On a complex-symplectic mirror pair

TL;DR

This paper rigorously constructs mirror symmetry between certain super-symmetric sigma-models using complex-symplectic structures on loop spaces, linking moduli of generalized complex structures with doubled geometry.

Contribution

It provides a rigorous mathematical framework for mirror symmetry via Poisson structures and quantization on loop spaces of twisted tori.

Findings

Constructed mirror symmetry as an intertwiner of N=2 super-conformal structures.

Identified moduli of equivariant generalized complex structures with doubled geometry moduli.

Established a connection between super-symplectic structures and mirror symmetry in complex dimension 2.

Abstract

We study the canonical Poisson structure on the loop space of the super-double-twisted-torus and its quantization. As a consequence we obtain a rigorous construction of mirror symmetry as an intertwiner of the N=2 super-conformal structures on the super-symmetric sigma-models on the Kodaira-Thurston nilmanifold and a gerby torus of complex dimension 2. As an application we are able to identify global moduli of equivariant generalized complex structures on these target spaces with moduli of equivariant orthogonal complex structures on the doubled geometry.