On Local Calabi-Yau Supermanifolds and Their Mirrors

TL;DR

This paper investigates local Calabi-Yau supermanifolds with bosonic sub-varieties using mirror symmetry, deriving conditions for super-CY structures and providing explicit examples, including local A_r super-geometries and fermionic cases.

Contribution

It introduces a super-geometry framework for local Calabi-Yau manifolds, deriving super-CY conditions and analyzing explicit examples with mirror symmetry.

Findings

Derived super-CY conditions for supermanifolds.

Provided explicit examples of local A_r super-geometries.

Commented on purely fermionic super-CY manifolds.

Abstract

We use local mirror symmetry to study a class of local Calabi-Yau super-manifolds with bosonic sub-variety V_b having a vanishing first Chern class. Solving the usual super- CY condition, requiring the equality of the total U(1) gauge charges of bosons \Phi_{b} and the ghost like fields \Psi_{f} one \sum_{b}q_{b}=\sum_{f}Q_{f}, as \sum_{b}q_{b}=0 and \sum_{f}Q_{f}=0, several examples are studied and explicit results are given for local A_{r} super-geometries. A comment on purely fermionic super-CY manifolds corresponding to the special case where q_{b}=0, \forall b and \sum_{f}Q_{f}=0 is also made.\bigskip