Almost Commutative Manifolds and Their Modular Classes

For the study of noncommutative algebras, it is convenient to impose a constraint for the noncommutativity of the algebras, which is often called a commutation rule. A commutative algebra is an algebra with a commutation rule $fg=gf$ for all element $f,g$ of the algebra. A superalgebra is a $\mathbb{Z}/2\mathbb{Z}$-graded algebra which has a commutation rule $fg=-gf$ if homogeneous elements $f,g$ are odd, and $fg=gf$ otherwise. A $\rho$-commutative (or almost commutative, $\epsilon$-commutative) algebra is an algebra graded by an arbitrary abelian group $G$ with its commutation rule controlled by a map called a commutation factor [Scheunert1979, Bongaarts1994, Ciupala2005, Bruce2020]. This condition characterizing noncommutativity covers not only commutative algebras or superalgebras, but quaternions, quantum planes, and noncommutative tori, etc. $\rho$-Lie algebras are also defined as the $\rho$-commutative version of Lie algebras.

Meanwhile, we have a characteristic class called the modular class of a Q-manifold. A Q-manifold or a dg-manifold is a supermanifold with an odd vector field squared to zero as a derivation on functions. This condition for a vector field appears in various cases. Lie algebroids, $L_{\infty}$-algebras, and the de Rham or Dolbeault complexes are formulated as Q-manifolds [Kontsevich1999]. In the aspect of mathematical physics, the classical BRST formalism is one of the application of Q-manifolds [Mnev2019]. Theories of characteristic classes of Q-manifolds are studied by Kotov [Kotov2007], Lyakhovich-Mosman-Sharapov [Lyakhovich2009], and Bruce [Bruce2017]. The modular class is one of them which generalizes the modular class of a higher Poisson manifold.

The main goal of this paper is to introduce a concept of the modular classes for $\rho$-Q-manifolds and give examples of them. Supermanifolds are constructed by replacing the local functional algebras of the underlying manifold with a graded-commutative $\mathbb{Z}/2\mathbb{Z}$-algebra. We apply the same procedure to a manifold but with a $\rho$-commutative algebra. We call the resulting manifold a $\rho$-manifold. Vector fields on a $\rho$-manifold are $\rho$-derivations on the $\rho$-commutative functional algebra. An odd vector field squared to zero in this case defines a $\rho$-Q-manifold. The $\rho$-commutative version of a Berezin volume form is defined, which allows us to define the modular class of a $\rho$-Q-manifold the same way as a Q-manifold. We give examples of $\rho$-Q-manifolds and their modular classes, e.g., the tangent or cotangent bundle of a $\rho$-manifold, the degree shift of them, and the BRST differential on noncommutative tori.

In Section 2, we review the theory of $\rho$-commutative algebras and $\rho$-Lie algebras. We mainly give definitions and examples of them here. In Section 3, we review the matrix algebra whose entries are in a $\rho$-commutative algebra. We introduce the determinant and the Berezinian in this version by [KobayashiNagamachi1984, Covolo2016]. In Section 4, we discuss the specific algebra which is a subalgebra of the set of formal power series with $G$-graded indeterminates. This section is devoted to the preparation for argument of $\rho$-manifolds. In Section 5, we define a $\rho$-manifold and a $\rho$-Q-manifold. Here we discuss the de Rham complex and the degree-$i$ Schouten bracket of a $\rho$-manifold. In Section 6, we introduce the Berezinian bundle of a $\rho$-manifold and its orientability using the result of Section 3. In section 7, we define the modular class of a $\rho$-Q-manifold and calculate them in several situations, for example, the degree-$i$ Schouten bracket and noncommutative tori.