Topic
Mirror symmetry
About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.
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04 Mar 1997
TL;DR: In this paper, the authors present the lectures of the third Regional Geometry Institute at Park City in 1993, which dealt with topics in algebraic geometry where one can reach the level of current research while starting with the basics.
Abstract: This volume contains the lectures presented at the third Regional Geometry Institute at Park City in 1993. The lectures provide an introduction to the subject, complex algebraic geometry, making the book suitable as a text for second- and third-year graduate students. The book deals with topics in algebraic geometry where one can reach the level of current research while starting with the basics. The topics covered include the theory of surfaces from the viewpoint of recent higher-dimensional developments, providing an excellent introduction to more advanced topics such as the minimal model program. Also included is an introduction to Hodge theory and intersection homology based on the simple topological ideas of Lefschetz and an overview of the recent interactions between algebraic geometry and theoretical physics, which involve mirror symmetry and string theory.
32 citations
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TL;DR: In this paper, the authors constructed a mirror Lefschetz fibration for a log Calabi-Yau surface with a distinguished complex structure, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$ and the wrapped Fukaya categories of $m$ is also isomorphic with respect to the total space of the almost toric fibration arising in the work of Gross-Hacking-Keel.
Abstract: Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M \to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$, and the wrapped Fukaya category $\mathcal{W} (M)$ is isomorphic to $D^b \text{Coh}(Y \backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.
32 citations
TL;DR: In this article, the authors investigate the cosmological properties of an ''induced gravity'' brane scenario in the absence of mirror symmetry with respect to the brane and find that brane evolution can proceed along one of four distinct branches.
Abstract: We investigate the cosmological properties of an `induced gravity' brane scenario in the absence of mirror symmetry with respect to the brane. We find that brane evolution can proceed along one of four distinct branches. By contrast, when mirror symmetry is imposed, only two branches exist, one of which represents the self-accelerating brane, while the other is the so-called normal branch. This model incorporates many of the well-known possibilities of brane cosmology including phantom acceleration (w < ?1), self-acceleration, transient acceleration, quiescent singularities, and cosmic mimicry. Significantly, the absence of mirror symmetry also provides an interesting way of inducing a sufficiently small cosmological constant on the brane. A small (positive) ?-term in this case is induced by a small asymmetry in the values of bulk fundamental constants on the two sides of the brane.
32 citations
TL;DR: In this article, it was shown that certain superpotential and Kahler potential couplings can be computed from Hodge theory and mirror symmetry in the context of supersymmetric compactions with branes or bundles.
Abstract: We show that certain superpotential and Kahler potential couplings
of $\mathcal{N} = 1$ supersymmetric compactifications with branes or bundles can
be computed from Hodge theory and mirror symmetry. This applies to
F-theory on a Calabi–Yau four-fold and three-fold compactifications of
type II and heterotic strings with branes. The heterotic case includes a
class of bundles on elliptic manifolds constructed by Friedmann, Morgan
and Witten. Mirror symmetry of the four-fold computes non-perturbative
corrections to mirror symmetry on the three-folds, including D-instanton
corrections. We also propose a physical interpretation for the observation
byWarner that relates the deformation spaces of certain matrix factorizations
and the periods of non-compact four-folds that are ALE fibrations.
32 citations
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TL;DR: In this article, Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow has been given, which is able to prove the symplectic topological version of SYZ mirror conjecture.
Abstract: In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres)
32 citations