Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds

TL;DR

This paper constructs various non-compact Calabi-Yau manifolds as gauge-theoretic line bundles over Hermitian symmetric spaces, including cases with exceptional group symmetries, revealing their geometric structures and singularities.

Contribution

It introduces a gauge-theoretic framework to systematically construct non-compact Calabi-Yau manifolds over Hermitian symmetric spaces, including new cases with exceptional symmetries.

Findings

Constructed line bundles over complex quadrics and Grassmannians.

Identified the role of resolution parameters in controlling singularities.

Extended the construction to manifolds with exceptional group symmetries.

Abstract

We construct the non-compact Calabi-Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CP^{N-1}. Imposing an F-term constraint on the line bundle over CP^{N-1}, we obtain the line bundle over the complex quadric surface Q^{N-2}. On the other hand, when we promote the U(1) gauge symmetry in CP^{N-1} to the non-abelian gauge group U(M), the line bundle over the Grassmann manifold is obtained. We construct the non-compact Calabi-Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. Each of these manifolds contains the resolution parameter which controls the size of the base manifold, and the conical singularity appears when the parameter vanishes.