On Local Calabi-Yau Supermanifolds and Their Mirrors

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We use local mirror symmetry to study a class of local Calabi-Yau super-manifolds with bosonic sub-variety Vb having a vanishing first Chern class. Solving the usual super- CY condition, requiring the equality of the total U(1) gauge charges of bosons Φb and the ghost like fields Ψf one ∑bqb=∑fQf, as ∑bqb=0 and ∑fQf=0, several examples are studied and explicit results are given for local Aᵣ super-geometries. A comment on purely fermionic super-CY manifolds corresponding to the special case where qb=0, ∀ b and ∑fQf=0 is also made.