Smooth relative connections on quiver bundles

TL;DR

This paper develops a theory of smooth relative connections on twisted quiver bundles, providing obstructions, conditions for existence, and links to quiver representation theory, especially for tree-type quivers.

Contribution

It introduces a comprehensive framework for smooth relative connections on twisted quiver bundles, including obstructions and criteria for existence, and connects flat connections to quiver representations.

Findings

Obstructions to smooth relative connections identified.

Necessary and sufficient conditions for tree-type quivers established.

Framework linking flat quiver bundles to representation theory developed.

Abstract

We develop a theory of smooth relative connections over the real path algebra $\mathbb{R}Q$ on smooth twisted quiver bundles. We give obstructions to the existence of a smooth relative connection on twisted quiver bundles. For tree-type quiver bundles, we establish a necessary and sufficient condition for the existence of a smooth relative connection. Additionally, we provide a framework for the representation theory of flat quiver bundles, relating the existence of flat relative connections to the underlying quiver representations.