2014/12/31 by Hyun Seok Yang · 8 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Duality (order theory) #Gauge theory #Geometry #Mathematical physics #Mathematics #Mirror symmetry #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Physics #Pure mathematics #Symmetry (geometry) #Symplectic geometry #Symplectic manifold #Symplectic representation #Symplectomorphism #Theoretical physics #Variety (cybernetics) #hep-th #math-ph #math.DG #math.MP
paper · pdf · open access · doi:10.1016/j.nuclphysb.2017.07.003
published in Nuclear Physics B 922, 264-279 (Elsevier BV) · v4; 20 pages, version to appear in Nucl. Phys. B
arxiv created 2017/07/10 · openalex publication_date 2017/07/17 · arxiv updated 2017/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Given a six-dimensional symplectic manifold ( M , B ) , a nondegenerate, co-closed four-form C introduces a dual symplectic structure 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.