Use cases of Embedded submanifolds

 

Use cases of Embedded submanifolds part6(Machine Learning)

Abstract : We describe the multisymplectic analysis of the constraints of first-order embedded submanifolds inherited from diffeomorphisms of the ambient manifold. The ambient diffeomorphism deformations of first-order embedded submanifolds are examined. We find that the covariant Noether currents, corresponding to the inherited ambient diffeomorphism symmetry, satisfy a non-Abelian deformation algebra, the structure functions being the Cartan structure functions on the ambient manifold. We define the covariant kinematical phase space of pseudoholomorphic embeddings (the symplectic 2-submanifolds of a symplectic manifold) explicitly as a subbundle of the covariant kinematical phase space of embeddings. The induced algebra of Noether currents satisfies the same algebra as before, the symmetry thus being preserved on this subclass of embeddings. The graded multisymplectic manifolds of the covariant Hamiltonian BRST formalism, developed by the author, are explicitly constructed for the symmetry of embeddings and pseudoholomorphic embedding.The ``(supersymmetry) multiplet and its dualities’’ postulated by Witten arise naturally as the local fibre coordinates on the graded phase space. The BRST algebra for the pseudoholomorphic class is computed. The structure functions implicit in Witten’s treatment of the topological sigma model arise as the Cartan structure functions in a Darboux basis on the ambient symplectic manifold

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