Preview

PDF, English Download (3MB) | Terms of use

Abstract

We formulate extended mirror symmetry of Calabi-Yau threefolds with D- branes as an equivalence between variations of mixed Hodge structure under the mirror map. After an introduction to Hodge theoretic closed string mir- ror symmetry, we review the relation between D-branes, normal functions and extensions by algebraic cycles on the side of the B-model. We define an exten- sion of the A-model variation of mixed Hodge structure whose flat connection is derived from an enhancement of the quantum product by holomorphic disks ending on Lagrangian submanifolds. Our construction is based on the Solomon- Tukachinsky axioms for open Gromov-Witten invariants together with the open WDVV equations and matches the predictions from extended mirror symmetry. For the particular case of homology spheres, we define an extension of Iritani’s Gamma-integral local system and propose an extended version of the Gamma conjecture. We demonstrate the validity of the conjecture for the standard pair of branes in case of the quintic and prove a corresponding extended Mir- ror Theorem. Using the extended holomorphic anomaly equations, we explore novel invariants from one-loop amplitudes for cycles of van Geemen-type, whose A-model geometry is at present unknown.