We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We emphasize the simplifying features of dimension three and supply more robust methods that do not rely on such special characteristics and hence naturally generalize to other dimensions. The moduli spaces for Calabi–Yau d-folds are somewhat different from the “special Kähler manifolds” which had...

In the present study, we examined how people code object location when an object is hidden in an enclosed space. On object location tasks involving disorientation, viewers must code the location of an object in relation to the spatial environment because they cannot directly track their changing relation to the object. Recently, we showed that viewers also code their perspective relative to the...

Starting with the work of Yau [Y1], Donaldson [D1], and Uhlenbeck-Yau [UY], the notion of stability has revealed itself under many guises to be closely related to the existence of canonical metrics in Kähler geometry. The equivalence between Hermitian-Einstein metrics on vector bundles and Mumford stability was proved by Donaldson and Uhlenbeck-Yau in [D1] and [UY], while the existence of Kähle...

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-CalabiYau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite globa...

A smoothing theorem for normal crossings to Calabi-Yau manifolds was proved by Y. Kawamata and Y. Namikawa ([KaNa]). This paper is a study of the observation that the Picard groups and Chern classes of these Calabi-Yau manifolds are constructible from the normal crossings in such smoothings. We provide and prove the formula for the construction in its full generality and various applications ar...

These are introductory lecture notes on complex geometry, Calabi–Yau manifolds and toric geometry. We first define basic concepts of complex and Kähler geometry. We then proceed with an analysis of various definitions of Calabi–Yau manifolds. The last section provides a short introduction to toric geometry, aimed at constructing Calabi–Yau manifolds in two different ways; as hypersurfaces in to...

In this note we explicitly work out the precise relationship between Ext groups and massless modes of D-branes wrapped on complex submanifolds of Calabi-Yau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describing Calabi-Yau manifolds at large radius, directly in BCFT using standard methods. Naively these ...