Mirror symmetry

About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.

Flat Connections in Open String Mirror Symmetry

Abstract: We study a flat connection defined on the open-closed deformation space of open string mirror symmetry for type II compactifications on Calabi-Yau threefolds with D-branes. We use flatness and integrability conditions to define distinguished flat coordinates and the superpotential function at an arbitrary point in the open-closed deformation space. Integrability conditions are given for concrete deformation spaces with several closed and open string deformations. We study explicit examples for expansions around different limit points, including orbifold Gromov-Witten invariants, and brane configurations with several brane moduli. In particular, the latter case covers stacks of parallel branes with non-Abelian symmetry.

Open Gromov-Witten invariants and mirror maps for semi-Fano toric manifolds

Abstract: We prove that for a compact toric manifold whose anti-canonical divisor is numerically effective, the Lagrangian Floer superpotential defined by Fukaya-Oh-Ohto-Ono is equal to the superpotential written down by using the toric mirror map under a convergence assumption. This gives a method to compute open Gromov-Witten invariants using mirror symmetry.

Arithmetic Properties of Moduli Spaces and Topological String Partition Functions of Some Calabi-Yau Threefolds

Abstract: This thesis studies certain aspects of the global properties, including geometric and arithmetic, of the moduli spaces of complex structures of some special Calabi-Yau threefolds (B-model), and of the corresponding topological string partition functions defined from them which are closely related to the generating functions of Gromov-Witten invariants of their mirror Calabi-Yau threefolds (A-model) by the mirror symmetry conjecture. For the mirror families (B-model) of the one-parameter families (A-model) of KP2 , KdPn , n = 5, 6, 7, 8 with varying Kahler structures, the bases are the moduli spaces of complex structures of the corresponding mirror Calabi-Yaus. We identify them with certain modular curves by studying the Picard-Fuchs systems and periods of the corresponding mirror families. In particular, the singular points on the moduli spaces correspond to the cusps and elliptic points on the modular curves. We take the BCOV holomorphic anomaly equations with boundary conditions as the defining equations for the topological string partition functions. Using polynomial recursion and the above identification, we interpret the boundary conditions as regularity conditions for modular forms and express the equations purely in terms of the language of modular form theory. This turns the problem of solving the equations into a combinatorial problem. We also solve for the first few topological string partition functions genus by genus recursively in terms of almost-holomorphic modular forms. Assuming the validity of mirror symmetry conjecture, we prove a version of integrality for the Gromov-Witten