Compactifications of moduli spaces inspired by mirror symmetry

Abstract

We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some ``extra structure'' described by means of classes in H^2. The expectation that this moduli space is well-behaved in these ``extra structure'' directions leads us to formulate a simple and compelling conjecture about the action of the automorphism group on the Kähler cone. If true, it allows one to apply Looijenga's ``semi-toric'' technique to construct a partial compactification of the moduli space. We explore the implications which this construction has concerning the properties of the moduli space of complex structures on a ``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a similarity which might have been noticed between certain work of Mumford and of Mori from the 1970's produces (with hindsight) evidence for mirror symmetry which was available in 1979. [The author is willing to mail hardcopy preprints upon request.]

25 pp., LaTeX 2.09 with AmS-Fonts

References in corpus (4)

• Phases of $N=2$ Theories In Two Dimensions
• Topological Field Theory and Rational Curves
• Mirror Manifolds in Higher Dimension
• Automorphisms and the Kähler cone of certain Calabi-Yau manifolds

Cited by in corpus (60)

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• Perturbative Corrections to Kahler Moduli Spaces
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• The Geometry Underlying Mirror Symmetry
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• D-branes and Normal Functions
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• The cone of curves and the Cox ring of rational surfaces given by divisorial valuations
• Collapsing hyperkähler manifolds
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