An algorithmic construction of F-manifold structures on smooth projective Calabi-Yau complete intersections

Abstract

The goal of this dissertation is to provide an explicit algorithmic construction of formal F-manifold structures on the primitive middle-dimensional cohomology of a smooth projective Calabi-Yau complete intersection variety X. We use a dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra A associated to the primitive middle-dimensional cohomology of X and we explain how a solution of the quantum master equation attached to A gives rise to a formal F-manifold structure on the cohomology. Our main technical result is to construct an inductive system of differential equations equivalent to the quantum master equation, which is amenable to computer algorithms. Using this inductive system we provide an explicit algorithm for solutions of the quantum master equation, computer codes, and concrete examples.