Fr\"olicher-Nijenhuis bracket on manifolds with special holonomy

TL;DR

This paper explores the use of the Fr"olicher-Nijenhuis bracket to study manifolds with special holonomy, introducing new cohomologies and algebraic structures with concrete computational examples.

Contribution

It introduces Fr"olicher-Nijenhuis cohomologies for special holonomy manifolds and constructs associated $L_$-algebras, providing explicit calculations.

Findings

Defined new cohomologies analogous to Dolbeault cohomology

Constructed $L_$-algebras for associative submanifolds

Provided concrete computations of the cohomologies

Abstract

In this article, we summarize our recent results on the study of manifolds with special holonomy via the Fr\"olicher-Nijenhuis bracket. This bracket enables us to define the Fr\"olicher-Nijenhuis cohomologies which are analogues of the $d^c$ and the Dolbeault cohomologies in K\"ahler geometry, and assigns an $L_\infty$-algebra to each associative submanifold. We provide several concrete computations of the Fr\"olicher-Nijenhuis cohomology.