Deformed super brackets on forms of a manifold

TL;DR

This paper investigates deformations of the super bracket on differential forms of a manifold, exploring conditions under which the deformed bracket remains a super bracket and providing concrete examples of parameter variations.

Contribution

It introduces a specific deformation of the super bracket on differential forms and analyzes the conditions for it to preserve the super bracket structure, including explicit examples.

Findings

Deformation of the super bracket depends on a 1-form parameter.

Conditions for the deformed bracket to remain a super bracket are characterized.

Explicit example shows the deformation parameter can alter the bracket's properties.

Abstract

Given a manifold, we have a super bracket on the graded algebra of differential forms by {A,B} = (-1)^{a} d(A \wedge B). We study when {A,B}_{t} = (-1)^{a} d(A \wedge B) + F(a,b) A \wedge t \phi \wedge B becomes super bracket for a 1-form \phi. And we show a concrete small example where the situation of parameter t is not identical.