Compactifications of moduli spaces inspired by mirror symmetry

TL;DR

This paper investigates the compactification of moduli spaces of Calabi-Yau sigma-models inspired by mirror symmetry, proposing a conjecture on automorphism actions and exploring implications for mirror pairs.

Contribution

It introduces a conjecture about automorphism group actions on the Kähler cone and applies Looijenga's semi-toric technique to construct a partial compactification of the moduli space.

Findings

Proposes a conjecture enabling partial compactification of moduli spaces.

Connects mirror symmetry with classical work of Mumford and Mori.

Provides evidence for mirror symmetry based on historical mathematical parallels.

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 K\"ahler 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.]