Mirror symmetry

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

Open string amplitudes and large order behavior in topological string theory

Abstract: We propose a formalism inspired by matrix models to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds. We find closed expressions for various open string amplitudes beyond the disk, and in particular we write down the annulus amplitude in terms of theta functions on a Riemann surface. We test these ideas on local curves and local surfaces, providing in this way generating functionals for open Gromov-Witten invariants in the spirit of mirror symmetry. In the case of local curves, we study the open string sector near the critical point which leads to 2d gravity, and we show that toric D-branes become FZZT branes in a double-scaling limit. We use this connection to compute non-perturbative instanton effects due to D-branes that control the large order behavior of topological string theory on these backgrounds

The modularity of K3 surfaces with non-symplectic group actions

Abstract: We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank The purpose of this note is to study the Galois representations associated to these K3 surfaces The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14 We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type We will establish the modularity of the Galois representations associated to them Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception

Evolution Equations for Special Lagrangian 3-Folds in C3

Abstract: This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m − 1)-submanifold P in R n , and constructed a family of special Lagrangian m-folds N in C m , which are swept out by the image of P under a 1-parameter family of affine maps φ t : R n → C m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C3. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C3.Our main results are a number of new families of special Lagrangian 3-foldsin C3, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R3 or % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuGqLXgBG0evaeXatLxBI9gBam% XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB% Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf% euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9% q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba% WaaqaafaaakeaaruWrPDwAaGGbciab-nfatbaa!3D86! 1× R2. Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.

Special Lagrangian submanifolds of log Calabi-Yau manifolds

Abstract: We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kahler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|−KY| is a smooth divisor with D2=d, then X=Y∖D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y= P2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kahler rotation, X can be compactified to the complement of a Kodaira type Id fiber appearing as a singular fiber in a rational elliptic surface πˇ:Yˇ→P1.

Determinantal Quintics and Mirror Symmetry of Reye Congruences

Abstract: We study a certain family of determinantal quintic hypersurfaces in $${\mathbb{P}^{4}}$$ whose singularities are similar to the well-studied Barth–Nieto quintic. Smooth Calabi–Yau threefolds with Hodge numbers (h 1,1,h 2,1) = (52, 2) are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi–Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi–Yau threefolds in $${\mathbb{P}^{4}\times\mathbb{P}^{4}}$$ which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over $${\mathbb{P}^{2}}$$ and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over $${\mathbb{P}^{2}}$$ completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.