Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Moduli spaces of Hom-Lie algebroid connections

Ayush Jaiswal

TL;DR

The paper addresses the problem of constructing moduli spaces for irreducible Hom-Lie algebroid connections under Hom-gauge symmetry on a compact manifold. It develops a Sobolev-space framework to analyze the gauge action and uses Fredholm and ellipticity arguments to obtain a local model by a Hilbert slice, establishing a Hausdorff Hilbert-manifold structure for the moduli space $\widehat{B}(E,\mathbb{L})_l$ and a principal bundle description for $\widehat{p}:\widehat{A}(E,\mathbb{L})_l\to\widehat{B}(E,\mathbb{L})_l$ with structure group $\text{H-Gau}(E)_{l+1}^r$. The approach generalizes classical Lie algebroid connection results to the Hom-Lie setting, integrating differential graded structures through the Hom-Chevalley–Eilenberg–de Rham complex and Sobolev techniques. The results provide a rigorous foundation for Hom-gauge theoretic invariants and may extend to broader deformation and moduli problems in Hom-Lie geometry.

Abstract

We have studied irreducible Hom-Lie algebroid connections for Hom-bundle and prove that the H-gauge theoretic moduli space has a Hausdorff Hilbert manifold structure. This work generalizes some known results about simple semi-connections and Lie algebroid connections for complex vector bundles on compact complex manifold.

Moduli spaces of Hom-Lie algebroid connections

TL;DR

The paper addresses the problem of constructing moduli spaces for irreducible Hom-Lie algebroid connections under Hom-gauge symmetry on a compact manifold. It develops a Sobolev-space framework to analyze the gauge action and uses Fredholm and ellipticity arguments to obtain a local model by a Hilbert slice, establishing a Hausdorff Hilbert-manifold structure for the moduli space and a principal bundle description for with structure group . The approach generalizes classical Lie algebroid connection results to the Hom-Lie setting, integrating differential graded structures through the Hom-Chevalley–Eilenberg–de Rham complex and Sobolev techniques. The results provide a rigorous foundation for Hom-gauge theoretic invariants and may extend to broader deformation and moduli problems in Hom-Lie geometry.

Abstract

We have studied irreducible Hom-Lie algebroid connections for Hom-bundle and prove that the H-gauge theoretic moduli space has a Hausdorff Hilbert manifold structure. This work generalizes some known results about simple semi-connections and Lie algebroid connections for complex vector bundles on compact complex manifold.
Paper Structure (11 sections, 12 theorems, 97 equations)

This paper contains 11 sections, 12 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hom-Lie-algeba
  4. Hom-Lie algebroid
  5. Hom-Lie algebroid connection
  6. H-Gauge group action
  7. Sobolve H-Gauge group action
  8. Sobolev completion of space of $\mathbb{L}$-connections
  9. Sobolev completion of space of H-gauge transformations
  10. Sobolev H-gauge action on the Sobolev space of $\mathbb{L}$-connection
  11. Main theorem

Key Result

Theorem 2.9

For a given $\mathbb{L}$-connection $\nabla$, there is a degree 1 differential operator $d^\nabla:\mathcal{A}^{\mathbin{\vcenter{\hbox{$\m@th\bullet$}}}}_\mathbb{L}(X,E)\rightarrow \mathcal{A}^{\mathbin{\vcenter{\hbox{$\m@th\bullet$}}}}_\mathbb{L}(X,E)$, uniquely determined such that The operator $d^\nabla$ is given by for $\xi_i(0\leq i\leq p)\in \Gamma(X,\mathbb{L})$.

Theorems & Definitions (41)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Example
  • Definition 2.5
  • Example
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 31 more