Semi-Classical Quantum Fields Theories and Frobenius Manifolds

We show that a semi-classical quantum field theory comes with a versal family with the property that the corresponding partition function generates all path integrals., satisfies a system of second order differential equations determined by algebras of classical observables. This versal family gives rise to a notion of special coordinates that is analogous to that in string theories. We also show that for a large class of semi-classical theories, their moduli space has the structure of a Frobenius super-manifold.