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Quantisation of the Laplacian and a Curved Version of Geometric Quantisation

par Meyer, Julien
Président du jury Gutt, Simone
Promoteur Fine, Joel
Publication Non publié, 2016-08-29

Thèse de doctorat

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Résumé :

Let (E,h) be a holomorphic, Hermitian vector bundle over a polarized manifold. We provide a canonical quantisation of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian. In the case when the bundle E is the trivial line bundle, we quantise solutions to the heat equation on the manifold. Furthermore we show that geometric quantisation can be seen as the differential of a natural map between two Riemannian manifolds. Motivated by this fact we compute its next order approximation, namely its Hessian.