TY - JOUR

T1 - Weyl quantization of degree 2 symplectic graded manifolds

AU - Grützmann, Melchior

AU - Michel, Jean Philippe

AU - Xu, Ping

N1 - Funding Information:
Research partially funded by the NSF grants DMS-2001599 , DMS-1707545 , DMS-1406668 , DMS-1101827 , the Luxembourgian NRF , AFR grant PDR-09-063 and the Belgian Interuniversity Attraction Poles (IAP) Program initiated by the Belgian Science Policy Office (framework “Dynamics, Geometry and Statistical Physics” (DYGEST)).
Publisher Copyright:
© 2021

PY - 2021/10

Y1 - 2021/10

N2 - Let S be a spinor bundle of a pseudo-Euclidean vector bundle (E,g) of even rank. We introduce a new filtration on the algebra D(M,S) of differential operators on S. As a main property, the associated graded algebra grD(M,S) is proved to be isomorphic to the algebra O(M) of smooth functions on M, where M is the degree 2 symplectic graded manifold canonically associated to (E,g). Accordingly, we establish the Weyl quantization of M as a map WQħ:O(M)→D(M,S), and prove that WQħ satisfies all the desired properties of quantizations. As an application, we obtain a bijection between Courant algebroid structures (E,g,ρ,〚⋅,⋅〛) equivalently characterized by Hamiltonian generating functions on M, and skew-symmetric Dirac generating operators D∈D(M,S). The operator D2 gives a new invariant of (E,g,ρ,〚⋅,⋅〛) which generalizes the square norm of the Cartan 3-form of a quadratic Lie algebra. We study this invariant in detail in the particular case of E being the double of a Lie bialgebroid (A,A⁎).

AB - Let S be a spinor bundle of a pseudo-Euclidean vector bundle (E,g) of even rank. We introduce a new filtration on the algebra D(M,S) of differential operators on S. As a main property, the associated graded algebra grD(M,S) is proved to be isomorphic to the algebra O(M) of smooth functions on M, where M is the degree 2 symplectic graded manifold canonically associated to (E,g). Accordingly, we establish the Weyl quantization of M as a map WQħ:O(M)→D(M,S), and prove that WQħ satisfies all the desired properties of quantizations. As an application, we obtain a bijection between Courant algebroid structures (E,g,ρ,〚⋅,⋅〛) equivalently characterized by Hamiltonian generating functions on M, and skew-symmetric Dirac generating operators D∈D(M,S). The operator D2 gives a new invariant of (E,g,ρ,〚⋅,⋅〛) which generalizes the square norm of the Cartan 3-form of a quadratic Lie algebra. We study this invariant in detail in the particular case of E being the double of a Lie bialgebroid (A,A⁎).

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U2 - 10.1016/j.matpur.2021.08.001

DO - 10.1016/j.matpur.2021.08.001

M3 - Article

AN - SCOPUS:85114088470

SN - 0021-7824

VL - 154

SP - 67

EP - 107

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -