Almost Commutative Manifolds and Their Modular Classes

TL;DR

This paper introduces the concept of almost commutative (or $ ho$-commutative) manifolds, generalizing supergeometry by incorporating graded algebras with controlled commutativity, and explores their modular classes and examples.

Contribution

It develops the theory of $ ho$-commutative manifolds, extending supergeometry concepts to a broader algebraic framework with new examples.

Findings

Definition of $ ho$-commutative manifolds and their structures

Construction of $ ho$-commutative versions of supergeometric objects

Examples including $ ho$-commutative Schouten bracket and noncommutative torus

Abstract

An almost commutative algebra, or a $\rho$-commutative algebra, is an algebra which is graded by an abelian group and whose commutativity is controlled by a function called a commutation factor. The same way as a formulation of a supermanifold as a ringed space, we introduce concepts of the $\rho$-commutative versions of manifolds, Q-manifolds, Berezin volume forms, and the modular classes. They are generalizations of the ones in supergeometry. We give examples including a $\rho$-commutative version of the Schouten bracket and a noncommutative torus.