Related papers: Mirror Transform and String Theory
We review various constructions of mirror symmetry in terms of Landau-Ginzburg orbifolds for arbitrary central charge $c$ and \CY\ hypersurfaces and complete intersections in toric varieties. In particular it is shown how the different…
We use a relative Fourier-Mukai transform on elliptic K3 surfaces $X$ to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of $X$ reproduces relative T-duality and provides an infinitesimal isometry…
We give a brief survey of some of the geometry of mirror symmetry, written in 2004 for the "Encyclopaedia of Mathematical Physics". Probably a little bit out of date now in a few places, but hey.
We study the effect of mirror symmetry for K3 surfaces on D-brane probe physics. The case of elliptically fibered K3 surfaces is considered in detail. In many cases, mirror can transform a singular fiber of Kodaira's type ADE into sets of…
We study new classes of metric transformations in the context of scalar-tensor theories, which involve both higher derivatives of the scalar field and derivatives of the metric itself. In general, such transformations are not invertible as…
The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg…
This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The…
We give an introductory review of Fourier-Mukai transforms and their application to various aspects of moduli problems, string theory and mirror symmetry. We develop the necessary mathematical background for Fourier-Mukai transforms such as…
(Talk given at the Oskar Klein centenary symposium 19-21 September 1994 in Stockholm, Sweden, to appear in the proceedings.) The first half of this talk is a non-technical discussion of some general aspects of string theory, in particular…
The B-side of Kontsevich's Homological Mirror Symmetry Conjecture is discussed. We give first a self-contained study of derived categories and their homological algebra, and later restrict to the bounded derived category of schemes and…
We review the applications of mirror symmetry to the study of the moduli spaces of two-dimensional conformal field theories with $N{=}(2,2)$ supersymmetry, particularly those constructed from Calabi--Yau manifolds. (Lecture delivered at the…
Physical reasons suggested in \cite{Ha-Ha} for the \emph{Quantum Gravity Problem} lead us to study \emph{type-changing metrics} on a manifold. The most interesting cases are \emph{Transverse Riemann-Lorentz Manifolds}. Here we study the…
This talk is dedicated to various aspects of Mirror Symmetry. It summarizes some of the mathematical developments that took place since M. Kontsevich's report at the Z\"urich ICM and provides an extensive, although not exhaustive,…
The dominance of string theory in the research landscape of quantum gravity physics (despite any direct experimental evidence) can, I think, be justified in a variety of ways. Here I focus on an argument from mathematical fertility, broadly…
In this informal expository note, we quickly introduce and survey compactifications of strata of holomorphic 1-forms on Riemann surfaces, i.e. spaces of translation surfaces. In the last decade, several of these have been constructed,…
It is shown that the notion of W_\infty-algebra originally carried out over a (compact) Riemann surface can be extended to n complex dimensional (compact) manifolds within a symplectic geometrical setup. The relationships with the…
This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace…
The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners.…
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an…
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…