From symplectic to spin geometry

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From symplectic to spin geometry Jean-Philippe Michel University of Luxembourg Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 1 / 16

Quantization for spinless system classical quantum link symplectic mfld (pre-)hilbert space (T M, ω) F 1 2 c = Γ c ( Λ n T M 1 2 ) vertical polarization co. graded P. alg. ass. filtered alg. Pol(T M) D 1 2, 1 2 (M) Pol k D 1 2, 1 2 k /D 1 2, 1 2 k 1 preserve {, } preserve [, ] σ p+q 1 ([P, Q]) = {σ p (P), σ q (Q)} Vect(M) Pol 1 Vect(M) D 1 2, 1 2 1 GQ(J X ) = X i i + 1 2 ( ix i ). Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 2 / 16

Quantization for spin system Let (M, g) be a pseudo-riemannian spin manifold with dim M = 2n. Remind that Cl(V, g) = V / u v + v u + 2g(u, v). Γ(S) F 1 2 c Pol(T M) Γ(Cl(M, g)) D 1 2, 1 2 P i 1 i k j 1 j κ (x)γ j 1... γ j κ i1... ik Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 3 / 16

Quantization for spin system Let (M, g) be a pseudo-riemannian spin manifold with dim M = 2n. Remind that Cl(V, g) = V / u v + v u + 2g(u, v). Γ(S) F 1 2 c Sb=Pol(T M) Ω C (M) D 1 2, 1 2 P i 1 i k j 1 j κ (x)γ j 1... γ j κ i1... ik Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 3 / 16

Quantization for spin system Let (M, g) be a pseudo-riemannian spin manifold with dim M = 2n. Remind that Cl(V, g) = V / u v + v u + 2g(u, v). M = T M ΠTM Γ(S) F 1 2 c Sb=Pol(T M) Ω C (M) D 1 2, 1 2 P i 1 i k j 1 j κ (x)γ j 1... γ j κ i1... ik Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 3 / 16

Quantization for spin system Let (M, g) be a pseudo-riemannian spin manifold with dim M = 2n. Remind that Cl(V, g) = V / u v + v u + 2g(u, v). M = T M ΠTM Γ(S) F 1 2 c Sb=Pol(T M) Ω C (M) D 1 2, 1 2 P i 1 i k j 1 j κ (x)γ j 1... γ j κ i1... ik Questions : symplectic form on M? Polarization and construction of S? Quantum action of vector fields? Poisson bracket from the commutator, which graduation? Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 3 / 16

1 From symplectic spinors to spinors Algebraic point of view Geometrization 2 Actions of vector fields On the supercotangent bundle M On the spinor bundle S 3 D as quantization of Sb Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 4 / 16

Quantization of symplectic even and odd vector spaces Let V be a vector space, with coordinates (ξ i ), dim V = 2n. Remind that W(V, ω) = V / u v v u ω(u, v). Even (V, ω) odd (ΠV, g) graded alg. S(V ) ΛV symplectic form ω ij dξ i dξ j g ij dξ i dξ j symmetries (S 2 (V ), {, }) sp(v, ω) (Λ 2 V, {, }) o(v, g) Moyal -product (S(V ), ) W(V, ω 1 ) (ΛV, ) Cl(V, g 1 ) Group action Mp(V, ω) = exp(sp(v, ω)) Spin(V, g) = exp(o(v, g)) Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 5 / 16

(Voronov 90) Even (V, ω) odd (ΠV, g) Darboux coord. ω = dp i dx i g = ε i dξ i dξ i ; ε i = ±1 Riemannian : ε i = 1 Lagrangian V = x i p i ( V ) C = P P 1+iJ P = 2 V, J 2 = 1 Herm. str. Polarized fct. S( x i ) ΛP Quant. fct. S( x i )(1 + p i ) ΛP (1 + P ) Rep. space H = S( x i ) det 1 2 H = ΛP Ber 1 2 End(H) W(V, ω 1 ) End(H) Cl(V, g 1 ) Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 6 / 16

(Voronov 90) Even (V, ω) odd (ΠV, g) Darboux coord. ω = dp i dx i g = ε i dξ i dξ i 2i ; ε i = ±1 Riemannian : ε i = 1 Lagrangian V = x i p i ( V ) C = P P 1+iJ P = 2 V, J 2 = 1 Herm. str. Polarized fct. S( x i ) ΛP Quant. fct. S( x i )(1 + p i ) ΛP (1 + P ) Rep. space H = S( x i ) det 1 2 H = ΛP Ber 1 2 End(H) W(V, ω 1 ) End(H) Cl(V, g 1 ) Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 6 / 16

Symplectic supermanifold Supermanifold := loc. ringed space N = (N, O) s.t. O U C (R n ) ΛR m. Batchelor Theorem ( 79) : N (N, E), ΠE = (N, Γ(ΛE )). Example : M = T M ΠTM supercotangent bundle over (M, g). Rothstein Theorem ( 91) : (N, ω) (N, ω 0, E, g, ). (M, ω) over (M, g), given by ω = dα, α = p i dx i + 2i g ijξ i d ξ j. Darboux coordinates (non-tensorial!) : ξ a = θ a i ξi and p i = p i + 2i ω iab ξ a ξb, with θ a i θb j g ij = η ab and ω iab = ξ a i ξ b. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 7 / 16

Classical aspects of spin mechanics The momentum of a rotation is : J Xij = p i x j x j p i + i ξ iξ j, hence the spin is S ij = i ξ iξ j, generating (Ω 2 x(m), {, }) o(p, q). Example Rotating particle e.o.m.= Papapetrou equations = Hamiltonian flows of g ij p i p j : ẋ j j ẋ i = 1 2 gik R(S) jk ẋ j, ẋ k k S ij = 0. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 8 / 16

GQ of the supercotangent bundle Locally : merging of the GQ of T R n and ΠR n. Explicit formulae in Darboux coord. Globally : we need a polarization in the fibers ΠT x M, almost Hermitian structure on (M, g). Polarized functions : Λ(T 1,0 M), holomorphic diff. forms. Γ(Cl(M, g)) End(Λ(T 1,0 M) Ber 1 2 ), spinor bundle S! Remark : coincide with usual construction of S over pseudo-hermitian manifold : almost Hermitian structure Spin c -structure, + existence Ber 1 2 (= K 1 2 ) Spin-structure, induced action of u(n) on S is the natural one on Λ(T 1,0 M) Ber 1 2. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 9 / 16

Vect(M) action on the supercotangent bundle Tensorial action : L X = X i i p j ( i X j ) pi + ξ j ( j X i ) ξ i. Hamiltonian action : L X α = 0, given by? Lemma ) X = X i i + Y ij ξ j ξi p j i X j pi + 2i (Rlij k ξ kξ l X j ( i Y kl )ξ k ξ l pi, where Y is an arbitrary 2-form on M, depending linearly on X. 1 Y = 0 Vect(M)-action iff is flat. Lift by of Ham. action on T M, "moment map" : J X = p i X i. 2 L X β = f β, where β = g ij ξ i dx j, dβ odd sympl. form on ΠTM. Then, conf(m, g)-action, Y ij = [i X j]. Moment map : J X = p i X i + 2i ξj ξ k [k X j]. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 10 / 16

Vector fields action on the spinor bundle We expect that an action of X on Γ(S) is of the form X + Y ij γ i γ j, with Y as above. For the representation space Γ(S), GQ is a Lie algebra morphism on : Sb 0,0 Sb 1,0 Sb 0,1 Sb 0,2, where Sb k,κ = Pol k (T M) Ω κ C (M). Hence, Hamiltonian lift Lie derivative of spinors. Theorem GQ(J X ) = X, spinor covariant derivative ; GQ(J X ) = l X = X + 1 4 [kx j] γ j γ k, spinor Lie derivative (Kosmann 72). Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 11 / 16

Algebra of spinor differential operators D as a deformation of Sb Spinless case : σ p+q 1 ([P, Q]) = {σ p (P), σ q (Q)}. Dirac operator : γ i i, Lichnerowicz Laplacian : [γ i i, γ i i ] = + R 4, second order diff. op. usual filtration by order D 0 D 1 does not fit. Idea : Hamiltonian filtration with i of order 1 and γ i of order 1 2. [ i, j ] = R ijkl γ k γ l and [γ i, γ j ] = 2g ij. Theorem D admits a filtration s.t. D K /D K 1 Sb K = 2k+κ=K Sb k,κ and 2 (Sb, {, }) is the graded version of (D, [, ]) σ K +L 1 ([P, Q]) = {σ K (P), σ L (Q)}. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 12 / 16

D as deformation of the conf(m, g)-module Sb Let us suppose that (M, g) is conformally flat, loc. g ij = Fη ij. D, adjoint action L X A = [l X, A], preserves Ham. and usual filtration. Sb, Ham. action preserves Ham. graduation, on Sb,κ : X = L X κ ( n ( ix i ) + Rlij k ξ kξ l X j ( i Y kl )ξ k ξ l) pi. As σ(gq(j X )) = J X, we have σ([l x, ]) = {J X, }. (Sb, L) is the graded version of (D, L). Proposition T = κ Sb,κ F κ n endowed with the tensorial action L = L κ n on Sb,κ is the graded version of (Sb, L) and (D, L) for the usual filtration. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 13 / 16

Main result Theorem There exists a unique morphism of conf(m, g)-modules preserving the principal symbol for the usual filtration S δ T : T δ Sb δ, it is named the conformally equivariant superization. Theorem There exists a unique morphism of conf(m, g)-modules preserving the principal symbol for the Ham. filtration, Q λ,µ : Sb δ D λ,µ, if µ λ = δ. It is named the conformally equivariant quantization. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 14 / 16

Examples and applications Examples : 1 S 0 T (J X = p i X i ) = J X = p i X i + 2i ξj ξ k [k X j]. 2 Dirac operator from De Rham differential : Q S T (p i ξ i ) = γ i i. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 15 / 16

Examples and applications Examples : 1 S 0 T (J X = p i X i ) = J X = p i X i + 2i ξj ξ k [k X j]. 2 Dirac operator from De Rham differential : Q S T (p i ξ i ) = γ i i. Hamiltonian description of the Killing-Yano tensors, i.e. diff. κ-forms T s.t. [i0 T i1 i κ] = 0, T T 1 n 0,κ s.t. {p iξ i, S 0 T (φ(t ))} = 0 T is a Killing-Yano tensor. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 15 / 16

Thanks! References : Quantification conformément équivariante des fibrés supercotangents, Jean-Philippe Michel, thèse, tel-00425576 version 1. Conformal geometry of the supercotangent and spinor bundles, Jean-Philippe Michel, arxiv :1004.1595v1. Conformally equivariant quantization of supercotangent bundles, Jean-Philippe Michel, in preparation. Jean-Philippe MICHEL (UL) NCTS-CPT Workshop Hsinshu, 24-02-2011 16 / 16

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