Toric Calabi-Yau supermanifolds and mirror symmetry

TL;DR

This paper explores mirror symmetry in supermanifolds derived from toric varieties, revealing a relation between super-Calabi-Yau conditions and superpotential properties in the mirror B-model.

Contribution

It establishes a connection between super-Calabi-Yau conditions and quasi-homogeneity of the mirror superpotential, linking fermionic charge matrices to superpotential degree.

Findings

Super-Calabi-Yau conditions correspond to quasi-homogeneity in the B-model.

The superpotential degree relates to the determinant of the fermion charge matrix.

Mirror symmetry properties extend to fermionic extensions of toric varieties.

Abstract

We study mirror symmetry of supermanifolds constructed as fermionic extensions of compact toric varieties. We mainly discuss the case where the linear sigma A-model contains as many fermionic fields as there are U(1) factors in the gauge group. In the mirror super-Landau-Ginzburg B-model, focus is on the bosonic structure obtained after integrating out all the fermions. Our key observation is that there is a relation between the super-Calabi-Yau conditions of the A-model and quasi-homogeneity of the B-model, and that the degree of the associated superpotential in the B-model is given in terms of the determinant of the fermion charge matrix of the A-model.