Higher Order Connections in Noncommutative Geometry
Keegan J. Flood, Mauro Mantegazza, Henrik Winther
TL;DR
The paper develops a noncommutative differential-geometric framework in which higher order connections are characterized via Spencer operators and jet-jet exact sequences. It establishes a deep equivalence between higher order connections and quantization procedures by introducing natural differential operators, symbol spaces, and a full quantization that induces total symbols and star products on symbol algebras. By extending Spencer theory to noncommutative jets and proving exactness results for Spencer complexes and bicomplexes, the authors connect jet-geometry with deformation quantization, providing explicit constructions and functorial behavior. The work culminates in a concrete quantization formalism, including polynomials, formal and parametrized star products, and a quaternionic example that demonstrates the applicability to noncommutative phase spaces, with potential implications for quantum geometry and NC-field theories.
Abstract
We prove that, in the setting of noncommutative differential geometry, a system of higher order connections is equivalent to a suitable generalization of the notion of phase space quantization (in the sense of Moyal star products on the symbol algebra). Moreover, we show that higher order connections are equivalent to (ordinary) connections on jet modules. This involves introducing the notion of natural linear differential operator, as well as an important family of examples of such operators, namely the Spencer operators, generalizing their corresponding classical analogues. Spencer operators form the building blocks of this theory by providing a method of converting between the different manifestations of higher order connections. A system of such higher order connections then gives a quantization, by which we mean a splitting of the quotient projection that defines symbols as classes of differential operators up to differential operators of lower order. This yields a notion of total symbol and of star product, the latter of which corresponds, when restricted to the classical setting, to phase space quantization in the context of quantum mechanics. In this interpretation, we allow the analogues of the position coordinates to form a possibly noncommutative algebra.