Existence of n-fold idempotent propagators integrating iV

Determine conditions under which there exists a propagator kernel Q(x, y) on M × M that integrates the Hermitian connection iV on a complex line bundle over a manifold M, such that the n-fold convolution Q * Q * … * Q equals Q for some integer n ≥ 2, where * denotes integral-kernel composition (convolution) with respect to the base measure on M.

Background

In Section 5.2, the paper discusses a generalized propagator kernel Q that integrates the connection ipdq − gdp on T*R and satisfies Q * Q * Q = Q, rather than the usual projection property Q * Q = Q. Despite not being a projection, such kernels can still lead to well-defined and useful quantizations, particularly in contexts involving real polarizations.

Motivated by this observation, the paper raises a broader existence question: under what circumstances do there exist kernels integrating iV that are idempotent under convolution only after n ≥ 2 factors, generalizing the standard (n = 2) idempotency requirement. Establishing such conditions could extend quantization schemes to settings where conventional projection kernels do not exist.