Mirror symmetry

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

Homological Mirror Symmetry and Algebraic Cycles

Abstract: In this chapter we outline some applications of Homological Mirror Symmetry to classical problems in Algebraic Geometry, like rationality of algebraic varieties and the study of algebraic cycles. Several examples are studied in detail.

On the Quantum Cohomology of Homogeneous Varieties

Abstract: About a decade ago physicists set off something of a “big bang” in the universe of algebraic geometry. A new approach to enumerative geometry, involving mirror symmetry, solved some old questions and raised many new ones. To mention just a few of the areas influenced by these developments, we have new approaches to the study of Calabi-Yau manifolds and orbifolds, the notions of Gromov-Witten and related invariants, and new results about intersection theory on moduli spaces.

Theta functions and mirror symmetry

Abstract: This is a survey covering aspects of varied work of the authors with Mohammed Abouzaid, Paul Hacking, and Sean Keel. While theta functions are traditionally canonical sections of ample line bundles on abelian varieties, we motivate, using mirror symmetry, the idea that theta functions exist in much greater generality. This suggestion originates with the work of the late Andrei Tyurin. We outline how to construct theta functions on the degenerations of varieties constructed in previous work of the authors, and then explain applications of this construction to homological mirror symmetry and constructions of broad classes of mirror varieties.

Supersymmetric vortex defects in two dimensions

Abstract: We study codimension-two BPS defects in 2d $$ \mathcal{N} = \left(2,\ 2\right) $$ supersymmetric gauge theories, focusing especially on those characterized by vortex-like singularities in the dynamical or non-dynamical gauge field. We classify possible SUSY-preserving boundary conditions on charged matter fields around the vortex defects, and derive a formula for defect correlators on the squashed sphere. We also prove an equivalence relation between vortex defects and 0d-2d coupled systems. Our defect correlators are shown to be consistent with the mirror symmetry duality between Abelian gauged linear sigma models and Landau-Ginzburg models, as well as that between the minimal model and its orbifold. We also study the vortex defects inserted at conical singularities.

Calabi-Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory

Abstract: We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear sigma-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kahler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kahler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological Calabi-Yau space---the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kahler and complex structure moduli spaces of a Calabi-Yau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kahler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized by the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory.