  In this paper we develop a quantization method for flat compact manifolds
based on path integrals. In this method the Hilbert space of holomorphic
functions in the complexification of the manifold is used. This space is a
reproducing kernel Hilbert space. A definition of the Feynman propagator, based
on the reproducing property of this space, is proposed. In the $\mathbb{R}^n$
case the obtained results coincide with the known expressions.
