Examples of dHYM connections in a variable background

TL;DR

This paper explicitly constructs examples of deformed Hermitian Yang-Mills (dHYM) connections on ruled surfaces, coupling them with variable background Kähler metrics, and explores their limits and geometric properties.

Contribution

It provides new explicit examples of coupled dHYM and scalar curvature equations on ruled surfaces, connecting to known limits and geometric structures.

Findings

Constructed explicit dHYM solutions on ruled surfaces.

Analyzed the large radius limit connecting to Kähler-Yang-Mills system.

Discussed singularities, B-branes, and limits of the solutions.

Abstract

We study deformed Hermitian Yang-Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application we provide many new examples of dHYM connections coupled to a variable background K\"ahler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the K\"ahler-Yang-Mills system of \'Alvarez-C\'onsul, Garcia-Fernandez and Garc\'ia-Prada, and in this limit our solutions converge smoothly to those constructed by Keller and T{\o}nnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realisation as B-branes, the small radius limit and canonical representatives of complexified K\"ahler classes.