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A natural Lie super-algebra bundle on rank 3 WSD manifolds. (English) Zbl 1158.81023
Summary: We prove the existence of a natural \(*\)-Lie super-algebra bundle on any orientable WSD manifold of rank 3. We describe in detail the associated Lie super-algebra \(\mathcal L_{3,\mathbb C}\) of global sections. We show that \(\mathcal L_{3,\mathbb C}\) is a product of \(\mathbf{sl}(4,\mathbb C)\) with the full special linear super-algebras of some graded vector spaces isotypical with respect to a natural action of \(\mathbf{so}(3,\mathbb R)\). We give an explicit description of a geometrically natural real form of \(\mathcal L_{3,\mathbb C}\). This real form is made up of \(\mathbf{so}(3,\mathbb R)\)-invariant operators which preserve the Poincaré pairing on the bundle of forms.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
58C50 Analysis on supermanifolds or graded manifolds
17A70 Superalgebras
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
81T60 Supersymmetric field theories in quantum mechanics
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