Involution in quantized endomorphism bundle and reality of noncommutative gravity actions

TL;DR

This paper demonstrates that Fedosov trace functionals commute with involution in Hermitian-compatible connections, ensuring noncommutative gravity actions are real across all deformation powers.

Contribution

It proves the reality of noncommutative gravity actions by showing the Fedosov trace commutes with involution for compatible connections.

Findings

Fedosov trace functional commutes with involution

Noncommutative gravity actions are real in all deformation powers

Establishes a link between Hermitian metrics and reality of gravity actions

Abstract

It is shown that for arbitrary connection in the vector bundle compatible with some Hermitian metric, the corresponding Fedosov trace functional commutes with involution generated by this metric. This result is then used to prove that certain noncommutative gravity actions are real in all powers of deformation parameter.