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Smooth relative connections on quiver bundles

Pavan Adroja, Sanjay Amrutiya, Riddhi Patil

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.

Smooth relative connections on quiver bundles

Abstract

We develop a theory of smooth relative connections over the real path algebra 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.
Paper Structure (10 sections, 10 theorems, 93 equations)

This paper contains 10 sections, 10 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Twisted Quiver Bundles
  4. Smooth connection on twisted quiver bundles
  5. Obstructions to smooth relative connections on twisted quiver bundles
  6. Bianchi identity for quiver bundle
  7. Smooth relative connection on Tree-type quiver bundles
  8. Smooth relative connections on $\mathbb{A}_n$-type quiver bundles
  9. Smooth relative connections on tree-type quiver bundles
  10. Representations and flat connection

Key Result

Proposition 3.7

Let $\mathcal{R}=(\mathcal{E}, \phi)$ be an $M$-twisted smooth $Q$-bundle. Then, $\mathcal{R}$ admits a smooth relative connection whose induced connections on $M_a$ are $\nabla_a$ for all $a\in Q_1$ if and only if the image of $\{\beta_a\}_{a\in Q_1}$ vanishes in $\mathrm{Ext}^1_{\mathcal{A}}(\math

Theorems & Definitions (30)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 3.5
  • Definition 3.6
  • Proposition 3.7
  • ...and 20 more