E[ n 2 2 n 1 n+1 n+2 ] S[ 2 ] S[ 2 ] E[ ( ) ) f φ ( f /Dφ 2 ] The HS read as ( 0 0 /D with new symmetries: ) ( a α α + A ) ( b β + = B β ) ( 0 0 /D ) α = β + /D on S[ n 1 2 ] and /Dα+ = β on E[ n 2 2 ]
8 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 1 Introduction 2 Classication of HS HS of Laplacian HS of Dirac operator HS of the system Laplace + Dirac operator 3 Algebra of higher symmetries
9 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Higher symmetries of Laplacian [Eastwwod, Ann. Math. 2005][M., AIF 2014] Examples: Up to constants, rst order symmetries are given by (X + n 2 2n i X i ) = (X + n+2 2n i X i ), where X is conformal Killing vector eld, i.e. L X ě [ě] or (i X j)0 = 0. The Lie algebra of such vector elds is g = o(n + 2, C). Denition: a symmetric conformal Killing k-tensor K is a solution of the equation (i 0 K i1 i k ) 0 = 0. Theorem For all w R, there exists a unique g-equivariant linear isomorphism Q w : Γ(STM) D w (R n ), such that Q w (K) = K i1 i k i 1... ik + l.o.t.. The map Q n 2 induces a bijection between symmetric conformal Killing tensors 2 and HS of Laplacian.
10 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Higher symmetries of Dirac operator [Eastwwod-Somberg-Sou ek][m., Silhan] Let γ : T M[1] EndS, so that /D = γ i i. Examples: Up to constants, rst order symmetries are [Benn-Charlton, 1997]: X 1 2 γ(d X ) + n 1 2n ( i X i ), ě ij γ(ι ei K) ej κ n κ γ(d K) + κ + 1 2(n + 1 κ) γ (δk) where X is conformal Killing vector eld and K a conformal Killing κ-form. Denition: a conformal Killing (k, κ)-tensor K Γ(S k TM κ T M[1]) is a solution of Π( (i 0 K i1 i k )[j1 j κ]) = 0, with Π a specic projection. Theorem For all w R, there exists a unique g-equivariant linear isomorphism Q w : Γ(STM T M[1]) D(R n, S[w]), s.t. Q w (K) = K (i1 i ) k [j1 j κ] γj1 γ jκ i 1 i k + lot. The map Q n 1 induces a bijection 2 between conformal Killing tensors and HS of the Dirac operator.
11 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 HS of the system Laplace + Dirac operator Let ε S S be the invariant pairing on S. Examples: if Λ S[ 1 2 ] is a twistor spinor, iλ = 1 n γ i( /DΛ), the following are symmetries [Wess-Zumino, Nucl. Phys. B 1974]: { α Λ = β+ Λ /D, α Λ (φ) = ε(λ, φ), β + Λ ( /Dφ) = ε(λ, /D2 φ) + 2 φ ε( /DΛ, /Dφ), n /Dα + Λ = β Λ, { α + Λ (f ) = γi (Λ) i f + n 2 β Λ ( f ) = Λ f, f n ( /DΛ) f, S[ n 1 2 ]; E[ n 2 2 ]. Theorem ( ) a α The matrix of operators α + A is a HS i a is a HS of and A is a HS of /D, α = i a i α Λ i, with a i HS of and α Λ i as above, α + = i α+ Λ i a i, with a i HS of and α + Λ i as above.
12 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Composition of twistor spinors actions Lie (super-)algebra? Candidate: Conf. Killing vector elds Twistor-spinors. Hint from Rep. Theory: in odd dimension, TwSp TwSp = + C n+2 = space of conf. Killing odd forms; in even dimension, TwSp TwSp = C n+2 = space of all conf. Killing forms, whereas TwSp + TwSp = + C n+2 = space of conf. Killing odd forms; Fact: the composition α + Λ α Λ gives indeed rise to all HS of 1st order of /D. Algebra of HS is not generated by a Lie (super-)algebra!
13 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Composition of twistor spinors actions Lie (super-)algebra? Candidate: Conf. Killing vector elds Twistor-spinors. Hint from Rep. Theory: in odd dimension, TwSp TwSp = + C n+2 = space of conf. Killing odd forms; in even dimension, TwSp TwSp = C n+2 = space of all conf. Killing forms, whereas TwSp + TwSp = + C n+2 = space of conf. Killing odd forms; Fact: the composition α + Λ α Λ gives indeed rise to all HS of 1st order of /D. Algebra of HS is not generated by a Lie (super-)algebra! Idea: If dimension=3 (or 4 and /D : S + S ), then the 1st order symmetries of /D are all given by the Lie algebra o(n + 2, C) C and we have o(5) = sp(4) and o(6) = sl(4).
14 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Supergeometric reformulation ΠS = R 3 2 is a supermanifold with sheaf of functions O(ΠS ) = E S 2 S. The pairing ε is a distinguished element of 2 S = E. We dene ) ( : O (ΠS [ 1 2 ] [ n 2 2 ] O ΠS [ 1 2 )[ ] n 2 ] by the formula := ε + /D + ε = Proposition /D The algebra of HS of is isomorphic to the one of. ( 0 0 /D ).
15 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Twistor-spinors as odd vector elds Let (x i, θ a ) be coordinates on R 3 2 and ( i, θ a) the corresponding derivatives. ) We dene on O (ΠS [ 1 2 ] [w] L X = X i i 1 2 γ(d X ) b aθ a θ b ( w n 1 2n θa θ a)( i X i ), L + Λ = γi a b Λ aθ b i 2( w n + 1 n θa θ a)( /DΛ), L Λ = εab Λ a θ b. We have L X = L X, L + Λ = L Λ, L Λ = L+ Λ. Proposition The space c, L X L + Λ, L Λ is stable under the commutator in D w (R 3 2 ) and isomorphic to the Lie superalgebra spo(4 2).
16 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Main result Theorem The algebra of HS of of and J the Joseph-like ideal. ( 0 0 /D ) is isomorphic to U(spo(4 2))/J, with We have J is generated by g g = C (Casimir). C (Casimir ρ).
17 Jean-Philippe MICHEL (ULg) 34 rd Winter school in Geometry and Physics Srni Srni, january / 14 Conclusion HS of system of dierential operators: new symmetries, new realization of representations, applications to higher spin eld theories.
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