Numerical Hermitian Yang-Mills Connection for Bundles on Quotient Manifold

TL;DR

This paper develops a systematic method combining equivariant structures and Donaldson's algorithm to compute Hermitian Yang-Mills connections on bundles over quotient Calabi-Yau manifolds, aiding heterotic string model building.

Contribution

It extends previous methods to quotient manifolds, enabling the computation of connections for equivariant bundles on these spaces using a generalized algorithm.

Findings

Algorithm converges for tested examples

Accurate approximation of Hermitian Yang-Mills connections

Facilitates heterotic model building on quotient manifolds

Abstract

We extend the previous computations of Hermitian Yang-Mills connections for bundles on complete intersection Calabi-Yau manifolds to bundles on their free quotients. Bundles on quotient manifolds are often defined by equivariant bundles on corresponding covering spaces. Combining equivariant structure and generalized Donaldson's algorithm, we develop a systematic approach to compute connections of holomorphic poly-stable bundles on quotient manifolds. With it, we construct the connections of an $SU(3)$ bundle on $Z_5$ quotient of quintic and an $SU(5)$ bundle on $Z_3 \times Z_3$ quotient of Bi-cubic. For both of these examples, the algorithm converges as expected and gives a good approximation of Hermitian Yang-Mills connections, which will be important for heterotic model building in Calabi-Yau compactification.