Shinobu Hosono

Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

Abstract We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton corrected Yukawa couplings and the topological one-loop partition function to the case of complete intersections with higher dimensional moduli spaces. We will develop a new method of obtaining the instanton corrected Yukawa couplings through a study of the solutions of the Picard-Fuchs equations. This leads to closed formulas for the prepotential for the Kahler moduli fields induced from the ambient space for all complete intersections in nonsingular weighted projective spaces. As examples we treat part of the moduli space of the phenomenologically interesting three-generation models which are found in this class. We also apply our method to solve the simplest model in which a topology change was observed and discuss examples of complete intersections in singular ambient spaces.

Lectures on mirror symmetry

We give an introduction to mirror symmetry of strings on Calabi-Yau manifolds with an emphasis on its applications e.g. for the computation of Yukawa couplings. We introduce all necessary concepts and tools such as the basics of toric geometry, resolution of singularities, construction of mirror pairs, Picard-Fuchs equations, etc. and illustrate all of this on a non-trivial example.

Maximal degeneracy points of GKZ systems

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel’fand-Kapranov-Zelevinsky(GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exists certain special boundary points, which we called maximal degeneracy points, at which all but one solutions become singular. 3/2/96 † email: hosono@sci.toyama-u.ac.jp ‡ email: lian@max.math.brandeis.edu ⋄ email: yau@math.harvard.edu

Autoequivalences of derived category of a K3 surface and monodromy transformations

We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge isometries of the Mukai lattice. Motivated by homological mirror symmetry we also consider the mirror counterpart, i.e. symplectic version of it. In the case of ρ(X) = 1, we find an explicit formula which reproduces the number of Fourier-Mukai (FM) partners from the monodromy problem of the mirror K3 family. We present an explicit example in which a monodromy action does not come from an autoequivalence of the mirror side.

Lie algebra cohomology and N=2 SCFT based on the GKO construction

We interpretN=2 superconformal field theories (SCFTs) formulated by Kazama and Suzuki via Goddard-Kent-Olive (GKO) construction from a viewpoint of the Lie algebra cohomology theory for the affine Lie algebra. We determine the cohomology group completely in terms of a certain subset of the affine Weyl group. We find that this subset describing the cohomology group can be obtained from its classical counterpart by the action of the Dynkin diagram automorphisms. Some algebra automorphisms of theN=2 superconformal algebra are also formulated. Utilizing the algebra automorphisms, we study the field identification problem for the branching coefficient modules in the GKO-construction. Also the structure of the Poincaré polynomial defined for eachN=2 theory is revealed.

GKZ Systems, Gröbner Fans, and Moduli Spaces of Calabi-Yau Hypersurfaces

We present a detailed analysis of the GKZ (Gel’fand, Kapranov and Zelevinski) hypergeometric systems in the context of mirror symmetry of Calabi-Yau hypersurfaces in toric varieties. As an application, we will derive a concise formula for the prepotential about large complex structure limits.

c = 2 Rational Toroidal Conformal Field Theories via the Gauss Product

We find a concise relation between the moduli τ,ρ of a rational Narain lattice Γ(τ,ρ) and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

Kummer structures on a K3 surface: An old question of T. Shioda

We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures, posed by T. Shioda 25 years ago.

Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds

We study mirror symmetry of complete intersection Calabi-Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.