Existence of n-fold idempotent propagators integrating a connection

Determine conditions under which there exists a kernel Q on M × M that integrates the Hermitian connection i∇ on a prequantum line bundle over a symplectic manifold (in the sense of Definition 2.0.1, i.e., its covariant derivative in the second argument vanishes at the diagonal and it acts as a reproducing kernel), and whose convolution power satisfies Q * Q * … * Q = Q (n times) for some integer n ≥ 2.

Background

The paper defines a propagator Q integrating a connection ∇ as an integral kernel (a section over M × M of the appropriate Hom line bundle) that reproduces under convolution (Q * Q = Q) and whose covariant derivative in the second variable vanishes on the diagonal. This structure yields a physical Hilbert space and a quantization map.

In Section 5.2, within the context of real polarizations, the author exhibits a kernel on T*R that fails the strict idempotence Q * Q = Q but does satisfy a relaxed property Q * Q * Q = Q, while still leading to meaningful quantizations. Motivated by this, the author asks for general criteria ensuring the existence of such kernels that are idempotent only after a finite number of convolutions, i.e., Q^n (under convolution) equals Q for some n ≥ 2.