Mirror symmetry in emergent gravity

Given a six-dimensional symplectic manifold $(M,B)$, a nondegenerate, co-closed four-form C introduces a dual symplectic structure $\tilde{B}=⁎C$ independent of B via the Hodge duality ⁎. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of noncommutative $U\left(1\right)$ gauge fields by considering the Seiberg–Witten map for each symplectic structure. As a result, emergent gravity suggests a beautiful picture that the variety of six-dimensional manifolds emergent from noncommutative $U\left(1\right)$ gauge fields is doubled. In particular, the doubling for the variety of emergent Calabi–Yau manifolds allows us to arrange a pair of Calabi–Yau manifolds such that they are mirror to each other. Therefore, we argue that the mirror symmetry of Calabi–Yau manifolds is the Hodge theory for the deformation of symplectic and dual symplectic structures.