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One-dimensional $\sigma $-models with $N = 5, 6, 7, 8$ off-shell supersymmetries. (English) Zbl 1175.81119
Summary: We computed the actions for the one-dimensional (1D) $N = 5$ $\sigma$-models with respect to the two inequivalent $(2, 8, 6)$ multiplets. Four supersymmetry generators are manifest, while the constraint originated by imposing the fifth supersymmetry automatically induces a full $N = 8$ off-shell invariance. The resulting action coincides in the two cases and corresponds to a conformally flat 2D target satisfying a special geometry of rigid type. To obtain these results we developed a computational method (for Maple 11) which does not require the notion of superfields and is instead based on the nowadays available list of the inequivalent representations of the 1D $N$-extended supersymmetry. Its application to systematically analyze the $\sigma $-models off-shell invariant actions for the remaining $N = 5, 6, 7, 8$ ($k, 8, 8-k$) multiplets, as well as for the $N>8$ representations, only requires more cumbersome computations.

81Q60Supersymmetry and quantum mechanics
81R12Relations of groups and algebras in quantum theory with integrable systems
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