(0,2) Mirror Symmetry on homogeneous Hopf surfaces

TL;DR

This paper presents the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds, specifically on homogeneous Hopf surfaces with special metrics, using vertex algebra techniques and T-duality.

Contribution

It introduces the first (0,2) mirror symmetry examples on non-Kähler manifolds, constructed via vertex algebras and homogeneous geometry.

Findings

Examples of (0,2) mirror pairs on Hopf surfaces.

Reduction to Killing spinors on quadratic Lie algebras.

Embedding of N=2 superconformal vertex algebra.

Abstract

In this work we find the first examples of (0,2) mirror symmetry on compact non-K\"ahler complex manifolds. For this we follow Borisov's approach to mirror symmetry using vertex algebras and the chiral de Rham complex. Our examples of (0,2) mirrors are given by pairs of Hopf surfaces endowed with a Bismut-flat pluriclosed metric. Requiring that the geometry is homogeneous, we reduce the problem to the study of Killing spinors on a quadratic Lie algebra and the construction of embeddings of the $N=2$ superconformal vertex algebra in the superaffine vertex algebra, combined with topological T-duality.