Supermanifolds, Rigid Manifolds and Mirror Symmetry

TL;DR

This paper explores the relationship between Landau-Ginzburg orbifolds, sigma models, and supermanifolds, proposing that mirror symmetry involves super-varieties and analyzing their conformal invariance and chiral rings.

Contribution

It establishes a correspondence between mirror symmetry and supermanifolds, expanding the understanding of mirror pairs beyond bosonic varieties.

Findings

Mirror of a rigid manifold is a supermanifold.

Sigma models with super-target spaces can be conformally invariant.

Mirror symmetry is better viewed as a relation among super-varieties.

Abstract

By providing a general correspondence between Landau-Ginzburg orbifolds and non-linear sigma models, we find that the elusive mirror of a rigid manifold is actually a supermanifold. We also discuss when sigma models with super-target spaces are conformally invariant and describe their chiral rings. Both supermanifolds with and without Kahler moduli are considered. This work leads us to conclude that mirror symmetry should be viewed as a relation among super-varieties rather than bosonic varieties.