Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{ 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(1) 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(1) 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.
Categories: hep-th math-ph math.DG math.MP