Mirror symmetry in emergent gravity

Hyun Seok Yang

doi:10.1016/j.nuclphysb.2017.07.003

Abstract

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.

Highlights

Emergent gravity is based on a novel form of the equivalence principle known as the Darboux theorem or the Moser lemma in symplectic geometry stating that the electromagnetic force can always be eliminated by a local coordinate transformation as far as spacetime admits a symplectic structure, in other words, a microscopic spacetime becomes noncommutative (NC) [1, 2, 3]
We explore the relation between the NC U(1) gauge theory in six dimensions and the Kahler gravity on a non-compact CY threefold and identify the curvature of a holomorphic line bundle with the Kahler form for a CY manifold
There exist two independent NC ⋆-algebras to define a dynamical NC spacetime. They are separately obtained by quantizing the line bundles L and L describing the deformation of symplectic structures in Ω2(M) and ∗Ω4(M), respectively

Summary

Introduction

Emergent gravity is based on a novel form of the equivalence principle known as the Darboux theorem or the Moser lemma in symplectic geometry stating that the electromagnetic force can always be eliminated by a local coordinate transformation as far as spacetime admits a symplectic structure, in other words, a microscopic spacetime becomes noncommutative (NC) [1, 2, 3]. Due to the nontrivial four-form on a CY manifold, one can define the second dual holomorphic line bundle whose curvature is related to the Hodge-dual of the four-form and argue that the CY manifold emergent from the dual holomorphic line bundle is mirror to the CY manifold emergent from the ordinary holomorphic line bundle This relation has been found and studied in [13] and the similar question has been further discussed in the papers [14, 15, 16], the language is a bit different, involving the counting of cohomologies via topological strings or the duality transformations of D-branes of different codimensions.

Holomorphic Line Bundle and Kahler Manifolds

Doubling of Emergent Calabi-Yau Manifolds

Mirror Symmetry of Emergent Geometry

Discussion