Semi-Classical Quantum Fields Theories and Frobenius Manifolds

TL;DR

This paper explores the structure of semi-classical quantum field theories, revealing their connection to Frobenius super-manifolds and providing a framework where the partition function encodes all path integrals through differential equations.

Contribution

It introduces a versal family for semi-classical QFTs that generates all path integrals and links the moduli space to Frobenius super-manifolds, with an analogy to string theory coordinates.

Findings

Partition function generates all path integrals.

Moduli space has Frobenius super-manifold structure.

Differential equations determined by classical observables.

Abstract

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 and satisfies a system of 2nd 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.