Algebraic Principles of Quantum Field Theory II: Quantum Coordinates and WDVV Equation

Abstract: This paper is about algebro-geometrical structures on a moduli space $\CM$ of anomaly-free BV QFTs with finite number of inequivalent observables or in a finite superselection sector. We show that $\CM$ has the structure of F-manifold -- a linear pencil of torsion-free flat connection with unity on the tangent space, in quantum coordinates. We study the notion of quantum coordinates for the family of QFTs, which determines the connection 1-form as well as every quantum correlation function of the family in terms of the 1-point functions of the initial theory. We then define free energy for an unital BV QFT and show that it is another avatar of morphism of QFT algebra. These results are consequences of the solvability of refined quantum master equation of the theory. We also introduce the notion of a QFT integral and study some properties of BV QFT equipped with a QFT integral. We show that BV QFT with a non-degenerate QFT integral leads to the WDVV equation---the formal Frobenius manifold structure on $\CM$---if it admits a semi-classical solution of quantum master equation.