Extending Mirror Conjecture to Calabi-Yau with Bundles

TL;DR

This paper extends mirror symmetry concepts to Calabi-Yau manifolds with bundles, linking Hodge structure variations to holomorphic map counts and proposing a new perspective on bundles via supersymmetric cycles.

Contribution

It introduces a novel notion of mirror for Calabi-Yau manifolds with stable bundles, connecting bundle cohomology to supersymmetric cycles and holomorphic map enumeration.

Findings

Defines mirror of Calabi-Yau with bundles in string theory context

Relates Hodge structure variation to holomorphic map counting

Suggests studying bundles via supersymmetric cycles on the mirror

Abstract

We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi-Yau manifolds in terms of supersymmetric cycles on the mirror.