Mirror Manifolds in Higher Dimension

TL;DR

This paper extends the concept of mirror manifolds beyond threefolds, introducing new geometric structures and methods for constructing mirror maps and calculating instanton corrections in higher-dimensional Calabi--Yau manifolds.

Contribution

It generalizes mirror symmetry techniques to higher dimensions, providing robust methods that do not depend on special threefold properties and exploring new moduli space geometries.

Findings

New geometric structures for higher-dimensional Calabi--Yau moduli spaces

Procedures for constructing mirror maps in arbitrary dimensions

Agreement of instanton correction calculations with traditional methods

Abstract

We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We emphasize the simplifying features of dimension three and supply more robust methods that do not rely on such special characteristics and hence naturally generalize to other dimensions. The moduli spaces for Calabi--Yau $d$-folds are somewhat different from the ``special K\"ahler manifolds'' which had occurred for $d=3$, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi--Yau manifolds. Our results agree with those obtained by more traditional mathematical methods in the limited number of cases for which the latter analysis can be carried out. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra.