On a complex-symplectic mirror pair
Marco Aldi, Reimundo Heluani
TL;DR
The paper constructs a canonical Poisson structure on the super-loop space of the double twisted torus and its quantization, enabling a rigorous formulation of mirror symmetry as an intertwiner of $N=2$ super-conformal structures on sigma-models with target spaces related by T-duality. By lifting to the $8$-dimensional doubled geometry $X\times T^2$, it recasts symplectic and complex data as equivariant complex structures and derives an $N=2$ algebra on the doubled Hilbert space with central charge $c=12$. It then identifies moduli of equivariant generalized complex structures on the gerby torus $Y$ with moduli of equivariant orthogonal complex structures on the doubled geometry, and shows an isomorphism of state spaces $\mathscr{H}\otimes \mathscr{H}_{T^2} \simeq \mathscr{H}_Y \simeq \mathscr{H}_N$, realizing mirror symmetry as an intertwiner between N=2 structures. Overall, the work provides a rigorous, algebraic-geometry–inspired framework for mirror symmetry in a non-Kähler, doubled-geometry context with explicit moduli correspondences.
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.