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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TL;DR: In this article, the rank 4 locus of a general skew-symmetric 7 × 7 matrix gives the Pfaffian variety in P20 which is not defined as a complete intersection.
Abstract: The rank 4 locus of a general skew-symmetric 7 × 7 matrix gives the Pfaffian variety in P20 which is not defined as a complete intersection. Intersecting this with a general P6 gives a Calabi–Yau manifold. An orbifold construction seems to give the 1-parameter mirror-family of this. However, corresponding to two points in the 1-parameter family of complex structures, both with maximally unipotent monodromy, are two different mirror-maps: one corresponding to the general Pfaffian section, the other to a general intersection of G(2,7) ⊂ P20 with a P13. Apparently, the Pfaffian and G(2,7) sections constitute different parts of the A-model (Kahler structure related) moduli space, and, thus, represent different parts of the same conformal field theory moduli space.
137 citations
TL;DR: In this article, the mirror isomorphism between Calabi-Yau hypersurfaces in toric varieties has been studied, and it has been shown that the moduli spaces of the nonlinear sigma models whose targets are the different birational models of a Calabi Yau space should be connected by analytic continuation.
Abstract: For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has proposed a possible mirror partner (which is also a family of Calabi-Yau hypersurfaces). We explain a natural construction of the isomorphism between certain Hodge groups of these hypersurfaces, as predicted by mirror symmetry, which we call the monomial-divisor mirror map. We indicate how this map can be interpreted as the differential of the expected mirror isomorphism between the moduli spaces of the two Calabi-Yau manifolds. We formulate a very precise conjecture about the form of that mirror isomorphism, which when combined with some earlier conjectures of the third author would completely specify it. We then conclude that the moduli spaces of the nonlinear sigma models whose targets are the different birational models of a Calabi-Yau space should be connected by analytic continuation, and that further analytic continuation should lead to moduli spaces of other kinds of conformal field theories. (This last conclusion was first drawn by Witten.)
134 citations
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TL;DR: In this paper, orientifolds of Calabi-Yau 3-folds were considered in the context of Type IIA and Type IIB superstrings, and mirror symmetry was used to sum up worldsheet instanton contributions to the superpotential for Type IIa superstrings.
Abstract: We consider orientifolds of Calabi-Yau 3-folds in the context of Type IIA and Type IIB superstrings. We show how mirror symmetry can be used to sum up worldsheet instanton contributions to the superpotential for Type IIA superstrings. The relevant worldsheets have the topology of the disc and ${\bf RP^2}$.
133 citations
TL;DR: In this paper, it was shown that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition.
Abstract: We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.
132 citations
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TL;DR: A survey of recent developments in the theory of toric varieties can be found in this article, including new constructions of Toric varieties and relations to symplectic geometry, combinatorics and mirror symmetry.
Abstract: This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic geometry, combinatorics and mirror symmetry.
132 citations