Mirror Symmetry in Emergent Gravity

Abstract

Given a six-dimensional symplectic manifold $(M,B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\tilde{B}=\ast C$ independent of $B$ via the Hodge duality $\ast$. 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.

v4; 20 pages, version to appear in Nucl. Phys. B

References in corpus (8)

• Symmetry, Gravity and Noncommutativity
• Emergent Gravity from Noncommutative Spacetime
• Multi-matrix models and emergent geometry
• Noncommutative Electromagnetism As A Large N Gauge Theory
• Quantization of Emergent Gravity
• Calabi-Yau manifolds from noncommutative Hermitian U(1) instantons
• Highly Effective Action from Large N Gauge Fields
• Noncommutative Instantons on CP^n

Cited by in corpus (6)

• Calabi-Yau manifolds from noncommutative Hermitian U(1) instantons
• Contravariant Gravity on Poisson Manifolds and Einstein Gravity
• Dark Energy and Dark Matter in Emergent Gravity
• Quantized Kähler Geometry and Quantum Gravity
• Hermitian-Einstein metrics from noncommutative $U\left(1\right)$ instantons
• Towards a T-dual Emergent Gravity