On an Axiomatization of Path Integral Quantization and its Equivalence to Berezin's Quantization

We axiomatize path integral quantization of symplectic manifolds. We prove that this path integral formulation of quantization is equivalent to an abstract operator formulation, ie. abstract coherent state (or Berezin) quantization. We use the corresponding path integral of Poisson manifolds to quantize all complete Riemann surfaces of constant non$\unicode{x2013}$positive curvature and some Poisson structures on the sphere.