Period integrals of Calabi-Yau manifolds are a very prominent example of the strong relationship between algebraic geometry and theoretical physics. In physics, these objects might be familiar in the context of String compactifications. As String Theory is anomaly-free only in 10 spacetime dimensions, one needs to compactify six dimensions on some internal space (which in turn has to be Calabi-Yau), to obtain an effective four dimensional theory. This low energy effective field theory is completely determined by the internal geometry - to be precise by the period integrals of the chosen Calabi-Yau manifold.
Beside in String Theory, Calabi-Yau manifolds recently appeared in another - unrelated - physical context; the computation of scattering amplitudes, as period integrals of Calabi-Yau manifolds define a set of generating functions for a large class of multi-loop Feynman Integrals.
For both applications, it is important to (numerically) compute period integrals up to a high accuracy. Conveniently, this is done by deriving a set of differential equations, called the Picard-Fuchs equations, whose solutions are precisely given by the period integrals.
Starting from the string theoretic point of view, I will introduce in this talk the notion of period integrals for three-dimensional Calabi-Yau manifolds and discuss some important properties thereof. Having set the stage, I will present first results of an ongoing research project, which aims towards an efficient computation of Picard-Fuchs ideals for general toric Calabi-Yau manifolds.
